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Going for Growth

Understanding Growth (Positive Feedback)

Through a Series of Models


by Nick Louca, September, 2009

(c) 2009 by Ed Seykota




Clip: http://gothamschools.org/wp-content/uploads/2008/10/kids-growth.jpg

We Experience Positive Feedback Early in Our Lives.


Other names for Positive Feedback are "reinforcing",

"amplifying", "self multiplying", "compounding",

"runaway", "snowballing" and "explosive growth."1



Positive Feedback motivates exponential growth - such as in epidemics, nuclear explosions, early mitotic (cell division) growth and interest bearing bank accounts.


In this study, I develop a series of models and run various tests to reinforce my understanding of Positive Feedback Growth and the structures and behaviors that associate with it.


I find the process of researching positive feedback and growth stimulating and fun.


1.     Defining  Positive Feedback


Positive Feedback is a mode of behavior in which system elements promote each others' growth.  Typically, (1) the Rate flows into the Level and increases it and (2)the Rate increases as the Level increases.  The  Positive Feedback loop is a self-reinforcing loop (see system structure below).




Structure of  Positive Feedback System


Other Definitions:


A policy is a mechanism that controls a Rate as a function of information from the Level(s).


A linear policy is a mechanism that sets the Rate to be linearly proportional to the Level.  The constant of proportionality is the Gain of the system or the reciprocal of the Time Constant.  For example, if the Level is in gallons, and the "Time Constant" is 5 minutes, the Inflow Rate = Level / 5 minutes.  For a Level = 20 gallons, the Inflow Rate = 20/5 = 4 gallons per minute.  Some modelers call the Gain the Growth Factor. In this example the Gain = 1/5 per minute.


A first order system is one with one Level.  The order of a system indicates the number of Levels in the system.


Another example:  You earn interest in your bank account.  This interest (the Rate) increases your balance (the Level).  The interest (proportional to the balance) also increases; this increases the balance even faster.  See Bank Account Model below for more about how this model works.


Note: Technically, we have a "Base Level" in a positive loop that is similar to the Target in a negative loop.  In a negative loop the Level tracks the Target and decreases the delta (gap); in a positive loop the Level moves away from the Base Level and increases the delta.  Typically, in positive loops the Base Level is zero so we omit it from our models.


I use the equations below in the Positive Feedback models I develop during my study.





Initial Value of Level (units) = a number (units)

Level (units)

= Level (units) + Inflow Rate (units/time) * dt (time)

Inflow Rate (units/time)

= Level (units) * Gain (units/units/time) or

= Level (units) / Time Constant (time)

Gain (units/units/time)

= a number (units/units/time)

= 1 / Time Constant (1 / time)

Time Constant (time)

= a number (time)

= 1 / Gain (time)

Solution Interval, dt (time) = a number (time)



1.1      Positive Feedback Behavior


Positive Feedback systems are self-reinforcing systems.  They grow.  First order linear systems (the simplest kind) exhibit exponential growth.


The behavior of the Positive Feedback loop depends on the initial value of the Level which may be either positive, negative or zero.  The positive feedback acts to increase the absolute value of the Level.  In cases where the initial value of the Level is zero then the system remains at zero.




Positive Feedback - From Different Initial Conditions




Positive Feedback From Different Initial Conditions



See Excel file.

2.     Developing Positive Feedback Models


I develop a series of models and carry out a number of experiments to get deeper insight into the  Positive Feedback loop.


2.1   The Wheat Chessboard Model


I brainstorm ideas with Ed of various  Positive Feedback models I may develop.  Ed suggests that I develop a model of the Wheat Chessboard fable.  I find this model intriguing as I imagine that this model can demonstrate the power of exponential growth.


In the Wheat Chessboard fable, a man invents the game of chess and gives it to the King.  The King likes the game.  In return, he offers the man any gift he desires.  The man is a wise mathematician.  He requests to receive one grain of wheat on the first square of a chessboard, two grains on the second square, four grains on the third square and so forth.  The amount of wheat grains double for each subsequent square on the chessboard.  The King agrees to this plan - before he thinks it through.


When the King finally finds out how much wheat his agreement states he must deliver (about half a trillion tons, see below), he summons the wise man for a re-negotiation.  After all, he is the King.  The wise man agrees to the re-negotiation.  After all, he is a wise man.


