ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E21-3

Income inequality in terms of a Gini coefficient :A Kaleckian perspective
by
Shinya FUJITA

April 2021

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Income inequality in terms of a Gini coefficient: A Kaleckian perspective

Shinya FUJITA †Abstract: This study presents a theoretical framework that connects
functional income distribution and personal income distribution from the
Kaleckian perspective and investigates the effects of changes in mark-up rate
and of monetary policy on income inequality in terms of the Gini coefficient.
The results demonstrate that a rise in the mark-up rate with weak bargaining
power of workers raises the Gini coefficient irrespective of a profit-led or
wage-led demand regime. The analyses also demonstrate that a decline in theinterest rate with monetary easing negatively affects the Gini coefficient if
and only if the bargaining power of workers is sufficiently strong.
Keyword: personal income distribution, Gini coefficient, functional income
distribution, demand regime, Kaleckian model, monetary policy
JEL classification: D31, E11, E25, E52

1† Address: Graduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya,

Aichi, 464-8601, Japan. E-mail: fujita@soec.nagoya-u.ac.jp. This work is supported by JSPS
KAKENHI Grant Number 19K01543.

1. Introduction
Income distribution is gathering attention again after Piketty (2014). Post Keynesians have traditionally provided many suggestions for functional
income distribution, especially between capitalists who own capital andworkers who do not. It has clarified the effects of factors such as labour market
institutions and financialization on the distribution between capitalists andworkers (Stockhammer, 2004; Mohun, 2006; Dallery, 2009; Dünhaupt, 2017;
Kohler et al., 2019). Moreover, following the tradition of Keynes, postKeynesians have tackled the question of how to reduce unemployment causedby the lack of effective demand. There is a clear disparity between t heemployed and the unemployed; the problem of unemployment is thus also
closely linked to inequality, and how many people will be hired from the poolof the unemployed is a matter of economic growth. Kaleckian models consider
the relationship between functional income distribution and effective demand,and show which combination of functional income distribution anddemand/growth regimes will increase growth and reduce unemployment(Rowthorn, 1981; Dutt, 1994; Bhaduri and Marglin, 1990; Taylor, 2004; Lavoi e
and Stockhammer, 2013; Onaran and Obst, 2016).
Functional income distribution has a significant impact on personal income
distribution (Glyn, 2009; Schlenker and Schmid, 2015; Wolff, 2015; Bengtsson
and Waldenström, 2018). However, in reality, workers own capital and receivea part of the profits as financial income, which implies that inequality infunctional income distribution does not directly lead to inequality in personal
income distribution. 1 Moreover, as described above, it is necessary to focus oninequality between the employed and the unemployed by using a growthmodel. Thus, this study clarifies the factors of income inequality at thepersonal level by using a Kaleckian model to comprehensively capture therelationship between functional distribution and growth. Using this model,
we consider inequality not only between capitalists and workers who own
some capital, but also between workers and the unemployed.
In line with the post-Keynesian tradition, some models investigate personal
income distribution. 2 Molero-Shimarro (2017) used a Bhaduri–Marglin–typemodel to investigate the relationship between functional income distribution
and personal income distribution. The model shows that a decrease in the2

wage share under a profit-led regime improves inequality via a reduction in
unemployment, while an increase in the wage share under a wage-led regime
contracts inequality by not only suppressing the high-income group but also
getting the poor out of unemployment. However, the model of MoleroShimarro (2017) is too simple. For instance, the effect of a change in the wage
share on workers’ income, including both wages and profits, is abstracted
when workers are assumed to own capital. Palley (2017) presents a Bhaduri–
Marglin model with two classes, capitalist-managers and workers, andinvestigates the factors promoting income inequality. In the model, althoughboth capitalist-managers and workers acquire profits as well as wages,
workers are distinguished from capitalist-managers by their small share ofcapital ownership, by gaining a larger share of wage income, and by theirrelatively low propensity to save. Palley shows that increases in the workers’
share of capital and workers’ share of wage income have an impact of reducing
income inequality between the two classes. However, in this model, workers’
capital share is given. Since capital/asset inequality is one of the main factors
for income inequality (Piketty, 2014; Alvaredo et al., 2018), it is necessary toconsider how workers’ capital share is endogenously determined. Using a
Bhaduri–Marglin

model,

EdererandRehm(2020a)investigate

themechanism of wealth distribution between capitalists and workers. Their
model assumes that capitalists acquire a proportion of wages, whereas
workers save and hold assets. In many EU countries, since the workers’
propensity to save is larger than zero, workers own a constant proportion of
wealth in the long run, which conflicts with the view of Piketty (2014) that
income inequality will continue to expand due to asset occupancy. Dutt (2016)
uses a model with the highest income group (the ‘top’) and other income groups(the ‘rest’), and investigates the effect of financialization, such as rising
dividend payout ratio on growth and distribution. In the model, the top ischaracterised by some assumptions: their labour supply is proportionate tocapital stocks, that their wages are a constant multiple of the wages of therest, and that they acquire brokerage fees as financiers and dividends fromfirms. In the model, various scenarios about growth and distribution can beobtained by using two types of closing models, the Marxian one of constant
output–capital ratio and the post-Keynesian one of endogenous output–capital3

ratio. Ryoo (2018) presents the long-run Kaldorian model as an alternative to
Piketty’s neoclassical explanation of inequality. His model assumes that both
capitalists and workers gain financial income, such as dividends and interest.
Ryoo (2018) indicates a scenario in which financialization factors such as
decreasing retention ratio and rising stock buybacks lead to a decrease insavings and thus an increase in the profit share, which are adjustment
variables for investment and savings, which in turn causes income inequality
between capitalists and workers. The model thus fully considers financial
variables but ignores the demand/growth regime.
In contrast to the above studies, we examine the effect of changes in themark-up rate in pricing and that of monetary policy on inequality in personal
income distribution. This study makes the following contributions: First, byusing the Kaleckian model with firms’ debt accumulation (Lavoie, 1995; Hein,
2007; Sasaki and Fujita, 2012; Nishi, 2015), we consider that a rise in themark-up rate, instead of a decline in the wage share, has an impact on income
distribution, which in turn affects demand and growth, and thus the Gini
coefficient, by changing the number of the unemployed. The mechanism of
improving inequality by reducing unemployment has rarely been addressedin previous studies. Expressing income inequality using the Gini coefficient is
key to this study. If we do not use the Gini coefficient, we ignore inequality
between the middle- and low-income groups (i.e. disparity between theemployees and the unemployed). In fact, as we will see in the next section, arise in the Gini coefficient in some countries comes from the low-income group.
Second, we assume four agents: capitalists who own capital, indirect(managerial) workers who own a part of capital, direct (productive) workers
without capital, and the unemployed. Changes in the mark-up rate and ofmonetary policy affect the Gini coefficient by changing each income share andpopulation share. Third, we show that an increase in the mark-up rate
positively affects the Gini coefficient. An increased mark-up rate does not leadto improving inequality under any demand regime, which differs from theresult of Molero-Shimarro (2017). Finally, this study analyses the impact oflower interest rates on inequality due to central bank’s monetary easing. 3
Specifically, we show that a decrease in the interest rate is effective for
improving inequality if and only if the mark-up rate is sufficiently small,4

