International Kuznets Curve (?):
A Schumpeterian Model of the World Economy
Tetsugen HARUYAMAMarch 2021
Discussion Paper No. 2112

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANInternational Kuznets Curve (?):
A Schumpeterian Model of the World Economy
Tetsugen HARUYAMA∗
March 2021

Abstract
The Industrial Revolution boosted growth in Britain, and as it diffuses to other countries, growthin the world economy accelerated. At the same time, inter-country inequality increased. On the other
hand, the recent literature and the 2018 Maddison dataset, the Penn World Table 9.1 and the World
Inequality Database suggest that inter-country inequality fell in recent decades. This implies thatinter-country inequality takes an inverted U shape, i.e. an International Kuznets Curve. The aim of
this paper is to build a Schumpeterian growth model of the world economy to explain this hypothesis.
In the model there is a continuum of countries, and transitional dynamics involve a series of take-offs ofcountries which are initially backward. Simulations show that the International Kuznets Curve emerges
in a transitional process of the world economy converging to the long-run equilibrium.

∗

Graduate School of Economics, Kobe University. Email: haruyama@econ.kobe-u.ac.jp

1GDP per capita in Different Regions (int'l $)50000
Africa40000

East Asia
Eastern Europe

30000

Figure 1: Source: the 2018
Maddison dataset.

Latin America

20000

Western Asia
Western Europe

10000

Western Offshoots

0
1870

1

1900

1940

1980

2016

Introduction

The Industrial Revolution in Britain in the 18-19th century was an unprecedented shock to the world
economy. Its massive effects spread throughout the world only gradually, and its full effects do not seemto have been realized even today. Indeed, Crafts and O’Rourke (2014) wrote, “[The Industrial Revolution]
set in train a variety of long-run adjustment processes which are still ongoing, ....”Such adjustments manifest themselves in the form of changes in the distribution of the living standard,measured by GDP per capita, across countries in the world. Following Britain, continental European
countries took off in terms of productivity improvement and new technologies introduced. The diffusionof industrialization raised income of those economies, but diffusion itself was a slow process with many
countries left behind. Indeed, Figure 1 shows diverging fortunes of 7 regions in the world. It is not difficult
to imagine that if data of GDP per capita of all countries in the world were to exist since the Industrial
Revolution, such inequality measures as the Gini coefficient would have steadily increased throughout the
19th century at least. In fact, this is emphatically implied by Bourguignon and Morrisson (2002) who
use data of 15 countries and 18 groups in the period of 1820-1992. They demonstrate that inter-country
inequality, which assumes individuals having the same income within each country/group, steadily anddramatically increased from a very low initial level. The increasing trend continued well into the 20th
century.1
In the 20th century, on the other hand, economic take-offs spread to more and more countries, including
Japan, what used to be known as Asian Four Tiger economies, and India and China in more recent decades.
The fact that more countries have been industrialized, while a growth pace of developed economies slowed
down generated a downward force on the degree of inequality between countries. Indeed, Milanovic (2016)shows that the Gini coefficient of GDP per capita between countries, calculated using the 2018 Maddison
dataset, has started declining around in 2000. Indeed, as the next section shows, the time series ofinequality indices, such as the Gini coefficient, show an inverted U shape with a peak in 2000. In addition,
1
In the literature, several concepts of inequality exist. For example, see Anand and Segal (2008). The present paper
concerns inequality between countries with GDP per capita of country as a unit of observation. We call it here between
inequality and inter-country inequality interchangeably.

2Inter-country
IncomeInequality
Figure 2: An image of the International Kuznets Curve hypothesis.

Stagnation PeriodTime

The Industrial
Revolution

the same calculation using the Penn World Table 9.1 and the World Inequality Index indicate that thoseinequality indices have a clear downward trend since 1970 and 1950, respectively. A similar observation ismade in Firebaugh (2003) and his other studies. Combined with the finding of Bourguignon and Morrisson(2002), this suggests the existence of a turning point of inter-country inequality.
The turnaround of inequality trends suggests a hypothesis of a Kuznets curve in terms of inter-country
inequality in the world. Namely, as the world economy grows breaking out of the Malthusian period,
inequality between countries initially increases, followed by the reversal of the inequality trend, as envisaged in Figure 2. It is no doubt that the Industrial Revolution and the subsequent gradual diffusion ofindustrialization in the form of sequential take-offs is a key driving force behind the emergence of what
we call the International Kuznets Curve.2
Then, the following questions arise. How do we explain the International Kuznets Curve hypothesis?
What is the mechanism of a gradual diffusion industrialization? What are necessary conditions for sequential take-offs to occur? What are the implications of sequential take-offs for dynamics of inequality
among countries?
The main aim of the present paper is to offer answers to those questions by building a Schumpeterian
growth models of the world economy. A starting point of our analysis is to take the Industrial Revolutionin Britain as given. That is, we assume that in the initial period there is a single country (Britain in thiscase) which is already industrialized and examine a diffusion process of industrialization to other countries.
By industrialization, we mean the start of producing new goods and more sophisticated products based on
new technologies. In particular, using the Schumpeterian technological progress framework, we focus on
knowledge accumulation in the diffusion process of industrialization. This is motivated by Jones and Romer(2010) who underline the importance of idea flows and technology adoption in the catch-up process3 andthe survey result of Caselli (2005) that differences in total factor productivity account for at least half of
2

The word international is meant to indicate between countries, as opposed to world or global. Bourguignon and Morrisson(2002) use world to denote inequality between and within countries combined. In Milanovic (2016), world inequality means
between inequality, and global indicates between and within inequality combined.
3
Pointing out the result of Caselli and Feyrer (2007) that the marginal product of capital is remarkably similar acrosscountries, Jones and Romer (2010) note “While the textbook transition dynamics - driven by diminishing returns to capital
accumulation - are elegant and easy to explain, they are most likely not especially relevant to catch-up growth in practice.”(p.232)3

differences in income per capita.
To be more specific, we build a model of a world economy consisting of the base economy (Britain)
and a continuum of countries. In the base economy, new knowledge is created, introducing new and betterindustrial products into the economy. On the other hand, other economies are initially backward andspecialized in agricultural production. In the initial equilibrium, the base economy and other economies
exchange high-tech and agricultural goods through international trade. Heterogeneity of countries are
introduced in terms of research productivity, and the base economy has the highest productivity level. Ina long-run equilibrium, a group of economies with relatively higher productivity levels engage in innovative
activities and export high-tech goods, and a group of other countries remain specialized in agricultural
production. We call the former the North, and the latter the South, and such division of countries into
two groups is determined by equilibriating forces of international trade. Indeed, the North can consist of
the base economy only if no diffusion of industrialization occurs in long run.
A more interesting aspect of the model lies in transitional dynamics. Starting from the initial condition
where the base economy only is innovating, if industrialization spreads to other economies, it occurs only
gradually so that Southern economies sequentially take off to join the Northern club.4 As the world
economy converges to steady state, the average Norther wages relative to the Southern wage also increase,tending to increase inter-country income inequality. In addition, the world economy can spiral aroundthe long-run equilibrium, so that a phase of sequential take-offs and a phase of sequential falling-behindalternate, as it eventually converges to steady state. An important factor in generating different transitional
patterns is what we call the take-off externality. This is a positive externality which captures widening
opportunities for learning in research. The more countries that engage in R&D, the more likely that
researchers learn different perspectives with diverse experiences. In line with this reasoning, the externality
raises R&D productivity of all countries, as more and more of economies are industrialized. In the model,
the take-off externality has an effect of “pulling” Southern economies out of backwardness. Indeed, if the
take-off externality is weak (i.e. a weak “pulling” effect), the world economy is stuck in the initial condition
with the base economy only in the North. If the externality is in the intermediate range, the diffusion ofindustrialization can occur (possibly with spirals) under a certain condition. With the strong externality,
sequential take-offs occur (possibly with spirals) with no further qualification.
Focusing upon the last two cases, we conduct numerical simulations. A common pattern that emerges
is that the diffusion of industrialization is a slow process with the initial world growth rate being low. The
Gini coefficient is also low. As more and more countries take off, growth accelerates, and it decelerates
as the world economy approaches steady state. That is, growth follows a sigmoid pattern. Regarding
inequality, the International Kuznets Curve emerges, measured by the Gini Coefficient. The mechanism
is driven by two opposing effects. Obviously, sequential take-offs increases inter-country inequality. On
the other hand, take-offs are also a catch-up process. As more countries move out of backwardness, they
achieve higher income levels. The Gini may start falling because the more countries that are industrialized,
the less number of countries that remain backward.
Our paper is related to the literature on the long-run growth and the Big Divergence (see Pomeranz(2000) and Pritchett (1997)). In Galor and Mountford (2008) and Galor and Mountford (2006) examine
the role of international trade in generating differential effects on fertility in developed and developing4

For example, see Williamson (2010) for take-offs of peripheral countries.

4

countries. In particular, trade discourages fertility in developed countries, but encourages it in developing
countries via unskilled intensive production. This translates into divergence of per capita living standards.
O’Rourke et al. (2019) is related to those two studies, but different in that the terms of trade is endogenized
and technology transfer are introduced. Galor et al. (2009) considers the effect of land ownership concentration which affects the transition from an agricultural to an industrial economy through human capital
accumulation. Lucas (2009) consider a simple model of knowledge accumulation and diffusion. The focusof those studies, however, is on the role of fertility and human capital accumulation, which quite differsfrom our Schumpeterian approach. Much closer to our study is Howitt and Mayer-Foulkes (2005). They
adopt a Schumpeterian framework and examine implications of the arrival of modern (inherently different
and scientific) R&D opportunities. Because conducting it is much costly activities, some countries starts
investing in it, while others do not. As a result, countries are sorted into two groups. Divergence occurs
between groups, and convergence within a group. However, there is no role of international trade, andall countries are connected only via knowledge spillovers. The role of international trade is stressed in
Baldwin et al. (2001) who focus upon two countries in a core-periphery model. In Matsuyama (2004),
many symmetric countries sort themselves into the rich and the poor. In a similar vein, Matsuyama (2013)
also considers the Lorenz curve among ex ante symmetric countries. The last three studies are insightful
in that they identify market forces which divide ex ante symmetric economies into groups of different
income levels, though it means that they do not identify characteristics which make those countries richer
or poorer. In contrast, our paper starts with heterogeneity in R&D productivity among countries. Becauseof this assumption, a transitional process of sequential take-offs endogenously divides the world into theNorth and the South only gradually, allowing us to examine a dynamic process of changing inter-country
inequality.
Our paper is structures as follows. Section 2 briefly reviews data related to inequality, and develops
a simple model which explains an intuition behind the result. Section 3 formally develops the model,
and long-run equilibrium is considered. In Section 5, transitional dynamics are examined with a focus on
the role of the learaning-opportunity externality. Simulation is conducted to show the emergence of theInternational Kuznets Curve. The last section concludes.

