Universidade de Aveiro Departamento de Matemática
2020
Olena Kostylenko
Análise Dinâmica e Controlo Ótimo
com Aplicações à Economia
Dynamic Analysis and Optimal Control
with Applications to Economics
Universidade de Aveiro Departamento de Matemática
2020
Olena Kostylenko
Análise Dinâmica e Controlo Ótimo
com Aplicações à Economia
Dynamic Analysis and Optimal Control
with Applications to Economics
Tese de Doutoramento apresentada à Universidade de Aveiro para
cum-primento dos requisitos necessários à obtenção do grau de Doutor em
Ma-temática, Programa Doutoral em Matemática Aplicada, MAP-PDMA 2015–
2019, das Universidades do Minho, Aveiro e Porto, realizada sob a
orien-tação científica do Prof. Doutor Delfim Fernando Marado Torres e da Prof.
Doutora Helena Sofia Ferreira Rodrigues.
Ph.D. thesis submitted to the University of Aveiro in fulfilment of the
re-quirements for the degree of Doctor of Philosophy in Mathematics, Doctoral
Programme in Applied Mathematics MAP-PDMA 2015–2019, of
Universi-ties of Minho, Aveiro and Porto, under the supervision of Professor Delfim
Fernando Marado Torres, Full Professor, Department of Mathematics,
Uni-versity of Aveiro, and Professor Helena Sofia Ferreira Rodrigues, Assistant
Professor, School of Business, Instituto Politécnico de Viana do Castelo,
Researcher at CIDMA, Department of Mathematics, University of Aveiro.
The research was supported by the Portuguese Foundation for
Sci-ence and Technology (FCT – Fundação para a Ciência e a Tecnologia),
through the PhD fellowship PD/BD/114188/2016, and within the Center
for Research and Development in Mathematics and Applications (CIDMA),
project UID/MAT/04106/2019.
o júri / the jury
presidente / president
Doutor Victor Miguel Carneiro de Sousa Ferreira
Professor Catedrático, Universidade de Aveiro
vogais / examiners committee
Doutor Alberto Adrego Pinto
Professor Catedrático, Universidade do Porto
Doutor Iván Carlos Area Carracedo
Professor Titular, Universidade de Vigo
Doutor Russell Gerardo Alpizar Jara
Professor Associado, Universidade de Évora
Doutora Maria Teresa Torres Monteiro
Professora Auxiliar, Universidade do Minho
Doutor José Maria Gouveia Martins
Professor Adjunto, Politécnico de Leiria
Doutora Cristiana João Soares da Silva
Investigador Doutorado (nível 1), Universidade de Aveiro
Doutor Delfim Fernando Marado Torres (Orientador)
Professor Catedrático, Universidade de Aveiro
agradecimentos /
acknowledgements
I would like to express my gratitude to my supervisor Prof. Delfim Torres
and co-supervisor Ph.D. Helena Sofia Rodrigues for the continuous
sup-port of my Ph.D. study, for their friendliness and high professionalism, for
sharing their knowledge with me, for their patience, advice and motivation.
I would like to express my deep gratitude to my parents for the education
they gave me. I am grateful to my family for believing in me, for providing
me through moral and emotional support in my life.
I express my sincere gratitude to Bogdan Postolnyi. All this adventure and
my study here would not have happened without him. I am grateful for
believing in me, for his help and support along the way.
A very special gratitude goes to the Portuguese Foundation for Science and
Technology (FCT – Fundação para a Ciência e a Tecnologia) for providing
the funding for my Ph.D. grant PD/BD/114188/2016.
I would like to thank also the Center for Research and Development in
Mathematics and Applications (CIDMA – University of Aveiro) for the
fi-nancial support to participate in international conferences, project
UID/-MAT/04106/2019.
Palavras-chave
Modelos epidemiológicos, Contágio financeiro, Processos cíclicos, Ondas
harmónicas, Controlo ótimo.
Resumo
Os modelos matemáticos, em combinação com a teoria do Controlo Ótimo,
são uma ferramenta poderosa para a realização de investigação em
Epide-miologia. Com a modelação matemática e as ferramentas computacionais,
é possível estudar o desenvolvimento da dinâmica de doenças e testar
vá-rias teová-rias de combate das mesmas e da sua disseminação. Contudo,
a sua aplicação pode estender-se a outras esferas da atividade humana.
Nesta tese de doutoramento, os princípios fundamentais da
epidemiolo-gia e da teoria do Controlo Ótimo são aplicados ao estudo e análise da
dinâmica de processos de propagação de uma crise na economia.
Uma crise, tal como um vírus financeiro, pode espalhar-se de um país para
outro, da mesma forma que as doenças infeciosas se transmitem de uma
pessoa para outra. Nesse sentido, o processo de disseminação do vírus
financeiro entre as economias nacionais dos países, por meio da rede de
suas atividades económicas, pode ser modelado com base nos conceitos
de epidemiologia matemática. Controlo, gestão estratégica e minimização
das consequências negativas são essenciais para a aplicação da teoria do
Controlo Ótimo.
Inicialmente, estudou-se a natureza cíclica da crise. Foi investigada a
iden-tificação de ondas de flutuação a longo prazo na dinâmica da produção dos
indicadores macroeconómicos de uma determinada economia nacional,
utilizando um modelo de input-output baseado em análise de regressão.
A propagação do contágio financeiro foi analisada usando modelos
ma-temáticos baseados em equações diferenciais ordinárias que descrevem
a dinâmica subjacente a uma doença financeira chamada crise, incluindo
interações económicas entre países. A disseminação do contágio
finan-ceiro para outros países pode ser enfraquecida pelo apoio finanfinan-ceiro
opor-tuno, tanto do Banco Central, na forma de Reservas de Capital, como pelo
empréstimo de recursos do Fundo Monetário Internacional, que
desem-penham um papel central na gestão internacional de crises financeiras.
Desta forma, foram analisadas estratégias ótimas para o uso desses
con-trolos e o efeito correspondente na redução ou eliminação da crise durante
um surto entre as economias nacionais dos países.
Como exemplo demonstrativo da dinâmica do processo epidemiológico de
disseminação do contágio financeiro entre países, foi aplicada a
mode-lação epidemiológica em redes. As suas implementações recorreram a
software de modelação matemática como o MATLAB, BOCOP e NetLogo.
Keywords
Epidemiological models, Financial contagion, Cyclical processes,
Har-monic waves, Optimal control.
Abstract
Mathematical models, in combination with the theory of Optimal Control,
are a powerful tool for performing research in Epidemiology. With
mathe-matical modelling and computational tools, it is possible to study the
devel-opment of disease dynamics and test several theories for combating
dis-eases and their spread. However, it can also be applied in other spheres of
human activity. In this Ph.D. thesis, the fundamental principles of
epidemi-ology and optimal control theory are applied to the study of the dynamics
of the crisis spread processes in economy.