which is consistent with the empirical result of Furceri et al. (2018).
The remainder of this paper is organised as follows. Section 2 explains thecurrent income inequality situation in some developed countries. Section 3
presents the model. Section 4 draws a Lorenz curve based on the propositions
obtained from the model and visualises the effects of changes in the mark-up
rate and of monetary policy on the Gini coefficient. In addition, we discuss
some factors that increase the Gini coefficient in a real economy. Section 5
concludes.
2. Current status of income inequality
This section briefly confirms how income inequality in terms of the Gini
coefficient expands in developed countries. The countries we consider in thissection are France, Germany, Italy, Sweden, Japan, and the United States.
According to research in comparative political economy and Règulation
approach, capitalist countries are institutionally diverse, and institutionalvariety has a crucial impact on economic performance, including growth,
unemployment, and inequality (Hall and Soskice, 2001; Amable, 2003;
Pontusson, 2005; Schneider and Paunescu, 2012; Hein, Meloni, and Tridico,2020). We select six countries that are considered to have different
institutional structures and performance. 4 In addition, regarding the Gini
coefficient, we used two common databases. The Luxembourg Income Study(LIS) is the most commonly used in international comparisons of income
inequality, but of the six countries considered here, it only provides complete
data for the United States and Germany. Another database is Solt’s (2019)
Standardized World Income Inequality Database (SWIID). This databasestandardises various international statistics on income inequality based on
LIS method, and the data are complete by country.
Table 1 shows the Gini coefficient of pre-redistribution (pre-tax and pretransfer) income and post-redistribution (post-tax and post-transfer) income.
Basically, there is no significant difference between the two databases, so wewill discuss inequality based on SWIID below. The Gini coefficient of pre redistribution income is rising in all countries, and that of post-redistribution
income is also on the rise as a result. However, there are shades of differencein each country. Post-redistribution income inequality in France and Italy has
5not widened significantly compared to other countries: In France, the Gini
coefficient of pre-redistribution income increased by only 0.014 points fromthe 1990s to the 2010s, which is in contrast to an increase of 0.058 points inGermany, 0.038 points in Sweden, and 0.032 points in the United States.
(Table 1 here)
Which income group is the main cause of the increase in the Gini
coefficient? Table 1 also shows the pre-tax income share of the bottom 50
percent from the World Inequality Database (WID), and the incidence of low
pay (i.e. percentage of full-time workers earning less than two-thirds of fulltime median wages) from the OECD. We find that the pre-tax income share of
the bottom 50 percent in France remained constant from the 1990s to the2010s. This seems to have prevented an excessive increase in the Gini
coefficient of pre-redistribution income. By contrast, in Germany, Sweden, andthe United States, the pre-tax income share of the bottom 50 percent has
dropped significantly. Moreover, regarding the incidence of low-pay, only
Germany showed an upward trend. From these facts, we find that in Germany,
the expansion of the low-wage worker group widened pre-redistribution
income inequality, and as a result, the Gini coefficient in post-redistribution
income increased (Baccaro and Pontusson, 2017). For the United States andSweden, it is often pointed out that regressive tax reform has become the maincause of widening inequality (Saez and Sucman, 2019; Gustafsson and
Jansson, 2008). However, we should not overlook the fact that preredistribution income of the lower groups is declining in these countries, if not
to as extreme an extent as in Germany.
As mentioned above, when we discuss income inequality, it is not enough toconsider inequality between the highest income group and others. We have tofocus not only on the inequality between intermediate-income workers andlow-income workers but also on the disparity between low-income workers andthe unemployed. The Gini coefficient is a useful indicator in that it can cover
the income inequality of all these economic agents. 5 In the following sections,
we build a framework that can capture changes in the Gini coefficient.
3. Model3.1 Basic settings
6The economy is assumed to be closed, without a government. This section
contains the following types of agents: firms (capitalists), indirect workers,
and direct workers. It is assumed that firms carry out various activities, suchas pricing, production, investment, etc., but that all the profits obtained there
are paid to the owner of the capital. For simplicity we assume wealth to takethe form of physical capital. 6 Owing to being the main owners of capital,
capitalists are the main recipients of profits. 7 Indirect (managerial) workers
are indirectly involved in production, such as via business management
within firms, and their employment does not change with respect to short term fluctuations in output. In addition, indirect workers are assumed to havehigher position within the firms and obtain a wage premium thereby. They
are relatively richer and save a part of their income, which, in turn, isborrowed by firms. This implies that indirect workers acquire interests that
accrue from their capital. Direct (productive) workers are directly involved in
the production of goods, and their employment is adjusted for short -term
fluctuations in output. This type of worker is characterised by unstable
employment and relatively low wages. As a result, direct workers do not saveand do not own capital.
Moreover, a single good can be used for both production and consumption.
Firms produce this good by using fixed capital stocks and two types of labour:
indirect and direct labour. We assume non-substitutability between capitaland the two types of labour, which differ from the neo-classical supposition of
Piketty (2014, 2015). Value added is distributed to two types of workers as
wages and to firms as profits; then, firms distribute profits as dividends tocapitalists and pay interest to indirect workers due to their capital. In
addition, we assume that oligopolistic firms adopt cost-plus pricing, which
also has excess capacity, and that the supply of output can be immediately
adapted to demand. Technological progress is not considered; both thepotential output–capital ratio and the output–labour ratio are assumed to beconstant. Finally, we abstract capital depreciation.
First, we consider costs. We represent the number of total workers as 𝑁,
the number of direct workers, which is variable according to output, as 𝑁𝑣 ,and the number of indirect workers, which is fixed according to output, as 𝑁𝑓 . 8Thus, we obtain 𝑁 = 𝑁𝑣 + 𝑁𝑓 . The employment of direct workers depends on7

the real output, 𝑌, and thus, their labour productivity is expressed by 𝑎𝑣 =𝑌/𝑁𝑣 (> 0). The amount of indirect workers depends on the full capacity output,
𝑌̅ , and accordingly, their productivity is expressed by 𝑎𝑓 = 𝑌̅/𝑁𝑓 (> 0) . We
represent the nominal wage of direct workers as 𝑤 and that of indirect
workers as 𝑤𝑓 ; the latter is larger than the former because of the wage
premium. For simplicity, we assume that the nominal wage of indirect workers
is 𝜎(> 1) times that of direct workers, that is, 𝑤𝑓 = 𝜎𝑤. 9
We assume that firms’ costs consist only of wages, and accordingly, total
costs are equal to

𝑤𝑁𝑣 + 𝑤𝑓 𝑁𝑓 . Thus, we express the unit cost as(𝑤𝑁𝑣 + 𝑤𝑓 𝑁𝑓 )⁄𝑌 . Moreover, by denoting the capacity utilisation rate as 𝑢(≡𝑌/𝑌̅), the unit cost can be rewritten as follows:

Unit cost =𝜎𝑓𝑤 (1 + 𝑢 )
𝑎𝑣

,𝑓 ≡

𝑎𝑣
.
𝑎𝑓

(1)

Firms are assumed to adopt normal cost pricing; here, normal cost pricing
indicates that the price is marked by normal costs (i.e. total costs divided bynormal output). This pricing is popular among large-scale corporations
because, in reality, firms tend to use normal output rather than real output
for cost calculation (Lee, 1998; Coutts and Norman, 2013). We represent as 𝑌𝑠
normal output, that is, output produced by the constant normal rate ofcapacity utilisation 𝑢𝑠 (≡ 𝑌𝑠 /𝑌̅). Then, firms set the price by multiplying themark-up rate 𝜇(> 0) on total costs per normal output:
𝑝 = (1 + 𝜇)𝑤(1 + 𝜎𝑓/𝑢𝑠 )
. (2)
𝑎𝑣