2

Preliminary

2.1

Data

The Industrial Revolution triggered a series of take-offs of economies reaching higher levels of income.
Because it was a gradual process, inter-country inequality steadily increased. Indeed, Bourguignon andMorrisson (2002) show a clear evidence of increasing inequality since 1820. They created dataset of
15 single countries and 18 country groups based on the Maddison data and others, and in particular,
income decile data within each country or group. By so doing, they examine (i) inter-country inequality,
which takes individuals having the same income within each country/group, (ii) within-country inequality,
which comes from inequality within each country/group, and (iii) global inequality, consisting of (i) and(ii). Figure 3 shows the between-country inequality indices, the Theil Index and the Mean Log Deviation,
which steadily increase. The Gini coefficient, calculated from the “raw” data of Bourguignon and Morrisson(2002) is also shown for reference. Although not shown here, their calculation indicates that the within-5Inequality Between Countries
Gini Index (raw)
Theil Index
Mean Log Dev

0.5

Gini
Theil Index
Mean Log Dev

0.70
0.65

0.4

0.60
0.55

0.3

0.50
0.2

0.45
0.40

0.1

0.35
1950

1820 1850 1870 1890 1910 1929 1950 1960 1970 1980 1992

Figure 3: Between inequality is derived fromthe Theil index and the Mean Log Deviationof GDP per capita in the world. The Gini Coefficient is calculated using “raw” data of GDP
per capita. Source: Bourguignon and Morrisson(2002)
1.0

1970

1980

1990

2000

2010

Figure 4: The Gini coefficient, the Theil index
and the Mean Log Deviation of GDP per capita
in the world are calculated using the “raw” datafrom the 2018 Maddison dataset (137 countries
in 1950-2016).

1.6

Gini
Theil Index
Mean Log Dev

0.9

1960

Gini
Theil Index
Mean Log Dev

1.4
1.2

0.8

1.0

0.7

0.8

0.6

0.6

0.5
1970

1980

1990

2000

0.4

2010

Figure 5: The Gini coefficient, the Theil index
and the Mean Log Deviation of GDP per capita
in the world are calculated using the “raw” datafrom the Penn World Table 9.1 (156 countries
in 1970-2017).

1950

1960

1970

1980

1990

2000

2010

Figure 6: The Gini coefficient, the Theil index
and the Mean Log Deviation of GDP per capita
in the world are calculated using the “raw” datafrom the World Inequality Database (136 countries in 1950-2017).

6

2020

inequality is high even in 1820 with its decreasing trend. It means that the relative importance of the
between-country increased in the total inequality in the period considered.5
Now the question is what happened after a sharp increase in inter-country inequality triggered aroundthe period of the Industrial Revolution. In Figure 3, the inequality indices appear to level off after 1950,
which suggests a certain equalizing mechanism is working after the WWII. To explore this issue, we turnto datasets which allow us to use GDP per capita of individual countries (rather than groups). Figure 4
uses the 2018 Maddison data with 137 countries in 1950-2016.6 Countries are selected so that no missing
values exist.7 All three indices show a steady increase around 2000, followed by a clear downturn in 2000.
The Gini coefficient appears to exhibit a less pronounced fall compared with the other two indices. This
supports the International Kuznets Curve hypothesis, and is consistent with the findings of Milanovic(2016). In Figures 5 and 6, the same indices are calculated using the Penn World Table 9.1 and the World
Inequality Database. They show clear negative trends since 1970 and 1950. In fact, a similar result is also
reported in, e.g. Firebaugh (2003). Given the increasing trends of the inequality indices in Figure 3, thosedatasets suggest that a peak of the inequality indices is likely to occur in earlier years than those figures
show. Although there are discrepancies regarding the timing of the peak in Figures 4-6, they all indicatethat the inter-country inequality took a U term from an increasing trend at least since the period of theIndustrial Revolution.
A rise and fall in the inequality indices reflects changes in the income distribution itself. Figures 7-9
show the evolution of its shape in 5-year intervals, using Kernel density estimates. There are two common
features. First, the mode is found at low income levels of income initially, and it gradually shifts rightward.
Second, the distribution itself gets flattened out. This feature is much stronger in the Maddison data in
Figure 7, and might have contributed to an inverted U shape in Figure 4. These visual traits can be cast inchanges in skewness and kurtosis. In deed, those indices show a clear negative trends in all datasets usedin Figures 7-9 (not shown here).8 A falling skewness means that the shape of the distribution is tilting
rightward, and a lower kurtosis implies that countries with extreme income levels are less in number inboth tails (low and high). These pictures are consistent with an increasing number of countries catching
up with developed countries, including Japan, what used to be called Asian Four Tigers, and BRICS in
more recent decades, though there are countries falling back as well.9
Then, a more fundamental question is what drives the process of changing inequality. Although a range
of factors are suggested (for example, convergence of institutional factors, human capital accumulation,
labour mobility, the emergence of service sectors), we believe that the creation/use/diffusion of technology
5

The importance of between-country inequality may also be highlighted by Milanovic (2002) who writes that betweencountry inequality explains 88 percent of global inequality measured by the Gini coefficient.
6
Bourguignon and Morrisson (2002) also used its older version.
7
We prioritize a balanced dataset covering as many countries as possible. A primary concern of using an unbalanced
dataset is bias towards more equality. Countries with missing values are typically developing countries, and if they are
excluded, inequality indices tend to understate the degree of inter-country inequality. In addition, as the number of countries
included in a dataset changes (typically increases), such bias is not constant. In the Penn World Data 9.1, the number ofcountries with no missing data is 55 in 1950. It increases by 36 in 1960 and 42 in 1970, preceded and followed by a steadyincrease before reaching 182 in 2005. Using all countries, the calculated Gini coefficient jumps in 1960 and 1970, and in fact,
it gives a much stronger image of an inverted U relationship between the index and income.
8
Using Bourguignon and Morrisson (2002), one can show that skewness and kurtosis of GDP per capita have a positive
trend up to 1950, followed by a negative trend.
9
There are countries which fell back from a relatively high income group, as opposed to catching-up. An often-cited
example is Latin American countries in the late 20th century. For example, see Edwards et al. (2007) and Jones (1997).

7

Kernel Density: Maddison Dataset
2015
2010
2005
2000
1995
1990
1985
1980
1975
1970
1965
1960
1955
1950

0.4
0.3
0.2
0.1

Figure 7: Kernel density estimates of the distribution of
GDP per capita of 137 countries. Source: the 2018 Maddison dataset

0.0
5

6

7

8
9
Log of GDP per capita

1011

Kernel Density: Penn World Table 9.1
1970
1975
1980
1985
1990
1995
2000
2005
2010
2015

0.35
0.30
0.25
0.20
0.15
0.10

Figure 8: Kernel density estimates of the distribution of
GDP per capita of 156 countries. Source: the Penn World
Table 9.1

0.05
0.00

6

7

8
9
10
Log of GDP per capita

1112

Kernel Density: World Inequality Dataset
1950
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
2015

0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
6

7

8

9
10
Log of GDP per capita

11

12

8

13

Figure 9: Kernel density estimates of the distribution of
GDP per capita of 136 countries. Source: the World Inequality Database

Figure 10: The Lorenz Curve for the two income
groups consists of two lines. Three cases are
drawn.

Figure 11: The Gini Coefficient takes an inverted U shape.

is the key to the understanding of a dynamic process of changing inequality between countries.10

2.2International Kuznetz Curve: An Intuitive AccountIn this section, we develop an intuition of the main mechanisms of our model which drives the emergenceof the International Kuznets Curve. Consider a world economy with a continuum of countries, which aregrouped into North and South. Countries are homogeneous within each group with income per capita yi ,i = N, S and yS < yN . Assume that there are m countries in North, and 1 − m in South, so that theworld average income is yW = (1 − m) yS + myN . Even at this stage, one can sense what is in store. If all
countries belong to South, i.e. m = 0, the Gini coefficient is zero because of the indentical income. The
same is true if all countries are in North, i.e. m = 1. This implies that the Gini coefficient is greater thanzero for 0 < m < 1 because yS < yN , and there must be the value of m at which its maximum is achieved.
To confirm this, let us consider the Lorenz curve for this world economy:
 py
S


yW
L (p) = (1 − m) y + [p − (1 − m)] y
S
N


yW

0≤p≤1−m1−m<p≤1

where p is the percentile of countries which are arranged starting from South. It is illustrated in Figure
10. Given the Lorenz curve, the Gini coefficient is now defined as
Z 1
G =1−2

L (p) dp =0m (1 − m) (yN − yS ).
(1 − m) yS + myN

Figure 11 depicts G as a function of m.
10

See Becker et al. (2011) for the role of human capital accumulation.

9

Suppose that initially m = 0, i.e. all countries are backward with income yS . Let us introduce the Big
Divergence. Industrialization increases the number of Northern economies m from zero. As it spreads,
m gradually increases with a rise in the average world income yW . Along with it, the Gini coefficientG also increases. This is due to the fact that more of initially backward economies take off reaching ahigher income level yN , while other economies are left behind with a lower income yS . This is what wecall the take-off effect which tends to raise G.11 On the other hand, an increase in m also means that
more of Southern countries are catching up with Northern economies, achieving a higher income. Put
it differently, there are less and less number of poorer countries. This process tends to reduce G, and iscalled the catch-up effect. When m is small, the catch-up effect is small relative to the take-off effect, sothat inequality widens between countries, as in the Big Divergence period. However, as more economies
are industrialized, the catch-up effect turns dominant, and G reaches its maximum at m̃ in Figure 11. As
m increases further, the Gini coefficient starts falling, giving rise to the International Kuznets Curve.12
This is the intuition for the key result of our model, developed in the following sections.