A crisis, like a financial virus, can spread from one country to another,
like infectious diseases that spread from one person to another. In this
regard, the process of spread of a financial virus between the national
economies of countries through the network of their economic activities
can be modelled based on mathematical epidemiology concepts. Control,
optimal management strategy, and minimization of negative consequences
are essential for the application of optimal control theory.
At first, it was studied the cyclical nature of a crisis. The identification of
fluctuation waves in the long-term production dynamics of the
macroeco-nomic indicators of a particular real national economy was investigated,
and an input-output model with an apparatus based on regression analysis
was used.
The spread of financial contagion was investigated using mathematical
models based on ordinary differential equations that describe the dynamics
underlying a financial disease called a crisis, including economic
interac-tions between countries. The spread of financial contagion to other
coun-tries can be weakened by timely financial support both from the Central
Bank, in the form of Capital Reserves, and by borrowing money from the
International Monetary Fund, which play a central role in the international
management of financial crises. This way, it was analyzed optimal
strate-gies for using these controls and the corresponding effect on reducing or
eliminating the crisis during an outbreak among the national economies.
As a demonstrative example of the dynamics of the epidemiological
pro-cess of the spread of financial contagion between countries, the
epidemio-logical modelling in networks was applied. Its implementations used
math-ematical modelling software such as MATLAB, BOCOP, and NetLogo.
Contents
Introduction
1
I
State of the Art
9
1
An economy and crisis
11
1.1 An economy: nature and purpose . . . .
11
1.2 Economic development . . . .
12
1.3 Cyclical economy . . . .
14
1.3.1
Cycle stages and types . . . .
15
1.3.2
Phases of economic development in a cycle . . . .
16
1.4 Globalization . . . .
16
1.5 Systemic risk
. . . .
17
1.5.1
Dimensions of systemic risk and its features . . . .
18
1.5.2
Stages of the evolution of a systemic crisis
. . . .
18
1.5.3
Impact on the real sector . . . .
19
1.5.4
Contagion . . . .
20
2
Mathematical modelling and dynamic system analysis in economics
23
2.1 Mathematical modelling in economics . . . .
23
2.2 Choice of models . . . .
25
2.2.1
Dynamic and static models . . . .
25
2.2.2
Discrete and continuous models . . . .
25
2.2.3
Linear and non-linear models . . . .
26
2.2.4
Deterministic and stochastic models . . . .
26
2.2.5
Input-Output models . . . .
27
2.3 Dynamic systems . . . .
28
2.4 Dynamic system analysis . . . .
30
3
Mathematical modelling in epidemiology
35
3.1 Introduction to epidemic modelling . . . .
35
3.2 Mathematical formulation of epidemic models . . . .
36
3.3 Compartmental models . . . .
37
3.3.2
SIR model
. . . .
39
3.3.3
SIS model
. . . .
47
3.3.4
SIRS model . . . .
48
3.3.5
SEIR model
. . . .
49
3.3.6
Advantages and disadvantages of population models
. . . .
50
3.4 Modelling types . . . .
51
4
Optimal control theory
53
4.1 Introduction . . . .
53
4.1.1
Historical facts of optimal control theory formation . . . .
53
4.1.2
The purpose and applications of optimal control theory . . . .
54
4.1.3
Optimal control of epidemiological models . . . .
55
4.2 Formulation of an optimal control problem . . . .
55
4.3 Three formulations of the OC problem: Bolza, Lagrange and Mayer forms . . . . .
56
4.4 Pontryagin maximum principle . . . .
57
4.5 Analysis of a controlled economy . . . .
58
4.6 Analysis of the controlled SIS epidemic . . . .
60
4.7 Software . . . .
62
II
Original Results
63
5
Parametric identification of the dynamics of inter-sectoral balance: modelling and
forecasting
65
5.1 Problem statement
. . . .
65
5.2 Algorithm . . . .
66
5.3 Implementation . . . .
70
5.4 Conclusions . . . .
75
6
Banking risk as an epidemiological model: an optimal control approach
77
6.1 The SIR mathematical model . . . .
77
6.2 SIR model simulations with real data . . . .
78
6.3 Optimal control
. . . .
81
6.4 Conclusions . . . .
86
7
The spread of a financial virus through Europe and beyond
87
7.1 Methodology . . . .
87
7.1.1
Network . . . .
87
7.1.2
Epidemiological model
. . . .
88
7.1.3
Data . . . .
89
7.2 Results . . . .
90
7.2.1
Network . . . .
90
7.2.2
Epidemiological SIR model . . . .
91
7.3 Conclusions . . . .
95
8
The risk of contagion spreading and its optimal control in the economy
99
8.1 Data . . . .
99
8.2 Epidemiological model
. . . .
100
8.3 Network . . . .
103
8.4 Optimal control
. . . .
104
8.5 Numerical simulation . . . .
107
8.6 Conclusion . . . .
110
Conclusion and future directions
111
Appendix
115
List of Figures
1.1 An economy as a set of interdependent production relations and consumer
activi-ties.
. . . .
12
1.2 Structure of the world economy . . . .
12
1.3 Economic equilibrium. (a) The intersection of the supply and demand curve
indi-cates an equilibrium price. (b) An imbalance in which the price is too high, and
therefore consumer demand is lower than supply. (c) An imbalance in which the
price is understated, and therefore consumer demand exceeds supply. . . .
13
1.4 Stable equilibrium and unstable equilibrium . . . .
14
1.5 Phases of the economic cycle. . . .
15
2.1 Input–Output transactions table. . . .
27
3.1 Compartmental model. . . .
38
3.2 Scheme of the basic dynamical SIR model. . . .
40
3.3 (a) An epidemic curve is monotonically decreasing. Parameters
β = 1
,
γ = 0.0821
,
N
= 2
,
I(0)
= 2
. (b) An epidemic curve is non-monotonically decreasing.
Parame-ters
β = 0.009
,
γ = 0.35
,
N
= 100
,
I
(0)
= 1
. . . .
43
3.4 The SIR schematic model with vital dynamics
. . . .
46
3.5 Scheme of the basic dynamical SIS model with constant population.
. . . .
48
3.6 Scheme of the basic dynamical SIRS model with constant population.
. . . .
49
3.7 Scheme of the basic dynamical SEIR model with constant population.
. . . .
50
5.1 Modelling curves of gross outputs (left) and fluctuation of gross output (right) of the
entire economy. . . .
71
5.2 Modelling curves of gross outputs (left) and fluctuation of gross output (right) for the
agricultural sector – top figures; for industry – figures in the middle; for the service
sector – bottom figures.
