Macroeconomic Dynamics- Accounting System Dynamics Approach-( On-going Draft Version 2 )Kaoru Yamaguchi Ph.D.Graduate School of BusinessDoshisha UniversityKyoto, JapanSeptember 9, 2010ContentsI Accounting System Dynamics 131 System Dynamics 151.1 Language of System Dynamics . . . . . . . . . . . . . . . . . . . 151.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.1 Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.2 Stock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.3 Stock-Flow Relation . . . . . . . . . . . . . . . . . . . . . 181.2.4 Integration of Flow . . . . . . . . . . . . . . . . . . . . . . 201.3 Dynamics in Action . . . . . . . . . . . . . . . . . . . . . . . . . 211.3.1 Constant Flow . . . . . . . . . . . . . . . . . . . . . . . . 221.3.2 Linear Flow of Time . . . . . . . . . . . . . . . . . . . . . 221.3.3 Nonlinear Flow of Time Squared . . . . . . . . . . . . . . 251.3.4 Random Walk . . . . . . . . . . . . . . . . . . . . . . . . 271.4 System Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.4.1 Exponential Growth . . . . . . . . . . . . . . . . . . . . . 281.4.2 Balancing Feedback . . . . . . . . . . . . . . . . . . . . . 301.5 System Dynamics with One Stock . . . . . . . . . . . . . . . . . 331.5.1 First-Order Linear Growth . . . . . . . . . . . . . . . . . 331.5.2 S-Shaped Limit to Growth . . . . . . . . . . . . . . . . . 341.5.3 S-Shaped Limit to Growth with Table Function . . . . . . 351.6 System Dynamics with Two Stocks . . . . . . . . . . . . . . . . . 361.6.1 Feedback Loops in General . . . . . . . . . . . . . . . . . 361.6.2 S-Shaped Limit to Growth with Two Stocks . . . . . . . . 371.6.3 Overshoot and Collapse . . . . . . . . . . . . . . . . . . . 381.6.4 Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . 391.7 Delays in System Dynamics . . . . . . . . . . . . . . . . . . . . . 411.7.1 Material Delays . . . . . . . . . . . . . . . . . . . . . . . . 411.7.2 Information Delay . . . . . . . . . . . . . . . . . . . . . . 421.8 System Dynamics with Three Stocks . . . . . . . . . . . . . . . . 441.8.1 Feedback Loops in General . . . . . . . . . . . . . . . . . 441.8.2 Lorenz Chaos . . . . . . . . . . . . . . . . . . . . . . . . . 451.9 Chaos in Discrete Time . . . . . . . . . . . . . . . . . . . . . . . 481.9.1 Logistic Chaos . . . . . . . . . . . . . . . . . . . . . . . . 481.9.2 Discrete Chaos in S-shaped Limit to Growth . . . . . . . 4934 CONTENTS2 Demand and Supply 532.1 Adam Smith! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.2 Unifying Three Schools in Economics . . . . . . . . . . . . . . . . 552.3 Tatonnement Adjustment by Auctioneer . . . . . . . . . . . . . . 572.4 Price Adjustment with Inventory . . . . . . . . . . . . . . . . . . 622.5 Logical vs Historical Time . . . . . . . . . . . . . . . . . . . . . . 652.6 Stability on A Historical Time . . . . . . . . . . . . . . . . . . . 662.7 A Pure Exchange Economy . . . . . . . . . . . . . . . . . . . . . 682.7.1 A Simple Model . . . . . . . . . . . . . . . . . . . . . . . 682.7.2 Tatonnement Processes on Logical Time . . . . . . . . . . 702.7.3 Chaos Triggered by Preferences . . . . . . . . . . . . . . . 732.7.4 O-Equilibrium Transactions on Historical Time . . . . . 752.8 Co-Flows of Goods with Money . . . . . . . . . . . . . . . . . . . 763 Accounting System Dynamics 833.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.2 Principles of System Dynamics . . . . . . . . . . . . . . . . . . . 853.3 Principles of Accounting System . . . . . . . . . . . . . . . . . . 863.4 Principles of Accounting System Dynamics . . . . . . . . . . . . 903.5 Accounting System Dynamics in Action . . . . . . . . . . . . . . 923.6 Making Financial Statements . . . . . . . . . . . . . . . . . . . . 1043.7 Ratio Analysis of Financial Statements . . . . . . . . . . . . . . . 1063.8 Toward A Corporate Archetype Modeling . . . . . . . . . . . . . 108II Macroeconomic System 1134 Macroeconomic System Overview 1154.1 Macroeconomic System . . . . . . . . . . . . . . . . . . . . . . . 1154.2 A Capitalist Market Economy . . . . . . . . . . . . . . . . . . . . 1164.3 Modeling a Market Economy . . . . . . . . . . . . . . . . . . . . 1184.4 Bank Loans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.5 Securities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.6 Retained Earnings . . . . . . . . . . . . . . . . . . . . . . . . . . 1245 Money and Its Creation 1255.1 Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.2 What is Money? . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.3 A Fractional Reserve Banking System . . . . . . . . . . . . . . . 1285.4 Money under Gold Standard . . . . . . . . . . . . . . . . . . . . 1325.5 Money out of Bank Debt . . . . . . . . . . . . . . . . . . . . . . . 1405.6 Money out of Government Debt . . . . . . . . . . . . . . . . . . . 144CONTENTS 56 Interest and Equity 1556.1 What is Interest? . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.2 Money and Interest under Gold Standard . . . . . . . . . . . . . 1566.3 Money and Interest under Bank Debt . . . . . . . . . . . . . . . 1586.4 Money and Interest under Government Debt . . . . . . . . . . . 1596.5 Interest and Sustainability . . . . . . . . . . . . . . . . . . . . . . 1607 Aggregate Demand Equilibria 1637.1 Macroeconomic System Overview . . . . . . . . . . . . . . . . . . 1637.2 A Keynesian Model . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.3 Aggregate Demand (IS-LM) Equilibria . . . . . . . . . . . . . . . 1717.4 Modeling Aggregate Demand Equilibria . . . . . . . . . . . . . . 1747.5 Behaviors of Aggregate Demand Equilibria . . . . . . . . . . . . 1807.6 Price Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1887.7 A Comprehensive IS-LM Model . . . . . . . . . . . . . . . . . . . 1957.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1988 Integration of Real and Monetary Economies 2038.1 Macroeconomic System Overview . . . . . . . . . . . . . . . . . . 2038.2 Changes for Integration . . . . . . . . . . . . . . . . . . . . . . . 2048.3 Transactions Among Five Sectors . . . . . . . . . . . . . . . . . . 2118.4 Behaviors of the Integrated Model . . . . . . . . . . . . . . . . . 2188.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2299 A Macroeconomic System 2319.1 Macroeconomic System Overview . . . . . . . . . . . . . . . . . . 2319.2 Production Function . . . . . . . . . . . . . . . . . . . . . . . . . 2319.3 Population and Labor Market . . . . . . . . . . . . . . . . . . . . 2379.4 Transactions Among Five Sectors . . . . . . . . . . . . . . . . . . 2399.5 Behaviors of the Complete Macroeconomic Model . . . . . . . . . 2479.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258III Open Macroeconomic System 26110 Balance of Payments and Foreign Exchange Dynamics 26310.1 Open Macroeconomy as a Mirror Image . . . . . . . . . . . . . . 26310.2 Open Macroeconomic Transactions . . . . . . . . . . . . . . . . . 26410.3 The Balance of Payments . . . . . . . . . . . . . . . . . . . . . . 26910.4 Determinants of Trade . . . . . . . . . . . . . . . . . . . . . . . . 27410.5 Determinants of Foreign Investment . . . . . . . . . . . . . . . . 27710.6 Dynamics of Foreign Exchange Rates . . . . . . . . . . . . . . . . 27910.7 Behaviors of Current Account . . . . . . . . . . . . . . . . . . . . 28210.8 Behaviors of Financial Account . . . . . . . . . . . . . . . . . . . 28410.9 Foreign Exchange Intervention . . . . . . . . . . . . . . . . . . . 28810.10Missing Feedback Loops . . . . . . . . . . . . . . . . . . . . . . . 2906 CONTENTS10.11Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29311 Open Macroeconomies as A Closed Economic System 29511.1 Open Macroeconomic System Overview . . . . . . . . . . . . . . 29511.2 Transactions in Open Macroeconomies . . . . . . . . . . . . . . . 29611.3 Behaviors of Open Macroeconomies . . . . . . . . . . . . . . . . . 30111.4 Where to Go from Here? . . . . . . . . . . . . . . . . . . . . . . 30511.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307IV New Macroeconomic System 32312 Designing A New Macroeconomic System 32512.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32512.2 Macroeconomic System of Money as Debt . . . . . . . . . . . . . 33012.3 System Behaviors of Money as Debt . . . . . . . . . . . . . . . . 33412.4 Macroeconomic System of Debt-Free Money . . . . . . . . . . . . 33912.5 System Behaviors of Debt-Free Money . . . . . . . . . . . . . . . 34212.