CRESCITA E SVILUPPO SOSTENIBILE

ECONOMIA, DIRITTI, CULTURE, STORIA

Direttore

Nicola Maria BSapienza Università di Roma

Comitato scientifico

Antonello Folco BSapienza – Università di Roma

Francesca CLibera Università Internazionale degli Studi Sociali “Guido Carli”

Giovanni LCoordinamento delle Organizzazioni non governative per la Cooperazione Internazionale alloSviluppo

Michael W PInstitute of Advanced Studies

Felice SIstituto Italo–Latino Americano

CRESCITA E SVILUPPO SOSTENIBILE

ECONOMIA, DIRITTI, CULTURE, STORIA

La collana affronta con un approccio multidisciplinare le tematiche propriedella crescita e dello sviluppo sostenibile. Le analisi teoriche e le verificheempiriche elaborate dalle discipline declinate nella denominazione dellacollana hanno da tempo evidenziato l’intreccio ineludibile fra l’economia,la storia, le culture e i diritti che scandiscono l’evoluzione di lungo peri-odo dello sviluppo economico. Obiettivo è non restringere le ricerche aeventi e fatti congiunturali, ma ampliare le analisi ai processi strutturali checaratterizzano le dinamiche sociali.

Questo volume è stato stampato con il contributo della facoltà di Scienze Politiche dell’Universitàdegli Studi di Teramo.

Vesna Jablanovic

A Contributionto the Chaotic Economic Growth Theory

Copyright © MMXVIARACNE editrice int.le S.r.l.

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First edition: November

Contents

Introduction

Chapter IThe Chaotic Version of the Harrod–Domar Model

.. Introduction, – .. The model, – .. The Logistic Equation, –.. Empirical Evidence, – .. Conclusion, .

Chapter IIA Simple Chaotic Version of the Solow Growth

.. Introduction, – .. The Model, – .. The Logistic Equation, –.. Empirical Evidence, – .. Conclusion, .

Chapter IIIThe Human Capital Augmented Solow Model and Chaos

.. Introduction, – .. The Model, – .. The Logistic Equation, –.. Empirical Evidence, – .. Conclusion, .

Chapter IVThe Human Capital AugmentedSolow Growth Model, Budget Deficit and Chaos

.. Introduction, – .. The Model, – .. The Logistic Equation, –.. Empirical Evidence, – .. Conclusion, .

Chapter VThe Simple Chaotic AK model

.. Introduction, – .. The Model, – .. The Logistic Equation, –.. Empirical Evidence, – .. Conclusion, .

Chapter VIThe Simple Chaotic AK model and Government Spending

.. Introduction, – .. The Model, – .. The Logistic Equation, –.. Empirical Evidence, – .. Conclusion, .

Contents

Chapter VIIThe Simple Chaotic AK model with increasing returns

.. Introduction, – .. The model , – .. The Logistic Equation, –.. Empirical Evidence, – .. Conclusion, .

Bibliography

Introduction

Global growth, currently estimated at . percent in , is projected at. percent in and . percent in . October World EconomicOutlook (WEO). In advanced economies, a modest recovery is expected tocontinue. The slowdown of the Chinese economy will continue to weigh ongrowth prospects. Growth in emerging market and developing economies( percent of global growth) declined, while a modest recovery continuedin advanced economies.

Three key factors continue to influence the global economy:

a) The gradual slowdown of economic activity in China (slowdown inimports and exports).

b) Lower prices for energy and other commodities, andc) an expansive monetary policy in the United States. Also, several other

major advanced economy central banks continue to ease monetarypolicy.

Global demand and investment are also weighing heavily on globaltrade and manufacturing activity. Oil prices have declined markedly sinceSeptember . This decline in oil prices was driven by higher oil supplyin OPEC memebers, and may provide a stronger boost to demand in oilimporters than currently envisaged, including through consumers’ possibleperception that prices will remain lower for longer.

Global growth is projected at . percent in and . percent in .Growth in advanced economies is projected to rise by . percentage pointin to . percent, and hold steady in . Growth in emerging marketand developing economies is projected to increase from percent in —the lowest since the - financial crisis — to . and . percent in and , respectively October World Economic Outlook (WEO).