. . . .
72
5.3 Modelling the harmonic waves for the economy of France. . . .
73
5.4 Modelling curves of gross domestic product (left) and fluctuation of gross domestic
product (right) of all economy . . . .
74
6.1 Summary statistics for
β
and
γ
based on the data of [159]. . . .
79
6.2 The SIR contagion risk model
(6.1)
–
(6.2)
with parameters
β
and
γ
as in Figure 6.1,
S
(0)
= 168
,
I(0)
= 1
,
R(0)
= 0
and
T
= 100
. . . .
80
6.3 The SIR contagion risk model
(6.1)
–
(6.2)
with parameters
β
and
γ
as in Figure 6.1,
S
(0)
= 168
,
I(0)
= 1
and
R(0)
= 0
: illustration of the time needed to achieve the
contagion-free equilibrium. . . .
81
6.4 Contagion risk model with and without control. From the first column to the second
column: Portugal and Spain as initially infected countries (Scenario 1 and
Sce-nario 2, respectively.) . . . .
83
6.5 Contagion risk from the United Kingdom (Scenario 3) with and without control. . .
84
6.6 The optimal control
u(t)
for different values of the weight
b
. . . .
85
7.1 Fully connected network, where each country is represented as a node and edges
indicate the existence of a link between countries.
. . . .
89
7.2 Summary statistics of
γ
and
β
parameters for the 16 European and Non-European
developed countries considered in our study. . . .
90
7.3 Virus spreading in the network of countries with parameters
β
and
γ
taken from
Figure 7.2; initially infected country is Portugal (PT). Nodes in white mean
“Sus-ceptible”; nodes in black mean “Infected”; nodes in grey mean “Recovered”.
. . .
92
7.4 Virus spreading in the network of countries with parameters
β
and
γ
taken from
Fig-ure 7.2; initially infected country is the United States (USA). Nodes in white mean
“Susceptible”; nodes in black mean “Infected”; nodes in grey mean “Recovered”. .
93
7.5 Virus spreading in the network of countries with parameters
β
and
γ
taken from
Figure 7.2; initially infected country is Switzerland (CH). Nodes in white mean
“Sus-ceptible”; nodes in black mean “Infected”; nodes in grey mean “Recovered”.
. . .
94
7.6 The SIR contagion risk model
(7.2)
–
(7.3)
with parameters
β
and
γ
for Portugal,
United States, and Switzerland, taken from Figure 7.2; the initial conditions are
S
(0)
= 15
,
I(0)
= 1
,
R(0)
= 0
.
. . . .
95
7.7 The SIR contagion risk model with parameters
β
and
γ
taken from Figure 7.2.
Countries belong to the Group 2. . . .
96
7.8 The SIR contagion risk model with parameters
β
and
γ
taken from Figure 7.2.
Countries belong to the Group 3. . . .
97
7.9 The SIR contagion risk model with parameters
β
and
γ
taken from Figure 7.2.
Countries belong to the Group 1. . . .
98
8.1 Scheme of the basic dynamical SIR model. . . .
100
8.2 Summary statistics of
β
and
γ
parameters for the 13 European and Non-European
developed countries considered in our study. . . .
101
8.3 The SIR contagion risk model. From upper-left to lower-right: Portugal, France,
USA and Canada as initially infected country. . . .
102
8.4 Virus spreading in the network of 13 countries with parameters
β
and
γ
as in
Fig-ure 8.2. From upper-row to lower-row: Portugal (
T
= 0
,
T
= 35
,
T
= 162
,
T
= 270
months) . . . .
104
8.5 Virus spreading in the network of 13 countries with parameters
β
and
γ
as in
Fig-ure 8.2. From upper-row to lower-row: France (
T
= 0
,
T
= 5
,
T
= 49
,
T
= 50
months)
. . . .
105
8.6 Virus spreading in the network of 13 countries with parameters
β
and
γ
as in
Fig-ure 8.2. From upper-row to lower-row: the USA (
T
= 0
,
T
= 2
,
T
= 15
,
T
= 16
months). . . .
106
8.7 Contagion risk from Portugal (Scenario 1): with optimal control (dotted line) vs
without control (continuous line). . . .
108
8.8 Contagion risk from France (Scenario 2): with optimal control (dotted line) vs
with-out control (continuous line). . . .
109
8.9 Contagion risk from the USA (Scenario 3): with optimal control (dotted line) vs
without control (continuous line). . . .
110
8.10 State and Control Variables with control and without control for
R
0= 10
. Parameter
values:
β = 1
,
γ = 0.1
,
T
= 5
,
b
= 15
,
u
max= 0.06
,
i
0= 0.01
. . . .
119
8.11 Multipliers. . . .
119
8.12 Dynamic constraints. . . .
120
8.13 Dynamic constraints. Each row corresponds to one of the scenarios (Portuguese
scenario, Spanish scenario or British scenario) with the application of optimal control.
131
List of Tables
1
World output (% Real GDP).
. . . .
2
1.1 Juglar on the inevitability of crises, as well as disease. . . .
17
5.1 The coefficients of trends determination. . . .
70
5.2 Contribution of harmonics into the oscillatory process. . . .
70
5.3 The quality of modelling trajectories of issues. . . .
71
5.4 The coefficients of trends determination. . . .
74
5.5 Contribution of harmonics into the oscillatory process, where the “—” means that
the harmonic is insignificant in the sector according to Student’s t-test. . . .
74
5.6 The quality of modelling trajectories of issues. . . .
74
6.1 Values of the cost functional
J (6.4)
. . . .
84
7.1
Grouping of countries depending on γ parameter.
. . . .
91
Acronyms
GDP
: Gross Domestic Product
GNP
: Gross National Product
IMF
: International Monetary Fund
ODE
: Ordinary Differential Equation
PDE
: Partial Differential Equation
I-O
: Input-Output Model
SIR
: Susceptible-Infected-Recovered
R
0: Basic Reproduction Number
DFE
: Disease Free Equilibrium
EE
: Endemic Equilibrium
ABM
: Agent-based Modelling
OC
: Optimal Control
NLP
: Nonlinear Problem
GO
: Gross Output
OLS
: Ordinary Least Squares
BIS
: Bank for International Settlements
CBS
: Consolidated Banking Statistics
Introduction
Motivation
Everything in the world is dynamic, or, in other words, nothing in the world is static, especially in the
long term. Every day, all of us observe the changes that are happening in ourselves, in the world around
us. For instance, looking back into the past, we can notice that the previous century was full of wide-scale
technological and scientific achievements which led to economic changes. It was also characterized by
the transition to mass mechanical production and the scientific and technological revolution that led to the
post-industrial stage of capitalism in the world economy [134].