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348PrefaceMy O-Road Journey for A Better WorldFutures StudiesIn early 1980s, I was told by one of the graduate colleagues at the University ofCalifornia, Berkeley, that if I continue the research involving Marx and Keynesin addition to neoclassical theory, I would never get a good job oer in the UnitedStates. He was right. It was a time for Reganomics which has eventually evolvedto the era of globalization in 1990s. Paying little attention to his thoughtfulsuggestion, I pursued my Ph.D. thesis on the subject Beyond Walras, Keynesand Marx - Synthesis in Economic Theory Toward a New Social Design, which,alas, became a start of my o-road journey. Main part of the thesis was luckilypublished with the same title [58], yet it has been left unnoticed among mainstream economists.When I started teaching at the Dept. of Economics, University of Hawaiiat Manoa, I almost lost my energy to continue the research on neoclassicalmathematical theory for academic survival, because the theory seemed to betotally detached from the economic reality. It was in those lost days when myintroduction to the futures studies and Prof. Jim Dator, then secretary generalof the World Futures Studies Federation, took place by chance in Hawaii in1987. Upon arrival to Japan next year, I immediately joined the Federation,and became very active on futures studies for more than ten years since then.Among the activities of futures studies I have been involved, a major onewas the organization of futures seminar series in Awaji Island, Japan, with anobjective to establish a future-oriented higher institution dubbed the NetworkUniversity of the Green World (http://www.muratopia.org/ NUGW). The sem-inars had been held for seven years from 1993 through 1999, then suspendeddue to the lack of fund. In the book based on the rst seminar in 1993, I haveproclaimed thatThus, what has been missing in industrial-age scientic research, andhence in the academic curricula of present-day higher institutions, isa study of interrelated wholeness and interdependences [61, p.200].In order to ll the missing niche, I have tried, with a help by the seminarparticipants, including Novel laureate Jerome Karle, to establish a new wholistic78 CONTENTSeld of study dubbed FOCAS, meaning Future-Oriented Complexity and Adap-tive Studies, in vain. Yet, my conviction on the need for such futures studiesfor higher education continued to remain as worth being upheld. Faced withthe threat of our survival due to climate changes and environmental disasters,future-oriented studies of interrelated wholeness and interdependence is, I be-lieve, more urgently needed for solving these complex problems, since solutionsoered by fragmented professionals at the current higher institutions might bethe causes of another problems as Asian wisdom connotes. For our survival andsustainability, we need future-oriented higher education which provides wholis-tic visions and solutions to the present complex problems caused by fragmentedscience and technology of the present-day higher education. This convictionbecame a fruit of reward for me at the cost of abandoning neoclassical economicresearch in a traditional academic stream.System DynamicsThroughout the future-oriented activities later on, I was luckily led to the sys-tems view, specically a method called system dynamics, by chance. It seemedto me a totally new eld of study that makes a heavy use of computer simula-tion for analyzing dynamic behaviors of system structures in physics, chemistry,engineering, environmental studies, business and economics, and public policies,to name a few, in a uniform fashion. In short, its methodology can uniformlycover many fragmented elds of studies, and in this sense it seemed for me tobe able to share a similar interdisciplinary vision with future-oriented studies.After many years frustration on the futures studies, Ive jumped in the led byattending its international conference in Istanbul, Turkey, in 1997. Since then Ihave been continually attending the system dynamics conferences up to the thepresent day.It didnt take much time to realize that, due to its interdisciplinary nature,system dynamics is also facing the similar diculty in nding an academicposition as a discipline in the highly fragmented current higher educationalsystem, as future-oriented studies have bee suering similarly. In other words,system dynamics and futures studies can have no comfortable places in thecurrent universities. The only dierence is the use of computer in the former,and the use of our brain in the latter.Hence, it seemed to me that future-oriented studies and system dynamicsconstitute two major elds of futures higher education, using our brain on theone hand and computer on the other hand for a study of interrelated wholenessand interdependence in order to attain human and environmental sustainability.In fact, it has been repeatedly argued at the international conferences whethersystem dynamics is merely a tool or discipline. For me it seemed to be not tothe point and accordingly a fruitless argument,On the contrary, the following description by Prof. Jay Forrester, a founderof system dynamics, on the nature of system dynamics looked to me to thepoint.CONTENTS 9Such transfer of insights from one setting to another will help tobreak down the barriers between disciplines. It means that learn-ing in one eld becomes applicable to other elds. There is nowa promise of reversing the trend of the last century that has beenmoving away from the Renaissance man toward fragmented spe-cialization. We can now work toward an integrated, systemic, ed-ucational process that is more ecient, more appropriate toward aworld of increasing complexity, and more compatible with a unity inlife [17].It is a useless eort to search for an appropriate academic citizenship atthe current fragmented higher education. In this sense, it seems to be a rightchoice to introduce the visions and methods of system dynamics to the K-12 education where academic fragmentation does not yet break down into thelearning process. The reader may visit a creative learning Web site for itssuccessful introduction at http://clexchange.org.I felt I have nally been led to a right truck, after more than a decade-long o-road journey, toward a better world. If I had stayed at the economicsprofession, I would have never encountered system dynamics as most economistsare currently still unaware of it. What I have learned from system dynamics isthe importance of system design.My continuing o-road journey got refurbished with this spirit of systemdesign. In the falls of 1999 and 2000, I had a chance to visit MIT where Iwas introduced systems thinking and system dynamics for the rst time as ifI was a rst-hand learning student by Prof. John Sterman and his doctoratestudents as well as Prof. Jay Forrester and his undergraduate team of Road Mapproject (educational self-learning system dynamics program through Web). Thisbecame my o-road journey of no return from system dynamics in my profession.Accounting System DynamicsInstead of being forced to stay in the economics profession, I was luckily given achance to teach system dynamics at two management schools in Japan; rst atthe Osaka Sangyo University in Osaka, then Doshisha Business School in Kyoto.System dynamics obtained its rst citizenship in this way as academic subjectto be taught in the fragmented higher educational system in Japan.Eventually, as a faculty member of management and business schools, Istrongly felt it necessary to cover accounting system in my system dynamicsclass, Yet, my search for SD-based accounting system turned out to be unsuc-cessful, giving me an incentive to develop a SD method of modeling nancialstatements and accounting system from a scratch. I started working on theSD-based accounting system in the summer of 2001 when I was spending rel-atively a quiet time on a daily rehabilitation exercise in order to recover fromthe physical operation on my shoulder in June of the same year. This retreatenvironment provided me with an opportunity to read books on accounting in-tensively. My readings mainly consisted of the introductory books such as [27],10 CONTENTS[34], [35], [51] and [52], since my knowledge of accounting was limited in thosedays1. Through such readings, I have been convinced that system dynamics ap-proach is very eective not only for understanding the accounting system, butmodeling many types of business activities. This conviction fruitfully resultedin my presentation on the principle of accounting system dynamics at the 21stinternational conference of the System Dynamics Society in New York in 2003[63], which became a turning point in my o-road journey.Rekindled in Berkeley, CaliforniaIn the same summer of 2003, I was luckily oered an 8 months sabbatical leave,and come back to Berkeley in almost 18 years since I left in 1986, this time as avisiting scholar at the Haas School of Business, not the Economics Department.My old friend, Nobie Yagi, from Berkeley days kindly provided his second houseon his site for my familys stay, which gave me a good opportunity to talkwith him almost daily. He received Ph.D. in nance and options trading fromBerkeley around the same time as I did.Conversation with him, together with my research environment at the busi-ness school rekindled my interest in economics, specically macroeconomics andnance again. Even so, in those days I have already taken an o-road journeyaway from main stream economics, and decided to investigate it from my o-road side way. Specically, I resolved to start reconstructing macroeconomictheories on the basis of the principle of accounting system dynamics which wascompleted in the same summer.Since then, being led by the inner logic of accounting system dynamics andmacroeconomics, I have spent almost my entire o-road journey on a step-by-step construction of macroeconomic models, which turned into a series of pre-sentation of papers such as [64], [65], [66] and [67]. This series of macroeconomicmodeling was completed in 2008 as [68] with a follow-up analytical renementmethod of price adjustment mechanism in [69] next year.An Oasis in Wellington, New ZealandSecond good luck visited me on my o-road journey as two months short sab-batical leave in 2009 at the Victoria Management School, Victoria University ofWellington, New Zealand. Prof. Bob Cabana, a well-known leading scholar insystem dynamics, kindly hosted my visit. This good luck enabled me to reviewthe above paper series uniformly for the publication of this book. Almost dailyconversation with him over lunch, as well as a lovely research environment inWellington, encouraged me to keep working on the draft. Without this stopoverin New Zealand as an oasis in my o-road journey, the draft would not havebeen completed.1In addition to these books, a paper dealing with corporate nancial statements [2] wasrecently in 2002. However, current research for modeling nancial statements is independentlycarried out here with a heuristic objective in mind.CONTENTS 11National ModelI was a late comer to the research community of system dynamics. While