1. Introduction, part 2

11 November 1999

To contents

  1. Dynamic phenomena
  2. Dynamic systems
  3. Models
  4. Dynamic models
  5. Decisions
  6. Random processes
  7. Chaos
  8. Self organizing systems
  9. The challenge of this project
  10. The great challenge of mankind
  11. Exercises.
  12. References.

1.1 Dynamic phenomena

The really interesting phenomena in economy have to do with the dynamics of the economic system. Economic growth and inflation are both dynamic phenomena. Business cycles are related to employment and unemployment. The living standard, the income distribution, the formation of assets and debts and the depletion of natural resources all develop as time passes.

The dynamic behavior comes from the accumulation of fixed capital, stocks, assets, debts and so on. Population size, education standard, infrastructure and natural resources are other examples. All these resources are the result of ongoing processes that continuously add new resources and possible also deplete old resources. E. g. the amount of production capacity depends upon how much was previously invested in fixed capital, education, infrastructure etc.

Dynamic processes have a memory. The state of a dynamic system depends upon the previous history of the system. This means that the same actions applied to the system under different circumstances will give different results. The sequence of actions and the timing has to be right.

A very simple example is that a well has to be drilled before the water pump is installed. Another example is that central banks try to increase or decrease interest rates at the right time in order to stabilize the economy.

1.2 Dynamic systems

Dynamic processes exhibit a very big variation. They really show that a system is something different than just the sum of it parts.

A very simple example is a mechanical system consisting of a spring and a mass. The spring has a length, l, and a spring constant, k, (ratio between force and elongation). The mass has its position, x, and weight, m. If the mass is hung at one end of the string and the other end is suspended in the ceiling, we get a simple oscillator. Pulling the mass and then releasing it will cause the mass to swing up and down. Now there are such properties as amplitude and frequency, cycles per second. These dynamic properties can not be found in the individual parts.

Figure 1.2:1 Spring and mass Figure 1.2:2 Oscillating system

The example above was a mechanical system. There are other kinds of systems, such as electrical, chemical, biological, ecological, social and economical systems. Some systems can be treated with mathematics, other systems are less suited for such a strict treatment. A common feature is that the behavior of all systems depend upon the properties of the components and the rules of interaction between the parts of the system.

Mechanical, electrical and chemical systems follow relatively well known laws and their behavior can be calculated to a certain degree, depending upon the complexity of the system. Biological and ecological systems follow more complicated laws. Social systems follow rules set by the culture and human nature. Different cultures give very different societies, although the components: human beings and their environment are the same.

The laws of economical systems can be seen as a combination of simple rules for flows and matter and the complicated behavior of humans in their cultural context.

Motions or flows accumulate to change the state of the dynamic system. In the oscillator example, the velocity of the mass moves it to positions as time passes. Cell growth makes organisms. Births and deaths changes the size of the population. Investments give new buildings and machinery. Thoughts form ideas.

1.3 Models

My definition of a model is: "A model is a simplification of the real world that describes those aspects that you want to study".

There are many kinds of models:

  1. Toys: Dolls, toy cars, construction kits and games. Toys have some properties of the real world and can be used for experiments and investigations.
  2. Engineering models: Architects make models of houses, bridges or town districts. Engineers test strength and function on models.
  3. Mathematical models: These are descriptions based on mathematical equations. The calculated quantities can describe how a bridge, a vehicle or a power station performs under certain circumstances.

Other authors have described models in their way (my translations from Swedish):

John von Neumann. from the book Chaos by Gleick 1987, ref. 1.

"Science does not try to explain… . It is mainly building models. A model means a mathematical construction which together with some verbal interpretations give a description of the observed phenomena. That mathematical construction is merely and exclusively justified because it is expected to work."

Lennart Ljung, professor of control theory at Linköpings Tekniska Högskola:

" A model is basically founded in common sense - but when that does not suffice, mathematics enters. In order to make a good mathematical model, it is important to experiment and keep in mind that models shall give good predictions. Model validation is fundamental - to gain confidence in the model - turn and twist the properties of the model. A model is experience in a package."

This work intends to show how economic systems can be described by mathematical models.

1.4 Dynamic models

A dynamic model is not a dynamic system, it is a simplified picture of a dynamic system. Part 1 of this work treated static models of economic systems, which might well be and probably are dynamic systems. A static model can show a picture of the state at a given time instant, a snapshot. Dynamic models can show the transitions from one state to another.

Engineering sciences have developed advanced methods for analysis and simulation of dynamic systems. The general theory of dynamic systems can be very abstract and requires advanced mathematics, ref. 2. The theory of control systems is closer to the practical applications. It is a theory for dynamic systems in general and also about the design of feed back loops for optimal performance of a system.