The modern world is also actively changing. Its dynamic changes are caused by scientific and
tech-nological discoveries and the information revolution. In the 21st century, everything is filled with a new,
innovative sense. There is a rapid growth of global transformation processes, which are caused primarily
by economic factors and lead to economic changes primarily in the world economy, which in turn is
trans-formed into a global economy. This process, called globalization, is based on the growing
interconnect-edness and interdependence of the modern world, which characterizes the development of international
relations in such different areas of the world community as financial and economic, information and
tech-nological, social and political, . . . [33, 143]. This leads to an increase in the role of external factors in the
evolution of all the countries participating in the process. Globalisation processes lead to both qualitative
and quantitative changes in the global economy, in which the interdependence of national economies is
increasing. They also lead to economic problems emergence and the aggravation of global problems for
humankind.
Strengthening international economic interdependence has become one of the leading causes for the
beginning and aggravation of the crisis in 2007–2009. In 2007, the US economy entered a mortgage
crisis. In 2008, the sub-prime mortgage crisis provoked a global liquidity crisis of world banks. It had a
large-scale, systemic nature and the most devastating consequences in the world economy [104]. Banks
stopped lending, indicating a severe shock to the financial system, and this quickly affected the real sector
and led to a decline in Gross Domestic Product (GDP) growth.
The crisis has become a significant source of knowledge about the process of synchronization of
economic relations in a crisis period. A simultaneous decline in all segments of the financial market,
which subsequently had a negative impact on the dynamics of the real sector, was observed in almost
all countries. A significant decrease in GDP was observed both in developed countries and in developing
countries (Table 1).
Source: Statistical data taken from IMF official website [105].Table 1 – World output (% Real GDP).
2005 2006 2007 2008 2009 World 4.6 5.2 5.3 2.8 -0.6 Advanced Economies 2.7 3.0 2.7 0.2 -3.2 United States 3.1 2.7 1.9 0.0 -2.6 Euro Area 1.7 3.0 2.9 0.5 -4.1 • France 2.0 2.4 2.3 0.1 -2.5 • Portugal 0.8 1.4 2.4 0.0 -2.6 • Ireland 6.0 5.3 5.6 -3.5 -7.6 Japan 1.9 2.0 2.4 -1.2 -5.2 United Kingdom 2.2 2.8 2.7 -0.1 -4.9 Canada 3.0 2.8 2.2 0.5 -2.5Emerging and Developing Economies 7.3 8.2 8.7 6.0 2.5 Central and Eastern Europe 5.9 6.5 5.5 3.0 -3.6
• Poland 3.6 6.2 6.8 5.0 1.7
Commonwealth of Independent States 6.7 8.8 9.0 5.3 -6.5
• Ukraine 2.7 7.3 7.9 2.1 -15.1
Developing Asia 9.5 10.4 11.4 7.7 6.9
• China 11.3 12.7 14.2 9.6 9.1
Latin America and the Caribbean 4.7 5.6 5.7 4.3 -1.7
• Brazil 3.2 4.0 6.1 5.1 -0.2
Middle East and North Africa 5.3 5.8 6.0 5.0 2.0
• Morocco 3.0 7.8 2.7 5.6 4.9
Sub-Saharan Africa 6.3 6.4 7.0 5.5 2.6
• Angola 20.6 18.6 20.3 13.3 0.7
The crisis showed that even if countries are loosely coupled economically during stable periods, the movements of their macroeconomic indicators in crisis periods are unidirectional, which is confirmed by Table 1. This similarity in reactions is associated with the occurrence of systemic risk and usually happens because a crisis in one country provokes a crisis in another, or, as it is often said, crisis spreads to other countries [121].
Thus, the global crisis showed how closely financial stability, in the context of the national economy, and the state of the entire global financial system are interconnected, and how quickly financial contagion can spread between countries and generate a global epidemic in the world financial network. The crisis made it clear that the acceleration of integration processes between countries has not only positive aspects in the form of more dynamic development of the world economy but can also cause the transmission of financial imbalances to almost all national segments of the global economic space.
From the 20th century until the onset of the crisis, scientists had been actively exploring the relationship between the real and financial sectors of the economy, considering the financial sector as a “conductor” of macroeconomic fluctuations. However, the global crisis has made scientists rethink the role of the financial sector in the economy, and the instability of the financial market has become regarded as the primary source of economic problems. This forced scientists to pay more attention to systemic risk. This could lead to the spread of shocks from one segment of the financial sector to other segments of this sector, as well as the transmission of the financial contagion to the real sector, leading to worsening macroeconomic dynamics and a sharp economic slowdown. The magnitude of the
adverse effects associated with the implementation of systemic risk in the financial sector is also indicated by the fact that it can infect not only the national economy but also spread to other countries.
It was not the first economic crisis that humanity has overtaken. Before that, there were already more than a dozen crises (1857 – 1858 the first worldwide economic crisis, 1873 – 1896 the long depression, 1929 – 1939 the great depression, and others). Scientists from all over the world have been trying to find effective anti-crisis measures that would help overcome the crisis and put economics into the growth phase. The study of this phenomenon has always come down to the fundamental question of how to prevent new crisis in the future. However, the fact that the crisis arose in 2008 and expanded to a global scale indicates that this issue has not been sufficiently studied. The onset of the global crisis focused not on the merits, but on the shortcomings of the methods, techniques and macroeconomic models used by scientists to warn and prevent crises. The existing macroeconomic models were insufficient for a high-quality and high-precision solution to such a problem in full, as well as its long-term forecast and predictions. Therefore, nowadays, the prevention of the global economic crisis and risk minimization are essential tasks in the field of economics.
The global crisis touched every country, and it has led to the financial sector’s collapse in the world economy, the effects of which can be seen and experienced even today [44, 58, 104]. Moreover, crises continue to occur at different economic levels, both at the micro and macro levels [122], which make the economy an interesting object of study, pointing out that the economy has not been studied yet completely [121]. Thus, the economy needs to be investigated in order to prevent possible negative consequences, since the systemic risks accumulate in the world financial system and become a general threat to the new global crisis [67]. The significance of global systemic financial risks is increased by their complexity in the identification, estimation, and developing methods for their calculation and minimization [47], that is complicated by the inability to observe the economic crisis in its purest form.
Mathematical models are important tools for studying and investigating in economics as well as in other sciences. For any dynamical object (technical, economic, environmental, and others), the most critical is the problem of limited resources and their optimal use. Thereby, there is a need to build mathematical models that adequately describe the existing trends and provide high-precision predictive properties of the dynamical systems. For this reason, it is necessary to evaluate the quality of mathematical models of dynamical processes from the perspective of imitative and predictive properties [120].