mystep-by-step macroeconomic modeling was advancing, some researchers havekindly suggested at the conferences that I should review the research papers onthe National Model project that was led by Prof. Jay Forrester with severalPh.D. students at MIT.Unfortunately, the national model itself was not available and its related pa-pers were scattered around. Under such situation, my survey managed to coverthe following papers [13], [14], [15], [16], [18], [19], [20], [21], [22], [23], [24], [25],[28], [32], [46], [47]. Yet, the review of these papers only gave me an impressionas if I were, with my eyes closed, touching various parts of an elephant withoutknowing what the elephant looks like. During the 23rd international conferenceof System Dynamics Society in New York, July, 2005, in which I presented aSD-based Keynesian model, I have strongly felt that my research cannot ad-vance without understanding a whole picture of National Model, because mymodeling approach, I feared, might have been already taken by the NationalModel project team.Without losing time, in September of the same year, I visited Prof. JayForrester in his oce at MIT. We spent almost two hours on discussing abouthis National Model. He told me that the national model is still going on, andI may have no chance to take a look at it until its completed. Even so, theconversation turned out to be a very fruitful to me, out of which I got convincedthat my modeling approach on the basis of accounting system dynamics is quitedierent from his modeling method. This conviction gave me an energy tocontinue my o-road journey in my own way. At the same time, I truly hopedthat the national model would be completed in the near future.In the spring of 2007, I was invited to review the Ph.D. dissertation byDavid Wheat at the University of Bergen, Norway, whose title is The FeedbackMethod: - A System Dynamics Approach to Teaching Macroeconomics [56].His model, written by Stella software, seemed to me to be a simple version ofthe National Model. In this sense it became the rst complete macroeconomicmodel ever presented to the public, and his eort should be congratulated. Inthe following year, at the 26th international conference of System DynamicsSociety, Athens, Greece, I have presented a complete macroeconomic model,written by Vensim, on the basis of accounting system dynamics [68].At a Vista Point over a Green WorldI now feel as if Im standing at a magnicent vista point in my o-road journeywhere I can glimpse the peak of a better world Ive been searching for. A betterworld is a green world, which I try to describe in the last chapter 12 of this book.To be specic, this turns out to be a world of MuRatopian economy I pursued inmy dissertation in 1980s. It is founded this time on a basis of debt-free moneysystem. I stop my o-road journey at this point. It is my hope that the readerwill continue this o-road journey to the summit, so that eventually it becomes12 CONTENTSa main street for a better and green world.With many thanks to those who guided me and oered a cordial help duringmy o-road journey.August, 2010Awaji Island, JapanKaoru YamaguchiAcknowledgmentsAt the 28th International Conference of the System Dynamics Society, held inSeoul, Korea, July 29, 2010, I have oered the SD Workshop: An Introduction toMacroeconomic Modeling Accounting System Dynamics Approach in whichthe draft of this book was distributed, for the rst time, to the participantsto obtain their feedbacks. Then, two months later, it is distributed to theparticipants at the 6th Annual AMI Monetary Reform Conference in Chicago,the United States, Sept. 30 - Oct.3, 2010. Following are those who gave mecomments and suggestions.. , .I do gratefully acknowledge their cordial feedbacks. All remaining errors,however, are mine.October, 2010Kaoru YamaguchiPart IAccounting SystemDynamics13Chapter 1System DynamicsThis chapter1introduces system dynamics from a dynamics viewpoint for be-ginners who have no formal mathematical background. First, dynamics is dealtin terms of a stock-ow relation. Under this analysis, a concept of DT (deltatime) and dierential equation is introduced together with Runge-Kutta meth-ods. Secondly, in relation with a stock-dependent ow, positive and negativefeedbacks are discussed. Then, fundamental behaviors in system dynamics areintroduced step by step with one stock and two stocks. Finally, chaotic behavioris explored with three stocks, followed by discrete chaos.1.1 Language of System DynamicsWhat is system dynamics? The method of system dynamics was rst createdby Prof. Jay Forrester, MIT, in 1950s to analyze complex behaviors in socialsciences, specically, in management, through computer simulations [11]. Itliterally means a methodology to analyze dynamic behaviors of system. Whatis system, then? According to Jay Forrester, a founder of this eld, A systemmeans a grouping of parts that operate together for a common purpose [12],page 1-1. For instance, following are examples of system he gave: An automobile is a system of components that work together to providetransportation. Management is a system of people for allocating resources and regulatingthe activity of a business. A family is a system for living and raising children.1This chapter is based on the paper presented at the 17th International Conference of theSystem Dynamics Society: Systems Thinking for the Next Millennium, New Zealand, July 20- 23 ,1999; specically in the session L7: Teaching, Thursday, July 22, 1.30 pm - 3.00 pm,chaired by Peter Galbraith.16 CHAPTER 1. SYSTEM DYNAMICSAccording to Edward Deming, a founder of quality control, A system is anetwork of interdependent components that work together to try to accomplishthe aim of the system [6], p. 50.Both denitions share similar ideas whose keywords are: interdependentparts or components, and common purpose or aim.StockFlowVariableFigure 1.1: Language of System Dynam-icsTo describe the dynamics of sys-tem thus dened, Forrester cre-ated a language of system dynam-ics consisting of four letters: Stock,Flow,Variable and information Ar-row. They are illustrated in Fig-ure 1.1. Flow is always connectedto Stock. Arrow connects Variable,Flow and Stock.This is a very genius idea. Todescribe a system, all we need isfour letters. If you learn some sim-ple grammar, you could combinethese 4 letters to write a sentence,paper, or even a big book. Ourbody is an excellent example of system that consists of about 30 thousandsgenes, yet genes are composed of only four types of DNA. Compared with this,English has 26 letters and Japanese has 55 phonetic letters.Textbooks and SoftwaresSeveral textbooks are now available for learning how to write sentences with theabove four letters such as [48] and [53] for business and management modeling,and [8] and [10] for sustainable environment. The reader is strongly recom-mended to learn system dynamics with these textbooks. I regularly use [48],[53] and [8] in my MBA and policy classes.Another way to learn quickly is start using SD softwares such as Stella, Ven-sim and PowerSim with manuals. For the modeling in this book, I have selectedVensim for two reasons; its graphical capability for creating a model and avail-ability of its free version such as Vensim PLE and Vensim Model Reader. It isrecommended that the reader runs the model attached to this book simultane-ously.Economics students might need a little bit more rigorous approach to mod-eling in relation with dierence and dierential equations, which is, however,not well covered in the above introductory textbooks, This is why I decided toadd another introduction to system dynamics method in this chapter.1.2. DYNAMICS 171.2 Dynamics1.2.1 TimeFor beginners system dynamics seems to be an analysis of systems in terms offeedback mechanisms and interdependent relations. In particular this is truewhen graphics-oriented softwares of system dynamics become available for PCsand Macs such as Stella, Vensim and PowerSim, enabling even introductorystudents to build a complicated dynamic model easily without knowing a mech-anism of dynamics and dierential equations behind the screen.Accordingly, the analysis of dynamics itself has been de-emphasized in alearning process of system dynamics. Dynamic analysis, however, has to be afoundation of system dynamics, through which systems thinking will be moreeectively learned. This is what I have experienced when I encountered a systemdynamics as a new research eld.Dynamic analysis needs to be dealt along with a ow of time; an irreversibleow of time. What is time, then? It is not an intention to answer this philo-sophically deep question. Instead, time is here simply represented as an onedimensional real number, with an origin as its initial starting point, that owstoward a positive direction of the coordinate.In this representation of time, two dierent concepts can be considered. Therst concept is to represent time as a moment of time or a point in time, denotedhere as ; that is, time is depicted as a real number such that = 1, 2, 3, . . . .The second one is to represent it as a period of time or an interval of time,denoted here as t, such that t = 1st, 2nd, 3rd, . . . , or more loosely t = 1, 2, 3,. . . (a source of confusion for beginners). Units of the period could be a second,a minute, an hour, a week, a month, a quarter, a year, a decade, a century, amillennium, etc., depending on the nature of the dynamics in question.In system dynamics, these two concepts of time needs to be correctly dis-tinguished, because stock and ow - the most fundamental concepts in systemdynamics - need to be precisely dened in terms of either or t as discussedbelow.1.2.2 StockLet us now consider four letters of system dynamics language in detail. Amongthose letters, the most important letter is stock. In a sense, system