3

The Model

3.1

Consumers

Time is continuous. There are “many” heterogeneous economies, indexed by i. It consists of the baseeconomy, i = 0, which has already been industrialized, and it plays the role of Britain in the diffusion
process of industrialization. The other economies i ∈ (0, M ], 0 < M < ∞, are initially backward and havecapability to take off, though not all of them will be industrialized in equilibrium, as we will show. In eacheconomy, there is a representative consumer who supplies constant L units of labour service inelastically.
At this stage, it is instructive to clarify some details about the distribution of workers in the world. To deal
with the initial situation where the base economy only is industrialized, we capture the economy at i = 0
using the Dirac Delta function. A consequence of this assumption is that there are (1 + M ) L workers in
the world as a whole, and the cumulative distribution of workers is illustrated in Figure 12. Without lossof generality, M = 1 is assumed in what follows.
A representative consumer in all countries share the following instantaneous utility function:ui (t) = β ln Xi (t) + (1 − β) ln zi (t)where 0 < β < 1 andZ N (t)ln Xi (t) =ln qj (t) xij (t) dj,

0 < N (0) ≤ N (t) ≤ 1.

(1)

0

zi denotes agricultural goods, and Xi is the composite goods consisting of industrial goods xij where thesubscript j ∈ [0, N (t)] is the industry index. We use N to denote the number of varieties of industrial
goods available for consumption and qj is the quality level of xij . In our model, two types of innovations
11

A the risk of abusing terms, we use the term take-off effect here. Note that it is different from the take-off externality
which increases R&D productivity as more economies take off, discussed earlier.
12
Skewness in our simple example is given by Skew = √ 1−2m . It is monotonically decreasing in m with Skew ∈ [−∞, ∞]m(1−m)and Skew = 0 for complete symmetry m = 1/2.

10Figure 12: The cumulative distribution of workers.

coexist. The first type of innovations raises quality levels qj of industrial goods geometrically whenever it
occurs:c (t)qj (t) = γj j ,γj > 1,cj (t) = 1, 2, 3......
As will be explained in more detail, innovative firms optimally determine γj .13 The second type ofinnovations expands the variety of industrial goods. It means that some industries do not initially exist
until the first innovation occurs. In this sense, the first innovation in those industries is a variety innovation
with the initial quality qj = γj , and follow-on innovations in the same industries are of quality type. We
assume that variety innovations can occur until the upper limit of N = 1 is reached.14
Given the Cobb-Douglas structure in (1), demand for each product is given byxij (t) =βEi (t),pj (t) N (t)zi (t) =(1 − β) Ei (t)py (t)(2)

where Ei is consumption expenditure. It can also be easily verified that an intertemporal utility maximizing
consumer choses the dynamic path of Ei which satisfies
Ėi (t)= r (t) − ρEi (t)(3)

where r is the interest rate and ρ is the common subjective rate of time preference. Now, define E =RM
E0 + 0 Ei di as the total consumption expenditure.15 From (3), we can derive the Euler condition for
13This assumption follows Aghion and Howitt (1992).
Early studies which model the coexistence of quality and variety innovation are Aghion and Howitt (1996) and Li (2000).
However, the approach we adop is different from them and other similar studies.
R +ε
RM
15
Formally, E = −ε E0 δ (0) di + 0 Ei di where δ (.) is the Dirac Delta function. It is reduced to the above expression, as
ε → 0.
1411

the world economy16Ė (t)= r (t) − ρ.
E (t)

3.2

(4)

Manufacturing of Products

In characterizing equilibrium below, we will show that the world is divided into the North and the South.
In the North, industrial goods are produced, and the South is specialized in the production of agricultural
goods. Both types of goods are traded.17
Let us consider agricultural goods, which are produced under perfect competition. It is assumed thata worker is required to produce one unit of the goods. Assuming that all countries have access to the same
technology, their price is given by py = wS where wS is wage in South.
Industrial goods are produced by local monopoly firms which invented those goods. First, consider
industrial products created through quality innovation. Suppose that a firm located in country k produces
an industrial product in an industry j. Further suppose that a different firm located in country i invented
a higher-quality product in the same industry. The firm charges the pricepj (t) = γj wk (t) .

(5)

To understand this, note that given the Cobb-Douglas preferences in (1), quality innovation cannot be
drastic in the sense that the marginal cost of the previous incumbent firm always binds the pricing decisionof an innovating firm. That is why in (5) there is wage in country k where the previous incumbent firm
is located. Note that k = i can also happen. Regarding γj , it is interpreted as the quality premium overthe binding marginal cost wk .
Let us turn to variety innovations. When it occurs in country i in industry j, its quality level starts
from qj = γj . In addition, we assume that copycat firms emerge in a randomly chosen country k and can
produce the same product with qj = 1. Due to this simplifying assumption, the price charged by a firm isequivalent to (5).
It is assumed that one unit of industrial goods is produced with one worker. Therefore, a monopoly
firm in country i earns profits in industry j 18
βE (t)wi (t).
πijk (t) = 1 −γj wk (t) N (t)

3.3

(6)

Uncertain Innovation Processes

There are two types of uncertainty in an innovation process. The first uncertainty is related to the choiceof an industry where innovation implemented. In most of existing Schumpeterian models, firms engage in
R&D in a particular industry, and if successful, innovation is implemented in that industry. In this sense,
R&D is “directed”. We depart from this perspective. It is assumed that R&D is “undirected” in the sense
16

Rewrite (3) as Ėi = Ei (r − ρ). Integrating it over i gives (4).
For example, see Section 6.1 of Galor and Mountford (2008) for the importance of international trade in the period
18-19th centuries.
18
Note that the subscript i in (6) refers to the country where the product is produced, whereas i in (1) and (2) denotes
the country where goods are consumed.
17

12

that an industry where innovation is implemented is randomly chosen from all industries i ∈ [0, 1] after
innovation occurs. This assumption captures in a simple way an unpredictable nature of R&D outcomes.19
An advantage of this assumption is that one type of R&D is enough to model both of variety andquality innovations, because a given innovation can be either of variety or quality type. For example,
consider an industry which has not been chosen for innovation implementation yet. If this industry is
selected for the first time, a variety innovation is implemented, creating a new variety product in a new
industry which did not exist before. On the other hand, if an industry where innovation was implemented
before is randomly selected, innovation takes a quality form.
The second uncertainty of innovation is related to the timing of successful innovation. This is thestandard type of uncertainty in the literature, and is captured by a Poisson process of innovation. To be
more specific, define Ri as the number of research workers used in country i. Then, innovation occurs withan arrival rate of ai Ri in country i where ai is R&D productivity. ai is the only factor that differentiates
a continuum of countries in the model. Despite such simplicity, our model will exhibit rich dynamics inequilibrium, dividing the world into the North and the South with an evolving distribution of GDP per
capita across countries.

3.4

The Value of Innovation

Let us define the value of innovation for an innovator in country i. First consider the ex post value after
innovation occurs and is implemented in an industry j (randomly selected). A successful innovator k earns
profits πijk in (9) until it is replaced with a higher quality product (i.e. the industry is selected randomly
for the new successful innovation). Therefore, the ex-post value of innovation vijk is given byr (t) vijk (t) = πijk (t) + v̇ijk (t) − g (t) vijk (t)whereZ
g (t) =i0 ∈N

ai0 (t) Ri0 (t) di0

(7)

(8)

and N is the set of countries in North. The right-hand side of (7) consists of a profit flow, capital gains
and the term reflecting the risk of losing the value because of an extra innovation hitting the industry j.
g in (8) is the Poisson rate of an innovation arriving in industry j, which is equivalent to the total sum ofPoisson rates across the North. Note that vijk is the same for both of variety and quality innovations.
Next, consider the ex ante value of innovation. We derive it in two steps. First, we assume that an
innovating firm in a country i optimally chooses the quality step γj . To make this explicit, we rewrite a
profit flow in (6) as
βE (t)wi (t)(9)
πik (t) = 1 −γi wk (t) N (t)where γj is replaced with γi . As is shown below, the optimally chosen γi will be the same whicheverindustry is selected for implementation. Hence, it is innocuous to replace the subscript j with i in profits(9). This also allows us to rewrite the ex post value of innovation as vik .
19

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 another example of acommercial product which was originally created for different purposes. A similar assumption is used in Kortum (1997).

13

Second, recall that following a successful innovation, an industry where it is implemented is randomly
selected. This implies that innovation takes a quality form with the probability of N , and a variety form
with 1−N . Now suppose that an industry j is chosen and the previous incumbent firm is located in country
k. Note that whichever industy is selected, profits are the same as long as the the previous incumbent firm
is from the same country k. In this sense, a random selection of an industry is effectively equivalent torandomly choosing a Northern country where the previous incumbent firm in a chosen industry is located.
Bearing this in mind, define ni to be a measure of industries where production of industrial goods takesRplace in country k such that N = i∈N nk dk. Then, in the case of quality innovation, the probability thatan incumbent firm is from country k (operating in a selected industry j) is
nk (t),N (t)σk (t) =k ∈ N.

(10)

In the case of variety innovation with quality qj = γij , copycat firms exist, which can produce the same
product with the quality level qj = 1 with the marginal cost of production being wage. We assume thatthe probability of copycat firms being located in a country k is also given by σk .20
Now we are in position to derive the ex post value of innovation vi . Remembering that vik is the samefor variety and quality innovations, vi is defined byZZZvi (t) = N
σk (t) vik (t) dk + (1 − N )σk (t) vik (t) dk =σk (t) vik (t) dk.
(11)
k∈N

k∈N

k∈N

The first term on the right-hand side of the first equality is the expected value of quality innovation with
N being the probability of innovation being implemented as improving quality. Similarly, the second termis the expected value of variety innovation where 1 − N is the probability of innovation being of variety
type. This allows us to write the ex ante asset equation asr (t) vi (t) = πi (t) + v̇i (t) − g (t) vi (t)
where

(12)

Z
πi (t) =σk (t) πik (t) dk.