The higher the monopoly power or the stronger the bargaining power of thefirm, the higher the mark-up rate (Kalecki, 1971; Sen and Dutt, 1995). Fromequation (2), we obtain the real wage of direct workers in the case of normal
cost pricing:𝑤 𝑎𝑣
𝜎𝑓= , Δ ≡ (1 + 𝜇) (1 + ) > 1. (3)𝑝∆𝑢𝑠
The real wage of indirect workers is 𝜎 times that of direct workers. Note thatthe real wages of both worker types are constant, because we assume constantproductivity.
The income of direct workers, Π𝑣𝑤 , per capital stock, 𝐾, is composed only of8their labour income. Using equation (3), we obtain:
Π𝑣𝑤 𝑤𝑁𝑣 𝑢== . (4)𝑝𝐾
𝑝𝐾∆
In the following, we assume 𝑌̅/𝐾 = 1 for simplicity. Note that Π𝑣𝑤 /𝑝𝐾 moves
pro-cyclically because the amount of direct workers is proportional to the
output.
Income of indirect workers, Π𝑓𝑤 , per capital stock is composed of labour
income and financial income:Π𝑓𝑤 𝑤𝑓 𝑁𝑓 + 𝑖𝐿 𝜎𝑓==+ 𝑖𝜆, (5)𝑝𝐾
𝑝𝐾∆
where 𝐿 represents indirect workers’ capital borrowed by firms and 𝑖(> 0)
represents the constant interest rate, which is controlled by the central bank.
In addition, the endogenous variable 𝜆(≡ 𝐿/𝑝𝐾) represents the debt–capitalratio of firms, that is, the capital share of indirect workers. The first term on
the right-hand side (RHS) in equation (5) is constant because the number ofindirect workers does not change according to the output, whereas the second
term endogenously changes according to the capital share of indirect workers.
We define the wage share 𝜋 as the total labour income divided by the value
added as follows:
𝜎𝑓
𝑤𝑁𝑣 + 𝑤𝑓 𝑁𝑓 1 + 𝑢
𝜋≡
=
. (6)
𝑝𝑌
∆
Unlike most studies, in our model wage share is an endogenous variable andmoves counter-cyclically, because of labour hording.
Income of capitalists, Π𝑐 , per capital stock is the difference between value
added and total labour income:Π𝑐 𝑝𝑌 − Π𝑣𝑤 − Π𝑓𝑤
∆−1
𝜎𝑓
=
=(
)𝑢 −
− 𝑖𝜆.
𝑝𝐾
𝑝𝐾
∆
∆

(7)

As equation (7) shows, Π𝑐 /𝑝𝐾 changes pro-cyclically and moves opposite to
the capital share of indirect workers.
Next, we turn to savings and investments. We assume that indirect workers
save a constant fraction, 𝑠𝑤 , of their income, and capitalists save a constantfraction, 𝑠𝑐 , of their income. Total savings, 𝑆, in real terms, comprises bothindirect workers’ savings and capitalists’ savings. Using equations (5) and (7),the aggregate saving function is obtained as follows:
9

𝑆
Δ−1
𝜎𝑓
= 𝑠𝑐 () 𝑢 − (𝑠𝑐 − 𝑠𝑤 ) ( + 𝑖𝜆) , 𝑠𝑐 ∈ (0,1), 𝑠𝑤 ∈ (0,1). (8)𝐾
Δ∆
The Kaleckian models feature many variants of the desired investment
function. Charles (2008) assumes that investment is an increasing function ofretained profits. Hein (2006) uses a normal Kaleckian-type investment
function that positively relates the investment per unit of capital to the profit
rate and the rate of capacity utilisation. Our model simply assumes that real
investment, 𝐼, per unit of capital responds positively to firms’ profits:𝐼∆−1
𝜎𝑓= 𝛼 + 𝛽 [()𝑢 −− 𝑖𝜆] , 𝛼 > 0, 𝛽 > 0. (9)𝐾∆∆
3.2 Solving endogenous variables
The disequilibrium between investment and savings is instantaneously
adjusted through changes in capacity utilisation. The equilibrium rate ofcapacity utilisation is obtained from equations (8) and (9):𝑢=𝛼∆ − (𝑠𝑤 + 𝛽 − 𝑠𝑐 )(𝜎𝑓 + Δ𝑖𝜆). (10)(𝑠𝑐 − 𝛽)(Δ − 1)This equilibrium is stable if the investment is sufficiently insensitive tovariations in capacity utilisation, that is, if 𝑠𝑐 − 𝛽 > 0.
The rate of capacity utilisation is debt-led if 𝑠𝑤 + 𝛽 − 𝑠𝑐 < 0. This is because
increases in the capital share of indirect workers (i.e. debt–capital ratio) andthe interest rate in turn raise the consumption demand of indirect workers byraising financial income. In contrast, if 𝑠𝑤 + 𝛽 − 𝑠𝑐 > 0, the rate of capacity
utilisation is debt-burdened because decreases in the capital share of indirect
workers and interest rate stimulate investment demand. Thus, depending oneach agent’s propensity to save the coefficient of investment to profits, various
scenarios occur. This study focuses on monetary easing, in which the central
bank is expected to stimulate effective demand by guiding the interest rate toa lower level. This implies that the economy exhibits a debt-burdened
situation, where a fall in the interest rate increases the capacity utilisation
rate. Thus, we present the following assumption:
Assumption 1. The Keynesian stability condition,𝑠𝑐 − 𝛽 > 0 , is always

satisfied, whereas 𝑠𝑤 + 𝛽 − 𝑠𝑐 > 0 always holds, so that the economy exhibits
10debt burden.
The growth rate of the capital share of indirect workers is given by𝜆̇ ≡ 𝐹(𝜆) =𝑠𝑤 𝛽𝑖 2−𝑠𝑐 αΔ + 𝑠𝑤 𝛽𝜎𝑓
𝑠𝑤 𝜎𝑓
𝜆 + [𝑠𝑤 𝑖 +]𝜆 +, (11)(𝑠𝑐 − 𝛽)Δ
𝑠𝑐 − 𝛽Δ

where dots over variables denote time derivatives. 10 Equilibrium is definedas 𝜆̇ = 0. By solving 𝐹(𝜆) = 0, we obtain multiple equilibria (𝜆∗1 , 𝜆∗2 ). The low
equilibrium value, which is represented by 𝜆1∗ , is locally stable, whereas thehigh value, 𝜆∗2 , is locally unstable. 11
This study focuses on the mark-up rate and interest rate as distributive
parameters that affect the stable equilibrium value 𝜆1∗ . First, we consider theeffect of a change in the mark-up rate on the equilibrium values. Note that achange in the mark-up rate is synonymous with a change in Δ because
𝑑Δ/𝑑𝜇 > 0. By simple calculation, we obtain 𝑑𝜆1∗ /𝑑Δ < 0,12 which means thatan increase in the mark-up rate always reduces the debt–capital ratio. On theother hand, as the following equation shows, the effect of a change in themark-up rate on the rate of capacity utilisation is ambiguous:
−

+

𝑑𝑢
=
𝜕Δ

⏞∗
𝑑𝜆
−𝛼 + (𝑠⏞𝑤 + 𝛽 − 𝑠𝑐 ) [𝜎𝑓 + 𝑖𝜆1∗ − Δ(Δ − 1)𝑖 ( 1 )]𝑑Δ
(Δ − 1)2. (12)

If 𝛼 is sufficiently small (large) or 𝑖 is sufficiently large (small) that thenumerator of the RHS in equation (12) is positive (negative), we obtain
𝑑𝑢/𝑑∆> 0 ( 𝑑𝑢/𝑑∆< 0 ), which means that an increase in the mark-up rate
raises (reduces) the rate of capacity utilisation. Here, we define the demand
regime as follows.
Definition. We call the economy a profit-led demand regime when 𝑑𝑢/𝑑∆> 0
holds. In contrast, we call the economy a wage-led demand regime when𝑑𝑢/𝑑∆< 0 holds.
Regarding the effect of a change in the mark-up rate on the rate of capital
accumulation, we obtain the following equation:

11

−

⏞1∗
𝑑(𝐼/𝐾)
1∆ − 1 𝑑𝑢 𝜎𝑓𝑑𝜆= 𝛽( 2𝑢++ 2 −𝑖).
𝑑∆∆∆ 𝑑∆ ∆𝑑∆(13)

If the economy exhibits a profit-led demand regime, we have 𝑑(𝐼/𝐾)/𝑑∆> 0,
and thus, increasing the mark-up rate accelerates capital accumulation.
However, in the case of the wage-led demand regime, we obtain either𝑑(𝐼/𝐾)/𝑑∆> 0 or 𝑑(𝐼/𝐾)/𝑑∆< 0. Summarising the results leads to the following
proposition.
Proposition 1. An increase in the mark-up rate reduces the equilibrium rateof the capital share of indirect workers, irrespective of the demand regime. In
addition, if 𝛼 is sufficiently small (large) or the interest rate is sufficiently
large (small), a rise in the mark-up rate increases (decreases) the equilibrium
rate of capacity utilisation. Moreover, when the economy is a profit-led
demand regime, an increase in the mark-up rate always increases the rate ofcapital accumulation.
Next, as for the interest rate, we obtain 𝑑𝜆1∗ /𝑑𝑖 > 0, which means that a fallin the interest rate always reduces the capital share of indirect workers. In
addition, as long as assumption 1 is satisfied, equation (10) shows that adecrease in the interest rate with a decreasing capital share of indirect
workers raises the rate of capacity utilisation. Moreover, in general, thesaving propensity of capitalists is higher than that of indirect workers, andthus 𝑠𝑐 − 𝑠𝑤 > 0 holds. 13 In this case, equation (8) indicates that a fall in theinterest rate accelerates capital accumulation by decreasing the capital shareof indirect workers and increasing the rate of capacity utilisation. Thus, we
obtain the following proposition.
Proposition 2. A decrease in the interest rate reduces the equilibrium rate of

the capital share of indirect workers, increases the rate of capacity utilisation,and increases the rate of capital accumulation if Assumption 1 is satisfied.
4. Income inequality in terms of the Gini coefficient
4.1 Setting the baseline case12

This section investigates the effects of changes in the mark-up rate and theinterest rate on the Gini coefficient, which is graphically represented by the
Lorenz curve. To draw the Lorenz curve, we must find the cumulative
population and cumulative income. Therefore, we add new assumptions.
Assumption 2. There are four groups in the economy: in ascending order ofper capita income, the unemployed with no income, direct workers, indirect
workers, and capitalists. There are no income disparities within each group.
In addition, capitalists always exist as a constant proportion of the economy.
One way to reduce income inequality is to rescue people from the unemployed
pool. Therefore, as in Assumption 2, this section considers an unemployedperson with no income. For the sake of simplicity, the population share ofcapitalists is assumed to be constant. Moreover, the employment growth rates
of both direct and indirect workers are equal to the rate of capital
accumulation. 14 However, the population share of the unemployed is unclear
because it depends not only on the rate of capital accumulation but also on
the population (labour supply) growth.
Based on these assumptions, we first consider the cumulative population
ratio. Here, we assign arbitrary value to each share as a baseline case 15: theshare of the unemployed in the total population is 10 percent, the share of theemployee including direct and indirect workers is 80 percent, and the shareof capitalists is 10 percent. Moreover, the ratio of indirect to direct workers
follows 𝑁𝑓 /𝑁𝑣 = 𝑓/𝑢, which shows that the ratio decreases when the capacity
utilisation rate increases. Now, in the baseline case, suppose 𝑓 = 0.5 and 𝑢 =0.6 ; then 𝑁𝑓 /𝑁𝑣 is equal to 0.833 and the share of direct workers isapproximately 55 percent (i.e. the share of indirect workers is 25 percent).
Needless to say, the ratio of direct to indirect workers varies, because thecapacity utilisation rate is endogenously determined.
Next, we consider the cumulative income ratio. The income share of theunemployed is zero under Assumption 2. The income share of direct workers,
the share of indirect workers, and the share of capitalists are represented by
Π𝑣𝑤 1= , (14)𝑝𝑌∆
13

Π𝑓𝑤 𝜎𝑓 𝑖𝜆=+ , (15)𝑝𝑌∆𝑢 𝑢
Π𝑐
1 𝜎𝑓 𝑖𝜆=1− −− . (16)𝑝𝑌∆ ∆𝑢 𝑢As equation (14) shows, the income share of direct workers is constant,because the number of direct workers increases as much as the value addedin the economy. Equation (15) shows that the income share of indirect workers
moves counter-cyclically, whereas equation (16) indicates that the income
share of capitalists varies pro-cyclically. However, both the income shares ofindirect workers and capitalists are affected by a change in the capital shareof indirect workers.
In Figures 1 and 2, where the horizontal axis shows the cumulative
population ratio and the vertical axis shows the cumulative income ratio, theLorenz curve in the baseline case is depicted by a solid line. The Gini
coefficient is the area surrounded by the perfect equality line (line OB) andthe Lorenz curve, divided by the area of the triangle OAB.
(Figure 1 here)
4.2 The effect of a change in the mark-up rate
Let us investigate the effect of a change in the mark-up rate on the Gini
coefficient. First, we suppose that 𝛼 is sufficiently small or the interest rateis sufficiently small, and thus a profit-led regime occurs. As equation (14)shows, a rise in the mark-up rate reduces the income share of direct workers.
Moreover, Proposition 1 shows that a rise in the mark-up rate reduces thecapital share of indirect workers and raises the rate of capacity utilisation,
which in turn reduces the income share of indirect workers by decreasing boththe first and second terms of the RHS in equation (15). As a result, capitalist
income share increases. In Figure 1, since the income share of capitalists
increases with their constant population share, point c in the baseline casemoves to c’.
Furthermore, Proposition 1 shows that a rise in the mark-up rate stimulates
capital accumulation, and thus growth rates of employment among both direct
and indirect workers under the profit-led regime. If the rate of capital
accumulation exceeds the growth rate of labour supply due to the increasing
14