could bedescribed as a collection of stock. What is stock, then? It could be an object tobe captured by freezing its movement imaginably by stopping a ow of time, ormore symbolically by taking its still picture. The object that can be capturedthis way is termed as stock in system dynamics. That is, stock is the amountthat exists at a specic point in time , or the amount that has been piled upor integrated up to that point in time.Let x be such an amount of stock at a specic point in time . Then stockcan be dened as x() where can be any real number.18 CHAPTER 1. SYSTEM DYNAMICSStocks thus dened may be classied according to their dierent types ofnature as follows.Physical Stock Non-Physical Stock Natural Stock Capital Stock Goods-in-Process and Use Information Psychological Passion Indexed FiguresTable 1.1: Classication of Stock Natural stock consists of those that exist in our natural environment suchas the amount of water in a lake, number of trees and birds in a forestand world population. Capital stock is a manufactured means of production such as buildings,factories and machines that have been used to produce nal goods. Goods-in process are those that are in a process of production, which aresometimes called intermediate goods, and goods-in-use are nal productsthat have been used by consumers such as cars and computers. Information (and knowledge) is non-physical stock that is stored in variousforms of media such as papers, books, videos, tapes, diskettes, CDs andDVDs. Psychological passion is emotional stock of human beings such as love,joy, happiness, hatred and anger that have been stored somewhere in ourbrain tissues. Indexed gures are specic forms of information stock that are (scienti-cally) dened to describe the nature of environment and human activitiessuch as temperature, prices, deposits and sales values.1.2.3 Stock-Flow RelationSince Newton, it has been a challenge in classical mechanics to describe a changein stock. One of the methods widely employed is to capture the amount of stockat various discrete points in time, = 0, 1, 2, 3, and consider a change instock at the next point as the amount of the stock at the present point and itsincrement between the present and next points; that is, and +1. Let us callsuch an interval of time between these two points a unit interval. The lengthof unit could be, as already mentioned above, a second, a minute, an hour, aday, a week, a month, a year, or whichever unit to be suitable for capturing themovement of the stock in question. Hence, a period of time t could be denedas a t-th unit interval or period, counting from the origin; that is, = 0.Flow is dened as an increment (or decrement) of stock during a unit interval,and denoted here by f(t). Flow that can only be dened at each discrete periodof time is called discrete ow.1.2. DYNAMICS 19It is important to note that ow dened in this way is the amount betweentwo points in time or a unit interval, while stock is the amount at a specicpoint in time. In other words, which is used for dening stock implies a pointin time and t which is used for dening ow means a t-th unit interval betweena point in time and its next period + 1.Stock xFlow f(t)Defined for aperiod of timeDefined at amoment in timeFigure 1.2: Stock-Flow RelationIn this way, any dynamic move-ment can be operatively understoodin terms of stock and ow. Thisstock-ow relation becomes funda-mental to a dynamic analysis. Itis conceptually illustrated in Figure1.2.It is essential to learn from thegure that ow is a part of stock,and in this sense physical or quan-titative unit of ow and stock hasto coincide. For instance, ow of oilcannot be added to the stock of water. As a system dynamics model becomescomplicated, we tend to forget this essential fact.Stock-ow relation can be formally written asx( + 1) = x() +f(t) and t = 0, 1, 2, 3, (1.1)To avoid a confusion derived from dual notations of time, and t, we needto describe stock-ow relation uniformly in terms of either one of these twoconcepts of time. Which one should, then, be adopted? A point in time couldbe interpreted as a limit point of an interval of time t. Hence, t can portrayboth concepts adequately, and can be chosen.Since t represents a unit interval between and + 1, the amount of stockat the t-th interval x(t) could be dened as a balance at a beginning point ofthe period or an ending point + 1 of the period; that is,x(t) = x() : Beginning balance of stock (1.2)orx(t) = x( + 1) : Ending balance of stock (1.3)When the beginning balance of the stock equation (1.2) is applied, the stock-ow equation (1.1) becomes as follows:x(t + 1) = x(t) +f(t) t = 0, 1, 2, 3, (1.4)In this formula, stock x(t +1) is valued at the beginning of the period t +1;that is, ow f(t) is added to the present stock value to give a stock value of thenext period.When the ending balance of the stock equation (1.3) is applied, the stock-ow equation (1.1) can be rewritten as20 CHAPTER 1. SYSTEM DYNAMICSx(t) = x(t 1) +f(t) t = 1, 2, 3, (1.5)In this way, two dierent concepts of time - a point in time and a period oftime - have been unied. It is very important for the beginners to understandthat time in system dynamics always implies a period of time which has a unitinterval. Of course, periods need not be discrete and can be continuous as well.1.2.4 Integration of FlowDiscrete SumWithout losing generality, let us assume from now on that x(t) is an amount ofstock at its beginning balance. If f(t) is dened at a discrete time t = 1, 2, 3, . . .,then the equation (1.4) is called a dierence equation. In this case, the amountof stock at time t from the initial time 0 can be summed up or integrated interms of discrete ow as follows:x(t) = x(0) +t1i=0f(i) (1.6)This is a solution of the dierence equation (1.4).Continuous SumWhen ow is continuous and its measure at discrete periods does not preciselysum up the total amount of stock, a convention of approximation has beenemployed such that the amount of f(t) is divided into n sub-periods (which ishere dened as 1n = t) and n is extended to an innity; that is, t 0. Then,the equation (1.4) can be rewritten as follows:x(t) = x(t 1n)+ f(t 1n)n= x(t t) +f(t t)t (1.7)Let us further denelimt0x(t) x(t t)t dxdt (1.8)Then, for t 0 we havedxdt = f(t) (1.9)This formulation is nothing but a denition of dierential equation. Continuousow and stock are in this way transformed to dierential equation, and theamount of stock at t is obtained by solving the dierential equation. In otherwords, whenever a stock-ow diagram is drawn as in Figure 1.2, dierentialequation is constructed behind the screen in system dynamics.1.3. DYNAMICS IN ACTION 21The innitesimal amount of ow that is added to stock at an instantaneouslysmall period in time can be written asdx = f(t)dt (1.10)Here, dt is technically called delta time or simply DT. Then an innitesimal (orcontinuous) ow becomes a ow during a unit period t times DT.Continuous sum is now written, in a similar fashion to a discrete sum inequation (1.6), asx(t) = x(0) + t0f(u)du (1.11)This gives a general formula of a solution to the dierential equation (1.9). Thenotational dierence between continuous and discrete ow is that in a continuouscase an integral sign is used instead of a summation sign. A continuous stockis, thus, alternatively called an integral function in mathematics.In this way, stock can be described as a discrete or a continuous sum ofow2. This stock-ow relation becomes a foundation of dynamics (and, hence,system dynamics). It cannot be separable at all. Accordingly, among 4 lettersof system dynamics language, stock-ow relation becomes an inseparable newletter. Whenever stock is drawn, ows have to be connected to change theamount of stock. This is one of the most essential grammars in system dynamics.1.3 Dynamics in ActionWe are now in a position to analyze a dynamics of stock in terms of stock-owrelation. What we have to tackle here is how to nd an ecient summation andintegration method for dierent types of ow. Let us consider the most funda-mental type of ow in the sense that it is not inuenced (increased or decreased)by outside forces. In other words, this type of ow becomes autonomous anddependent only on time. Though this is the simplest type of ow, it is indeedworth being fully analyzed intensively by the beginners of system dynamics.Examples of this type to be considered here are the following: constant ow linear ow of time non-linear ow of time squared2In Stella the amount of stock is described, similar to the equation (1.7), asx(t) = x(t dt) + f(t dt) dt,while in Vensim it is denoted, similar to the equation (1.11), asx(t) = INTEG(f(t), x(0)).22 CHAPTER 1. SYSTEM DYNAMICS random walkAnother examples would be trigonometric ow, present value, and time-series data, which are left uncovered in this book.1.3.1 Constant FlowThe simplest example of this type of ow is a constant amount of ow throughtime. Let a be such a constant amount. Then the ow is written asf(t) = a (1.12)This constant ow can be interpreted as discrete or continuous. A discreteinterpretation of the stock-ow relation is described asx(t + 1) = x(t) +a (1.13)and a discrete sum of the stock at t is easily calculated asx(t) = x(0) +at (1.14)On the other hand, a continuous interpretation of stock-ow relation is rep-resented by the following dierential equation:dxdt = a (1.15)and a continuous sum of the stock at t is obtained by solving the dierentialequation asx(t) = x(0) + t0a du= x(0) +at. (1.16)From these results we can easily see that, if a ow is a constant amountthrough time, the amount of stock obtained either by discrete or continuousow becomes the same.1.3.2 Linear Flow of TimeWe now consider autonomous ow that is linearly dependent on time. Thesimplest example of this type of ow is the following3:3When time unit is a week, f(t)) has a unit of (Stock unit/ week) . Accordingly, in orderto have the same unit, the right hand side has to be multiplied by a unitary variable of unitconverter which has a unit of (Stock unit / week / week).f(t) = t unit converterThis process is called unit check, and system dynamics requires this unit check rigorouslyto obtain equation consistency. In what follows in this introductory chapter, however, thisunit check is not applied.1.3. DYNAMICS IN ACTION 23f(t) = t (1.17)Let the initial value of the stock be x(0) = 0. Then the analytical solutionbecomes as follows:x(t) = t0u du = t22 (1.18)Stock xFlow f(t)