(13)

k∈N

3.5

R&D

Inequality in the world economy in the present model arises due to heterogeneity of R&D productivity ai ,and this variable plays an important role in our analysis. We assume the following form:ai (t) = κN (t) eλm(t)−ηi γi−µ ,κ, λ, η, µ > 0(14)where m is the number of Northern countries. (14) captures the following four assumptions:
1. ai is decreasing in i for a given γi . This means that countries are arranged in a decreasing order of
20

The crux of this explanation on an innovation implementation is that a random selection of an industry j is effectively
equivalent to that of a country k where an incumbent is located. Given this, the probability that the previous incumbent
firm is located in country k, given that new innovation is of quality type is N σk . Similarly, (1 − N ) σk is the probability that
copycat firms are located in a country k, given that new innovation is of variety type. Therefore, the probability of selecting
a country k where the previous incumbent or copycats are located is N σk + (1 − N ) σk = σk .

14

R&D productivity. The degree of R&D heterogeneity is captured by η.
2. ai is a decreasing function of the size of quality innovation γi . Noting that a profit flow in (9) isincreasing in γi , this assumption makes it possible for research firms to optimally choose γi , weighing
benefits in the value of innovation against costs in R&D productivity. µ > 0 determines the marginal
cost of increasing the size of quality innovation.
3. ai is increasing in the number of innovative industries N . It represents a positive externality that isoften found in the variety-based approach of endogenous growth, e.g. [Romer (1990)].
4. ai is increasing in the number of Northern countries, m. This assumption captures in a simple way
a positive spillover effect arising from take-offs. In each and every Northern economy, different experiences (successes and failures) are accumulated, generating diverse perspectives about research.
Indeed, such varied perspectives are widely considered important in order to be successful in uncertain research. Opportunities for such learning are greater if there are more researchers and more
Northern countries. In this sense, a scope for learning new technologies will be greater as morecountries start R&D. λ is a parameter which governs the degree of such externality, which we callthe take-off externality. It plays an important role in determining the nature of equilibrium.
Turning to the optimal decision of R&D, we assume that research firms decide on the number of workers
employed and the size of quality innovation. That is, a representative firm in country i solves the following
problemvi (t)zRmax
Ri (t),γi}|{βE (t)wi (t)dkk∈N σk (t) 1 − γ w (t)
N (t)
i k
ai (t)Ri (t) − (1 − si ) wi (t) Ri (t)r (t) + g (t) − v̇i (t) /vi (t)where si is the rate of subsidy. The first-order conditions arevi (t) ai (t) = (1 − s) wi (t)for Ri > 0(15)vi (t) ai (t) < (1 − s) wi (t)for Ri = 0

(16)

Z

wi (t)∂ai (t)dk = −ai (t)σk (t) 2∂γi
γi wk (t)
k∈N

Z



wi (t)σk (t) 1 −γi wk (t)
k∈N


dk

(17)

(15) and (16) are standard conditions under free entry. (17) defines the optimal size of quality innovation.21
Its left-hand side is the marginal benefit of increasing γi and and the marginal cost is on the right-handside.
Rearranging (17) gives
γi =
where

21

1
=
wN (t)

1 + µ wi
µ wN

(18)

Z

σi (t)
di.
i∈N wi (t)The second-order condition is satisfied for µ > 0, which is assumed.

15

(19)

South

North

0

1

Figure 13: The endogenous division of countries into the North and the South.
(19) is the the weighted harmonic mean of Northern wages. In (18), γi is proportional to wi , implying
firms in a higher wage country tend to choose a greater size of quality innovation. Precisely because ofthis feature, profit flows will be independent of the country and industry indices. To show this, we makeuse of (18) to rewrite (9) asβπ (t) = πik (t) =E (t) .
(20)
1+µIn addition, (20) allows us to write vi = v for all i, and hence simplify (12) asr (t) v (t) = π (t) + v̇ (t) − g (t) v (t) .

(21)

This equation shows that the ex ante value of innovation is identical for all countries. However, this does
not mean that all countries will engage in R&D. It is because R&D productivity is too low for somecountries, and the first-order condition (16) holds for firms in the South. On the other hand, (15) issatisfied for economies with higher R&D productivity in the North with active R&D. This suggests theexistence of the threshold economy between the North and the South.
To demonstrate this, consider (15) which is now given by v (t) ai (t) = (1 − s) wi (t) because of vi = v.
By assumption, ai is decreasing in i, and let m denote the economy with the lowest R&D productivity
among countries with active R&D with vam = wm . Note that by construction m is equivalent to themeasure of Northern countries. In addition, (16) or vai < wS holds in Southern countries. Given that ai
is continuous in i, we conclude that wm = wS . Intuitively, the threshold country has the same wage as in
the South, but its R&D productivity is just high enough to justify investment in research. The situation
is illustrated in Figure 13. using (15) and vi = v, we can determine relative wages among countries:
wi (t)ai (t)=
wm (t)am (t)

ηwi (t)(m(t)−i)= e 1+µwS (t)⇒(22)

where wm = wS . This shows that relative wages for a Northern country i get larger as more countries
take off (i.e., an increase in m). This result is intuitive because newly industrialized countries have lower
productivity levels relative to a Northern country i.

16To explore implications further, combining (18) and (22) yields
η(m(t)−i)

1 + µ e 1+µ
γi =
·
µ
ω (t)
where
wN (t)
ω (t) ≡=wS (t)(23)

−1wS (t)σi (t)diwi (t)
i∈N

Z

(24)

is interpreted as the Northern relative wages in terms of the Southern wage. (23) captures a mechanism
through which take-offs induce further take-offs. As more and more countries are industrialized (i.e. ahigher m), it is intuitive and indeed the case that the Norther relative wages ω tends to rise. This translates
into a lower quality step γi chosen by firms, which in turn increases R&D productivity. This tendency
prompts more countries to engage in R&D. This can be confirmed by substituting (23) into (14) to obtainai (t) ≡ N (t) αi (m (t) , ω (t))

(25)

where

αi (m (t) , ω (t)) ≡ κ

µ
1+µ

µ

µ



ω (t) eµηη
λ− 1+µm(t)− 1+µi

(26)

N αi (m, ω) is R&D productivity expressed in (ω, m) space. (25) shows that R&D productivity ai isincreasing in ω.22 Note that this take-off inducing mechanism embeded in (25) is realized through ω, andis different from the take-off externality which is captured by λm in (26).

3.6

Evolution of the Number of Industries

The number of innovative industries changes as the world economy grows. Recall that g is the arrival rateof innovation, and the proportion 1 − N of innovations creates new industries. Therefore, a flow of newly
created industries obeys the following equation:Ṅ (t) = g (t) (1 − N (t)) .

3.7

(27)

Labour Markets

First consider the South. There are (1 − m) number of countries, and each country supplies L labour
service, meaning that the total labour supply is (1 − m) L. Demand for agricultural goods is given by thesecond equation in (2). Integrating it over i gives the total demand in the world. Therefore, workers in
the South are fully employed if23
(1 − β) E (t).
(28)(1 − m (t)) L =wS (t)
Turning to the North, consider a country i. There are Ri number of workers in R&D. To calculate
labour demand for manufacturing, suppose that a monopoly firm operates in an industry j where the22

R&D productivity is increasing m, and it comes from the take-off externality introduced earlier.
Wage of the threshold economy is the same as in the South, i.e. wm = wS . This implies that agricultural goods canbe also produced there. Given its infinitesimal economy size relative to the world, we ignore its production of agricultural
goods.
23

17

previous incumbent was located in country k. Then, given that one worker is required to produce one
unit of industrial goods, labour demand is γiβE
wk N where γi wk is the price. Now recall that an industry is
randomly chosen to implement innovation after it occurs. Therefore, the average labour demand is given
RβE
by `i ≡ i∈N σk γiβE
wk N dk = γi wN N using (19). In addition, ni is the share of manufacturing industries in
the country, implying that the total labour demand arising from industrial goods production is ni `i . Now,
making use of (10) and (18), full employment of workers in country i is achieved when
L = Ri (t) + σi (t)3.8µβ E (t).
1 + µ wi (t)(29)

Growth Rate

In equilibrium, new innovations are generated and industrial goods are produced in the North. Those
goods are traded among Northern countries and exported to the South. Agricultural goods are producedin the South and exported to the North. The North and the South both benefit from specialization andeconomic growth.
To calculate the growth rate, multiply both sides of the labour market condition (29) by αi (m, ω) in(26). Then, integrate the resulting equation from 0 to 1 over i, add the corresponding equation of the baseeconomy and eliminate E by using (28) to derive the following expression
g (t) = N (t) G (m (t) , ω (t))

(30)

where


1 − m (t)µβ·G (m (t) , ω (t)) ≡ Lαm (m (t) , ω (t)) A (m (t)) −1+µ1−βZ m(t)
ηηm(t)(m(t)−i)
A (m (t)) ≡ e 1+µ
+
e 1+µ
di


(31)
(32)

0

A (m) is equivalent to the sum of relative R&D productivity levels in terms of the threshold country.24 Notethat the growth rate in (30) reflects the world resource constraints as well as firms’ incentives for production
and R&D and consumers’ optimizing behaviors. In particular, those incentive effects are realized throughthe two endogenous variables m and ω. But, in this subsection, we take m and ω parametrically to develop
intuitions about the effects of changes in those two variables on G (m, ω) in (31). This helps us understandhow the present model works.
First, (31) shows that G is increasing in the Northern relative wages ω through the R&D productivityof the threshold country αm (.). This works via the take-off inducing channel mentioned above, throughwhich higher Northern relative wages lead to a lower quality step chosen by firms, resulting in higher R&D
productivity. 25 That is, the presence of ω in G(.) comes from the endogenous determination of a quality
step size.
24

A (m) is derived using the Dirac Delta function δ (.), i.e.
Z +εZ m(t)
αi (m (t) , ω (t))αi (m (t) , ω (t))
δ (0) di +diA (m (t)) =αm (m (t) , ω (t))−ε αm (m (t) , ω (t))
0which is reduced to the expression in the main text.
25
See the discussion regarding (23) and (25) in Sec.3.5.