We only need a very small part of control theory in order to develop our dynamic models, mainly the concepts of state variables and systems of dynamic equations formulated as difference equations or differential equations. We will use very simple numerical integration methods to carry out the simulations.

Control theory usually draws a system like in figure 1.4:1 below. The box is the system. It is the part of the world that we want to study, in our case an economic system. Influences from the outside world, quantities inside the system that vary with time and influences from the system upon the outside world are called signals.

Input signals may be decisions about legislation, taxrates, to start a project and so on. It may also be an event that "just happens" to the system like a rise in oil prices or a catastrophe like a hurricane. The difference between a conscious decision and an uncontrollable event is a matter of perspective. The rice in oil price is an uncontrollable event for a single country but a decision for the whole world or possible an internal variable of the system.

Output signals may be influences upon the outside world like demand for import products. They may also be quantities inside the system that we want to observe like production volume or unemployment.

The internal signals are the payment and real flows that we have discussed in part 1.

Figure 1.4:1 A model of a dynamic system.

Input signal are called exogenous variables. Signals inside the system are called endogenous variables. The exogenous variables can not be calculated, they are preconditions for the simulations. The endogenous variables and their development in time is a consequence of the dynamic properties of the system and the time history of the exogenous variables. The output variables or output signals from the system are functions of the endogenous variables. See also appendix A2.

Because a dynamic system has memory, the output depends upon the input signal at the actual time and all previous times. In contrast, a static system has no memory, a certain input signal always gives the same output signal.

1.5 Decisions

Legislation taken by the government or interest rate decided by the central bank are used to control the course of the economy. These decisions are taken by human beings at special occasions and can never be predicted or modeled. The reasoning behind the decisions is based on judgment and sometimes rules of thumb but no strict laws. The situation seldom repeats and it is not possible to gather any statistics about the behavior. We have to treat them as exogenous to the economic system. Using the terminology of control theory, we call them input signals to the system. Many smaller decisions may also be treated as noise added to the input signal.

Decisions taken by many people or companies can be treated with statistics. Then individual decisions taken by each person can be neglected in the model and only averages are considered. Such averages are the propensity to spend, investment ratio, wage level or sales prices etc. These quantities may be regarded as exogenous but can also be modeled depending upon the state of the system.

1.6 Random processes

The individual events of a random process can not be predicted nor decided about. It is often possible to observe an average of a process, e.g. how many cars that pass a road during 24 hours or the number of car accidents during a year. The averages can be measured and put into the model as constants or model parameters of a sector.

1.7 Chaos

There are processes that may seem to be random but they follow well defined laws. They can be predicted to a high degree in the short run but not for longer times. The weather is an example of a process that can not be predicted more than a few days. Economic development is also very hard to predict for more than a year or two.

This behavior is typical for so called "nonlinear systems". The process is called a chaotic process. Chaos is not the same as random behavior. Chaos is unpredictable because the future state of the system is very sensitive to the initial state. The initial state can never be known with such accuracy that the whole process can be anticipated. The states after a certain time will be very different for small variations of the initial state, no matter how small the error of the estimate of the initial state is.

My belief is that some processes that often are regarded as random, such as the price of shares on the stock market, could be treated as a chaotic process instead. Economic cycles are said to have a period of four to five years but make sometimes jumps and are rarely repeated. This irregularities may be explained as chaotic behavior in combination with decisions that can not be predicted.

1.8 Self organizing systems

Large nonlinear systems consisting of many parts may exhibit special features, called self organization. Self organization is a large scale order over the whole system that is established spontaneously form the interaction of the individual parts of the system. The theory dealing with these systems is called synergetics and one of the pioneers in this area is Hermann Haken, ref. 3. Erik Skarman, ref. 2, says that self organizing systems have many state variables, in the order of Avogadoro's number 6.023*1026 = the number of atoms in a kilogramatom (e.g. 16 kg of oxygen or 56 kg of iron). Biologists talk about the number of cells in an organism.

Economical systems do not have that many parts, the population of the earth is now estimated to be 6*109 = six billion people. Even smaller systems may develop some overall order, think of a class in school or the culture of a country.

Self organizing systems are continuously receiving energy from and discharging energy to its surroundings. The earth receives energy from the sun, animals eat food, plants absorb light. Energy leaves the systems mainly in the form of low temperature heat.

On the contrary, an isolated system will eventually die when the temperature is the same in the whole system. The entropy or disorder of the system increases according to the second law of thermodynamics. This will happen with a system based on fossil energy, oil or nuclear energy.

1.9 The challenge of this project

The goal of this project is to develop a general theory of economic systems. The project combines theory from many disciplines such as economics, history, mathematics, statistics, dynamic systems and control theory. The goal might seem very ambitious. I believe that, by starting with very basic principles, it will be possible to make a complete and rather simple theory without too many loose ends. Every reader is invited to add new features to the theory and to explore the results that can be obtained even with very basic components.