Figure 1.3: Linear Flow of TimeAt the period t = 10, we havex(10) = 50. This is a true value ofthe stock. Stock and ow relationof the solution is shown in Figure??. The amount of stock at a timet is depicted as a height in the g-ure, which is equal to the area sur-rounded by the ow curve and time-coordinate up to the period t.Stock and Flow5037.52512.50 222222222211111111110 1 2 3 4 5 6 7 8 9 10Time (Week)unitStock x : Current 1 1 1 1 1 1 1 1 1 1 1 1"Flow f(t)" : Current 2 2 2 2 2 2 2 2 2 2 2Figure 1.4: Linear Flow and Stock24 CHAPTER 1. SYSTEM DYNAMICSDiscrete ApproximationIn general, the analytical (integral)solution of dierential equation isvery hard, or impossible, to obtain. The above example is a lucky exception.In such a general case, a numerical approximation is the only way to obtain asolution. This is done by dividing a continuous ow into discrete series of ow.Let us try to solve the above equation in this way, assuming that no analyticalsolution is possible in this case. Then, a discrete solution is obtained asx(t) =t1i=0i (1.19)and we have x(10) = 45 at t = 10 with a shortage of 5 being incurred comparedwith a true value of 50. Surely, the analytical solution is a true solution. Onlywhen it is hard to obtain it, a discrete approximation has to be resorted as analternative method for acquiring a solution. This approximation is, however,far from a true value as our calculation shows.a) Continuous Flow dt 0Two algorithms have been posed to overcome this discrepancy. First algorithmis to make a discrete period of ow smaller; that is, dt 0, so that discreteow appears to be as close as to continuous ow. This is a method employedin equation (1.7), which is known as the Eulers method. Table 1.2 showscalculations by the method for dt = 1 and 0.5. In Figure 1.4 a true value att = 10 is shown to be equal to a triangle area surrounded by a linear owand time-coordinate lines; that is, 10 10 1/2 = 50. The Eulers method is,graphically speaking, to sum the areas of all rectangles created at each discreteperiod of time. Surely, the ner the rectangles, the closer we get to a true area.Table 1.3 shows that as dt 0, the amount of stock gets closer to a true valueof x(10) = 50, but it never gets to the true value. Meanwhile, the number ofcalculations and, hence, the calculation time increase as dt gets ner.b) 2nd-Order Runge-Kutta MethodSecond algorithm to approximate a true value is to obtain a better formulafor calculating the amount of f(t) over a period dt so that a rectangular areaover the period dt becomes closer to a true area. The 2nd-order Runge-Kuttamethod is one such method. According to it, a value of f(t) at the mid-pointof dt is used.x(t +dt) = x(t) +f(t + dt2 )dt (1.20)1.3. DYNAMICS IN ACTION 25Table 1.2: Linear Flow Calculation of x(10) for dt=1 and 0.5t x(t) dx0 0 01 0 12 1 23 3 34 6 45 10 56 15 67 21 78 28 89 36 910 45where dt = 1 anddx = f(t)dt = t.t x(t) dx t x(t) dx0.0 0.00 0.00 5.0 11.25 2.500.5 0.00 0.25 5.5 13.75 2.751.0 0.25 0.50 6.0 16.50 3.001.5 0.75 0.75 6.5 19.50 3.252.0 1.50 1.00 7.0 22.75 3.502.5 2.50 1.25 7.5 26.25 3.753.0 3.75 1.50 8.0 30.00 4.003.5 5.25 1.75 8.5 34.00 4.254.0 7.00 2.00 9.0 38.25 4.504.5 9.00 2.25 9.5 42.75 4.7510.0 47.50where dt = 0.5 and dx = f(t)dt = 0.5t.In our simple linear example here, it is calculated asf(t + dt2 ) = t + dt2 (1.21)Applying the 2nd-order Runge-Kutta method, we can obtain a true value evenfor dt = 1 as shown in Table 1.3.1.3.3 Nonlinear Flow of Time SquaredWe now consider non-linear continuous ow that is dependent only on time.The simplest example is the following:f(t) = t2(1.22)Let the initial value of stock be x(0) = 0. Then the analytical solution isobtained as follows:x(t) = t0u2du = t33 (1.23)At the period t = 6, a true value of the stock becomes x(6) = 72. Figure 1.5illustrates a stock and ow relation for this solution. Stock is shown as a heightat a time t, which is equal to an area surrounded by a nonlinear ow curve anda time-coordinate up to the period t.Table 1.3: Discrete Approximation for x(10)f(t)\dt 1 1/2 1/4 1/8 1/16Euler 45 47.5 48.75 49.375 49.6875Runge-Kutta 2 5026 CHAPTER 1. SYSTEM DYNAMICSNonlinear Flow of Time8060402002 2 2222222222222222222221 1 1 1 1 1111111111111111110 1 2 3 4 5 6Time (Month)unitStock x : run 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1"Flow (t)" : run 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2Figure 1.5: Nonlinear Flow and StockDiscrete ApproximationA discrete approximation of this equation is obtained in terms of a stock-owrelation as follows:x(t) =t1i=0i2(1.24)At the period t = 6, this approximation yields a value of 55, resulting in a largediscrepancy of 17. In general, there exists no analytical solution for calculatinga true value of the area surrounded by a nonlinear ow curve and a time-coordinate. Accordingly, two methods of approximation have been introduced;that is, the Eulers and 2nd-order Runge-Kutta methods. Table 1.4 shows suchapproximations by these methods for various values of dt. As shown in thetable, both the Eulers and 2nd-order Runge-Kutta methods are not ecient toattain a true value for nonlinear ow even for smaller values of dt.f(t + dt2 ) = t2+tdt + dt24 (1.25)Clearly, in a case of nonlinear ow, the 2nd-order Runge-Kutta method, whetherit be Stella or Madonna formula , fails to attain a true value at t = 6; that is,x(6) = 72.1.3. DYNAMICS IN ACTION 27Table 1.4: Discrete Approximation for x(6)f(t)\dt 1 1/2 1/4 1/8 1/16Euler 55 63.25 67.5625 69.7656 70.8789Runge-Kutta 2 71.5 71.875 71.9688 71.9922 71.998Runge-Kutta 4 72.04th-Order Runge-Kutta MethodThe 4th-order Runge-Kutta method is a further revision to overcome the in-eciency of the 2nd-order Runge-Kutta method in a nonlinear case of ow asobserved above. Its formula is given as4x(t +dt) = x(t) + f(t) + 4f(t + dt2 ) +f(t +dt)6 dt (1.26)Table 1.4 and 1.5 show that the 4th-order Runge-Kutta method are able toattain a true value even for dt = 1. The reader, however, should be remindedthat this is not always the case as shown below.Table 1.5: 2nd- and 4th-Order Runge-Kutta Method (dt = 1)t x(t) Runge-Kutta 2 x(t) Runge-Kutta 40 0 0.25 0 0.331 0.25 2.25 0.33 2.332 2.5 6.25 2.66 6.333 8.75 12.25 9 12.334 21 20.25 21.33 20.335 41.25 30.25 41.66 30.336 71.5 721.3.4 Random WalkStochastic ow is created by probability distribution function. The simplest oneis uniform random distribution in which random numbers are created betweenminimum and maximum. Let us consider the stock price whose initial value is$ 10, and its price goes up and down randomly between the range of maximum$1.00 and minimum - $1.00.f(t) = RANDOM UNIFORM (Minimum, Maximum) (1.27)Figure 1.6 is produced for a specic random walk. It is a surprise to see howa random price change daily produces a trend of stock price.4Fore detaied explanation, see [9], section 2.8, pp.103 - 107 and 388 - 391.28 CHAPTER 1. SYSTEM DYNAMICS 2014.593.5-20 20 40 60 80 100 120 140 160 180 200Time (Day)DollarStock Value : CurrentStochastic Flow : CurrentFigure 1.6: Random Walk1.4 System Dynamics1.4.1 Exponential GrowthSo far the amount of ow is assumed to be created by autonomous outside forcesat each period t. Next type of ow we now consider is the one caused by theamount of stock within the system. In other words, ow itself, being causedby the amount of stock, is causing a next amount of ow through a feedbackprocess of stock: that is to say, ow becomes a function of stock. Wheneverow is aected by stock, dynamics becomes system dynamics.When ow is discrete, a stock-ow relation of this feedback type is describedas follows:x(t + 1) = x(t) +f(x(t)), t = 0, 1, 2, (1.28)In the case of a continuous ow, it is presented as a dierential equation asfollows.dxdt = f(x) (1.29)The simplest example of stock-dependent feedback ow is the following:f(x) = ax (1.30)Figure 1.7 illustrates this stock-dependent feedback relation.1.4. SYSTEM DYNAMICS 29Stock xFlow f(t)aFigure 1.7: Stock-Dependent FeedbackIts continuous ow is depictedas an autonomous dierential equa-tion:dxdt = ax (1.31)From calculus, an analytical solu-tion of this equation is known as thefollowing exponential equation:x(t) = x(0)eatwhere e = 2.7182818284590452354 (1.32)It should be noted that the initial value of the stock x(0) cannot be zero, sincenon-zero amount of stock is always needed as an initial capital to launch agrowth of ow.What happens if such an analytical solution cannot be obtained? Assum-ing that ow is only discretely dened, we can approximate the equation as adiscrete dierence equation:x(t + 1) = x(t) +ax(t), t = 0, 1, 2, (1.33)Then, a discrete solution for this equation is easily obtained asx(t) = x(0)(1 +a)t(1.34)A true continuous solution of the equation could be obtained as an approxi-mation from this discrete solution (1.34), rst by dividing a constant amount ofow a into n sub-periods, and secondly by making n sub-periods into innitelymany ner periods so that each sub-period converges to a moment in time.x(t) = x(0)[ limn(1 + an)n]t= x(0)[ limn(1 + 1na)na]at= x(0)eat(1.35)Let an initial value of the stock be x(0) = 100 and a = 0.1. Then, a truevalue for the period t = 10 is x(10) = 100e0.110= 100e1= 271.8281828459 .This is to obtain a