18

Next, let us turn to the number of Northern economies m, whose effects on G are realized via threechannels. First, a higher m raises growth through the term 1−m inside the parentheses in (31). This is due
the fact that the real consumption expenditure in terms of the Southern wage falls (see (28)), causing the
reduction of manufacturing labour demand. This makes more workers available in R&D in all Northerneconomies. Second, a higher m raises A (m), leading to a higher growth. This reflects the fact that as morecountries take off, the total number of research workers across the North simply increases. Third, a higher
m raises the R&D productivity of threshold country αm (.) for λ > η, but it falls for λ < η. To explain
it in more detail, suppose that the take-off externality is absent, i.e. λ = 0. Then, αm (.) unambiguously
falls with a higher m. This is due to the fact that a newly industrialized economy inevitably has a lower
productivity. Now, suppose λ > 0. Then, a higher m amplifies the take-off externality effect. If this effectis sufficiently strong such that λ > η, a flow of take-offs raises R&D productivity of countries with a higher
growth. Otherwise, growth tends to fall.
As is shown below, starting from the initial condition with m = 0, an equilibrium trajectory often
involves an increase in ω and m both. Given the mechanisms described above, growth gradually increases(especially for λ > η) as the world economy evolves with sequential take-offs of economies. Having
developed intuitions on growth, we will turn to the issue of how m and ω are determined in and off steadystate in the ensuing sections.

4

Equilibrium and Steady State

4.1

Equilibrium Conditions

In this paper, we are interested in transitional dynamics as well as steady state. To analyse both, we needto derive the time path of σi which is the share of manufacturing industries in a country i. Appendix A
showsσ̇i = Ai (m (t) , ω (t)) Ri (t) − G (m (t) , ω (t)) σi (t) .
(33)
Intuitively, (10) shows that the numerator is ni which rises in proportion to the Poisson rate of innovation
arriving in country i. This is captured by the first term in (33). On the other hand, the denominator of
σi is N . Given that N increases with the world-wide Poisson rate g, the share σi falls in proportion to G(see (30)).
In interpreting (33), we need to pay attentions to the threshold country. To describe the issue, noteRmthat the sum of manufacturing industry shares must be one, i.e. 1 = σ0 + 0 σi di. Time-differentiating itRmgives σm ṁ = − σ̇0 + 0 σ̇i di . Its RHS is reduced to 0 if (33) is used. This indicates the following threepossible cases:
1. σm = ṁ = 0
2. σm > 0 and ṁ = 0
3. σm = 0 and ṁ ≷ 0.
Note that (1) cannot arise because the threshold country conducts R&D, and hence σm > 0 holds insteady state. (2) holds in steady state only, and (3) is the case which is relevant to transitional dynamics.
ṁ > 0 indicates that sequential take-offs occur, and ṁ < 0 means that some Northern economies are19

de-industrialized, reverting to the South. In both cases, the industry share must be nil. This should
be intuitive. Consider ṁ > 0. More and more countries become industrialized, and the identity of thethreshold country changes as time passes. Simply, those threshold countries do not initially produce
industrial goods, i.e. σm = 0 (R&D success comes first before production). More formally, we add thefollowing condition:σ̇m = Am (m (t) , ω (t)) Rm (t)for ṁ > 0.

(34)

For ṁ < 0, those Northern countries which revert to the South do so after losing manufacturing industries.
Now, we are ready to state the following lemma:
Lemma 1. Equilibrium is defined by the following three differential equations:ω̇ (t)= F (m (t) , ω (t))ω (t)ω̇ (t)
ηṁ (t) −= H (m (t) , ω (t))
1+µ
ω (t)

∆ (m (t)) ṁ (t) − µṄ (t) = N (t) (1 − N (t)) G (m (t) , ω (t))

(35)
(36)
(37)

where
F (m (t) , ω (t)) = Lαm (m (t) , ω (t)) [A (m (t)) − B (1 − m (t))] + ρ(38)

H (m (t) , ω (t)) = Lαm (m (t) , ω (t)) [ω (t) (1 + m (t)) − A (m (t))]
1+η−λ∆ (m (t)) ≡1 − m (t)1βB ≡ µ+
1 − s (1 + µ) (1 − β)

(39)
(40)
(41)

See Appendix B.
The lemma shows that equilibrium in (m, ω) can be characterized by using (35) and (36) only, giventhat they are independent of N . Also note that we do not need to track the changes of (33) and (34).26
These are convenient features of this dynamical system especailly for our purpose of analyzing the time
path of m.

4.2

Steady State

Before we consider transitional dynamics, we briefly consider steady state. According to (37), the numberof industrial industries reaches the upper limit, i.e. N ∗ = 1 in the long run. Regarding m and ω, we
require ṁ = ω̇ = 0 in (50) and (51) in steady state. It means that the steady state value m∗ and ω ∗ are26

Those equations will be required in simulating the Gini coefficient in transition.

20

Figure 14: Steady state.

determined by the following two conditions∗∗

F (m , ω ) = 0⇒

H (m∗ , ω ∗ ) = 0

⇒


1
µ
ρ
1+µ
ω =
∗
(λ−η)m∗∗µLκe[B (1 − m ) − A (m )]∗A (m )
ω∗ =1 + m∗
∗

(42)
(43)

They are depicted in Figure 14 under the assumptions of
B > 1,

∗ω |F (0,ω∗ )=0 ≡

1+µ
µ

µ

ρ
< 1.
Lκ (B − 1)

(44)

These inequalities ensure that the F (.) = 0 curve has a vertical intercept at ω < 1, which in turn meansthat it starts from m = mF > 0 for ω = 1, monotonically increasing in m. The following result is is clear
from the figure.
Lemma 2. The slopes of F (m∗ , ω ∗ ) and H (m∗ , ω ∗ ) in (m, ω) space are positive, and the former is always
greater than the latter:
(−)

(+)

z
}|
{
z
}|
{
Hm (m∗ , ω ∗ )Fm (m∗ , ω ∗ )−<−Hω (m∗ , ω ∗ )Fω (m∗ , ω ∗ )
|
{z
}
|
{z
}
(+)

(−)

Note that this inequality holds irrespective of the values of parameters under (44).
Another feature worth mention is that there is the maximum value that m can take in equilibrium. It
is denoted by mF in Figure 14. Its exact value can be found by setting the denominator of (42) to zero.
This implies that not all countries can be industrialized, and Southern countries i ∈ [mF , 1] are trappedin underdevelopment. This result is clear from the figure.
Proposition 1. Countries i ∈ [mF , 1] are trapped in underdevelopment.
21

The following proposition summarizes comparative statics results of parameters:
Proposition 2. In steady state, the world growth rate are given by27
µβ
1 − m∗∗
g = G = L A (m ) −·1+µ 1−β(45)Comparative statics results:
1. A fall in ρ and a rise in L, κ, λ, β and s increase all of m∗ , ω ∗ and g.
2. An increase in µ and η generates ambiguous effects on m∗ , ω ∗ and g.
Results in (1) are intuitive. Those parameter changes are all conducive to innovation in the sense thatthey incentivize firms to invest more in R&D. Perhaps more interesting is how those parameters affect thecountries in the underdevelopment trap.
Proposition 3. A fall in η and a rise in µ, β and s raise mF , while it is independent of κ, λ, L and ρ.
A lower η means a fall in the degree of heterogeneity of R&D productivity among countries. It is
intuitive that the more homogeneous the countries are, the smaller the range of the underdevelopmenttrap. A greater µ raises the marginal cost of raising the quality step size (see (14)). The result comesfrom the relationship between the quality step size and R&D productivity explained in discussing the
take-off inducing effect of ω pertaining to (23) and (25). Indeed, as (26) shows, a higher µ raises R&D
productivity, and this effect leads to the shrinkage of the trap. What is somewhat surprising is that R&D
productivity improvement κ does not make the range of the trap smaller. Neither does λ.

4.3

Gini

We first calculate GDP per capita, denoted by yi . In the South, all workers are employed in the agricultural
sector, earning wS with no saving. Therefore, its GDP per capita is equivalent to yS = wS . In the North,workers are used in the production of manufacturing goods and R&D, earning wage wi . In addition,
monopoly firms earn profits πi . Hence, GDP per capita is given byyi = wi + σi

β E
,
1+µL

i ∈ N.

(46)

To rewrite the industry share of industrial goods σi , use (33) and (45) to derive
σi =

e

η
i
− 1+µ

R m∗ − η k ,1 + 0 e 1+µ dk

i ∈ N.

(47)

It depends on the exogenous measure of industry heterogeneity η. It also depends on µ that affects theendogenous determination of the quality step size, which in turn determines the level of R&D productivity.
− η iUsing (25), (46) and (47), one can easily confirm yi = e 1+µ y0 . This means that GDP per capita of theNorthern economies relative to the total Northern GDP per capita is also equivalent to (47).
27

The derivation of (45) is given in Appendix C.

22

Figure 15: The Gini in theηm∗North with C ≡ e 1+µ .

We follow Lambert and Decoster (2005) in calculating the Gini coefficient, which we denote by G. An
advantage of their method is to allow us to decompose inequality into several components, including theGini within the North only, denoted by GN . First let us consider the Northern Gini without the baseeconomy for simplicity. In Appendix D, we show that the Lorenz curve is given by
 η ∗
i
m
1+µ
1
+
Φ
e
−
1
dΦ
0
 η ∗
i
LN (p) = R h1m1+µ− 1 dΦ
0 1+Φ eRph(48)

where Φ is the distribution function of yi in the North. Hence, the Gini coefficient of the North (without
the base economy) is
Z 1
GN = 1 − 2

LN (p) dp =0

C −1
3 (C + 1)ηwhere C ≡ e 1+µm∗.