In advanced economies, where inflation rates are still well below centralbanks’ targets, expansive monetary policy remains essential. Short–termfiscal policy should be more supportive of the recovery. Investments willincrease capital stocks.

In emerging market and developing economies, policy priorities arevaried. Policymakers need to lift growth and to ensure continued conver-gence toward advanced economy income levels. They need to press on

Introduction

with structural reforms (deepening local capital markets, improving fiscalrevenue etc.).

Global growth disappointed in , slowing to . percent. Growth isprojected to reach . percent in , as a modest recovery in advancedeconomies continues according to the World Bank’s January GlobalEconomic Prospects. «More than percent of the world’s poor live in the de-veloping countries where growth slowed in », said World Bank GroupPresident Jim Yong Kim. In the East Asia and Pacific region, economicgrowth is projected to continue to slow to . percent in . Growthin China is forecast to ease further to . percent in from . per-cent in . Growth in the region excluding China was . percent in. Developing and high–income economies in Latin America and theCaribbean contracted . percent in . In South Asia, economic growthis projected to reach . percent in (Global Economic Prospects). Indiais projected to grow at a faster . percent. In the Europe and CentralAsia region, economic growth is projected to rise to percent in from . percent in the year just ended as oil prices fall more slowly orstabilize, the Russian Federation’s economy improves, and Ukraine re-covers, according to the January issue of Global Economic Prospects(http://www.worldbank.org/en/publication/global-economic-prospects/).

Figure : The World Economy: Population () . million; Surface area () .thousand sq.km; Gross national product () $ . billions ; Gross national productper capita $ . World Development Indicators , World Bank Group, The WorldBank https://issuu.com/world.bank.publications/docs/?e=/.

Introduction

In the Middle East and North Africa region, economic growth is forecast toaccelerate to . percent in (the January issue of Global EconomicProspects.). Sub–Saharan Africa faces a challenging near–term outlook. Afterslowing to . percent in , activity is expected to pick up to . percentin .

Figures , and Table describe global economic activity.

Figure : Population million , World Development Indicators , World Bank Group,The World Bank https://issuu.com/world.bank.publications/docs/?e=/.

Figure : Surface area, thousand sq.km, . World Development Indicators ,World Bank Group, The World Bank https://issuu.com/world.bank.publications/docs/?e=/.

Introduction

Figure : Gross national product, purchasing power parity, $ billions, World DevelopmentIndicators , World Bank Group, The World Bank https://issuu.com/world.bank.publications/docs/?e=/.

Figure : Gross national product per capita, purchasing power parity, $ billions,WorldDevelopment Indicators , World Bank Group, The World Bank https://issuu.com/world.bank.publications/docs/?e=/.

Introduction

Figure : Gross domestic product % growth - World Development Indicators ,World Bank Group, The World Bank https://issuu.com/world.bank.publications/docs/?e=/.

Figure : Gross domestic product per capita % growth – World DevelopmentIndicators , World Bank Group, The World Bank https://issuu.com/world.bank.publications/docs/?e=/ .

Introduction

Figure : Gross domestic product % growth – World Development Indicators, World Bank Group, The World Bank (https://issuu.com/world.bank.publications/docs/?e=/).

Figure : Gross domestic product % growth –: World (http://www.imf.org/).

Introduction

Figure : World Gross domestic product based on purchasing–power – parity valuationof country GDP current international dollar, – (http://www.imf.org/).

Figure : World investment (% of GDP) and World gross national savings (% of GDP),–. (http://www.imf.org/)

.

Introduction

Figure : World inflation, –. (http://www.imf.org/.

Figure : GDP, constant prices, % change: World, Advanced economies, Euro area, G,European Union, Emerging market and developing economies, – (http://www.imf.org/).

Introduction

Table : Labor productivity, GDP per person employed (% growth) (http://www.worldbank.org/table/.).