compounding increase in 10% for 10 periods. The amountof initial stock is shown to be increased by a factor of 2.7 when a growth rate is10%. Table 1.6 shows numerical approximations by the Eulers, 2nd-order and4th-order Runge-Kutta methods. It is observed from the table that even the4th-order Runge-Kutta method cannot obtain a true value of the exponential e.30 CHAPTER 1. SYSTEM DYNAMICSTable 1.6: Discrete Approximationf(t)\dt 1 1/2 1/4 1/8 1/16Euler 259.374 265.33 268.506 270.148 270.984Runge-Kuttta 2 271.408 271.7191 271.8004 271.8212 271.8264Runge-Kutta 4 271.8279 271.828169 271.828182 271.828182 271.828182In Figure 1.7, the amount of ow is shown to be determined by its previousamount through the amount of stock. Structurally this relation is sketched asa following ow chart: an increase in ow an increase in stock anincrease in ow . At an annual growth rate of 10%, for instance, it takes onlyseven years for the initial amount of stock to double, and 11 years to triple,and 23 years to become 10 folds. In fty years, it becomes about 150 timesas large. This self-increasing relation is called a reinforcing or positive feedback(see Figure 1.7.) Left-hand diagram in Figure 1.14 illustrates such a positivefeedback growth.Constant Doubling TimesOne of the astonishing features of exponential growth is that a doubling timeof stock is always constant. Let x(0) = 1 and x(t) = 2 in the equation (1.32),then it is obtained as follows.ln2 = at = t = 0.693147a (1.36)For instance, when a = 0.02, that is, an annual growth rate is 2%, thendoubling time of the stock becomes about 35 years. That is to say, every 35years the stock becomes twice as big. When a = 0.07, or an annual growthrate is 7%, stock becokes dobled about every 10 years. Consider an economygrowthing at 7% annually. its GDP becomes 8 folds in 30 years. This enormouspower of exponential is usually overlooked or under estimated. See the sectionof Misperception of Exponential Growth on pages 269 - 272 in [48].Examples of Exponential Growth (Reinforcing Feedback)In system dynamics, this exponential growth is called positive or reinforc-ing feedback. Figure 1.8 illustrates some examples of these reinforcing stock-dependent feedback relation. Left-hand diagram illustrates our nancial systemin which our bank deposits keeps increasing as long as positive interest rate isguaranteed by our banking system. This nancial system creates the environ-ment that the rich becomes richer exponentially.1.4.2 Balancing FeedbackIf system consists of only exponential growth or reinforcing feedback behaviors,it will sooner or later explode. System has to have a common purpose or the1.4. SYSTEM DYNAMICS 31aim of the system as already quoted in the beginning of this chapter. In otherwords, it has to be stabilized to accomplish its aim.To attain a self-regulating stability of the system, another type of feedbackis needed such that whenever a state of the system x(t) is o the equilibriumx, it tries to come back to the equilibrium, as if its being attracted to theequilibrium. If system has this feature, it will be stabilized at the equilibrium.In economics, it is called global stability. Free market economy has to have thisprice stability as a system to avoid unstable price uctuations.Stock xFlow f(t)AdjustingTimex*Figure 1.9: Balancing FeedbackStructurally the stability is at-tained if stock-ow has a relationsuch that an increase in ow a decrease in stock a decreasein ow. This stabilizing relation iscalled a balancing or negative feed-back in system dynamics. Figure1.9 illustrates this balancing feed-back stock-dependent relation.Mathematically, this is to guar-antee the stability of equilibrium.Let x be such an equilibriumpoint, or target or objective of thestock x(t)). Then stabilizing behavior is realized by the following owf(x) = xx(t)AT (1.37)where AT is the adjusting time of the gap between x and x.Fortunately, this dierential equation can be analytically solved as follows.First rewrite the equation (1.37) asd(x(t) x)x(t) x = dtAT (1.38)Then integrate both side to obtainln(x(t) x) = tAT +C, (1.39)where ln denotes natural logarithm. This is further rewritten asBankDepositsInterestInterestRatePopulationBirthBirth RateFigure 1.8: Examples of Exponential Feedback32 CHAPTER 1. SYSTEM DYNAMICSx(t) x = e tATeC(1.40)At the initial point in time, we haveeC= x(0) x at t = 0 (1.41)Thus, the amount of stock at t is analytically obtained asx(t) = x(xx(0))e tAT(1.42)Examples of Balancing FeedbackFigure 1.10 illustrates some examples of these balancing stock-dependent feed-back relation.InventoryProductionDesiredInventoryInventryAdjustingTimeWorkersEmploymentDesiredWorkersEmploymentAdjusting TimeFigure 1.10: Examples of Balancing FeedbackExponential DecayWhen x = 0, system continues to decay or disappear. In other words, stockbegins to decrease by the amount of its own divided by the adjusting time.f(x) = x(t)AT (1.43)Stock xAdjustingTimeOutflow f(t)Figure 1.11: Exponential DecayThis decay process is called ex-ponential decay. Whenever expo-nential decay appears, ow has onlynegative amount. In this case insystem dynamics we draw ow outof stock so that it becomes more in-tuitive to understand the outow ofstock. Figure 1.11 illustrates suchstock-outow relation.At an annual declining rateof 10%, for instance, the initialamount of stock decreases by halfin seven years, by one third in 11years, and by one tenth in 23 years, balancing to a zero level eventually. Right-hand diagram in Figure 1.14 illustrates such a negative feedback decay.1.5. SYSTEM DYNAMICS WITH ONE STOCK 33Examples of Exponential DecayFigure 1.12 illustrates this stock-dependent feedback relation.PopulationDeathDeath RateCapitalAssetsDepreciationDepreciationPeriodFigure 1.12: Examples of Exponential Decay1.5 System Dynamics with One Stock1.5.1 First-Order Linear GrowthWe have now learned two fundamental feedbacks in system dynamics; reinforc-ing (exponential or positive) feedback and balancing (negative) feedback. Let usnow consider the simplest system dynamics which have these two feedbacks si-multaneously. It is called rst-order linear growth system. First-order impliesthat the system has only one stock, while linear means that its inow andoutow are linearly dependent on stock. Figure 1.13 illustrates our rst systemdynamics model which has both reinforcing and balancing feedback relations.StockInflow OutflowInflowFractionOutflowFractionFigure 1.13: First-Order Linear Growth ModelTable 1.7 describes its equation.Left-hand diagram of Figure 1.14 is produced for the inow fraction value of0.1 and outow fraction value of zero, and has a feature of exponential growth.Right-hand diagram is produced by the opposite fractional values, and has afeature of exponential decay. It is easily conrmed that whenever inow frac-tion is greater than outow fraction, the system produces exponential growthbehavior. When outow fraction is greater than inow fraction, it causes an34 CHAPTER 1. SYSTEM DYNAMICSTable 1.7: Equations of the First-Order Growth ModelInow= Stock*Inow FractionUnits: unit/YearInow Fraction= 0.1Units: 1/Year [0,1,0.01]Outow= Stock*Outow FractionUnits: unit/YearOutow Fraction= 0.04Units: 1/Year [0,1,0.01]Stock= INTEG ( Inow-Outow, 100)Units: unitStock2,0001,5001,00050000 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30Time (Year)unitStock : GrowthStock10075502500 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30Time (Year)unitStock : DecayFigure 1.14: Exponential Growth and Decayexponential decay behavior. In this way, the rst-order linear system can onlyproduces two types of behaviors: exponential growth or decay.Stock xFlow f(t)ax*Figure 1.15: S-shaped Growth ModelThis model can best describepopulation dynamics. Suppose theworld birth rate (inow fraction)is 3.5%, while its death rate (out-ow fraction) is 1.5%. This impliesthat world population grows expo-nentially at the net growth rate of2%.1.5.2 S-Shaped Limit toGrowthIn the rst-order linear model, thesystem may explode if inow frac-1.5. SYSTEM DYNAMICS WITH ONE STOCK 35tion is greater than outow fraction. Population explosion is a good exam-ple. To stabilize the system, the exponential growth ax(t) has to be curbedby bringing another balancing feedback which plays a role of a break in a car.Specically, whenever x(t) grows to a limit x, it begins to be regulated as ifpopulation is control ed and speed of the car is reduced. This is a feedbackmechanism to stabilize the system. It could be done by the following ow:f(x) = ax(t)b(t), where b(t) = xx(t)x (1.44)Apparently b(t) is bounded by 0 b(t) 1, and reduces to zero as x(t)approached to its limit x. Figure 1.15 is such model in which both reinforcing(exponential) and balancing feedback are brought together. It is called S-shapedgrowth in system dynamics, 100 unit4 unit/Year75 unit3 unit/Year50 unit2 unit/Year25 unit1 unit/Year0 unit0 unit/Year0 10 20 30 40 50 60 70 80 90 100Time ()Stock x : Current unit"Flow f(t)" : Current unit/YearFigure 1.16: S-Shaped Limit to Growth 11.5.3 S-Shaped Limit to Growth with Table FunctionAnother way to regulate the growth of x(t) is to increase b(t) to the value of aas x(t) grows to its limit x as shown below.f(x) = (a b(t))x(t) (1.45)Specically, any functional relation that has a property such that b(t) ap-proaches a whenever x(t)/x approaches 1 works for this purpose.One of the simplest function isb(t) = ax(t)x (1.46)36 CHAPTER 1. SYSTEM DYNAMICSIn this case the above function becomesf(x) = (a b(t))x(t) = a(xx(t)x)x(t) (1.47)which becomes the same as the above S-shaped limit to growth.Stock xInflow OutflowInflowFractionOutflow FractionTable: OutflowFractionx* "Table: Outflow Fraction"0.100 1Figure 1.17: S-Shaped Limit to Growth Model with Table Function100 unit20 unit/Year75 unit15 unit/Year50 unit10 unit/Year25 unit5 unit/Year0 