mark-up rate, the population share of the unemployed continues to decrease.
In Figure 1, we depict this case at a specific point in time where theintersection point of the Lorenz curve and the horizontal axis moves to pointe, and thus the Gini coefficient decreases. However, when a rise in the markup rate increases the rate of capacity utilisation, the ratio of indirect to direct
workers decreases according to 𝑁𝑓 /𝑁𝑣 = 𝑓/𝑢. The larger the value of 𝑓, thedramatically lower the ratio with decreasing capacity utilisation. As a result,point d in the baseline case moves to d’. A decrease in the ratio of indirect todirect workers positively affects the Gini coefficient because the income of theformer is larger than that of the latter.
In Figure 1, a new Lorenz curve with increasing mark-up under the profitled regime is represented by a dashed line. Although an increase in the markup rate may reduce the Gini coefficient by increasing employment, thedeclines in both income shares of direct and indirect workers are remarkable.
Thus, unless unemployment decreases significantly, increasing the mark-up
rate can expand income inequality, even under a profit-led demand regime.
Next, we consider how a mark-up rate change affects the Gini coefficientunder a wage-led demand regime. A rise in the mark-up rate always reduces
the income share of direct workers. On the other hand, unlike the profit -ledregime, the effects of a rise in the mark-up rate on the income shares ofindirect workers and capitalists are ambiguous. This is because a rise in themark-up rate increases ∆ but decreases the capacity utilisation rate, andthus has an ambiguous effect on the first term of the RHS in equation (15).
Moreover, a rise in the mark-up rate decreases both the capacity utilisation
rate and capital share of indirect workers, and thus has an unclear effect on
the second term. There are three patterns in which a rise in the mark-up rateaffects the income shares: the income share of indirect workers increases, butthat of capitalists decreases (Pattern 1); the income shares of both indirect
workers and capitalists increase (Pattern 2); and the income share of i ndirect
workers decreases, but that of capitalists increases (Pattern 3). Patterns 2 and3 have in common that capitalists benefit from a significant decrease in theincome share of direct workers, for example due to a large increase in themark-up rate. In Figure 1, we depict Pattern 2, where point c moves to pointc” and point d to d”.
15Even if a rise in the mark-up rate occurs under a wage-led regime, it does
not necessarily stagnate capital accumulation, because we find either𝑑(𝐼/𝐾)/𝑑∆> 0 or 𝑑(𝐼/𝐾)/𝑑∆< 0 in Section 2.2. In the latter case, the growthrate of employment is likely to be smaller than the growth rate of labour
supply, and thus the population share of the unemployed continues to increase,as depicted by point h. However, a rise in the mark-up rate decreases the rateof capacity utilisation, which in turn increases the ratio of indirect workers to
direct workers. As a result, it contributes slightly to a decrease in the Gini
coefficient.
A new Lorenz curve with an increasing mark-up rate under the wage-led
regime is depicted by the dotted line in Figure 1. This case, with increasing
Gini coefficient, depends on the supposition that an increase in the mark-up
rate reduces the capital accumulation below the growth rate of labour supply
and produces Pattern 2. However, there may be a possibility that an increasein the mark-up rate reduces the Gini coefficient under the wage-led regime.
For instance, this case holds under the condition that a rise in the mark-up
rate significantly increases the growth rate of employment despite the wageled regime.
4.3 The effect of monetary easing
Let us consider the effect of a decrease in the interest rate on the Gini
coefficient. From the viewpoint of the theory of endogenous money supply,
central banks can control the policy interest rate instead of money supply:
monetary easing by the central bank negatively affects interest rates.
How does a change in the interest rate affect the Gini coefficient? Even if
the interest rate changes, the income share of direct workers does not change.
As Proposition 2 shows, a fall in the interest rate reduces the capital share ofindirect workers and raises the rate of capacity utilisation, which decreases
the income share of indirect workers (see equation (15)). As a result, theincome share of capitalists increases. In Figure 2, since the income share ofcapitalists increases with their constant population share, point c in thebaseline case moves to c’’’.
(Figure 2 here)
From Proposition 2, a fall in the interest rate accelerates capital
16

accumulation and employment growth. If monetary easing is effective enoughand capital accumulation is higher than labour supply growth, the population
share of the unemployed will decrease. In Figure 2, we depict the case at a
specific point in time where the population share of the unemployed decreases
to point e. Moreover, when the rate of capacity utilisation increases with
monetary easing, the ratio of indirect workers to direct workers decreases. In
an extreme case, it may change point d to d’’’.
As the dashed line Lorenz curve in Figure 2 shows, monetary easing
negatively affects the Gini coefficient by increasing employment, but
positively affects the coefficient by increasing the income share of capitalists
and decreasing the ratio of indirect to direct workers. So, how can the negative
effect on the Gini coefficient be greater? Let us now consider a case where thepopulation share of the unemployed has decreased due to a fall in the interest
rate, but point d remains in the same position by chance as a result of thedecrease in the ratio of indirect to direct workers, which is represented by adotted line. In this case, the Gini coefficient decreases by doubling the triangle
area [(10 − 𝑒)/100](Π𝑣𝑤 /𝑝𝑌). Therefore, the negative effect of a fall in theinterest rate on the Gini coefficient depends not only on the degree of adecrease in the population share of the unemployed as well as the income
share of the direct workers. How can we increase the income share of direct
workers? It is obvious that the decreasing mark-up rate or decreasing wage
premium, 𝜎, leads to a decrease in ∆ and an increase in their income share.
Thus, to reduce income inequality, it is necessary not only to reduce theinterest rate but also to raise real wages so that the income share of direct
workers increases.
4.4 Discussion
Section 2 confirms that developed countries have experienced an expansion inincome inequality. This subsection briefly discusses the factors that widen
inequality and investigates their solutions in those countries.
The above simple analysis reveals that a rise in the mark-up rate is one of
the main factors that expands inequality, even under the profit-led demand
regime. From the Kaleckian viewpoint, it is emphasised that the bargaining
power of workers and market structure affect the level of the mark-up rate. 1617Let us consider the bargaining power of workers. Here, we identify thesources of bargaining power as the union density rate and collective
bargaining coverage. The former is the most traditional and useful index
reflecting workers’ bargaining power. However, if the collective agreement islegally guaranteed to apply to unionised workers, the union is still the
representative of all workers, and thus the union becomes powerful.
Figure 3 plots the combination of density rate and the bargaining coverageof each country in 1990s and in 2010s. It is clear from this figure that thebargaining power of workers cannot be measured by the density rate alone.
Although the density rate in France remains low, the collective bargaining
coverage is the highest among the countries considered here. As a result, it
seems that a fall in the income share of the lower income group was
suppressed, and thus the Gini coefficient increased only slightly.
(Figure 3 here)
In contrast, Germany experienced a significant decrease in density rate.
According to Thelen (2017), although labour market reforms in Germany
progressed in the early 2000s, workers who engaged in industries other thancore manufacturing were not organised. Even in the core industries, while theemployment of core workers continues to be protected, the increasing numberof short-term employees are excluded from the existing labour–management
relations. The coverage of collective bargaining declined significantly inproportion to a decrease in the density rate. Thus, the wages of the so-calledmini jobs have been reduced, and the Gini coefficient has increased
significantly. This is the reason why the legal minimum wage was introduced
in 2015.
The situation in Japan is similar to that in Germany. In Japan, both thedensity rate and coverage of collective bargaining are low, and thesedownward trends cannot be stopped. In particular, deregulation of the labo ur
market has progressed since the late 1990s, and non-regular workers, whichcurrently account for about 40 percent of total workers, have not been
sufficiently organised. On the other hand, the minimum wage is set separately
by each prefecture. Although the Gini coefficient is increasing, the incidenceof low-pay is declining due to the improvement in the minimum wage year
over year.
18Now, we turn to the market structure, which affects the mark-up rate. As
the ‘superstar hypothesis’ of Autor et al. (2020) suggests, some technologicallysuperior companies gain a higher market share, resulting in high mark-up
rates. Autor et al. (2020) statistically show that market share tends toincrease in six major US industries. However, unlike the United States,
Cavalleri et al. (2019) find that market share is stable in the EU. In Japan,
where some traditional manufacturing companies are leading the macro
economy, the hypothesis that weakening the bargaining power of workers
increases the mark-up rate seems more valid. Furthermore, we have to see
product market regulation as a factor that influences the market structure.
Roughly speaking, when product market regulation is strong, the market
tends to be oligopolistic. However, by checking product market regulation
indicators (PMR) in the OECD, we do not find that regulation has beentightened: according to the comparable 1998 and 2013 indicators, the PMR
was smaller in 2013 than in 1998 in any country. 17
Finally, we examine the conditions for monetary easing to improve income
inequality. In developed countries, untraditional monetary policy has caused
the base rates (US Fed funds rate, ECB main refinancing rate, and Japanese
overnight call rate) to fall to extremely low levels. However, whether suchpolicy succeeds in improving inequality depends on the mark-up rate. In somecountries with low bargaining power of workers, such as Germany, the UnitedStates, and Japan, even if monetary easing improves employment, the effectof suppressing inequality is offset by the effect of lower interest rates, leading
to a higher income share for capitalists. Thus, in order to improve pre-tax andpre-transfer income inequality in those countries, it is necessary to intervenein the wage determination process by raising the minimum wage, which iscalled ‘direct redistribution’ in Piketty (2015).
5. Conclusion
This study builds a theoretical framework that connects functional income
distribution and personal income distribution from the Kaleckian viewpoint,
and investigates the effects of changes in the mark-up rate and of monetary
policy on income inequality, which is expressed in terms of the Gini coefficient.
Our results can be summarised as follows.
19

