A Simple Model of Accumulated Earnings Inequality in a Human Society

1
A Simple Model of Accumulated Earnings Inequality
in a Human Society

• Tim F. Liao
• Sociology, University of Illinois

15 May 2007, 7th Understanding Complex Systems
Conference
2
Abstract

• In this analysis, we study the complex
  phenomenon of earnings inequality in society. To
  simplify the matter, we consider a single-sex (or
  really unisex) population that has a certain life
  expectancy at the arbitrary working age of 15.
  Each individual is able to search the landscape
  to gather and cumulate earnings. We allow their
  categorization into three classes, the lower, the
  middle, and the upper. We also consider the
  Mathew effect in earnings accumulation (or the
  rich get richer and the poor get poorer). The
  model monitors the Gini index and the three-class
  stratification over time to understand potential
  accumulated earnings polarization and
  stratification.

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Introduction

• In the social sciences in general, and sociology
  in particular, developments are relatively slow
  in using the complex systems approach to study
  social complex systems
• This is so, in spite of some of the early
  path-breaking publications that have been
  accessible by sociologists (e.g., Thomas
  Schelling, Dynamics Model of Segregation,
  Journal of Mathematical Sociology, 1971)
• Motivation of the current research, long-term
  interest and this conference
• Inequality is a fundamental social science
  question
• Yet its understanding cannot be complete without
  being considered as a complex system

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Schellings Segregation Model Initial Random
Setup
50
40
30
20
10
0
0
10
20
30
40
50
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Schellings Segregation Model Figure after 2
Moves per Agent
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Research Questions

• Would equality rules generate inequality?
• Under a simple set of equality rules, would
  earnings inequality emerge?
• Under the same set of rules, would earnings class
  emerge?
• What is the impact of resource constraints on
  inequality?
• To what degree would the Matthew effect compound
  inequality?

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The Model

• The agent based model I developed is adapted from
  Uri Wilenskys (1998) wealth distribution model,
  which is in turn adapted from Joshua Epstein and
  Robert Axtells (1996) Sugarscape, where
• Agent forage in a renewable resource landscape
• Agent are diverse in vision and metabolism
• Sugar grows back
• Some agents die
• Carrying capacity may be reached

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The Concepts

• Population working age population aged 15 and
  older with a chosen value of life expectancy
• Life expectancy at 15 a integer value determined
  by a random Poisson process
• Life cycle an agent dies if life expectancy is
  reached or if all resources owned are consumed
  then a new agent is born
• Resources randomly set in the landscape, with a
  chosen percent of best land
• Consumption individuals use resources gathered
  at a certain rate
• Accumulated earnings there is no inheritance
  though individuals can save unused resources for
  later consumption

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Concepts

• Best land The land that is given the highest
  amount of resources possible
• Matthew effect eminent scientists get more
  credit for similar work than relatively unknown
  researchers (Merton) here, upper-class
  individuals double the result of their search and
  gather labor, or rate of returns
• Class
• Lower
• Middle
• upper

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Measuring Inequality Gini Index Lorenz Curve
Y cumulated proportion of wealth X cumulated
proportion of population G 0 everybody has
same earnings G 1 All is earned by one
individual
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Ginis Definition

• Let ?F(yi) indicate the distribution for yi
• Gini is defined as
• It can also be written as
• Of all equivalent computational formulae, the
  most revealing is (Dagum)

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The Model Parameters

• First set of Parameters
• pop-size700
• max-vision2
• metabolism-max15
• Matthew-effectno/yes
• life-expectancy-1550
• percent-best-landvarious
• grain-growth-interval1
• num-grain-grown4
• Run time10,000

Second set of Parameters pop-size500
max-vision2 metabolism-max15 Matthew-effect
no/yes life-expectancy-1510 percent-best-land
5 grain-growth-interval3 num-grain-grown3
Run time1,000 1,000
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Limited Resources with No Matthew Effect Class
Plots

• 6 best land
• 20 best land

• 34 best land
• 50 best land

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Limited Resources with Matthew Effect Class Plots

• 6 best land
• 20 best land

• 34 best land
• 50 best land

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Limited Resources with No Matthew Effect
Wealth Range

• 6 best land
• 20 best land

• 34 best land
• 50 best land

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Limited Resources with Matthew Effect Wealth
Range

• 6 best land
• 20 best land

• 34 best land
• 50 best land

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Limited Resources with No Matthew Effect
Wealth Distribution

• 6 best land
• 20 best land

• 34 best land
• 50 best land

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Limited Resources with Matthew Effect Wealth
Distribution

• 6 best land
• 20 best land

• 34 best land
• 50 best land

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Limited Resources with No Matthew Effect Gini
Index

• 6 best land Gini?4.5
• 20 best land Gini?5.3

• 34 best land Gini?5.5
• 50 best land Gini?5.5

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Limited Resources with Matthew Effect Gini Index

• 6 best land Gini?5.5
• 20 best land Gini?6.5

• 34 best land Gini?6.5
• 50 best land Gini?6.5

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Most Limited Resources and an Evaluation of the
Matthew Effect
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Most Limited Resources and an Evaluation of the
Matthew Effect

• Gini 3.9?4.7

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Conclusions

• This agent based model assisted in understanding
  inequality in complex systems that cannot be
  achieved with conventional approaches
• Equal initial conditions lead to inequality
• Do equal initial conditions lead to
  classification? May be
• Poorest conditions would lead to inequality, but
  only mildly richer conditions will do so more
• The Matthew effect, when conceived as doubling
  the rate of returns, in generating inequality, as
  measured by the Gini index, is about 10