unit0 unit/Year3 3 3 3 3 3 3 3 3 3 3 3 3 33333 3 3 3 32 2 2 2 2 2 2 2 2 2 2222222 2 2 2 2 21 1 1 1 1 1 1 1 1111111111 1 1 1 1 10 10 20 30 40 50 60 70Time (Year)Stock x : Current unit1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1Inflow : Current unit/Year2 2 2 2 2 2 2 2 2 2 2 2 2 2 2Outflow : Current unit/Year3 3 3 3 3 3 3 3 3 3 3 3 3 3 3Figure 1.18: S-Shaped Limit to Growth 2 with Table FunctionIf mathematical function is not available, still we can produce S-shapedbehavior by plotting the relation, which is called table function. One of suchtable function is shown in the right-hand diagram of Figure 1.17. Left-handdiagram illustrates S-shaped limit to growth model.1.6 System Dynamics with Two Stocks1.6.1 Feedback Loops in General1.6. SYSTEM DYNAMICS WITH TWO STOCKS 37Stock xStock yFlow yFlow xFigure 1.19: Feedback Loops in GeneralWhen there is only one stock, twofeedback loops are at maximumproduced as in rst-order lineargrowth model. When the numberof stocks becomes two, at maximumthree feedback loops can be gener-ated as illus rated in Figure 1.19Mathematically, general feed-back loop relation with two stockscan be represented by a followingdynamical system in which eachow is a function of stocks x andy.dxdt = f(x, y) (1.48)dydt = g(x, y) (1.49)1.6.2 S-Shaped Limit to Growth with Two StocksBehaviors in system dynamics with one stock are limited to exponential growthand decay generated by the rst-order linear growth model, and S-shaped limitto growth. To produce another fundamental behaviors such as overshoot andcollapse, and oscillation, at least two stocks are needed. System dynamics withtwo stocks are called second-order system dynamics.Stock yFlow f(x,y)aStock xLimitFigure 1.20: S-shaped Growth ModelLet us begin with another typeof S-shaped limit to growth behav-ior that can be generated with twostocks x and y. When the totalamount of stock x and stock y islimited by the constant available re-sources such that x+y = b, and theamount of stock x ows into stocky as shown in Figure 1.20, stock ybegins to create a S-shaped limit togrowth behavior. A typical systemcausing this behavior is described asfollows.dxdt = f(x, y) (1.50)dydt = f(x, y)= ax(t)y(t)= a(b y(t))y(t) (1.51)38 CHAPTER 1. SYSTEM DYNAMICSwhere a = 0.001 and b = 100. This relation is also reduced todxdt + dydt = 0 (1.52) 100 unit4 unit/Year75 unit3 unit/Year50 unit2 unit/Year25 unit1 unit/Year0 unit0 unit/Year0 10 20 30 40 50 60 70 80 90 100Time ()Stock x : run unitStock y : run unit"Flow f(x,y)" : run unit/YearFigure 1.21: S-Shaped Limit to Growth 3Mathematically, the above equation (1.51) is similar to the S-shaped limitto growth equation (1.44). In other words, x(t) = b y(t) begins to diminishas y(t) continues to grow. This is a requirement to generate S-shaped limit togrowth.Examples of this type of S-shaped limit to growth are abundant such aslogistic model of innovation diusion in marketing.So far we have presented three dierent gures to illustrate S-shaped limitto growth. Mathematically, all of the S-shaped limits to growth turn out tohave the same structure. The same structures can be built in three dierentmodels, depending on the issues we want to analyze. This indicates the richnessof system dynamics approach.1.6.3 Overshoot and CollapseNext behavior to be generated with two stocks is a so-called overshoot andcollapse. It is basically caused by the S-shaped limit to growth model with tablefunction. However, coecient b(t) is this time aected by the stock y. Increasingstock x causes stock y to decrease, which in turn makes the availability of stocky smaller, which then increases outow fraction. This relation is described bythe table function in the right-hand diagram of Figure 1.22. Increasing fractioncollapses stock x. The model is shown in the left-hand diagram. Behaviors ofovershoot & collapse is shown in Figure 1.23.1.6. SYSTEM DYNAMICS WITH TWO STOCKS 39Stock xInflow x Outflow xInflowFractionOutflowFractionTable: OutflowFractionInitialStock yStock yOutflow yOutflow yFractionGraph Lookup - "Table: Outflow Fraction"100 1Figure 1.22: Overshoot & Collapse Model80 unit x40 unit x/Year100 unit y40 unit x20 unit x/Year50 unit y0 unit x0 unit x/Year0 unit y44444444444444 4 43 3 3333333333333322222222 2222222 211111111111111110 6 12 18 24 30 36 42 48 54 60Time (Year)Stock x : run unit x1 1 1 1 1 1 1 1 1 1 1 1 1Inflow x : run unit x/Year2 2 2 2 2 2 2 2 2 2 2 2Outflow x : run unit x/Year3 3 3 3 3 3 3 3 3 3 3Stock y : run unit y4 4 4 4 4 4 4 4 4 4 4 4 4 4Figure 1.23: Overshoot & Collapse BehaviorExamples of Overshoot and CollapseMy favorite example of overshoot and collapse model is the decline of the Mayanempire in [3].1.6.4 OscillationAnother behavior that can be created with two stocks is oscillation A simpleexample of system dynamics with two stocks which is illustrated in Figure 1.2440 CHAPTER 1. SYSTEM DYNAMICSStock xflow xStock yflow yabFigure 1.24: An Oscillation ModelIt can be formally represented asfollows:dxdt = ay, x(0) = 1, a = 1 (1.53)dydt = bx, y(0) = 1, b = 1 (1.54)This is nothing but a systemof dierential equations, which isalso called a dynamical system inmathematics. Its solution by Eulermethod is illustrated in Figure 1.25 in which DT is set to be dt = 0.125.Movement of the stock y is illustrated on the y-axis against the x-axis of time(left gure) and against the stock of x (right gure). In this Eulers solution,the amount of stock keeps expanding even a unit period is divided into 8 sub-periods for better computations. In this continuous case of ow, errors at eachstage of calculation continue to accumulate, causing a large deviation from atrue value. 630-3-60 2 4 6 8 10 12 14 16 18 20Time (Week)unitStock x : Euler Stock y : Euler 630-3-6-4 -3 -2 -1 0 1 2 3 4 5Stock xunitStock y : EulerFigure 1.25: Oscillation under Euler MethodOn the other hand, the 2nd-order Runge-Kutta solution eliminates this de-viation and yields a periodic or cyclical movement as illustrated in Figure 1.26.This gives us a caveat that setting a small number of dt in the Eulers methodis not enough to approximate a true value in the case of continuous ow. Itis expedient, therefore, to examine the computational results by both methodsand see whether they are dierentiated or not.Examples of OscillationPendulum movement is a typical example of oscillation. It is shown in [4] thatemployment instability behavior is produced by the same system structure whichgenerates pendulum oscillation.1.7. DELAYS IN SYSTEM DYNAMICS 41 210-1-20 2 4 6 8 10 12 14 16 18 20Time (Week)unitStock x : Rung-Kutta2Stock y : Rung-Kutta2 210-1-2-1.50 -1.10 -0.70 -0.30 0.10 0.50 0.90 1.30Stock xunitStock y : Rung-Kutta2Figure 1.26: Oscillation under Runge-Kutta2 Method1.7 Delays in System Dynamics1.7.1 Material DelaysFirst-Order Material DelaysSystem dynamics consists of four letters: stock, ow, variable and arrow, asalready discussed in the beginning of this chapter. To generate fundamentalbehaviors, these letters have to be combined according to its grammatical rules:reinforcing (positive) and balancing (negative) feedback loops, and delays. Sofar reinforcing and balancing feedback loops have been explored. Yet delays havebeen already applied in our models above without focusing on them. Delays playan important role in model building. Accordingly, it is appropriate to examinesthe meaning of delays in system dynamics in this section.InputOutput DelayFigure 1.27: Structure of DelaysDelays in system dynamics has astructure illustrated in Figure 1.27.That is, output always gets delayedwhen input goes through stocks.This is an inevitable feature in sys-tem dynamics. It is essential in sys-tem dynamics to distinguish two types of delays: material delays and informa-tion delays. Let us start with material delays rst.When there is only one stock, delays becomes similar to exponential decayfor the one-time input, which is called pulse. In Figure 1.28, 100 units of materialare input at time zero. This corresponds to the situation, for instance, in which100 units of goods are purchased and stored in inventory, or 100 letters aredropped in the post oce. Delay time is assumed to be 6 days in this example.In other words, one-sixth of goods are to be delivered daily as output.Second-Order Material DelaysIn the second-order material delays, materials are processed twice as illustratedin the top diagram of Figure 1.29. Total delay time is the same as 6 days.42 CHAPTER 1. SYSTEM DYNAMICSMaterial inTransitInflowDelay TimeInflowAmountOutflow 100 unit/Day4 unit/Day75 unit/Day3 unit/Day50 unit/Day2 unit/Day25 unit/Day1 unit/Day0 unit/Day0 unit/Day-2 0 2 4 6 8 10 12 14 16 18 20Time (Day)Inflow : Current unit/DayOutflow : Current unit/DayFigure 1.28: First-Order Material DelaysAccordingly, delay time for each process becomes 3 days. In this case, outputdistribution becomes bell-shaped. The reader can expand the delays to the n-thorder and see what will happen to output.Delay TimeMaterial inTransit 1Outflow 1Material inTransit 2Outflow 2InflowAmountInflow100 unit/Day6 unit/Day75 unit/Day4.5 unit/Day50 unit/Day3 unit/Day25 unit/Day1.5 unit/Day0 unit/Day0 unit/Day3333 333333333 3 3 3 3 3 3 3 3 22222222222 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1-2 0 2 4 6 8 10 12 14 16 18 20Time (Day)Inflow : run unit/Day 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1Outflow 1 : run unit/Day 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2Outflow 2 : run unit/Day 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3Figure 1.29: Second-Order Material Delays1.7.2 Information DelayFirst-Order Information DelaysInformation delays occur because information as input has to be processed byhuman brains and implemented as action output. Information literally meansin-form; that is, being input to brain which forms it for action. This is