The economic system is viewed as two layers, the monetary layer of the economy with payment flows and the real layer of the economy with flows of natural resources, labor, goods and services. The system is describes by four classes of equations:

Traditional economic theory covers the prices and strategies and call it market behavior. This theory will describe a "market" by one price equation and two strategy equations, one for the delivering sector and one for the receiving sector. The conservation equations and process equations are part of input-output economic theory. Input-output theory very often quantify flows in monetary terms, even if it is said that real flows are treated.

Part one of this work described how to build static models. Part number two (this part) describes how to build dynamic models by using the four classes of equations listed above.

Chapter 2 describes the basic principles of dynamic systems. It is an extension to the basic principles for static models that were described in Part 1.

Chapter 3 gives some examples of the elements of dynamic change (integration processes) such as the building up of stocks, investments to form fixed capital, savings, loans and interests that accumulate to financial assets and debts.

Chapters 4 - 10 investigate the basic behavior of dynamic models. A simple example treats a system with a production sector and households, consumers. It is not a model for a real society although some lessons can be drawn regarding the behavior of the production and consumption processes.

Chapters 11 and the following chapters add a financial sector. The introduction of financial assets, loans and interests makes it possible to build more detailed and realistic strategy equations. Economic constraints, such as liquidity, can be introduced. Profit maximization and financial transactions can be optimized in the short or long run. Investment strategies can be analyzed. Stocks and foreign currency are introduced together with foreign trade.

A complete model has to include the public sector. Now the economic activity of a whole country can be described. A nation wide model of the society can also be used as a framework for the analysis of historical economic data. This data can in turn give input to the calculation of the constants included in various equations.

The rest of the work consists of adding mode details to the theory and building models for the study of special problems: What will be the impact of different taxation systems, different tax rates, increased productivity, daily working hours or wage distribution?

The ultimate model will include many countries or even the global economy.

1.10 The great challenge of mankind

The great challenge for all of us is to find ways to survive and prosper during the coming centuries. How do we use the natural resources and protect the environment? One textbook of economics has the definition: "Economics is the study of the use of scarce resources to satisfy unlimited human wants." What happens if some resources are available in an unlimited amount and the economic power is limited to a small number of people or institutions while other resources are limited and a vast number of people have no economic power at all?

David C Korten has studied the present development in the book When Corporations Rule the World, ref. 6. He discusses the consequences of the global economy. His next book The Post Corporate World, Life After Capitalism, ref.7, suggests a system with smaller economic units with a strong feed-back from the people to local enterprises. He envisions a self organizing system of small communities and refers to the original ideas of Adam Smith, ref. 8.

My idea is that such ideas could be tested by using economic models. The models could reveal what special conditions have to prevail and the system's sensitivity to disturbances.

1.11 Exercises

I will try to add some exercises to each chapter. I do not publish any answers for the moment but readers are welcome with their suggestions to my e-mail. If the reader wishes, I can add the answers to this document. I reserve the right to add my own comments to the answers. If contributors wish, I can also, as far as I have time, return personal comments by e-mail.

  1. List the flows that add to and/or subtract from the following state variables:
    - Food in the refrigerator.
    - Money in your wallet.
    - Gasoline in the tank of your car.
    - Number of registered cars.
    - Stocks of goods.
    - Production capacity of a factory.
    - Oil reserves.
    - Carbon dioxide in the atmosphere.
    - Fertile land.
    - Inhabitants in a city.
    - Population of a country.
  2. What state variables are associated with the following flows:
    - Rain fall - evaporation.
    - Tree planting.
    - Fishing.
    - The education system.
    - Savings - loans.
    - Consumption of fixed capital.
    - Transportation.
    - Building railways.


1.12 References

  1. James Gleick: Chaos, Making a New Science, Viking Penguin Inc., New York 1987.
  2. Erik Skarman: Dynamiska system (Dynamic Systems), Studentlitteratur, Lund, Sweden 1997.
  3. Hermann Haken: Synergetics, An Introduction, Springer Verlag, 1983.
  4. Erland Lagerroth: Den nya utvecklingsläran, Oording ger det döda liv (The New Evolution Theory, Chaos Gives Life), Dagens Nyheter, Stockholm, Sweden, 2:nd August 1985.
  5. Erich Jantsch: The Self-Organizing Universe, Scientific and Human Implication of the Emerging Paradigm of Evolution, Pergamon Press 1985.
  6. David C Korten: When Corporations Rule the World, Kumarian Press, Inc. And Berret-Koehler Publishers, Inc. 1995.
  7. David C Korten: The Post Corporate World, Life After Capitalism, Kumarian Press, Inc. And Berret-Koehler Publishers, Inc. 1999.
  8. Adam Smith: An Inquiry into the Nature and Causes of the Wealth of Nations, [1776], New York: Modern Library, 1937.

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