First, a rise in the mark-up rate is likely to increase the Gini coefficient,regardless of whether there is a profit-led or wage-led demand regime. A risein the mark-up rate leads to a decline in the capital share of indirect workers,
which in turn increases the income share of capitalists while putting
downward pressure on the financial income of indirect workers. At the sametime, a rise in the mark-up rate always reduces the labour income of direct
workers, which positively affects the Gini coefficient.
Second, a decrease in the interest rate due to monetary easing has positive
and negative effects on the Gini coefficient. On the one hand, a lower interest
rate reduces the income share of indirect workers, increases the income shareof capitalists, and thus increases the Gini coefficient. On the other hand, afall in the interest rate increases employment growth and decreases the Gini
coefficient. The combination of these two contradictory effects determines
what the Gini coefficient will ultimately be. To increase the effect of lowering
the Gini coefficient, it is necessary to reduce the mark-up rate sufficiently. In
other words, monetary easing and reduction of the mark-up rate must be
implemented as a set to improve inequality. This will be an answer to thequestion of why the untraditional monetary easing in developed countries
does not have enough effect on the reduction of inequality.
The issues remaining after this study are now briefly described. First, thisstudy abstracts from workers whose consumption is based on debt
accumulation. Although household debt accumulation has slowed since thefinancial crisis in 2008, lower interest rates leave room to improve inequality
by curbing households’ interest payment burden. Second, by endogenizing theemployment rates, the shares of the unemployed and the employed canactually be calculated. In this case, the assumption of constant labour
productivity is relaxed.
References
Alvaredo, F., Chancel, L., Piketty, T., Saez, E. and Zucman, G. 2018. World

Inequality Report 2018 , Cambridge, MA, The Belknap Press of Harvard
University Press
Amable, B. 2004. The Diversity of Modern Capitalism , Oxford, Oxford
University Press
20

Atkinson, A.B. 2015. Inequality: What Can Be Done? Cambridge, MA,
Harvard University Press
Atkinson, A.B., and Bourguignon, F. 2000. Introduction, pp. 1–57 in Atkinson,
A.B. and Bourguignon, F. (eds.) Handbook of Income Distribution vol. 1 ,Amsterdam, Elsevier
Autor, D., Dorn, D., Katz, L.F., Patterson, C. and Reenen, J.V. 2020. The fallof the labor share and the rise of superstar firms, The Quarterly Journal of

Economics , vol. 135, no. 2, 645–709
Baccaro, L. and Pontusson, J. 2016. Rethinking comparative political
economy: the growth model perspective, Politics & Society , vol. 44, no. 2,
175–207
Bengtsson, E. and Waldenström, D. 2018. Capital shares and income
inequality: evidence from the long run, The Journal of Economic History ,vol. 78, no.3, 712–743
Bernanke, B.S. 2015. Monetary policy and inequality,
http://www.brookings.edu/blogs/ben-bernanke/posts/2015/06/01-monetarypolicy-and-inequality
Bhaduri, A. and Marglin, S.A. 1990. Unemployment and the real wage: theeconomic basis for contesting political ideologies, Cambridge Journal of

Economics , vol. 14, no. 4, 375–93
Carvalho, L. and Rezai, R. 2016 Personal income inequality and aggregate
demand, Cambridge Journal of Economics , vol. 40, no. 2, 491–505
Cavalleri, M.C., Eliet, A., McAdam, P., Petroulakis, F., Soares, A. and
Vansteenkiste, I. 2019. Concentration, market power, and dynamism in euroarea, ECB working paper series, No.2253
Charles, S. 2008. Corporate debt, variable retention rate and the appearanceof financial fragility, Cambridge Journal of Economics , vol. 32, no. 5, 781–
95
Coutts, K. and Norman, N. 2013 Post-Keynesian approach to industrial
pricing, pp. 443–84 in Harcourt, G. C. and Kriesler, P. (eds.) The Oxford

Handbook of Post-Keynesian Economics Volume 1: Theory and Origins ,Oxford, Oxford University Press
Dallery, T. 2009 Post-Keynesian theories of the firms under financialization,

Review of Radical Political Economics , vol. 41, no. 4, 492–515
21

Draghi, M. 2016. Stability, Equity and Monetary Policy. 2nd DIW Europe
Lecture,http://www.ecb.europa.eu/press/key/date/2016/html/sp161025.en.html
Dutt, A.K. 1984. Stagnation, income distribution and monopoly power,

Cambridge Journal of Economics , vol. 8, no. 1, 25–40
Dutt, A.K. 2016. Growth and distribution in heterodox models with man agers
and financiers, Metroeconomica , vol. 67, no. 2, 364–96
Dünhaupt, P. 2017. Determinants of labour’s income share in the era offinancialisation, Cambridge Journal of Economics , vol. 41, no. 1, 283–306
Ederer, S. and Rehm, M. 2020a. Will wealth become more concentrated inEurope? Evidence from a calibrated Post-Keynesian model, Cambridge

Journal of Economics , vol. 44, no. 1, 55–72
Ederer, S. and Rehm, M. 2020b. Making sense of Piketty’s ‘fundamental laws’in a Post-Keynesian framework: the transitional dynamics of wealthinequality , Review of Keynesian Economics , vol. 8, no. 2, 195–219

Furceri, D., Loungani, P. and Zdzienicka, A. 2018. The effects of monetary
policy shocks on inequality, Journal of International Money and Finance ,vol. 85, July, 168–86
Glyn, A. 2009. Functional distribution and inequality, pp. 101–26 in Salvedra,W., Nolan, B. and Smeeding, T.M. (eds.), The Oxford Handbook of Economic

Inequality , Oxford, Oxford University Press
Gustafsson, B. and Jansson, B. 2008. Top incomes in Sweden over threequarters of a century: a micro data approach, Cambridge Journal of

Economics , vol. 32, no. 6, 963–76
Hall, P. and Soskice, D. 2001. Varieties of Capitalism: The Institutional

Foundations of Comparative Advantage , Oxford, Oxford University Press
Hein, E. 2006. Interest, debt and capital accumulation: a Kaleckian approach,International Review of Applied Economics , vol. 20, no. 3, 337–52
Hein, E. 2007. Interest rate, debt, distribution and capital accumulation in apost Kaleckian model, Metroeconomica , vol. 56, no. 2, 310–39
Hein, E., Meloni, W. and Tridico, P. 2020. Welfare models and demand-led
growth regimes before and after the financial and economic crisis, Review

of International Political Economy ,https://doi.org/10.1080/09692290.2020.1744178
22

Kalecki, M. 1971. Selected Essays on the Dynamics of the Capitalist Economy ,Cambridge, Cambridge University Press
Kohler, K., Guschanski, A., and Stockhammer, E. 2019 The impact offinancialisation on the wage share: a theoretical clarification and empiricaltest, Cambridge Journal of Economics , vol. 43, no. 4, 937–74
Krugman, P. 2014. Notes on Easy Money and Inequality, The New York Times,
25 October 2014. https://krugman.blogs.nytimes.com/2014/10/25/notes-oneasy-money-and-inequality/
Lavoie, M. 1995. Interest rate in post-Keynesian models of growth anddistribution, Metroeconomica , vol. 46, no. 2, 146–77
Lavoie, M. 2009. Cadrisme within a Kaleckian model of growth anddistribution, Review of Political Economy , vol.21, no.3, 369–91
Lavoie, M. and Stockhammer, E. 2013. Wage-led growth: Concept, theories
and policies, pp. 13–39 in Lavoie, M. and Stockhammer, E. (eds.) Wage-Led