The meaning of mathematical modelling, as a research method, is determined by the fact that a model is a conceptual tool focused on the analysis and forecasting of dynamical processes using a set of differential or differential-algebraic equations that are based on a quantitative description of real phenomena. However, the parameters of these equations are usually unknown in advance. Hence, in practice, any direct problem (imitation, prediction, and optimization) is always preceded by an inverse problem (model specification and identification of parameters and variables included in it) [154, 165].
Modelling of economic systems is a perspective area of economic research, particularly in creating useful tools for solving various financial and economic problems of a market economy. Mathematical models are commonly used to test proposed scientific hypotheses and expected trends of economic development. In order to construct dynamical models in economics, it is necessary to formulate principles, based on economic theory, and consider equations that adequately determine the evolution of the investigated process. If hypotheses are correct, then the long-term conse-quences of management decisions making are investigated with the help of mathematical models of macroeconomic systems and processes.
Macroeconomic mathematical models have important practical significance. They are used to develop concepts of economic and social development, to study possible alternative economic policies, and for forecasting the generalized indexes of a national economy. Hence, the construction of mathematical models of macroeconomic systems and the development of their identification apparatus is the topic of ongoing interest. The application of mathematical models to the systemic risk can comprehensively characterize the modern picture of the financial world. Besides, it can be useful for improving existing models and developing new mechanisms and methods for modelling, forecasting, analysing the economy. It is essential in order to develop new methods of protection against global threats, which is necessary not
only in terms of national economic regulation but also the coordination of stabilization policies of individual countries.
State of the art
The contribution of scientists in the study of the world economy is huge and has increased rapidly over the past decade. The interest of scientists in this field predictably increases after the occurrence of economic collapses. For example, after the crisis in 2008 (when the fourth largest investment bank in the United States filed for bankruptcy, which had the effect of a “bomb” and within a few days, the US credit markets were subjected to a financial tsunami, which then overtook the whole world), the authors of the article [59] focused their interest on the Lehman Brothers Collapse and the subsequent international shock transmission. The authors of [73] explored nine crises that have occurred from 1997 – the Asian crisis (1997–1998) to 2013 – the European debt crisis (2010–2013). Financial crises from 2007 to 2014 are investigated and analysed in such works as [15, 55, 65, 183, 197]. After the global crisis, and even now, the interest of scientists began to focus more and more on research topics such as financial contagion; mechanisms for the spread of financial stress between institutions and also countries; the relationship on the systemic risk transmission of the financial market and the implications for the real economy; and identification of the degree of vulnerability of a national economy to shocks emanating from other regions.
In work [145], the authors investigated the crisis periods and the effect of financial infection between countries. As a result of their research, they identified and found confirmation that the United States was the source of financial contagion during the recent crises. The authors in their study focused on banking risks and the transfer of these risks between countries as the main channel for the spread of the crisis. The effect of the spread of financial contagion from the US financial market to other countries, which occurred during the global crisis of 2008, was also reflected in [197]. The authors of this paper investigated the correlation behavioural response between countries at different time scales. The authors identified that during the crisis, countries showed an increase in inter-market correlations compared to the pre-crisis period. The authors in article [65] investigated the effect of financial contagion from the US banking market on other countries. They discussed three channels of contagion – systematic, idiosyncratic and volatility, through which the banking sectors of the economies during the crisis were exposed to infection. The authors in [183] went further in their consideration of the transmission channels of financial infection and in addition to the banking sector channel also studied the contagion in the stock market channel and the channel of the sovereign Credit Default Swaps market. Also, one of the interesting types of research in this area is the work [4], where the authors studied information contagion due to the counterparty risk and examined its effects on banks ex-ante choices and systemic risk.
Mechanisms of mutual influence of countries in crises are studied using mathematical and instrumental methods of economic analysis. The main stage of modelling the epidemic of the financial crisis is the choice of a mathematical model that adequately describes the process. One of the common methods used to study this problem is correlation analysis, as example, see articles [9, 15, 55, 73, 102, 145, 197]. In [102], the local correlation is used to study contagion between international equity markets. In [197], the stock market contagion is investigated using the multiscale corre-lation test, while in [9] – using the asymmetric dynamic conditional correcorre-lation dynamics. Based on cross-correcorre-lations analysis, the authors in [55] analysed the financial crisis of 2008 and its contagion effect. In works [15, 73], in addition to the correlation test, the coskewness and covolatility tests are also used to study contagion.
Researchers also use other methods and models. For instance, the article [5] explored the behaviour of financial contagion in world economic markets using the jump-diffusion model, that helped to identify the patterns of jumps excitation in behavioural dynamics. In [73], the authors used a regime-switching model to date the crisis. The authors in [42] studied bank-run contagion using coordination games.
Several researchers draw an analogy between economic and natural viruses, between a network and biological epidemics [159, 189, 200]. Because economic interactions of countries form a network, the use of network theory can enrich our understanding of financial systems as complex systems, which is used to explain the process of virus contamination in the network. Financial contagion demonstrates the ability to pass from one country to another, from one sector of a country’s economy to another that is similar to the spread of infectious diseases among the population.
Because of this, it makes sense to consider financial contagion as an infection in the economy, describing the financial virus spread in the economy by similar models of differential equations as for the spread of a viral disease, and apply similar mathematical methods from epidemiology for studying the process of “disease” in the global economy, from which comes the need to study the optimal control of these processes. However, too few scientific papers focus on the infectiousness of the economy in terms of epidemiology, and the use of epidemiological models in the economy is scanty.
Scientists in their works quite often draw an analogy between a complex system of interconnections in an econ-omy and a network. For example, the authors investigated the propagation of different types of shocks, in isolation and in combination with the other shocks, through the input–output network for the US economy in paper [2]. The authors in [79] investigated the corporate default swap spreads using the vector autoregressive regression with correlation networks in their model. In [101] the authors made a detailed literature review on interbank networks with direct con-nections in the interbank market as well as indirect concon-nections in the form of total assets. The authors in [7] analysed a banking network model featuring risk transmission via different channels using the baseline network configuration with an interbank matrix computed via the closest matching algorithm. In [133], the authors used a network theory for studying input–output data of money flows and for step by step representation of cascading failure propagation in the world economic network. They proposed the methodology for ranking the economic importance of each industry and country in order to measure interdependencies in the world industrial network.
