Beyond the Concentration Index: An Atkinson Alternative for the Measurement of the Socioeconomic Ine

1
Beyond the Concentration Index An Atkinson
Alternative for the Measurement of the
Socioeconomic Inequality of Health
Guido Erreygers 2005
HEDG Seminar Series University of York 1st
March 2006
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Content and Aim of the paper

• The paper
• challenges the usefulness of the concentration
  index as an indicator of the socioeconomic
  inequality in health
• proposes an alternative to the health
  concentration index.
• Two alternatives
• An adaptation of the CI
• An Atkinson measure of socioeconomic inequality
  in health
• The Atkinson approach, 2 crucial steps
• From a comparison of health status levels to a
  comparison of the probabilities of
• attaining different health levels
• Definition of an equivalent income

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The Health Concentration Index

• Pure inequalities in health Gini coefficient
• Income-related health inequality Health
  Concentration Index

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Alchemic transmutation Van Doorslaer and Jones
(2003)
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Cardinality is not enough (1)

• Let t be a cardinal variable let l be a
  cardinal ratio-scale variable.
• Any cardinal variables t and tsatisfy
• However, only ratio-scale variables are such
  that

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Cardinality is not enough (2)

• How does the lack of ratio-scale properties
  affect the health concentration index?
• Assuming that health (h) is cardinal but
  not ratio-scale
• Unless a 0, a concentration index close to 1
  does not entail high income-related inequality in
  health
• The same applies to the Gini coefficient
  regarding pure health inequality.

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The Egg of Columbus

• If the problem is the health measure why not
  relate health ranks with income levels instead of
  relating health levels with income ranks?
• The index is scale independent
• Also,

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The Omelette

• Is the ranking of social states given by the new
  concentration index equivalent to that generated
  by the health concentration index?
• In general NO, but
• But isnt this half baked since is also
  based in the criticised measure of health?

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Brief reminder Atkinson (JET, 70)

• Social welfare as a simple sum of individual
  welfare levels
• Atkinson (JET,1970) individual welfare
  determined solely by income
• Atkinson measure
• 
  that income which, if given to
  everyone, would
• 
  generate exactly the same level of
  social welfare
• 
  as the one generated by the
  existing unequal
• 
  distribution of income
• where equivalent income
• Due to the concavity of the individual welfare
  function equivalent income cannot exceed the
  average income and therefore 0 lt A lt 1

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What Erreygers proposes

• Suppose that individual welfare is determined
  both by income and health status
• social welfare
  becomes
• Purpose of socioeconomic indicator of health
  inequality find out whether the distribution of
  health follows the same pattern as the
  distribution of income.
• In the Atkinson spirit compare the existing
  situation of income and health to another
  equivalent situation of income and health where
  income and health are distributed without
  correlation.
• We can compare probabilities of attaining health
  statuses

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Risk profiles useful definitions

• Individual health drawn from the set
• is the probability that person
  i has health status
• We would like to know whether the risk profiles
  
  differ systematically for different income groups
• Each subset of the population or reference group
  R has a specific risk profile
• 
  and
• The average risk profile
  is found by calculating the
  frequency with which health status occur in
  society.

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Expected welfare function equivalent income

• We consider the group-specific expected
  individual welfare of person i
• which is the individual welfare according
  to the risk profile of his reference group
  instead of his actual individual welfare
• We are looking for the equivalent income of
  person i
• the income level such that
• Income that would ensure that person i
  with his specific risk profile can expect to be
• as well off as he could be if his risk
  profile were that of society in general

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Properties of the expected welfare function

• 1. Decreasing marginal expected welfare of
  income
• Implication a richer person has a lower marginal
  expected welfare of income than a poorer one.
  
• 2. Social luck persistence
• constant risk profile preferences
  preference order of risk profile is
• independent of income
• non decreasing social luck the effect of
  improving your risk profile does
• not decrease with income
• 3. Constant risk impact the ratio of marginal
  expected welfare of income is constant for all
  income levels.

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An Atkinson measure

• The author proposes to construct and indicator
  which takes the individual income differences (Yi
  Xi) as the basic unit. This can be
• 
  
  with gt 0, lt 0, 0
• If risk profiles distributed at random
• 
  
• 
  
• If positive correlation between the distribution
  of good risk profile and the distribution of
  income
• 
  
• If negative correlation between the distribution
  of good risk profile and income
• 
  

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An alternative measure

• Let us weight income difference
  according to the rank of i in the
  distribution y. Such a measure could be
• It can be shown that this measure is a
  combination of , the Gini coefficient of
  the income distribution Y , and ,
  Concentration Index of the income distribution
  
• X using the ranks of the distribution Y,
  i.e.
• 
  
• 
  where
• Graphical terms as twice the surface between the
  Lorenz curve of the distribution of
• Y and the Concentration curve of
  distribution X, scaled by the ratio of the means.

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Just two thoughts

• An Atkinson alternative or just a different way
  of considering health?
• The new element just health in terms of
  probabilities?
• The Atkinson index anyway classical criticism to
  the social welfare approach.
• ..based on the familiar concept of social
  welfare theory.. and the familiar criticism to
  it?

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Discussion / Conclusions

• Erreygers criticism of the concentration index
  targets a well known, but unresolved, problem is
  his suggestion of a new concentration index the
  egg of Columbus also from an operational
  perspective?
• The suggested Atkinson measure performs well in
  terms of mathematical properties and has a clear
  link to welfare economics but doesnt it ignore
  well established criticisms of the social welfare
  approach such as the inadequacy of the individual
  utilitarian perspective (Rawls (1971), Sen
  (1989)) and its inconsistency with observational
  data (Cowell (1994))?
• How legitimate is it to criticize the health
  concentration index by wrongly assuming
  ratio-scale properties of health and to suggest a
  social welfare based measure that requires many
  well known unrealistic assumptions.
• The proposed Atkinson measure can only be
  computed once a specific utility function is
  assumed which assumptions would be reasonable in
  the two-dimensional case of health and income?