a processto adjust our perceived understanding in the brain to the actual situation outsidethe brain. Structurally this is the same as balancing feedback explained above1.7. DELAYS IN SYSTEM DYNAMICS 43to ll the gap between x and x, as illustrated in the left-hand diagram of Figure1.30.ActualInputAdjustment TimePerceivedOutputAdjusted GapOutput 25020015010050-2 0 2 4 6 8 10 12 14 16 18 20Time (Day)DmnlActual Input : runOutput : runFigure 1.30: First-Order Information DelaysOur perceived understanding, say, on daily sales order, is assumed here to be100 units, yet actual sales jumps to 200 at the time zero. Our suspicious brainhesitates to adjust to this new reality instantaneously. Instead, it slowly adaptsto a new reality with the adjustment time of 6 days. This type of adjustmentis explained as adaptive expectations and exponential smoothing in [48].Second-Order Information DelaysActualInputAdjustment TimePerceivedOutput 1Adjusted Gap 1PerceivedOutput 2Adjusted Gap 2Output 250200150100503 3 3 3 3 33 333 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 32 2 222222 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 21 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1-2 0 2 4 6 8 10 12 14 16 18 20Time (Day)valueActual Input : Current 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1Perceived Output 1 : Current 2 2 2 2 2 2 2 2 2 2 2 2 2 2Output : Current 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3Figure 1.31: Second-Order Information DelaysSecond-order information delays imply that information processing occursthrough two brains. This is the same as re-thinking process for one person or a44 CHAPTER 1. SYSTEM DYNAMICSprocess in which information is being sent to another person. This structure ismodeled in the top diagram of Figure 1.31.In either case, second-order adaptation process becomes slower than therst-order information delays as illustrated in the bottom diagram.Adaptive Expectations for Random WalkFirst-order information delays are called adaptive expectations or exponentialsmoothing because perceived output tries to adjust gradually to the actual in-put. When random walk becomes actual input, output becomes exponentiallysmoothing as illustrated in Figure 1.32.ActualInputAdjustment TimePerceivedOutputAdjusted GapOutputStock ValueStochasticFlow 2014.593.5-2 3333 33 333 3333 3333333 333332222 2 22222222 222 222222 22 221111111 1 1 111 1111111 1111 11 10 20 40 60 80 100 120 140 160 180 200Time (Day)valueOutput : run 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1Stock Value : run 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2Stochastic Flow : run 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3Figure 1.32: Adaptive Expectations for Random Walk1.8 System Dynamics with Three Stocks1.8.1 Feedback Loops in GeneralIt has been shown that system dynamics with two stocks can mostly produceall fundamental behavior patters such as exponential growth, exponential de-cay, S-shaped limit to growth, overshoot and collapse, and oscillation. Actualbehaviors observed in complex system are combinations of these fundamentalbehaviors. We are now in a position to build system dynamics model based onthese fundamental building blocks. And this introductory chapter on system1.8. SYSTEM DYNAMICS WITH THREE STOCKS 45dynamics seems appropriate to end at this point, and we should go to nextchapter in which how system dynamics method can be applied to economics.StockxStockyStockzFlow yFlow xFlow zFigure 1.33: Feedback Loop in GeneralYet, the exists another behav-iors which can not be produced withtwo stocks; that is a chaotic behav-ior! Accordingly, we stay here for awhile, and consider a general feed-back relation for the case of threestock-ow relations. Figure 1.33 il-lustrates a general feedback loops.Each stock-ow relation has itsown feedback loop and two mutualfeedback loops. In total, there are 6feedback loops, excluding overlap-ping ones. As long as we observethe parts of mutual loop relations,thats all loops. However, if we ob-serve the whole, we can nd twomore feedback loops: that is, a whole feedback loop of x y z x,and x z y x. Therefore, there are 8 feedback loops as a whole. Theexistence of these two whole feedback loops seems to me to symbolize a complexsystem in terms of loops; that is, the whole is more than the sum of its parts.A complex system is one whose component parts interact with suf-cient intricacy that they cannot be predicted by standard linearequations; so many variables are at work in the system that its over-all behavior can only be understood as an emergent consequenceof the holistic sum of all the myriad behaviors embedded within.Reductionism does not work with complex systems, and it is nowclear that a purely reductionist approach cannot be applied whenstudying life: in living systems, the whole is more than the sum ofits parts (emphasis is made by the author) [37], pp. 7-8.Mathematically, this general feedback loop relation can be represented by afollowing dynamical system in which each ow is a function of all stocks x, yand z.dxdt = f(x, y, z) (1.55)dydt = g(x, y, z) (1.56)dzdt = h(x, y, z) (1.57)1.8.2 Lorenz ChaosAs a special example of the general feedback loops by three stock-ow relations,let us consider well-known Lorenz equations which yield a chaotic movement.46 CHAPTER 1. SYSTEM DYNAMICSMathematical equations of the Lorenz chaos are written asdxdt = a(x y) (1.58)dydt = xz +bx y (1.59)dzdt = xy cz (1.60)where initial values are assigned as x(0) = 0, y(0) = 2 and z(0) = 0 andparameters are as originally set at a = 10, b = 28 and c = 8/3 by Lorenz. (SeeChapter 14 The Lorenz System in [33]).xyzy dotx dotz dotabcFigure 1.34: Lorenz Feedback LoopFigure 1.34 illustrates feedback loops of the Lorenz equations. Comparedwith a general case of the left-hand diagram, a link from the Stock z to theFlow x is missing. Accordingly, we have in total 6 feedback loops - a loss of twoloops ! (We dont know whether this loss of loops is related with a chaos to bediscussed below.)Figure 1.35 illustrates a phase diagram of Lorenz chaos in which movementsof the stock y and z are illustrated on the y-axis against the stock x on the x-axis.Calculations are done by the 4th-order Runge-Kutta method at dt = 0.0078125;that is, at each of 128 sub-periods in a unit period. With such a small sub-period, computational errors may arise less likely as explained above.Sensitive Dependence on Initial ConditionsIn the above Lorenz phase diagram, movement of stocks does not converge toa xed point or a limit cycle, or diverge to innity. Instead, wherever it starts,it seems to be eventually attracted to a certain region and continue uctuatingin it, with the information of its start being lost eventually. That region iscalled a strange attractor or chaos. One of the main features of chaos is asensitive dependence on initial conditions. This is numerically explained as1.8. SYSTEM DYNAMICS WITH THREE STOCKS 4780 50 52.5 30 25 10 -2.5 -10 -30 -30 -18 -14 -10 -6 -2 2 6 10 14 18 22XFigure 1.35: Lorenz Chaosfollows. Suppose a true initial value of the stock y in the Lorenz equations isy(0) = 1.0001 instead of y(0) = 1.0, and denote its true value by y. At theperiod t = 10.25 those two values of the stock are calculated as y(t) = 5.517,and y(t) = 5.518. The dierence is only 0.001 and they stay very close eachother. This makes sense, because both started at the very close distance of0.0001. To our surprise, however, at the period t = 27, they are calculated asy(t) = 2.546, and y(t) = 13.77; a large dierence of 16.316 is made. Smallamount of dierences at an initial time eventually turns out to cause a bigdierence later. In other words, stock values sensitively depend on their initialconditions. Figure 1.36 illustrates how values of the stock y begin to divergefrom a true value y around the period t = 18.y40200-20-400 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30Time (Second)hpy : y0001 y : y0000Figure 1.36: Sensitive Dependence on Initial Conditions48 CHAPTER 1. SYSTEM DYNAMICSWhy could it be possible? It is caused by the power of exponential magni-cation empowered by feedback loops. As illustrated in Figure 1.14, for instance,a simple calculation yields that an initial dierence of 0.001 is exponentiallymagnied to 22.02 by the time t = 100, more than twenty-two thousand factorslarger, because of a positive feedback loop. Chaos is a region called strangeattractor to which innitely many iterated and exponentially magnied valuesare conned. Hence, it is intuitively understood that exponentially magniedvalues in a chaos region sensitively depend on initial conditions; in other words,values whose initial conditions dier only very slightly cannot stay close andbegin to diverge eventually.This chaotic feature creates annoying problem in system dynamics: unpre-dictability in the future. It is almost impossible in reality to obtain true initialvalues due to some observation errors and round-o errors of measurement andcomputations. These errors are magnied in a chaotic system dynamics to apoint where predictions of the future and forecasting become almost meaninglessand misleading. If analytical solutions of dierential equations could be found,this would never happen, because solutions are continuous function of time andwe could easily predict or approximate the future behavior of the system evenif initial conditions are missed slightly. Without the analytical solutions, thefuture has to be iterated step by step, causing an exponential magnicationby feedback loop. Unfortunately as discussed above, it is almost impossibleto nd analytical solutions in a nonlinear dynamics and system dynamics. Insuch cases, if a true initial value fails to be specied, then we cannot predictthe future at all, even if we try to make calculations as precise