η and µ directly affect GN , which also indirectly depends on other parameters via m∗ . Its convenientηm∗only. GN is illustrated in Figure 15 where it unambiguously
feature is that GN depends on C ≡ e 1+µ
rises in C starting from 0 for C = 1 with an upper bound 1/3. A higher m∗ means that countries
that become industrialized have lower GDP per capita than those already in the North, and hence it
increases GN . But the marginal effect of m∗ on the Norther Gini is diminishing. An intuition is that as m∗increases, the industrial production is shared among an increased number of Northern countries, reducing
σi for incumbent Northern countries. As a result, the marginal impact of a take-off shrinks. For example,
suppose that an R&D productivity parameter κ rises, perhaps due to a major scientific discovery. We
know from Proposition 2 that more countries are industrialized with a higher m∗ . This translates to arise in GN according to Figure 15. Though it raises inequality among the North, its impact on inequality
diminishes as κ increases further.28
Turning to the Gini coefficient of the world economy (excluding the base economy), it can be decomposed into29
G = ξ (m∗ ) (1 + m∗ GN (m∗ )) − m∗
(49)
28

GN increases following one or combinations of the parameters in (1) of the Proposition rise, and the same implications
also applies to them.
29
See Appendix E for derivation.

23

whereξ (m∗ ) =

m∗ yN
yW

is the income share of the North, yN is the average GDP per capita in the North and yW is the average
GDP per capita in the world.30 m∗ in ξ (m∗ ) and m∗ attached to GN are interpreted as the direct effectof the take-off effect which increases inequality. On the other hand, the last m∗ in (49) represents the thecatch-up effect of decreasing inequality. There are of course the indirect effects of a greater m via yN , yW
and GN . As one can imagine, G (m∗ ) is highly non-linear in m∗ , and hence, it is difficult to draw a general
conclusion regarding the impacts of parameter changes on G (m∗ ).31
Note that the above results, e.g. Figure 15, are valid only in steady state. In transitional dynamics
with sequential take-off which we are interested in, the analysis of the income distribution of Northerneconomies are much more involved. To cope with it, we will resort to simulation later.

5

Transitional Dynamics

5.1

The Role of the take-off Externality

In this section, we focus upon off steady state equilibrium. For this, (35) and (36) are used to derive thefollowing system of differential equations:Lemma 3. Equilibrium is defined by the following two differential equations:ṁ (t) = W (m (t) , ω (t))(50)ω̇ (t) = U (m (t) , ω (t))

(51)

where
F (m (t) , ω (t)) − µH (m (t) , ω (t))ηµ∆ (m (t)) −
1+µ


η
W (m (t) , ω (t)) − H (m (t) , ω (t))
U (m (t) , ω (t)) ≡ ω (t)1+µW (m (t) , ω (t)) ≡

(52)

(53)

Note that both (42) and (43) are required in order for ṁ = 0 and ω̇ = 0 to hold simultaneously.
We investigate the local properties of steady state by linearizing the the system:
#
"
#
" # "
ṁ
W (m∗ , ω ∗ )m − m∗=+Jω̇
U (m∗ , ω ∗ )ω − ω∗30

ξ (m∗ ) is given by the following expression:ξ (m∗ ) =

R m∗

η
k
1+µ

R m∗


η
k
e 1+µ dk 1 +

η
m∗
1+µ
β 1−m∗
R m∗ − η i1+µ 1−β
0
1+µ
1+ 0
e
di
η
−
m∗
β 1−m∗
e 1+µR m∗ − η i1+µ 1−β1+ 0
e 1+µ di0
−

e

0
ym dk

=
R m∗ η k
R m∗ η k(1 − m∗ ) wS + 0 e 1+µ ym dk(1 − m) +e 1+µ dk 1 +
0

e

0

31

(54)



.

Having said this, note that G (0) = 0 and G (1) = GN (1). Given Figure 15, there can be a maximum of G in m∗ ∈ (0, mF ).

24

where"#Wm (m∗ , ω ∗ ) Wω (m∗ , ω ∗ )J=
Um (m∗ , ω ∗ ) Uω (m∗ , ω ∗ )(55)

Given this, we can establish the following lemma:
Lemma 4. The sign of the determinant of the system (54) depends on that of ∆ (m∗ ) −ηµ, which in
1+µturn depends on the take-off externality λ:32

Det (J)



< 0

> 0

ηµ
with relatively low λ
1+µηµfor ∆ (m∗ ) <with relatively high λ1+µfor ∆ (m∗ ) >Proof. See Appendix G.
ηµTo explain the lemma, suppose that λ = 0, i.e. no take-off externality. (40) shows that ∆ (m∗ ) > 1+µ
in this case. Now let us increase λ. Then, m∗ rises, and so does ∆ (m∗ ) initially. However, there is anηµupper limit mF which m∗ cannot exceed. This implies that as λ increases further, ∆ (m∗ ) − 1+µwill
eventually turn negative. As this lemma implies, the take-off externality λ plays a crucial role in theexistence of the diffusion of industrialization. To highlighte its role, we distinguish the following threecases in terms of ∆ (m∗ ):33

1. A Weak Take-off Externality:ηµ< ∆ (m∗ )1+µ

˜ < ∆ (m∗ ) < ηµ
2. A Moderately Strong Take-off Externality: ∆1+µ
˜
3. A Strong Take-off Externality: ∆ (m∗ ) ≤ ∆η˜ ω (m∗ , ω ∗ ). It is the value of ∆ which just makesFω (m∗ , ω ∗ ) = ∆H1+µ
˜ < 0 because Fω (m∗ , ω ∗ ) < 0 and Hω (m∗ , ω ∗ ) > 0. Diagrammatically,= ∞. Note that ∆˜ is defined such that
where ∆∂ ω̇∂ω (m∗ ,ω∗ )those three cases can be distinguished in terms of the relative slopes of the ṁ = 0 and ω̇ = 0 curves, as in
Figures 16-18.34

5.2

A Weak Take-off Externality

Here we consider Det(J) < 0, which arises if the take-off externality is non-existent or relatively weak.
The result is stated first.
Proposition 4. Suppose that the take-off externality is weak with low λ such that ∆ (m∗ ) >(m∗ , ω ∗ ) is a saddle point equilibrium, and the diffusion of industrialization cannot occur.
Proof. See Appendix G.
We do not consider the case of D (m∗ ) = 0 at which the determinant is undefined.
ηµThe determinant Det (J) is undefined for ∆ (m∗ ) =.
1+µ34
See Appendix F for proof.
32

33

25

ηµ
. Then,1+µThis proposition is best explained using Figure 16. In this case, the ṁ = 0 curve is steeper than theω̇ = 0 curve around the steady state. The directions of streamlines can be confirmed by the following
derivatives:35
(−)
(+)

ηµ
z
}|
{
}|
{ 
z

<
0
for
∆
(m
(t))
>
∗
∗
∗
∗
Fω (m , ω ) − µHω (m , ω )
dṁ
1+µ
=
ηµ
ηµ

dω (m∗ ,ω∗ )> 0 for ∆ (m (t)) <∆ (m (t)) −
1+µ
1+µ


(−)
(+)
∗
z
}|
{
}|
{
z
dω̇
ω
 η

=
Fω (m∗ , ω ∗ ) − ∆ (m∗ ) Hω (m∗ , ω ∗ )ηµdω (m∗ ,ω∗ ) ∆ (m (t)) −
1+µ
1+µ

(56)

(57)

ηµ
The both derivatives are negative for ∆ (m) > 1+µ, i.e. Det (J) < 0. Based on this information, a saddle
path is depicted in the figure.
The question we are interested in is whether or not industrialization diffuses to backward countries
from the base economy. To answer this question, let us define the initial condition. We suppose that attime 0, the base economy is a sole country which has already been industrialized (i.e. σ0 = 1), and othereconomies i ∈ (0, 1] are specialized in agricultural production, i.e. m (0) = 0. Regarding the relative wages,
w0 = wS , given that R&D productivity ai is a continuous function. In Figure 16, the initial condition is
given by (m (0) , ω (0)) = (0, 1).
Having identified the initial condition, now the above question is rephrased as follows: Is there atransitional path from (0, 1) to the interior long-run equilibrium E1 ? Appendix G shows the existence ofm̂ > 0 in the figure at which the ṁ = 0 curve hits the horizontal line at ω = 1. Because the saddle pathmust lie below the ṁ = 0 curve for m < m∗ , it must starts from a point on the ω = 1 line, e.g. m̃ ≥ m̂.
That is, to reach the saddle path from (0, 1), the world economy must somehow jump onto it. However, it
is natural to interpret m as a stock variable in the sense that industrialization takes place gradually andits diffusion is a progressive process where countries take off sequentially. Therefore, with a weak take-off
externality, the diffusion of industrialization does not occur and the world economy is stuck in the initialcondition (0, 1).
An intuition goes as follows. Industrialization requires sufficiently high R&D productivity, and a
dynamic process of its diffusion necessitates a sustained increase in R&D productivity to pull Southerneconomies with successively lower productivity levels out of backwardness. The strength of such a “pull”
mechanism depends the take-off externality, which is too weak in this case. The result also implies thatthe take-off inducing effect through the determination of a quality step size, discussed towards the end of
Sec.3.5 is not enough make set the diffusion of industrialization in motion.

5.3

A Moderately Strong Take-off Externality

We turn to the case where the strength of the take-off externality is in the intermediate range with amoderately strong “pulling” effect. The both ṁ = 0 and ω̇ = 0 curves are positively sploped, but thelatter is steeper now.
35

The derivatives are evaluated at steady state to make calculation and exposition easier. Note that their signs are validṁω̇for ddωand ddωas well, otherwise different demarcation lines (differential equations) would be required.
m∗
ω∗

26

Figure 16: The case of a weak
take-off externality. The steadystate is a saddle with no pathfrom (0, ω (0)) to E1 .

Figure 17: The case of a moderately strong take-off externality. The steady state can be asink or a source. A path from(0, ω (0)) to E2 exists only if thesteady state is a sink.

Figure 18: The case of a strong
take-off externality. The steadystate is a sink. A path from(0, ω (0)) to E2 always exists.