1991–1993 2011–2014

World 0.8 2.1Low and Middle Income 2.7 3.9High Income 1.2 0.7Euro Area 1.4 0.1United States 2.1 0.8China 8.2 7.0Russian Federation -10.3 1.5

What is chaos? Chaos can be presented by behavior of a dynamicalsystem that has a very large (possibly infinite) number of attractors. Chaostheory studies the behavior of dynamical system that is highly sensitive toinitial conditions (“butterfly effect”).

Chaos theory states that small changes can result in large differences andthat there is an underlying order in all that surrounds us. In short, chaosembodies three important principles:

a) Sensitivity to initial conditions or “butterfly effect”.b) Unproportionality of inputs andc) Nonlinearity.

Chaotic system is unpredictable, because in order to predict its behaviorinto the future we must know its currrent value precisely. Namely, a slightdifference, in the decimal place, resulted in prediction failure.

Chaos theory is used to prove that erratic and chaotic fluctuations canindeed arise in completely deterministic models. Chaos theory reveals struc-ture in aperiodic, dynamic systems. The number of nonlinear businesscycle models use chaos theory to explain complex motion of the economy.Chaotic systems exhibit a sensitive dependence on initial conditions: seem-ingly insignificant changes in the initial conditions produce large differencesin outcomes. This is very different from stable dynamic systems in whicha small change in one variable produces a small and easily quantifiablesystematic change.

Chaos theory started with Lorenz’s () discovery of complex dynamicsarising from three nonlinear differential equations leading to turbulence inthe weather system. Li and Yorke () discovered that the simple logisticcurve can exibit very complex behavior. Further, May () describedchaos in population biology. Chaos theory has been applied in economicsby Benhabib and Day (,), Day (, ,, ), Grandmont (),

Introduction

Goodwin (), Medio (,), Medio, A. and Lines, M (), Lorenz(), Shone, R. (), Jablanovic (,, , , , , ,, ), among many others.

The basic aim of the book A Contribution to the Chaotic Theory of EconomicGrowth is to compare the main theories of economic growth in order toisolate their chaotic structures. The book provides original solutions totheoretical questions still debated in contemporary economic literature andpoints out new directions for further research. Attention is given to thechaotic approach to the theory of economic growth.

The simple logistic map, which produces astoundingly complex dynam-ics resulting in chaos if a single parameter is varied, was used. Chaoticbehavior is discovered in the simple logistic model. The logistic modelis simply a quadratic function. A couple of economic growth models arereconstructed to include chaos in economic growth theory.

Further, this book shows that some important values of the parameter πof the created economic growth models are part of the Fibonacci sequence.

In the th century, Leonardo Fibonacci (Filius Bonacci) introduced asimple numerical sequence to the West after his travels throughout theMediterranean world and North Africa. In his book The Liber Abaci (),he introduced the advantages of the Hindu Arabic system of numerationover the Roman System. Fibonacci numerical sequence was known as earlyas the th century AD by Indian mathematicians.

Each new number in the Fibonacci sequence is simply the sum of the twobefore it: , , , , , , , , , , , , etc. The golden ratio (symbol isthe Greek letter “phi” ) of each successive pair of numbers in the sequence is aspecial number approximately equal to . . . . The division ofany two adjacent numbers gives the Golden Ratio (e.g. /= .).

Figure : The Golden Ratio, the Golden Rectangle and the Parthenon https://www.mathsisfun.com/numbers/golden-ratio.html.

Introduction

The Fibonacci sequence appears many times in geometry, art (Leonardoda Vinci, Debussy), architecture and other areas. For example, the Parthenonin Athens ( BC) “contains” the Golden Ratio. Golden rectangle has beenmade using the Golden Ratio ( + + + + +. . . ).

This book shows that the Golden Ratio appears in the seven createdeconomic chaotic growth models (the modified chaotic Harrod–Domarmodel, the modified chaotic Solow growth model, the chaotic AK growthmodel, . . . ). Namely, the important parameter (π) values of the createdeconomic chaotic growth models (, , , , ) follow the Fibonacci sequence.

This book will be of great interest to scholars of economic growth,macroeconomics, develompent economics, and historians of economicthought.