Growth: An Equitable Strategy for Economic Recovery , Basingstoke,
Palgrave Macmillan
Lee, F. 1998. Post Keynesian Price Theory , Cambridge, Cambridge University
Press
Mohun, S. 2006. Distributive shares in the US economy, 1964–2001,

Cambridge Journal of Economics , vol. 30, no. 3, 347–70
Molero-Shimarro, R. 2017. Growth and inequality revisited: the role ofprimary distribution of income. A new approach for understanding today’s
economic and social crises, Cambridge Journal of Economics , vol. 41, no. 2,
367–90
Nishi, H. 2015. Comparative evaluation of post-Keynesian interest rate rules,
income distribution and firms’ debts for macroeconomic performance,

Cambridge Journal of Economics , vol. 39, no. 1, 189–219
Onaran, O. and Obst, T. 2016 Wage-led growth in the EU15 member-states:
The effects of income distribution on growth, investment, trade balance andinflation, Cambridge Journal of Economics , vol. 40, no. 6, 1517–51
Palley, T. 2017. Wage- vs. profit-led growth: the role of the distribution ofwages in determining regime character, Cambridge Journal of Economics ,vol. 41, no. 1, 49–61

23

Piketty, T. 2014. Capital in the Twenty-First Century , Cambridge, MA,
Harvard University Press
Piketty, T. 2015. L’économie des Inégalités , Paris, La Découverte
Pontusson, J. 2006. Inequality and Prosperity: Social Europe vs. Liberal

America , Ithaca, Cornell University Press
Rowthorn, R.E. 1981. Demand, real wages and economic growth, Thames

Papers in Political Economy , Autumn
Ryoo, S. 2018. Top income shares and aggregate wealth income ratio in a twoclass corporate economy, Cambridge Journal of Economics , vol. 42, no. 3,
699–728
Saez, E. and Zucman, G. 2019. The Triumph of Injustice: How the Rich Dodge

Taxes and How to Make Them Pay , New York, W.W. Norton & Company
Sasaki, H. and Fujita, S. 2012. The importance of the retention ratio in a
Kaleckian model with debt accumulation, Metroeconomica , vol. 63, no. 3,
417–28Schlenker, E. and Schmid, K.D. 2015. Capital income shares and income
inequality in 16 EU member countries, Empirica , vol. 42, 241–68
Schneider, M.R. and Paunescu, M. 2012. Changing varieties of capitalism andrevealed comparative advantages from 1990 to 2005: a test of the Hall andSoskice claims, Socio-Economic Review , vol. 10, no. 4, 731–53
Sen, A. and Dutt, A.K. 1995. Bargaining power, imperfect competition and themarkup: optimizing microfoundations, Economic Letters , vol. 48, no. 1, 15–
20
Stockhammer, E. 2004. Financialisation and the slowdown of accumulation,

Cambridge Journal of Economics , vol. 28, no. 5, 719–41
Tavani, D. and Vasudevan, R. 2014. Capitalists, workers, and managers: Wageinequality and effective demand, Structural Change and Economic

Dynamics , vol. 30, September, 120–31
Taylor, L. 2004. Reconstructing Macroeconomics: Structuralist Proposals andCritiques of the Mainstream , Cambridge, MA, Harvard University Press
Taylor, L., Rezai, A., Kumar, R. and Carvalho, R. 2017. Wage increases,transfers, and the socially determined income distribution in the USA,

Review of Keynesian Economics , vol. 5, no. 2, 259–75

24

Thelen, K. 2014. Varieties of Liberalization and the New Politics of Social

Solidarity , New York, Cambridge University Press
Wolff, E. N. 2015. Inequality and rising profitability in the United States,
1947–2012, International Review of Applied Economics , vol. 29, no. 6, 741–
69
Appendix (a)The growth rate of the capital share of indirect workers is given by
𝐿̇ 𝐾̇
𝐿̇
𝑆
𝜆̇ = ( − ) 𝜆 =− 𝜆. (A1)𝐿 𝐾
𝑝𝐾 𝐾
We have 𝑝̇ /𝑝 = 0 from equation (2) and 𝐾̇ /𝐾 = 𝐼/𝐾 = 𝑆/𝐾 because the goods
market clears.
An increment of debt in a firm is the difference between investment andfirms’ (capitalists’) retained earnings. From equations (7) and (8), we have:𝐿̇𝜎𝑓= 𝑠𝑤 ( + 𝑖𝜆). (A2)𝑝𝐾
Δ
Substituting equations (8), (10), and (A2) in (A1) and rearranging yields:𝜆̇ ≡ 𝐹(𝜆) =𝑠𝑤 𝛽𝑖 2−𝑠𝑐 αΔ + 𝑠𝑤 𝛽𝜎𝑓
𝑠𝑤 𝜎𝑓
𝜆 + [𝑠𝑤 𝑖 +]𝜆 +. (11)(𝑠𝑐 − 𝛽)Δ
𝑠𝑐 − 𝛽ΔThe equilibrium value of the capital share of indirect workers 𝜆∗ is obtainedby solving 𝐹(𝜆∗ ) = 0 . Using the (𝜆, 𝜆̇) plane, as shown in Figure 4, theexistence of a positive equilibrium value and its local stability can be easily
verified. The function 𝐹(𝜆) shows a parabola with its vertex oriented
downward, and its intercept is always positive. The necessary and sufficientconditions for the existence of a positive equilibrium value are that thediscriminant of𝐹(𝜆) = 0

is positive. If𝛼is sufficiently large, thediscriminant is positive, which we assume in the following discussion.
(Figure 4 here)
Appendix (b)Figure 4 shows that our model has multiple equilibria (𝜆1∗ , 𝜆∗2 ), where the low
equilibrium, 𝜆1∗ , is locally stable and the high one, 𝜆∗2 , is locally unstable.
Additionally, note that the following condition is always satisfied in the stable
positive equilibrium:25

𝑑𝜆̇ 𝜕𝐹(𝜆)𝑠𝑤 𝛽𝑖 ∗
−𝑠𝑐 αΔ + 𝑠𝑤 𝛽𝜎𝑓==2
𝜆 + [𝑠𝑤 𝑖 +] < 0. (A3)𝑑𝜆𝜕𝜆𝑠𝑐 − 𝛽(𝑠𝑐 − 𝛽)ΔEquation (A3) holds if 𝛼 is sufficiently large or 𝑖 is sufficiently small. We
assume a stable equilibrium, 𝜆1∗ .
Appendix (c)Totally differentiating 𝐹(𝜆∗1 ; Δ) = 0 yields
𝜕𝐹(𝜆∗1 ; Δ))
𝜕Δ=−
, (A4)
𝜕𝐹(𝜆∗1 ; Δ)
𝑑Δ
(
)
𝜕𝜆

𝑑𝜆1∗

(

where the numerator is given by
𝜕𝐹(𝜆1∗ ; Δ)𝑠𝑤 𝛽𝜎𝑓𝑠𝑤 𝜎𝑓=−
𝜆1∗ − 2 < 0.
2
𝜕Δ
(𝑠𝑐 − 𝛽)ΔΔ
The sign of 𝜕𝐹(𝜆∗ ; Δ)/𝜕𝜆 is negative from equation (A4), and thus we obtain𝑑𝜆∗ /𝑑Δ < 0.
Totally differentiating 𝐹(𝜆∗1 ; 𝑖) = 0 yields
𝜕𝐹(𝜆1∗ ; 𝑖)
(
)
𝑑𝜆
𝜕𝑖
=−
, (A5)
𝜕𝐹(𝜆1∗ ; 𝑖)
𝑑𝑖
(
)
𝜕𝜆
∗

where the numerator is given by:
𝜕𝐹(𝜆∗ ; 𝑖)𝑠𝑤 𝛽 ∗2=2
𝜆 + 𝑠𝑤 𝜆1∗ > 0.
𝜕𝑖𝑠𝑐 − 𝛽 1
Thus, we obtain 𝑑𝜆∗ /𝑑𝑖 > 0.

