Multiplicative Growth and Inequality

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Rupertpassphotoneu Rupert Nagler (Author)



Tagged by Rupert Nagler about 4 years ago


Tagged by Rupert Nagler about 4 years ago

inequality & taxation 

Tagged by Rupert Nagler about 4 years ago

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Multiplicative Growth and Inequality.nlogo

Author: Rupert Nagler, Jan 2020,


Simulation of multiplicative vs additive growth and the impact on equality of wealth.

Our turtles assume they have all the same chance to get wealthy doing business. They are represented in their blue 2d-world as yellow circles. Their vertical position reflects their actual wealth while their horizontal position reflects their unique "who" number.

You will experience the difference between additive growth (as generated by labour income vs. consumption) and multiplicative growth (as generated by investments, interests, shares). Multiplicative growth will automaticly lead to an uneven distribution of wealth, while a wrong ergodic hypothesis will make you think - like most traditional economists - that everybody has equal chances in multiplicative economic growth.

You can explore the intrinsic effects why "the rich get richer" and the benefits of cooperation induced by a form of wealth tax. Lorenz Curve, Gini Coefficient and a histogram show the current distribution of the current wealth of each turtle.


All turtles play by the same rules; nobody cheats or has more influence or better connections. In each round a percentage "leverage" of the current wealth of each turtle is multiplied by a normally distributed random variable with mean "mult-mean" and standard deviation "mult-sdev". Added to the wealth is another normally distributed random variable with mean "add-mean" and standard deviation "addd-sdev".

After the wealth of all turtles has been adopted, some redistribution in the form of a wealth tax may be applied: If "tax-factor" is > 0 and current wealth is > "tax-limit" a wealth tax (wealth * tax-factor) is subtracted. Then the collected wealth tax is redistributed evenly to all turtles or to the poor turtles below tax-limit, depending on the switch "redist-all?". So you can simulate the effects of cooperation between players through risk-sharing.


  • Use the sliders to control the number of turtles "num-turtles" and the initial wealth "init-wealth".
  • If you switch "random-init-wealth?" to "off" each turtle starts with equal "init-wealth" wealth; if you switch "random-init-wealth?" to "on" each turtle starts with a random wealth between 1 and "init-wealth".
  • Set the fraction of current wealth to multiply in each round "leverage" (default: 1.0).
  • Set the multiplicative parameters "mult-mean", "mult-sdev" (defaults: 1.05, 0.3) for the generation of the random normally distributed variable, by which the fraction of current wealth will be multiplied.
  • Set the additive parameters "add-mean", "add-sdev" (defaults: 0.0, 0.0) for the generation of the random normally distributed variable, which will be added to current wealth.
  • Optional set "tax-factor", "tax-limit", and "redist-all?"
  • If you want bancrupt turtles to die, set "turtles-die?" to on.
  • To setup the simulation, press "setup".
  • To play one round press "go-1", to play as long as you wish, press "go".


  • You see all turtles sitting on the blue world area. Each turtle will go up or down vertically dependent of its current wealth after each tick.
  • In the wealth-plot you see min, max, mean and median of the turtles wealth on a log10 scale.
  • In the wealth-distribution histogramm you see the number of turtles in different classes of wealth.
  • In monitor "richest 1% own wealth%" you see the actual % of total wealth owned by the richest 1% of turtles
  • In the Lorenz Plot you see the actual shape of the Lorenz Curve.
  • In the Gini Plot you see the value of the Gini Coefficient over time.


  • Try different values for multiplicative growth "mult-mean", "mult-sdev" and additive growth "add-mean", "add-sdev",
  • Compare the wealth-distribution for no multiplicative growth (set "mean-mult" to 1.0 and "sdev-mult" to 0.0) to other values of multiplicative growth (eg. 1.01, 0.2)
  • Compare the wealth-distribution for no additive growth (set both "heads-add", "tails-add" to 0.0) to other values of additive growth (eg. 0.5, 0.2)
  • Try different "tax-factor"s and "tax-limit"s, switch "redist-all?" on/off.
  • What changes can you see in the histogram, Gini Plot and Lorenz Curve?