A useful literature review on the application of network theory in studying economics is provided in [112], where the authors demonstrated the relevance among scientists of the application of network theory that confirms the ef-fectiveness of this method for describing the system structure, studying the effect of insolvency of system subjects, depending on the structure of connectivity as well as for the analysis and of the penetration effect, investigation the contagion effect. The impact of connectivity on financial stability is studied in [28] using random regular graphs models. The distress propagation in a financial system is represented as a large network in article [13], while the study in [75] is focused on contagion in arbitrary financial networks. In the article [3], the authors analysed the relationship between the structure of the financial network and the density of relationships in it and the probability of system failures due to infection of the counterparty. In [14] it is presented a link prediction method with the maximum risk allocation on it for the formation of the density interbank. The authors also make a comparative analysis of both minimum-density and maximum entropy methods in the context of stress testing, showing that the solution with the first one overestimate the risk contagion, and the other one, on the contrary, underestimates it. Neural networks are also used as research methods. For example, the paper [20] is aimed at the prediction of bank efficiency using neural networks and multiple linear regression.
One of the difficult tasks in the economy is forecasting the future state of the economy. An essential tool for its prediction is the dynamical model of inter-industry equilibrium balance developed by scientist W. Leontief [131]. The so-called input–output model for which W. Leontief was awarded the Nobel Prize in Economics, became the basis of mathematical economics [132]. This model represents country’s macroeconomy in terms of interdependencies between different sectors or regions, describing the equilibrium behaviour both of national and regional economies [19, 37, 127, 135, 141, 179], and is a useful tool in the economic decision making processes [142]. The dynamic Leontief’s model is considered in papers [1, 107, 125, 193, 194]. The interbranch balance model of a macroeconomic system often occurs in mathematical economics writings of such scientists as [83, 84, 85, 115, 156, 166, 172, 173]. The natural resources balance, which is renewed in the functioning of macroeconomics, is simulated using Leontief’s model in the writings of such scientists as [63, 64, 70, 158]. Depending on different hypotheses regarding financial balance, the price trajectories can also be received with the help of this model.
The input–output model becomes fundamental for scientists and widely used in its different variations. The au-thors in [85] proposed the Leontief-based infrastructure input–output mode. The auau-thors in [56, 84, 172] and [173], used in their research the inoperability input–output model. The same model was used in [83], where the authors pay their attention to such sectors of the country economy as electricity, communications, and water. In [171] it is developed a novel linear Leontief model with an Armington flavour. The authors in [19] used a modification, which is
used to study changes in international or interregional trade patterns by including the effects of changes in transport networks – random–utility–based multiregional input–output models. The analysis of the input–output model is critical for developing government regulatory programs, that are an integral part of the economic and environmental policy for any developed market system. Such an analysis is a useful tool for studying the strategic directions of economic development, the expansion of dynamical inter-industry relations, and the interaction between economic sectors and the natural environment [63, 70].
The definition of financial contagion, which scientists consider in their works, varies considerably. Nevertheless, there is a similarity in all considered models: they always attempt to provide a structure, explaining why a shock in one country may be transmitted elsewhere [110]. As shown above, the applications of various networks behind the structure to the problems of financial and interbank systems are popular among scientists. However, the credit risk transfer, including such transactions as loan guarantees, is also an important source of potential contamination, which makes our work in demand.
The functioning of a world market economy, as well as any economic system, is not uniform and uninterrupted, as economic crises are inevitable as well as inefficient management and incomplete use of resources. The world economy is formed from countries’ macroeconomics. It is therefore important to study economic (financial, market and trade) relations between countries, as well as macroeconomic processes in individual countries. Thus, the economy is an open and multifaceted area for investigations. Scientists, from all over the world, have proposed their theories about the development of the financial crisis and methods to prevent it. However, this topic has not been studied yet completely. The relevance of its study is confirmed by the fact that crises continue to occur from time to time in different countries and different sectors of the economy, that says about the systemic nature of their functioning. This confirms and accentuates the need to study systemic risks both within a separate macroeconomic system and in terms of global economic dynamics, in order to determine a rational and effective strategy of the behaviour of industries, regions, countries and the world community in general.
The object of study
The objects of study are the world economy and the economic relations between the subjects of the global financial system for which the statistical information is available.
The subject of the research
The subjects of the research are
• at the macroeconomic level – the relationship between the sectors of the real macroeconomic system and harmonic waves propagating in it;
• at the global level – the process of contagion transmission between countries’ economies that have financial relations.
The purpose of the research
This research empirically intends to identify the regularities of change in the dynamics of economic development of the countries with advanced economies. In the current research, we investigate the regularities of transmission the effect of negative dynamics between countries and demonstrate the effectiveness of the application of optimal control theory for the regulation of the dynamics of general economic development. Relevant goals for achieving a common goal:
• To construct a mechanism for simulation and forecasting dynamics of a macroeconomic system.
6
• To develop methodologies for specification and identification of a dynamic model, that represents the interde-pendencies between different branches of the economy, and economic waves/cycles for analysis and forecast-ing of the macroeconomic system development.
• To explore the dynamic shock transmission behaviour in the banking market between banks of different coun-tries in term of epidemiology and optimal control theory.
• To investigate the process of spreading the crisis through network interconnections and test the contagion effect depending on banking systems’ exposures to particular countries.
• To test the effectiveness of applying optimal control theory to avoid or minimize significant losses in the world economy.
Research methodology
In this work, we use econometric methods for parametric identification of the input–output model, that depicts inter-sectoral relationships within an economy. We use the decomposition into the trend and periodic components for the specification of unknown functions of output sectors, where the periodic cyclical oscillations we also decompose into a cut Fourier series with defining the significant harmonics in it. We identify a financial contagion as epidemic disease and simulate it using an epidemiological model as a system of ordinary differential equations. We simulate the interactions between countries, through which the infection spreads out, basing on network theory. We propose a formulation of the optimal control problem of financial contagion in Bolza form, and we solve it by rewriting it in the Mayer form.
Scientific and practical novelty
The scientific and practical novelty of this work is as follows:
• The approach to the problem of specification and identification of a weakly-formalized dynamic system was designed.
• The procedure of matching parameters of the linear stationary Cauchy problem with the decomposition of its upshot on trend and periodic component was proposed.
• The approach for the allocation of significant harmonic waves, which are inherent for real macroeconomic dynamic systems, was designed.
• The method of modelling and investigation of financial contagion in the network using epidemiological models was introduced.
• The procedure for identifying the transmission and recovery rate was proposed. • The approach to minimising the negative effects of a financial network were designed.
Thesis Outline
In order to achieve the goals mentioned, the thesis consists of the following chapters. Chapter 1 introduces the development of the economy, with cyclical fluctuations in the process of its development. This chapter also provides basic information about the crisis and financial contagion in the economy. Chapter 2 gives an introduction to mathe-matical modelling and dynamic analysis and their roles in economics. Chapter 3 describes the role of mathemathe-matical modelling in epidemiology. This chapter also describes some of the well-known epidemiological population models. Types of modelling of epidemiological problems and programs for their simulation are also given in Chapter 3. Chap-ter 4 sets out the theory of optimal control, its goals, both in economics and in epidemiology. Methods for formulating
the optimal control problem and a method for finding optimal control, called the Pontryagin Maximum Principle, were presented. Chapter 4 provides an example of an optimal control problem for the economy, based on an interindustry balance (Leontief’s input-output model), an example of an optimal control problem for the epidemic model, and also provides a program for solving them.