26

France

Germany

Italy

SwedenJapan

US

Gini coefficient from

Gini coefficient from

Pre-tax income

Low-pay incidence

LIS

SWIID

share of bottom

from OECD

Pre-tax

Post-tax

Pre-tax

Post-tax

50% from WID

1990s

0.486

0.284

0.476

0.282

21.352

2000s

0.473

0.286

0.478

0.284

22.022

2010s

N.A.

N.A.

0.490

0.297

22.367

1990s

0.456

0.260

0.461

0.257

22.991

15.560

2000s

0.502

0.282

0.505

0.278

20.85117.789

2010s

0.512

0.292

0.519

0.289

18.77018.556

1990s

0.465

0.331

0.466

0.327

23.09910.056

2000s

0.485

0.328

0.485

0.327

22.399

9.298

2010s

0.549

0.336

0.513

0.335

21.328

8.443

1990s

0.461

0.236

0.469

0.235

27.046

2000s

0.445

0.248

0.483

0.245

25.624

2010s

N.A.

N.A.

0.507

0.261

24.778

1990s

0.419

0.299

2000s

0.442

0.310

2010s

0.451

0.319

N.A.

N.A.
15.43

N.A.

14.94713.241

1990s

0.478

0.365

0.476

0.355

15.29624.353

2000s

0.491

0.378

0.490

0.366

14.05724.175

2010s

0.508

0.383

0.508

0.380

12.62324.700

Table 1. Gini coefficients and other indicators in developed countries.
Source: Luxembourg Income Study Database (author’s calculation), Solt, F (2019) “Measuring Income Inequality across Countries andOver Time: The Standardized World Income Inequality Database.” SWIID version 8.2, November 2019, W orld Inequality Database,27

OECD stat. database.
Note: We omits the Gini coefficient in Japan because there are only data for 2008, 2010, and 2013. The WID has no data on the pre-tax
income share of the bottom 50 percent in Japan, and the OECD has no data on low-pay incidence in France and Sweden. All values areaverages.

28

Cumulative incomeBΠ𝑐 /𝑝𝑌cc’’
Lorenz curvec’Π𝑓𝑤 /𝑝𝑌dd’d’’Π𝑣𝑤 /𝑝𝑌

O

e

10

h

65

90

A

Cumulative population

Figure 1. Shift of Lorenz curve with increasing mark-up rate
Note: The solid line indicates the baseline case, the dashed line indicates the shift
under the profit-led demand regime, and the dotted lines indicate the shift under the
wage-led regime.

29

Cumulative incomeBΠ𝑐 /𝑝𝑌cc’’’

Π𝑓𝑤 /𝑝𝑌dd’’’Π𝑣𝑤 /𝑝𝑌

O

e

10

65

90

A

Cumulative population

Figure 2. Shift of Lorenz curve with decreasing interest rate
Note: The solid line indicates the baseline case, and both dashed and dotted lines
indicate the case of decreasing interest rates.

30

Bargaining
Coverage

100.0
90.0

France 2014

France 1997Germany 1995

Sweden 2015
Sweden 1995

Italy 1995

80.0
Italy 2014
70.0
Germany 2015

60.0
50.0
40.0

United States
1995

30.0

Japan 1995

20.0
Japan 2015
10.0
United States
2014

0.0
0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

Figure 3. Union density rate and collective bargaining coverage
Source：OECD.stat database.

31

100.0

Density rate

𝜆̇
𝐹(𝜆)

O

𝜆1∗

𝜆∗2

𝜆

Figure 4. Equilibria of the capital share of indirect workers and theirstability
The classical two-class model with capitalists and workers in Atkinson and
Bourguignon (2000, pp.5-9) shows that a rise in the profit share positively
affects the Gini coefficient, though this depends on the assumptions thatworkers acquire only wages, capitalists gain only profits, and per capita income
within a class is completely equal. Our model is extended version of their modelin several respects, which relates functional income distribution and the Gin icoefficient.
2 There are other interesting models. Carvalho and Rezai (2016) and Tavani
and Vasudevan (2014) study the effect of inequality in personal income
distribution on growth. Taylor et al. (2017) and Ederer and Rehm (2020b)
investigate the fiscal policy and tax system effects on growth and inequality.
3 Whether monetary easing has contributed to widening inequality or not has
become a major issue in recent years. See Krugman (2014), Bernanke (2015),
and Draghi (2016).
4 According to Hall and Soskice (2001), capitalistic countries are divided into
two types, coordinated market economies and liberal market economics,
depending on institutional structure. Amable (2003), using actor analysis andcluster analysis, classifies them into five types: the social democratic type, suchas Sweden and Finland; Europe continental type, such as Germany and France;
132

Mediterranean type, such as Italy and Spain; Anglo-Saxon type, such as theUnited States and the United Kingdom; and Asian type such as Japan andSouth Korea.
5 Needless to say, however, the Gini coefficient is not universal. See Atkinson(2015) and Alvaredo et al. (2018) for the explanation.
6 Monetary easing has the effect of increasing the price of financial assets,
which widens the gap between those who have assets and those who do not.
However, such a process does not seem to last due to the convergence of the
bubbles. Thus, we abstract from dynamics of the asset market.
7 For simplicity capitalists are assumed not to acquire any labour income.
8 In Tavani and Vasudevan (2014), the ratio of managerial workers to
productive workers is assumed to be always fixed because the former has tomonitor the latter.
9 Our model follows the formulations of Lavoie (2009) and Tavani and
Vasudevan (2014).
10 See Appendix (a) for the derivation of equation (11).
11 See Appendix (b) for the local stability of the equilibrium.
12 See Appendix (c) for the comparative statistics.
13 Carvalho and Rezai (2016) demonstrates that the higher the income, thelarger the propensity to save. Thus, this paper also assumes that the saving
propensity of capitalists is higher than that of indirect workers.
14 Since labour productivity of direct workers is assumed to be constant, thegrowth rate of their employment is equal to the growth rates of output and capital.
Moreover, we assume the constant potential output–capital ratio and labour
productivity of indirect workers, the growth rate of indirect workers’ employment
is equal to the rate of capital accumulation.
15 We can calculate the rate of capital accumulation by substituting theequilibrium values of the capital share of indirect workers and the capacity
utilization rate into equation (8). Thus, we can also calculate the intertemporal
numbers of direct and indirect workers if the initial values and parameters ofthe model are given. Moreover, if the growth rate of labour supply and its initialvalue are given, we can calculate the number of the unemployed. However, ourpurpose can be fully archived without such a calculation.
16 There are other factors that affect the mark-up rate: the interest rate andfinancialization. In our model, the interest rate is independent from the markup rate. Financialization raises financial payments such as dividends andinterests by instilling shareholder value orientation in firms, which in turn
pass on to a rise in the mark-up rate and reduces the wage share. This isempirically verified by Dünhaupt (2017) and Kohler et al. (2019).
17 The PMR from 1998 to 2013 is roughly composed of three categories: state
control, barriers to entrepreneurship, and barriers to trade and investment. In
each country covered in this paper, all of these indicators decreased, with one
exception: the US state control increased from 1.96 in 1998 to 2.70 in 2013.

33