  • better visualization ideas?
  • turtles get children and die of age
  • implement inheritance tax


  • plotting on a log scale,
  • using turtle world to show turtle ranking by position,
  • histogram with varying upper and lower bounds,



Credit: computation of Lorenz Curve and Gini index copied from: NetLogo Wealth Distribution model. Wilensky, U. (1998). Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

in-depth readings:

Wikipedia: Distribution of wealth, retrieved 12/2019

Wikipedia: Lorenz Curve, retrieved 12/2019

Wikipedia: Gini Coefficient, retrieved 12/2019

Wikipedia: Ergodic process, retrieved 12/2019

Ergodicity Economics, Ole Peters and Alexander Adamou, 2018

Entrepreneurs, Chance, and the Deterministic Concentration of Wealth, Joseph E. Fargione u.a., 2011

An evolutionary advantage of cooperation, Ole Peters and Alexander Adamou, 2018

Capital and Ideology, Thomas Piketty, 2019

Farmers Fable: Simulation benefits of cooperation, retrieved 12/2019

Gier, Marc Elsberg, novel, blanvalet 2019


Copyright 2020 Rupert Nagler. All rights reserved. Permission to use, modify or redistribute this model is hereby granted, provided that both of the following requirements are followed:

  • this copyright notice is included.
  • this model will not be redistributed for profit without permission from Rupert Nagler. Contact the author for appropriate licenses for redistribution for profit.

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; Multiplicative Growth and Inequality, Rupert Nagler Jan 2020

globals [
  gini-index-reserve ; actual Gini %
  lorenz-points ; list of Lorenz points
  rich1%own ; % of total wealth owned by richest 1%

turtles-own [
  wealth ; actual wealth of turtle
  tax ; actual amount of wealth tax turtle has payed

to setup
  ask patches [set pcolor 104]

to setup-turtles
  create-turtles num-turtles [
    set heading 0
    set color yellow
    set shape "circle"
    ifelse random-init-wealth? [; random distributuion
      set wealth random-float init-wealth
    ][; equal distribution
      set wealth init-wealth
    set tax 0
    ; place turtle on plain according id(own) on x-axsis and wealth on y-axsis
    setxy (who / num-turtles * 100) wealth

to go
  if not any? turtles [stop]

to playing ; each turtle throws coin
  ask turtles [ ; leverage is the fraction of wealth to bet
    set wealth winNorm (wealth * leverage) + wealth * (1 - leverage); compute new wealth on thrown coin

to-report winNorm [stake] ; function to compute new wealth according to coin throw with multiplicative and additive win
  let m random-normal mult-mean mult-sdev
  let a random-normal add-mean add-sdev
  report (stake * m) + a

to taxing
  if tax-factor > 0 [ ; do we have to compute taxes?
    let notax-turtles [self] of no-turtles ; empty unsorted list of turtles
    let sumtax 0
    ask turtles [ ; pay wealth tax
      ifelse wealth > tax-limit [ ; is there a tax to pay?
          set tax wealth * tax-factor
          set wealth wealth - tax ; turtle pays tax
          set sumtax sumtax + tax ; add to total tax collected
        ] [
          set tax 0
          set notax-turtles lput self notax-turtles ; add to list of notax-turtles
    let count-notax-turtles length notax-turtles ; number of notax-turtles
    ifelse redist-all? or (count-notax-turtles <= 0) [ ; do we have to redistribute to all turtles?
      let mtax (sumtax / count turtles) ; divide total tax between all turtles
      ask turtles [ ; redistribute tax to all turtles
        set wealth wealth + mtax ; redistribute
    ] [; divide total tax between all no-tax-turtles
      let mtax (sumtax / count-notax-turtles)
      ask turtle-set notax-turtles [ ; changes list into agentset
        set wealth wealth + mtax ; redistribute

to move-turtles ; according to new wealth
  ask turtles [
    if turtles-die? [ ; should bancrupt turtles die?
      if wealth < 1.0E-10 [die]
    set ycor (wealth + min [wealth] of turtles) / max [wealth] of turtles * 100

to update-lorenz-and-gini
  ; recompute value of gini-index-reserve and the points in lorenz-points for the Lorenz and Gini-Index plots
  let sorted-wealths sort [wealth] of turtles
  let total-wealth sum sorted-wealths
  let wealth-sum-so-far 0
  let index 0
  let c-turtles count turtles
  set gini-index-reserve 0
  set lorenz-points []
  ; now actually plot the Lorenz curve -- along the way, we also calculate the Gini index
  repeat c-turtles [
    set wealth-sum-so-far (wealth-sum-so-far + item index sorted-wealths)
    set lorenz-points lput ((wealth-sum-so-far / total-wealth) * 100) lorenz-points
    set index (index + 1)
    set gini-index-reserve gini-index-reserve + (index / c-turtles) - (wealth-sum-so-far / total-wealth)
  let clp length lorenz-points
  let rich1%ix clp - (clp  / 100.0) - 1
  set rich1%own 100.0 - item rich1%ix lorenz-points

There is only one version of this model, created about 4 years ago by Rupert Nagler.

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