27

Proposition 5. Suppose that the take-off externality is moderately strong with the value of λ such that˜ < ∆ (m∗ ) < ηµ . Then, a sufficient condition for (m∗ , ω ∗ ) to be a sink is
∆
1+µ
−

η
Wm (m∗ , ω ∗ )
> ω∗
,
∗
∗
Wω (m , ω )
1+µ

(58)

in which case the diffusion of industrialization occurs.
Proof. See Appendix H.
To draw a figure, let us evaluate (56) and (57) for this case:

dω̇
> 0.
dω (m∗ ,ω∗ )

dṁ
> 0,
dω (m∗ ,ω∗ )Making use of this information, we can draw Figure 17 where the steady state is totally stable andpossibly spiral. In particular, starting from the initial condition (0, 1), the world economy moves along apath towards long-run equilibrium E2 . In this process, industrialization gradually diffuses to economies
with high R&D productivity levels and then to lower productivity levels. A steady increase in m captures
the fact that more and more economies take off sequentially. As more economies join the Northern club,
the relative wages rise as well. An intuition is that the take-off externality is strong enough to pull m∗Southern countries out of backwardness in the long run. Note that in this process the externality boosts
R&D productivity via the take-off externality as more countries are industrialized, and this is the main
driver of this diffusion process.
On the other hand, once Ẽ2 is passed, the relative wages ω begins falling. In addition, m also starts
falling after m̃2 is reached. This is interpreted as follows. After Ẽ2 some Northern economies with lower
R&D productivity levels reduce investing in research, and their share of industrial industries shrinks,
creating downward pressure on their wages. Consequently, the Northern relative wages steady fall. The
threshold countries revert to the South after losing industrial industries, creating further downward pressure on the relative wages. Such fall-back results from the overshooting of ω and m. It is also possible thattake-off and fall-back can occur alternately as the world economy spiral around the long-run equilibrium.
However, given that (58) is a sufficient condition, the steady state can become totally unstable, in whichcase no transitional path exists from the initial condition to E2 and long-run equilibrium is equivalent to
the initial condition. This happens if, e.g. heterogeneity of countries η is so large that the pulling effectof the take-off externality is not enough to initiate the process of sequential take-offs. In Figure 17, thiscase would be represented by a much flatter ṁ = 0 curve (the LHS of (58) is its slope). In this case, forexample, if the world economy was initially located at Ẽ2 , it would be dragged outward, increasing the
distance to E2 .

5.4

A Strong Take-off Externality

This is the case where the take-off externality is strong to the extent that the ω̇ = 0 curve has a negative
slope, as in Figure 18.

28˜ Then,
Proposition 6. Suppose that the take-off externality is strong with λ such that ∆ (m∗ ) ≤ ∆.
∗∗(m , ω ) is a sink, and hence the diffusion of industrialization occurs.
Proof. See Appendix H.
To check the directions of streamlines, evaluate (56) and (57):dṁ
> 0,dω (m∗ ,ω∗ )

dω̇
< 0.
dω (m∗ ,ω∗ )There exists a path from the initial condition to the long-run equilibrium E3 . Although an inverted
U trajectory is illustrated in the figure, a monotonically increasing transitional path or a cycle are also
possible. A difference from the case of a moderately strong take-off externality is the absence of a condition
like (58). This is because the pulling effect of the externality is sufficiently strong.

5.5

Simulation: International Kuznets Curve

Does the International Kuznets Curve arise as the world economy moves along a transitional path? To
address this question, we resort to simulation, given the highly non-linear nature of the Gini Coefficient.
We introduced the base economy in order to set the initial condition, so that we can examine thediffusion of industrialization rather than a question of why the Industrial Revolution occurred in Britain.36
It is a useful device for our analytical purposes in the preceding sections. However, in conducting simulation
analysis, the size of the base economy relative to the world is too large. The Dirac Delta function pins
down the size of the base economy to one, while there is a measure one of other economies. It means thatfor computational purposes, the base economy is equivalent to half of the world economy. To exclude its
large effect on inequality measures, we consider the Gini Index of all economies except the base economy.
We present two simulations. One with a monotonically increasing Gini, and the other with the case of
the International Kuznets Curve. For the first one, we assume the following parameters:
λ = 6.0,

η = 3.0,

κ = 0.09,µ = 0.03,

β = 0.8,

ρ = 0.07,

L = 0.32,

s = 0.

Parameter values are chosen to make sure that steady state is a sink. Figure 19 demonstrates the resultswith m∗ = 0.14 and ω ∗ = 1.48. Panels (a) and (d) show (m, ω) which are monotonically increasing,i.e. countries are sequentially taking off earning higher wages. In Panel (e), two series are plotted. The
variety of industrial goods N , which is set to 0.03 in the initial period, increases as innovations occur. The
introduction of those goods into the world economy is indeed a slow process. It mirrors growth rate inPanel (f). σ0 in Panel (e) is the share of the industrial goods of the base economy. It is initially one, but
its share is eroded as industrialization spreads to other economies. The Gini Coefficient is shown in Panel(c), and increasing monotonically. Recall that (49) consists of three components: (1) the Gini in the North
GN , (2) the income share of the North ξ and (3) the number of Northern countries m. The first element
GN takes an inverted U trajectory. This contrasts with Figure 15 which shows that GN monotonicallyrises in m∗ in C. When m is small, income differences among the Northern economies are large to theextent that GN rises. But, as m increases, the industry production is shared among an increasing number
36

For example, see Clark (2014)

29

m
0.10

0.25

xi

0.3

0.20
0.15

0.2
0.05
0.00

0

25

50

75

100

125

150

1.5
1.4
1.3
1.2
1.1
1.0

0.10

0.1

0

25

50

75
Time

100

125

150

0.0
1.0
0.8
0.6
0.4
0.2
0.0

0.05
0

25

50

75

100

125

150

0.00

Gini
Gini in North
0

0.06

25

50

75

100

125

150

25

50

75
Time

100

125

150

g

0.04
N
0

0

25

50

75
Time

100

125

150

0.02
0.00

0

Figure 19: A monotonic increase in the Gini Coefficient of the world economy.
of Northern countries, tending to reduce income differences with a decreasing GN . Via these mechanisms,
the Northern Gini has an upward and downward effect upon the world Gini. The next element is theNorthern income share ξ. It monotonically increases in Panel (b), putting upward pressure on the world
Gini. The last factor m captures the catch-up effect, which tends to reduce the Gini. Given that its steadystate value is small, its effect is not large enough to offset an increasing tendency of G.
In Figure 20, the result of another simulation is shown with the following parameters
λ = 7.0,

η = 0.6,

κ = 0.09,µ = 0.03,

β = 0.8,

ρ = 0.07,

L = 0.3,

s = 0.

As before, the parameter values are such that the steady state is a sink. Calculation shows that m∗ = 0.48
and ω ∗ = 1.27, and it takes shorter time for the world economy to converge to the steady state. A maindifference from the previous simulation lies in the value of η, which captures heterogeneity of countries.
Because it is now less pronounced, the diffusion of industrialization is more wide-spread with a higher m∗ .
This in turn reinforces the downward pressure on the world Gini through the last term in (49). Note thatthe peak of the world Gini occurs before m reaches its steady state value. This means that the criticalmass of Norther economies is reached just before the steady state to reduce the world Gini. A fall in theGini in North, i.e. income convergence among the developed countries, also plays a role in the world Gini.

6

Conclusion

The Industrial Revolution boosted growth in Britain, and as it diffuses to other countries, growth in theworld economy accelerated. At the same time, inter-country inequality increased. On the other hand,
the recent literature as well as the major macroeconomic datasets suggest that inter-country inequality
fell in recent decades. This suggests that inter-country inequality takes an inverted U shape, i.e. the

30

0.5

m

0.4
0.3

0.15
0.10

0.2

0.1
0

10

20

30

40

50

60

0.0

0.05
0

10

20

30

40

50

60

1.0

0

10

20

30
Time

40

50

60

0.0

10

20

30

40

50

60

10

20

30
Time

40

50

60

g

0.50

0.4
0.2

1.0

0

0.75

0.6

1.1

0.00

1.00

0.8

1.2

Gini
Gini in North

0.20

0.4

0.2
0.0

0.25

xi

0.6

0

10

20

30
Time

40

50

N

0.25

0

0.00

60

0

Figure 20: The case of the International Kuznets Curve.
International Kuznets Curve hypothesis. The aim of this paper was to build a Schumpeterian growthmodel of the world economy to explain this hypothesis. Simulations show that the International Kuznets
Curve emerges in a transitional process of the world economy converging to the long-run equilibrium.
Especially, it is likely to occur if heterogeneity of countries is sufficiently small. Although the model cangenerate the International Kuznets Curve, there still remain room for improving simulation results.

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32

A

Derivation of (33)ni changes according toṅi = N Ai (m, ω)Ri (1 − ni ) − (g − N Ai (m, ω)Ri )ni
where the first term is a flow of new industries introduced and the last term is the number of industries
lost. Now differentiating (10) and rewriting the resulting equation with the above equation and (30) gives
(33).

B

Proof of Lemma 1

Using (4), (9), (21), (15) and (25), we have
Ė
v̇
β
E
+ρ+g− =
αi (m, ω).
Ev(1 − s)(1 + µ) wi(59)

for the threshold country. Time-differentiate (15) and rewrite the resulting equation with (25) and (22)to obtain
v̇
ω̇ Ṅ
ẇS
= − (λ − η) ṁ − µ −
+
.
v
ω N
wS

(60)

Similarly, time-differentiating (28) yields
Ė
ẇS
ṁ
=
−
E
wS
1−m

(61)

Using (27), (30), (28), (60) and (61), (59) can be rewritten as (35).
Rm− η (m−i)− η mdi. Time-differentiating it gives
Using (22), rewrite (19) as ω1 = σ0 e 1+µ + 0 σi e 1+µ
ω̇
− 2 =
ω



Z m
α (m, ω)α (m, ω)
η ṁ
σ̇0
+
σ̇i
di −
A0 (m, ω)Ai (m, ω)1+µ ω0where σ̇m = 0 is used. Substitute (33) into it, eliminate Ri using (29) and use (31) to derive (36).

