Measurement 12

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Measurement 12

• Inequality

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Concepts (1)

• utilitarianism individual satisfaction
• perfectionnism collective results from the
  standpoint of a planner
• liberalism individual freedom
• 3 difficulties
• preference attrition
• paternalistic arbitrariness
• Capabilities

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Concepts (2)

• The liberal critique (Rawls, Sen)
• - Distributive justice is for the society what
  truth is for science (Rawls, p.1)
• - Justice ? Equality of what? (Sen, p.1)
• - Procedural method Rousseaus social contract
• - Freedom is the ultimate criterion not
  utility, not even responsibility (because of
  varying capacities to exert)
• Dworkins cut what individuals should not be
  deemed accountable of?
• Resources (Dworkin) or opportunities (Sen,
  Roemer) equalization, or minimal functionings
  (Fleurbaey)

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Concepts (3)

• Critiques of the liberal standpoint
• Walzers Equity as pluralism
• In really existing societies, spheres of
  justice apply their specific distributive
  principle to the distribution of a specific good
  (citizenship, knowledge, money, public charges
• ? Separation of powers (Montesquieu, Pascal)
• ? Correlation between spheres,
  multidimensionality
• Fleurbaeys non-deserving poor?
  implementation problems
• ? More modest equality of minimal functionings

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Axiomatics (1)

• In the utilitarist tradition individualistic
  social welfare function W W(y1, , yN), where y
  is income
• Anonymity Symmetry if I switch the positions
  of i and j, W does not change
• Pareto principle If i is better-off and others
  do not change, W increases
• ? W Si1,..,N u(yi)/N or WFSi1,..,N
  u(yi)/N
• With F increasing (Fgt0) and u increasing (ugt0)
• u individual utility or social planner weighting
  scheme

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Axiomatics (2)

• 1st order stochastic dominance
• Cdf functions if FA gts.d. FB then WA gt WB for
  any W with anonymity and Pareto
• 2nd order stochastic dominance
• Additional assumption ult0
• (i.e. Pigou-Dalton like in inequality)
• ? Generalized Lorenz (integrals of F) dominance
• 3rd order ugt0 (decreasing transfers) etc.

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Axiomatics (3)

• Example of Atkinson-Kolm welfare function
• We (1/N).Siyi1- e/(1-e) 1/(1- e)
• F(z) z1/(1- e) u(y) yi1- e/(1-e)
• u(y) yi- e gt0 u(y)-e yi- e -1 lt0
• e0 average income (utilitarist)
• e8 minimum income (Rawlsian)
• e society aversion for inequality, or individual
  risk aversion under the veil of ignorance

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Axiomatics (4)

• Inequality index I(y1, , yN)
• A1 Anonymity
• A2 Pigou-Dalton principle transfers from rich
  to poor decrease inequality
• A3 Relative I(?y1, , ?yN) I(y1, , yN)
• A3 Absolute I(y1d,,yNd) I(y1,,yN)
• A1A2A3 Lorenz dominance and usual indexes
  (Gini, coefficient of variation, Theil, Atkinson)

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Axiomatics (5)

• We µ 1- Ie
• With Ie Atkinson-Kolm inequality index
• Ie1 - (1/N).Si(yi/µ)1- e/(1-e) 1/(1- e) for
  e?1
• I11-exp(1/N).Silog(yi/µ) for e1
• with µ is mean income
• ? We measures the  equivalent-income  of an
  equal distribution Ie is the share of total
  income I am ready to loose to reach an equal
  distribution with the same welfare as with y and
  prevailing inequalities

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Theil indexes (1)

• A4 Additive decomposability
• Define mutually inclusive groupings (social
  classes, etc.) When can I write?
• I Ibetween group meansIwithin groups
• Only with generalized entropy of the form
• GE(ß) 1/ß(ß-1) Si yi/µ (yi/µ)ß-1 -1

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Theil indexes (2)

• GE(ß) 1/ß(ß-1) Si yi/µ (yi/µ)ß-1 -1
• ß ?0 Theil-L (linked to Atkinsons I1), also
  named mean logarithmic deviation
• (weights simple population weights)
• ß ?1 Theil-T or simply Theil index
• (weights income weights)
• ß 2 gives (half the square of) the coefficient
  of variation (CV), ie (1/2) Var(y)/µ²
• Theil more sensitive to transfers at bottom
• Gini more sensitive to transfers at median

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B
C
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Lorenz dominance
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Multidimensionality

• 2 variables x and y (ex. income health)
• Dominance on x and dominance on y
• No problem, A gt B on both dimensions and on the
  whole
• Otherwise, same problems of aggregation over
  variables as over individuals how much of x is
  equivalent to y equivalent incomes

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Measurement errors

• Inequality indexes most sensitive to low and high
  incomes
• Especially low incomes for GE(ß) with ßlt0
• Especially high incomes for GE(ß) with ßgt1
• For Gini and 0ß1
• - Theil-T (ß1) more sensitive to high incomes
• (see example in homework)
• - Theil-L (ß0) more to low incomes but nos as
  much
• Simulations suggest that Gini or Theil-L could be
  preferred on those grounds

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Sampling

• Variance of inequality indexes
• For some, asymptotic formulas, but slow
  convergence
• In any case, bootsrapping seems preferable
  resampling data with replacement (provided that
  observations are independent)

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