My goal is to develop a model of the Chessboard fable.  I notice that the chessboard has 64 squares that are either empty or have wheat grains on them.  I cannot use any of these squares as my Level.  I decide to use the entire chessboard as my Level and the value of the Level becomes the total amount of grains on the chessboard. 




Wheat Chessboard Model



I also notice that the fable proceeds "one square at a time" rather than one minute at a time or one hour at a time.  The "time" axis is more precisely an "event" axis that I measure in squares, units of area, rather than units of time.  My solution interval in the Wheat Chessboard model is delta square (ds) instead of delta time (dt).  Note: The solution interval is the "time" or in Wheat Chessboard Model the "event" interval between each calculation.




Wheat Grains on Chessboard

Clips: http://en.wikipedia.org/wiki/Nonlinear_system



Next I set up the equations.


First, the Rate of putting wheat grains on the chessboard flows into the Chessboard, the Level.  I can model this as a simple integration, Rate into the Level.


Second, the Rate depends on the Level of wheat grains on the board.  The fable states that each square thereafter receives twice the amount of grains as appear on the previous square.


Translating this math to the chessboard as a whole, we have the Initial Value = 1 (grain).  The Rate for the next square = the total amount of grains on the chessboard plus 1 grain.  In terms of an equation for the Rate, we have: Inflow Rate (grains/square) = (Chessboard (grains) + 1 (grain)) * Growth Factor (1/square).


I note that for very large Levels of grain the extra grain becomes relatively small and the equation approaches a simple linear model where Inflow Rate (grains/square) = Level (grains) * Growth Factor (1/square).





Chessboard (grains) = Chessboard (grains) + Inflow Rate (grains/square) * dt    (square)

Initial Value of Chessboard (grains)

= 1 (grain)

Inflow Rate (grains/square)

= (Chessboard (grains) + 1 (grain) * Growth Factor (1/square)

Growth Factor (grains/square)/grains)

= 1 (1/square)

Solution Interval, ds (squares)

= 1 (square)






Wheat Chessboard Model Behavior

Note: the simulation of the Wheat Chessboard

model starts at square 1.




Wheat Chessboard Model Behavior

Notice that the shape of the curve from 0 to 10 squares is

similar to the shape of the curve from 0 to 20 squares.


2.1.1  Confusion


On my first attempt to set up my equations I multiply the Level by the Growth Factor to get the Inflow Rate.  In the multiplicative formulation for the Inflow Rate, the amount of grains  in the Level doubles with every recalculation (square).  The Level however is always off by one grain. 


I go through some confusion.  I discuss this with Ed.  Ed tells me that the Wheat Chessboard Model is not exactly linear.  Ed brings a Chessboard and asks me to describe the Policy.  The Policy is everything that goes on in the Flow Rate i.e. the  rate equation.


Ed explains that in the Inflow Rate = (Chessboard (grains) + 1 (grain)) * Growth Factor (1/square).  Ed also points out that the Wheat Chessboard system is not exactly linear.  In a linear system the Inflow Rate = Level (units) * Growth Factor (units/units/time).  In this formulation the Inflow Rate is a  fixed multiple of the Level. 


This is not the case in the Wheat Chessboard Model. The Growth Factor is not constant at the beginning as per the below chart. 




Growth Factor

Notice that the Wheat Chessboard Model is not linear at the beginning.




Answer:  To cover the chessboard requires equivalent to 264 grains = 18,466,744,073,709,600,000 grains of wheat.


This is approximately 730 years worth of global wheat production at the 2008/09 global production estimate.



The Math


1,000,000 grains

= 1 bushel

18,466,744,073,709,600,000 grains

= 18,466,744,073,709.60 bushels

Convert bushels to tons = bushels * 0.027216

= 502,046,586,710.08 tons

2008/2009  Global Production Estimate

= 687,000,000 tons

Years to produce wheat

= 730.78 years

Length of 1 grain = 0.005m = 92,233,720,368,547,800.00 meters
Length in miles (grains laid out one by one) = 57,311,359,059,124.90 miles

             Source: International Grains Council, Kansas Wheat Commission, Baking Industry Research Trust



The Wheat Chessboard model is an interesting experiment to examine and understand the proportions of exponential growth.  It does however seem unrealistic.