C

Derivation of (45)αi (m, ω) Ri. Using this and (29), we have
G




Ai (m, ω)µβα (m, ω)µβ
L = Ri 1 +E = Ri 1 +Ewi
G (1 + µ)
wm G (1 + µ)

In steady state (33) is reduced to σi =∀i

αi (m, ω)αm (m, ω)=. This means Ri = R. Making use of this result,wiwm
rewrite (29) by integrating the resulting equation from 0 to 1 and adding the corresponding equation ofwhere the second equality uses

33the base economy:(1 + m) L = (1 + m) R +

µβ
1+µ



σ0
+
w0

Z m
0

σi
di
wi



(M − m) LwS
1−β

Making use of (28) and (24), rewrite this further to geµβ (1 − m∗ )L
1 + µ (1 − β)ω ∗ (1 + m∗ ) L = ω ∗ (1 + m∗ ) R +Finally, substitute (43) into this equation. Rearrangement gives (45).

D

Derivation of (48)
η
− 1+µ
i

Using (25), (46) and (47), we have yi = e
the density of yi is given byη

y0 where y0 = wS e 1+µ
e

y
R ∗i
=
y0 + 0m yi0 di0m∗+ R m∗ −1 η i0
e

0

η
i
− 1+µ

1+µ

di0

β E
1+µ L . Therefore,R m∗ − η i0 , which is equivalent to (47).
1 + 0 e 1+µ di0
Now consider the North without the base economy. Its density is now given byηη(m∗ −i)ym e 1+µ
e 1+µ
=
=
R m∗
η
R
∗
∗
0
m
(m −i ) 0YN
yi0 di0
ym 0 e 1+µdi0yiwhere k = m∗ − i and YN = ym eη∗
1+µ m −1η
1+µm
= y0 −yis the total GDP per capita in the North without theη
1+µbase economy. The associated cumulative distribution is
η

k

ym

Rk
0

η
k0
e 1+µ dk0

YN

 η

k
ym e 1+µ −1=

ηYN 1+µm= yx−y= Φ (x)0 −ymkwhere x = yk = ym e 1+µ . This can beR inverted to give x (Φ) = ym + (y0 − ym ) Φ = ym [1 + (C − 1) Φ].
px(Φ)dΦ
gives (48).
Therefore, substituting this in L (p) = R01
0 x(Φ)dΦE

Sketch of Derivation of (49)

The cumulative distribution function of yi , excluding the base economy is0y < wS
H (y) = (1 − m) FS (y) + mFN (y) = 1 − m
y = wS
1 − m + mF (y) ym ≤ y ≤ y0
N
where FS (y) and FN (y) are the cumulative distribution function within each region. In particular,R∞S)
FS (y) = 0 δ (y − wS ) dy = 1 where δ (y − wS ) = dH(y−wand H (y − wS ) is the Heaviside function.
dy
The average income in the South, the North and the world is as follows:
Z y0µ S = wS ,Z y0
[1 − FN (y)] dy,µN = ym +ym[1 − H (y)] dy.

µ Y = wS +wS34

Note that the product of the mean income and the Gini in each region is
Z y0

Z y0
µS GS = 0,

[1 − FN (y)] FN (y) dy,µN GN =µY G =[1 − H (y)] H (y) dy.
wSymNow, after tedious rearrangement, one can obtainµY + µY G = (1 − m)2 wS + mµN [2 (1 − m) + m (1 + GN )]which can be rearrange into (49).

F

Relative Slopes of the ṁ = 0 and ω̇ = 0 Curves

First, note thatHω (m∗ , ω ∗ ) = Lαm (m∗ , ω ∗ ) (1 + m) > 0A (m∗ )∗∗∗∗0∗Hm (m , ω ) = Lαm (m , ω )− A (m ) < 0
1 + m∗µFω (m∗ , ω ∗ ) = − ρ < 0ω
Fm (m∗ , ω ∗ ) > 0
See Figure 14
> 0 if ∆ (m∗ ) > ηµ
∗ , ω ∗ ) − µH (m∗ , ω ∗ )
F
(m
m
m
1+µ
Wm (m∗ , ω ∗ ) =ηµ< 0 if ∆ (m∗ ) < ηµ∆ (m∗ ) − 1+µ1+µηµFω (m∗ , ω ∗ ) − µHω (m∗ , ω ∗ ) < 0 if ∆ (m∗ ) > 1+µWω (m∗ , ω ∗ ) =ηµ> 0 if ∆ (m∗ ) < ηµ∆ (m∗ ) − 1+µ
1+µ

dω
Wm
=−
>0
dm ṁ=0Wωηµfor ∆ (m∗ ) ≷1+µDifferentiating (51) gives
η
∗
∗
∗
∗
dω
1+µ Fm (m , ω ) − ∆Hm (m , ω )
=− η
.
∗
∗
∗
∗
dm ω̇=0
1+µ Fω (m , ω ) − ∆Hω (m , ω )(62)whose sign to be determined. For this, note that
(+)

∂
∂∆



dω
dm ω̇=0



η
=−
h
1+µ η

(−)

(−)

z

z
}|
{z
}|
{
Hω (m∗ , ω ∗ ) Fω (m∗ , ω ∗ )∗∗∗∗1+µ Fω (m , ω ) − ∆Hω (m , ω )

i2

}|
{
Fm (m∗ , ω ∗ )
Hm
−
− −
<0
Hω (m∗ , ω ∗ )Fω (m∗ , ω ∗ )(m∗ , ω ∗ )dωdωafter rearrangement. Therefore, lim∆↓∆˜ dm ω̇=0 = ∞ and lim∆↑∆˜ dm ω̇=0 = −∞. It means thatdω<0dm ω̇=035

˜in the case of a strong take-off externality (∆ (m∗ ) < ∆).
Turning to the remaining two cases, (62) can be rearranged into
(+)

}|
{
,z
dω
dω
=Ω
dm ω̇=U (m,ω)=0
dm ṁ=W (m,ω)=0
where
η
1+µ +

1



ηµ
∆ (m∗ ) − 1+µ





ηµ
∆ (m∗ ) − 1+µ



(+)

z}|{
Fm + µ
−H
m
|{z}
(−)

Ω≡
η
1+µ +

1
(−)

z}|{
Fω + µ
−H
ω
|{z}
(+)

dω
> 0. Now noting Lemma 2,
and assume dmω̇=U (m,ω)=0ηµ(a weak take-off externality): Ω < 1,1+µηµ∆ (m∗ ) <(a moderately strong take-off externality): Ω > 1.
1+µ∆ (m∗ ) >G

Proof of Proposition 4

After tedious rearrangement, the determinant of the Jacobian (55) can be reduced toDet (J) = Wm (m∗ , ω ∗ ) Uω (m∗ , ω ∗ ) − Wω (m∗ , ω ∗ ) Um (m∗ , ω ∗ )
(+)

(+)

(−)

z
}|
{z
}|
{
 }| 
{ z
∗
∗
∗
∗
∗, ω∗)∗, ω∗)
H
(m
F
(m
H
(m
,
ω
)
F
(m
,
ω
)
m
m
ω
ω
− −
= ω∗
−
ηµ
∗
∗
∗
∗
∗
Fω (m , ω )Hω (m , ω )∆ (m ) − 1+µ

(63)

ηµ
It is negative for ∆ (m∗ ) > 1+µ.
Turning to the existence of m̂ in Figure 16, note that the ṁ = 0 curve is given by F (m, ω) = µH (m, ω).
Define ω̂ such that F (m̂, 1) = µH (m̂, 1), which can be rearranged to
RHS(m̂)LHS(m̂)

}|
{
}|
{ z
ρ
(1 + m̂) − A (m̂)=µ+ 1.
Lαm (m̂, 1) [B (1 − m̂) − A (m̂)]
B (1 − m̂) − A (m̂)

zNote that LHS (0) < 1 due to (44), and LHS (m∗ ) > 1 because F (m∗ , ω ∗ ) = 0 and αm (m, ω) is increasingin ω. Also note that RHS (0) = 1 and RHS (m∗ ) < 1 because 1 + m̂ < A (m̂) and B (1 − m̂) > A (m̂)
for m̂ ≤ m∗ . Therefore, there exist 0 < m̂ < m∗ . We conclude from this that the saddle path must startfrom m ≥ m̂ > 0 at ω = 1, as in Figure 16, given that it must lie below the ṁ = 0 curve for m < m∗ .

36

H

Proof of Proposition 5

The trace of the Jacobian (55) is
(+)
(−)
h
i
z
}|
{
}|
{
z
η
∗
∗ , ω ∗ ) − ∆ (m∗ ) H (m∗ , ω ∗ )
∗
∗
∗
∗
ω
F
(m
ω
ω
1+µ
Fm (m , ω ) −µ Hm (m , ω )Tr (J) =
+
.
ηµ
ηµ
∗
∆ (m ) −∆ (m∗ ) −
1+µ
1+µ
|
{z
}
|
{z
}
(−)

(64)

(−)

For the case of a strong take-off externality, we have
(−)

(−)

(+)

}|
{ z }| {z}|{η zFω (m∗ , ω ∗ ) − ∆ (m∗ )Hω (m∗ , ω ∗ )≥ 0
1+µ

(65)

˜ Therefore, Tr (J) < 0. To consider the case of a moderately strong take-off
because ∆ (m∗ ) < ∆.
externality, rearrange (64) into
"
#
η
F
−
∆H
ω
ω
Fm − µHm1+µ1 + ω∗Tr (J) =ηµ∆ − 1+µFm − µHm
˜ < ∆ (m∗ ) < ηµ for this case, and hence
Note that ∆1+µFm − µHm< 0,ηµ˜∆ − 1+µ∆(m∗ )↓∆
Fm − µHmWω∗ ηlim ηµ Tr (J) =1+ω.
ηµ∆ − 1+µ
1 + µ Wm∆(m∗ )↑ 1+µlim

Tr (J) =Given that (65) is decreasing in ∆ (m∗ ), (58) is sufficient for Tr (J) < 0.

37