Chapter 5 is devoted to modelling and identification of the dynamics of the inter-sectoral balance of a macroeco-nomic system. An approach to the problem of specification and identification of a weakly formalised dynamical system is developed. In Chapter 5, the emergence of crisis processes is associated with the cyclical behaviour of economic development. Cyclicity is considered as an oscillatory process of harmonic waves of different lengths and frequen-cies. An approach for detection of significant harmonic waves, which are inherent to real macroeconomic dynamical systems, is developed. When the significant harmonic waves were identified and analysed, a forecast of the country’s economic development for the near future was made.
Chapter 6 presents the spread of infection between countries through the channel of the banking sector. In this chapter, we present a system of ordinary differential equations, simulating data from the largest European banks. Then, an optimal control problem is formulated in order to study the impact of a possible measure of the Central Bank in the economy. The proposed approach enables qualitative specifications of contagion in banking obtainment and adequate analysis and prognosis within the financial sector development and macroeconomy as a whole. We show that our model describes well the reality of the largest European banks. Simulations were done using MATLAB and BOCOP optimal control solver, and the main results are taken for three distinct scenarios.
In Chapter 7 we analyse the importance of international relations between countries on financial stability. The contagion effect in the network is tested by implementing an epidemiological model, comprising several European and non-European countries and using bilateral data on foreign claims between them. Banking statistics of consolidated foreign claims on ultimate risk bases, obtained from the Banks of International Settlements, allow us to measure the exposure of contagion spreading from a particular country to the other national banking systems. We show that the financial system of some countries, experiencing the debt crisis, is a source of global systemic risk because they threaten the stability of a larger system, being a global threat to the intoxication of the world economy and resulting in what we call a “financial virus”. Illustrative simulations were done in the NetLogo multi-agent programmable modelling environment and in MATLAB.
Chapter 8 presents the epidemiological model for investigating the behaviour of the contagion spreading when a particular country cannot fully fulfil its obligations. The research in this chapter focuses on the dynamic behaviour of contagion spreading in the global financial network. The effect of infection by a systemic spread of risks in the network of national banking systems of countries is tested. An optimal control problem is then formulated to simulate a control that may avoid significant financial losses. The results show that the proposed approach describes well the reality of the world economy and emphasizes the importance of international relations between countries on financial stability.
Finally, the main conclusions are reported and future directions of research are pointed out.
Part I
Chapter
1
An economy and crisis
This chapter describes the object of study in detail. The nature of its functioning, goals, and risks, arising in the process of its development, are also considered. Finally, this chapter will shows and confirms the significance and necessity of its study. This chapter is divided into five sections. Section 1.1 provides a definition of economy as an
interconnected relationships of vertical and horizontal activities. Section 1.2 presents a measure of economic development and the importance of the production – consumer balance for this. Section 1.3 describes the cyclical nature of the economy, represents cycle types, as well as the stages and phases of economic development in a cycle.
Section 1.4 explains the importance of globalisation in the economy and Section 1.5 explains the exposure of the economy to systemic risk that tends to spread at all levels of economic activity.
1.1
An economy: nature and purpose
Definition 1 (An Economy). An economy is the economic activity of society, as a result of which an economic system is formed, consisting of interconnected subsystems and the relations between them that arise in the process of production, distribution, trade and consumption of goods and services (Figure 1.1) [113].
Production is the manufacture, mining or cultivation of material goods. The supply of limited resources to var-ious enterprises and industries is called distribution. The use of goods and services to meet their needs is called consumption, respectively.
An economy is a multilevel system, generalised by objects and subjects of economic activity (Figure 1.2). At the microeconomic level, the economy is perceived as a separate enterprise or production, and the study of the processes of economic activity takes place at the level of individual enterprises. The macroeconomic level is the economy of a country where the study of economic activity takes place at the level of the national economy, which is expressed in the gross national product (GNP), national income, aggregate supply and demand, levels of inflation, as well as employment and unemployment, where all these values are considered in aggregated form. Macroeconomics, as a branch of economic theory, studies the entire economy as well as such patterns that govern the activity of the system [113]. World economy level reflects the totality of national economies and their diverse international economic interactions in the context of international exchange of goods and services. The world economy, on the one hand, is characterized by certain patterns of its development, and on the other hand, influences the development of each national economy.
The genesis of the economic system, as well as its functioning and development, are impossible without its subjects which are its driving force. Subjects of economic activity are states, enterprises and households, which in the course of their economic activities form a branched system of interconnections that gives it integrity and systemic
Figure 1.1 – An economy as a set of interdependent production relations and consumer activities.
Source: Figure is based on [113].Figure 1.2 – Structure of the world economy
nature. On the one hand, this determines the overall effectiveness of this system that exceeds the aggregate result of its constituent parts measurements, and on the other – the regularities that adjust its activities. Thus, the economy, as an open system with direct and inverse, horizontal and vertical connections (Figure 1.2), can successfully develop only with the effective management of these connections, both at the micro and macro levels. At the same time, the main goal of the economy is to satisfy consumer demand (Figure 1.1) and, therefore, effective resource management [113].
1.2
Economic development
The efficiency of the economy is evaluated using a number of important criteria and indicators that characterize the dynamics and state of the economy. One of the most common indicators used to track the state of a national
economy, as well as the level and pace of economic development, is the gross domestic product (GDP) [50, 123]. Definition 2 (Gross Domestic Product). GDP is a monetary measure of a system of national accounts that measures the market value of all the final goods and services produced in a country by residents in a specific time period and used for final consumption.
Thus, GDP covers the main areas, sectors and factors of economic development, while characterizing the overall economic activity both in individual countries and in the world, where stable GDP growth is a sign of an efficient and powerful economy. According to the long-term trend of economic development, it is growing. Graphically, this trend is a straight line with a positive slope.
The economy is characterized by such conditions as equilibrium (balance) and disequilibrium (imbalance). The first one is characterized by constant stable economic growth, where output increases in proportion to the growth of production factors. This means that there is a complete correspondence between supply and demand, production and consumption, as well as expenses and income. The second one characterizes the fluctuations in the dynamics of economic development. However, given the characteristics of the real economy, the equilibrium state cannot be long-lasting, since with every change in the availability of goods in the market, the market balance of supply and demand changes, which leads to a shortage or excess of goods that will interrupt the balance and affect the price (Figure 1.3) [49]. Disequilibrium that establishes a new equilibrium and changes the level of prices, as well as the
(a) (b)
(c)
Figure 1.3 – Economic equilibrium. (a) The intersection of the supply and demand curve indicates an
equilibrium price. (b) An imbalance in which the price is too high, and therefore consumer demand is
lower than supply. (c) An imbalance in which the price is understated, and therefore consumer demand
exceeds supply.