Systems such as the Wheat Chessboard model do not exist in isolation in the real world.  Exponential growth processes ultimately run into limits to growth.  I examine these limits and how they affect exponential growth later in this series. See the Excel model.




Clip: http://www.krankyscartoons.com/images/Growth_Versus_Sustainability.jpg

Anyone who thinks that steady growth can continue

indefinitely is either a madman, or an economist.

-- Kenneth Boulding


2.2    The Bank Account Model


The bank account model demonstrates exponential compounding.  I develop the bank account model to gain deeper insight about how this works.


I develop the Bank Account Model as per the below structure and set up my equations accordingly in Excel.




Bank Account Model Structure





Bank Account (US$) = Bank Account (US$) + Interest Payment Rate (US$/year) * dt (year)

Initial Balance of Bank Account (US$)

= 1000 (US$)

Interest Percentage (US$/US$/year)

= 10% (US$/US$/year)

Interest Payment Rate (US$/year)

=  Level (US$) * Interest Percentage Rate (US$/year)/US$)

Solution Interval, dt (year)

= 0.25 (year)



2.2.1  Examining Periodic and Continuous Compounding


As a first step I test the model with quarterly and continuous compounding.  For continuous compounding I assume that the bank pays interest into the account on a  daily basis.




Bank Account Model Behavior

Bank Account Model receives interest

(compounds) on a quarterly basis.



I observe that the bank account doubles in approximately 7 years.  Following a discussion with Ed on doubling time he suggests I explore how doubling time works and its relationship to Euler's number e = 2.71828...


I recall that in order to find the doubling time of a system we divide 0.7 by the growth rate or in the case of the Bank Account Model the interest rate.  In the Bank Account Model this is 10 % per year which gives a doubling time of = 0.7 / 0.1 (10 %per year) ~ 7 years.


Following some research I find that 0.7 represents the natural logarithm of 2.



Note: loge2 = 0.69314 = e0.69314 = 2 or ln (2) = 0.69314.


The doubling time is =  loge2 / Growth Rate. This equation is also known as the Law of 70.



I also find out Euler's number, "e" also represents the actual growth for a system or process that grows at 100% continuously compounding rate.  That is limn=(1 + 1/n)n = e = 2.71828.



Note:  We can also derive the final value of the Bank account by using the following formula: Pt = P0 * (1 + r/n



)tn where:


Pt = Principal at any time t;

P0 = Principal or Initial Value;

r = Interest Percentage;

n = times per year bank credits the account with interest; and

t = number of years.


For example:

P0 = $ 1000

r = 10%

n = 4 times per year

t = 10 years


P10 = $ 1000 * (1+0.1/4)(10*4) = US$ 2685.06

The account grows to $ 2,685.00 in ten years.


Note: If we were to receive an annual interest payment and annual compounding, we would need an interest rate of 10.381% to achieve the same ending balance.


(1 + r/n)n - 1 = (1 + 0.1/4)4 -1 * 100 = 10.381% per year.


If we were to receive interest instantaneously (account compounds hourly) we would need an interest rate of 10.517 % to achieve the same ending balance.


(1 + 0.1/8760)8760 - 1 * 100 = 10.517%


Note: 365 days * 24 hours = 8760 hours per year


We can use the continuously compounding interest rate to compute the account doubling time, per:


log102 / log10(1 + r) = 0.3010 / 0.0434 = 6.913 years.


Note: In the case of continuous compounding growth we may use the following formula to derive the final  value:


Pt = P0 * ert where:


Pt = Principal at any time t;

P0 = Principal or Initial Value;

r = Interest Percentage Rate; and

t = number of years.


P10 = US$ 1000 * 2.71828(0.1*10) = $ 2718.28

The account grows to $ 2718.28 in ten years when it compounds continuously.



I notice that despite hourly compounding the final value in the account after 10 years does not differ greatly from the value when it compounds quarterly.  When I discuss this with Ed, he recommends I develop a Compounding Frequency Table to explore how this works.



Compounding Frequency Table


Notice that at a continuously compounding rate of

69.314% the actual growth rate is 100%, e0.69314 = 2.




