Mobility: Measurements and Applications

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Mobility Measurements and Applications

• Yougui Wang
• Department of Systems Science, School of
  Management, Beijing Normal University, Beijing
  100875, P. R. China

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Outline

• Background and Motivation
• Absolute Mobility
• Income Mobility of Chinese Households
• Relative Mobility
• Rank Clocks of Top Firms
• Speed-Like Mobility Index

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Background and Motivation

• In field of complexity, size distributions,
  especially power-law distribution and Zipf plots
  attract much attention.
• In economics, inequality which focuses on the
  income distribution has long been an important
  topic.
• The generation mechanism of various universal
  distributions is being explored.

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Background and Motivation

• Under any one stable distribution, there is a
  micro-structure that undergoes a dynamic process.
• Most of efforts has been devoted to the static
  distribution, but neglecting the micro dynamic
  characteristics.
• More facts need to be revealed for understanding
  the formation of distributions

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The Absolute Mobility
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An Evolving Multi-agents System
Time
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Probability Distribution
In economics, Lorenz curve and Gini Coefficient
can be derived from income distribution.
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Individual Evolution
Variation in attribute indicate it moves randomly
over time .
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Aggregate Variation I All Individuals
Evolutions
We can not figure out the aggregate variation in
attribute by simply putting all individuals
evolutions together.
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Aggregate Variation IIChanges in Distribution
over Time
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Aggregate Variation IIChanges in Distribution
over Time
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Aggregate Variation IIChanges in Distribution
over Time
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Individual Evolution
The individual attributes are still varying even
though the distribution keep unchanged.
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Aggregate Variation Absolute Mobility

• where, N is the number of agent, xk0 and
  xk1 are the attributes of agent i at time 0 and
  1 respectively.

• G. S. Fields and E. A. Ok, The meaning and
  measurement of income mobility, Journal of
  Economic Theory, vol. 71, pp. 349377. 1996.

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Aggregate Variation Absolute Mobility
Absolute mobility covers changes of all
individuals attributes and is somewhat
independent of scale.
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Application The Income Mobility of Chinese
Households

• N. Ding, Y. Wang, Household income mobility
  in China and its decomposition, China Economic
  Review, vol. 19, pp. 373-380, 2008.

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Source of Data
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Data Sampling Areas
http//www.cpc.unc.edu/projects/china
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Data Analysis
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Income Mobility
1985-1997 Belgium 0.335 West Germany
0.392 USA 0.523.
Absolute mobility in China comes mainly from the
contribution of growth.
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The Relative Mobility
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Relative Perspective of Mobility
The ranks of individuals can be derived from
attributes of them.
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Zipf Plot
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Evolution of Rank for An Individual
Variation in rank for a typical agent
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Aggregate Rank ShiftRelative Mobility

• where, N is the number of agent, rk0 and
  rk1 are the rank of agent i at time 0 and 1
  respectively.

• Shlomo Havlin, The distance between Zipf
  plots, Physica A, vol. 216, pp. 148-150,1995.

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Application Firm Size Distribution and Rank
Shift
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Firm Size Distribution in China

• J. Zhang, Q. Chen, Y. Wang, Zipf distribution
  in top Chinese firms and an economic
  explanation, Physica A, vol. 388, pp. 2020-2024.
  2009.

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Firm Size Distribution in USA
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Firm Size Distribution in World
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Rank Clock of Top 100 Cities in USA

• Michael Batty, Rank clocks, Nature, vol.
  444, pp. 592596, 2006.

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Rank Clocks of Firms
Seven firms ranked in top 500 of the world during
the thirteen years
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Rank Clocks of Firms
Many firms ranked in 500 firms in USA
Eight firms ranked in top 500 in USA
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Rank-shift Clocks of Firm Size
Rank Shift Clocks of Firms
Top 500 in the world
Top 500 in USA
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A New Approach to Measuring Mobility

• Ning Ding and Yougui Wang, A Speed-like Index
  for Measuring the Economic Mobility, manuscript.

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Random Exchange Models
Money is always transferred from one to another
in the trading process.
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Model with Uniform Saving Rate

• A. Chakraborti and B. K. Chakrabarti, Eur.Phys.
  J. B, 17, 167 (2000)

Saving
Saving Rate
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Rank Variations for Typical Individuals
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Relative Mobility Index

• where, N is the number of agent, rk0 and
  rk1 are the rank of agent k at time 0 and 1
  respectively.

• Measurement of mobility is based on the data
  of two time points. But so far no one casts
  doubts about whether this kind of indexes can
  show the whole picture of mobility.

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Mobility in Random Exchange Models
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The Cause of Such Defect
Displacement Index Only the starting moment and
the ending moment are taken into account. All
changes occurred within the interval are
neglected.
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Remedying this Defect
Summing up all shifts The mobility exhibits the
aggregate movements during the whole process. It
should conclude all shifts in rank occurred
within the sampling interval.
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Remedying this Defect
Summing up all shifts The mobility exhibits the
aggregate movements during the whole process. It
should conclude all shifts in rank occurred
within the sampling interval.
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Remedying this Defect
Summing up all shifts The mobility exhibits the
aggregate movements during the whole process. It
should conclude all shifts in rank occurred
within the sampling interval. .
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Speed-Like Measurement Index

• where, N is the number of agents, rk (t) is
  the rank of agent k at time t, T denotes the
  starting time point and T the ending one.

• The rank shift in any two consecutive time
  points should not be excluded when we measure the
  mobility. This turns the index from
  displacement-like one to speed-like one.

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Measurement Results
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Summary

• Besides static distributions, mobility can also
  reflect the nature of an evolving system
• Two kinds of measurements of mobility (absolute
  and relative mobility) are presented with
  empirical data
• A new measurement index of mobility is proposed
  and proved to be a better one.
• 4. We suggest that a good test for a theoretical
  model is based on both empirical distribution and
  mobility.

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Thank you for listening!Any comments and
questions are welcome!