Source: Figure is based on [49].
equilibrium, and sets the previous equilibrium price and output, indicates that the equilibrium was stable (Figure 1.4).
Figure 1.4 – Stable equilibrium and unstable equilibrium
Thus, the functioning of a market economy, as well as any economic system, is not steady and uninterrupted. The state of equilibrium is periodically disturbed, since in real life a crisis and such stress-factors as the rise and fall of prices, inflation, unemployment and others, are inevitable. Within the framework of the evolutionary movement of the world economy, it is possible to note the repeated undulation of these processes in various sectors, financial markets, etc. Economic crises, as well as other economic stress factors, make oscillatory movements in medium and short time intervals and accompany the entire history of human society. From time to time, economic growth and successful industrial development alternate with periods of stagnation and recession, accompanied by a decrease in production and unemployment, that is, a decrease in all economic (business) activity, and, accordingly, macroeconomic indicators. Such periodic fluctuations and the recurrence of economic phenomena indicate the cyclical nature of economic development and the functioning of the world economy. Consequently, it is possible to consider economic cycles as GDP fluctuations around a long-term growth trend (Figure 1.5).
1.3
Cyclical economy
A characteristic feature of the functioning of national economies and the world economy as the entire is cyclicity. Their economic growth (upturn phase) alternates with the processes of stagnation and decline in productions (down-turn phase), i.e., a decline in all the economic activity. After a decline phase, again, the economic cycle continues, similarly with ups and downs. The process of repeating periods of economic recovery and economic downturn of economic activity, which occurs due to changes in interdependent indicators, is called economic cycles (Figure 1.5) [120].
Definition 3 (Economic cycle). The economic cycle is the regular fluctuation of the economy and business activity between periods of expansion (growth) and contraction (recession) [48].
The length of the economic cycle is determined by the time interval between the points of maximum economic development, or other two identical conditions of the economic conjuncture, i.e., from peak to peak, or from trough to trough.
Remark 1. The economy does not move in a circle, but in a spiral upward from the lower level of development to the highest.
Figure 1.5 – Phases of the economic cycle.
1.3.1
Cycle stages and types
There are four main types of economic cycles depending on the regularity and the frequency of their occurrence: • The Kitchin cycle (period 3 – 5 years) is a short-term business cycle of about 40 months [116] which is
associated with both aggregate demand and aggregate supply shocks, and it is believed that this is due to delays in the movement of information affecting decision-making by commercial firms [117];
• The Juglar cycle (period 7 – 11 years) is a fixed investment cycle in which fluctuations in fixed capital
invest-ment can be observed [117];
• The Kuznets cycle or Kuznets swing (period 15 – 20 years) is also known as a medium-range economic
wave which is associated with demographic processes and changes in construction intensity that it caused [126] Others scientists have interpreted it as an infrastructural investment cycle [72, 117];
• The Kondratieff wave K-wave (period 45 – 60 years), which was called long economic cycle, is associated with
the mechanism of accumulation and distribution of capital sufficient to create new elements of infrastructure. Long cycles consist of alternating intervals of high sectoral growth and intervals of relatively slow growth [117]. The economic cycle (Figure 1.5) consists of the following stages:
• Peak is the highest point of the cycle, when business activity is the highest, production runs at full capacity,
maximum production in the country’s economy, the highest level of employment and prices has been achieved. It is the stage of maximum growth, after which there are no future signs of economic growth. The end of the peak marks the beginning of contraction (recession).
• Recession (decline) is characterized by a decrease in production and a decrease in business and investment
activity, as well as negative economic growth.
• Trough is the lowest point of production and employment. This stage is more associated with the withdrawal
from the market of failed producers of goods and services, with falling prices and rising unemployment when business activity and production decline to the lowest level.
• Expansion occurs after reaching the lowest point of the cycle (trough). Characteristic of this stage is a gradual
increase in employment and production and introduction of innovations in the economy with a short payback period.
1.3.2
Phases of economic development in a cycle
An economic cycle (Figure 1.5) covers the following phases, that characterize the feature of the country’s economic development:
• Crisis or full recession, or economic downturn (the phase from boom to trend line) is characterized by a
violation of imbalances that are increasing. During this phase, demand and production are reduced due to oversupply. As a result, there is an increase in unemployment, interest rate and a deterioration in macroeco-nomic indicators. Amid all this, there is a wave of bankruptcies and the mass closure of enterprises. The crisis phase ends with the onset of depression;
• Depression or stagnation (downturn from the trend line to the bottom) is a cycle phase that manifests itself
in the stagnation of production. At this stage, there is a gradual adjustment of production through the sale of inventories, which were not realized during the crisis due to a sharp decline in demand. The unemployment rate is still high but stable. Characteristic of this phase is the fact that the interest rate falls to its minimum value. Aggregate demand is gradually increasing, and conditions are being prepared for future improvements in production activity;
• Recovery or early recovery (the phase from bottom to trend line) – there is a gradual growth of the economy.
Production increases due to increased demand, and there is a gradual increase in employment. As a result, production gradually reaches the previous highest level of the economy and enters a phase of increase; • Boom or late recovery (expansion phase from trend line to peak) is the cycle phase when the volume of
production increases rapidly and exceeds the volume of the previous cycle. New enterprises are being built, and employment is increasing, prices and interest rates are also increasing. The so-called boom begins, laying the groundwork for the next crisis.
The leading role here is assigned precisely to the phase of the crisis, with which the cycle begins and ends. In this phase, the main signs and contradictions of the cyclical process of reproduction are concentrated.
The cyclical nature of the economy ensures the regular emergence of crises. Therefore, they are inevitable in the process of economic development. The scientist Juglar compared crises with diseases that a person is exposed to many times in his life (Table 1.1 ).
Thus, the cyclic content is multidimensional, and also quite complex in structure. The cycle can be considered as one of the ways of self-regulation of the market economy, which ensures its progressive development.
1.4
Globalization
The global economy is a qualitatively new stage in the development of the world economy. Since the world economy consists of the national economies of countries that are in close connection with each other, it can be considered a global economic system, which is characterized by a more dynamic and efficient development than national economies. The world economy is the “world-system” (Figure 1.2) in which there is an objective process of growth of economic interdependence of countries, due to the active integration of their national markets of goods, services and capital.
Definition 4 (Globalization). The process of growth and scale-up of world trade, as well as other processes of international exchange, transforms the world into a united area (the state of one dominant world-system) with an open and integrated global economy, which is called globalization.