Compounding Frequency Chart

Note: when modeling continuous compounding the solution interval

becomes infinitesimally small dt = 1/ Compounding Frequency. 

For daily compounding dt = 1/365 days = 0.0027 years.



Notice that as the compounding frequency increases, actual growth converges around e.  In the case of 100% nominal interest rate over one period e represents the maximum compound rate of growth in a continuous compounding scenario.  In this case e =  limn=(1 + 1/n)n = 2.71828.


See Excel file.



2.2.2  Positive and Negative Interest Rates


The next experiment I carry out on the Bank Account Model is with different interest rates.


I test the Bank Account Model with various interest rates both positive and negative.  My experiment confirms that keeping money in the bank and earning interest is a surefire way to multiply your money without taking much risk.


This assumption comes with a caveat.  You have to sit out the first couple of hundred years and make sure your bank will be around for that long.  You might also consider the effect of negative interest rates and inflation.




Bank Account Model Structure




Bank Account Behavior at positive and negative interest

rates with quarterly interest payments.



I notice that when interest rates are negative the positive feedback loop becomes a negative feedback loop and I have asymptotic decay.  More specifically in the Bank Account Model I notice the erosion of money. See Excel file.



Matt cartoon


Clip: jeffreyhill.typepad.com/.../matt-cartoon-.html

I am less interested in the return on my money than the return of it.

-- Will Rogers



2.2.3  The Effects of Tax


Following discussion on the Bank Account Model with Ed we touch upon the subject of tax and how tax affects returns on investment.


I decide to develop a model to find out how tax affects investment.  I develop a simple model of a company and make certain assumptions as follows:

  1. the company invests $ 100,000 initial Capital (tools);

  2. it earns a 20 % Profit (US$/tools/year) on its initial Capital per year;

  3. the Company pays Tax (percent) at the end of each year.  I run the model on different tax percentage 30%, 15%, and 0%; and

  4. it re-invests 50%, Re-investment Fraction (percent) of Net Profit (US$/year) to purchase new tools (US$/tools) at a price $ 1.00.




Company Model Structure

Notice: Tax does not flow out of Capital or the Re-investment Rate





Capital (tools) = Capital (tools) + Reinvestment Rate (tools/year) * dt (years)

Initial Capital (tools)

= 100,000 (tools)

Profit Percentage (US$/tools/year)

= 20% (US$/tools/year)

Profit (US$/year) = Capital (tools) * Profit Percentage (US$/tools/year)
Tax Percentage (percent) = 30% , 15 % , 0 % (percent)

Tax (US$/year)

= Profit (US$/year) * Tax Percentage (percent)

Net Profit (US$/year) = Profit (US$/year) - Tax (US$/year)
Reinvestment Fraction (percent) = 50% (percent)
Price of Tools (US$/tools) = $ 1 (US$/tools)

Reinvestment Rate (tools/year)

=  Net Profit (tools) * Reinvestment Fraction (tools/year) / Price of tools (US$/tools)

Solution Interval, dt (years)

= 1 (years)




Company Model Behavior

The effects of tax over a 20 year period.



I notice that the effect of  tax makes a big difference over a 20 year period.  I am aware that this is very simple model. Even through this simple model I can imagine that there is much to study on taxes and how they affect corporate re-investment.


I intend to study the effects of taxes and develop economic models later in this series.  See Excel model.


3.     Conclusion


Positive Feedback loops are common in many real world systems around us.  The behavior we observe in positive feedback loops is most commonly positive exponential growth.


The main characteristics of Positive Feedback systems are:

  1. Positive Feedback loops act to increase the Delta (Gap);

  2. Linear First Order systems have a constant doubling time;

  3. The initial value of the Level might be positive, zero or negative. A positive feedback system tends to increase the absolute value of the Level.

Positive feedback loops associate with growth.  In the real world positive feedback loops do not exist in isolation and eventually run into limits to growth.  I intend to explore the type of behavior we might observe in systems where  a Positive Feedback loop interacts with a Negative Feedback loop later in this series.



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Ref: 1 Thinking in Systems, Ch 1, The Basics, Pg 30,  Donella Meadows


Clip: http://gothamschools.org/wp-content/uploads/2008/10/kids-growth.jpg