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Measurement 12. Inequality. Concepts (1). utilitarianism: individual satisfaction perfectionnism: collective results from the standpoint of a planner liberalism: individual freedom 3 difficulties: preference attrition paternalistic arbitrariness Capabilities. Concepts (2).

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measurement 12

Measurement 12


concepts 1
Concepts (1)
  • utilitarianism: individual satisfaction
  • perfectionnism: collective results from the standpoint of a planner
  • liberalism: individual freedom

3 difficulties:

  • preference attrition
  • paternalistic arbitrariness
  • Capabilities
concepts 2
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: Rousseau’s social contract

- Freedom is the ultimate criterion: not utility, not even responsibility (because of varying capacities to exert)

  • Dworkin’s cut: what individuals should not be deemed accountable of?
  • Resources (Dworkin) or opportunities (Sen, Roemer) equalization, or minimal functionings (Fleurbaey)
concepts 3
Concepts (3)

Critiques of the liberal standpoint:

Walzer’s: 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

Fleurbaey’s: “non-deserving poor”? + implementation problems

 More modest equality of minimal functionings

axiomatics 1
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 = Σi=1,..,N u(yi)/N or W=Φ[Σi=1,..,N u(yi)/N]

With Φ increasing (Φ’>0) and u increasing (u’>0)

u: individual utility or social planner weighting scheme

axiomatics 2
Axiomatics (2)

1st order stochastic dominance:

Cdf functions: if FA >s.d. FB then WA > WB for any W with anonymity and Pareto

2nd order stochastic dominance:

Additional assumption: u’’<0

(i.e. Pigou-Dalton like in inequality)

 Generalized Lorenz (integrals of F) dominance

3rd order: u’’’>0 (decreasing transfers) etc.

axiomatics 3
Axiomatics (3)

Example of Atkinson-Kolm welfare function:

Wε = [(1/N).Σiyi1- ε/(1-ε)] 1/(1- ε)

Φ(z) = z1/(1- ε) u(y) = yi1- ε/(1-ε)

u’(y)= yi- ε>0 u’’(y)=-εyi- ε -1 <0

ε=0: average income (utilitarist)

ε=+∞: minimum income (Rawlsian)

ε= society aversion for inequality, or individual risk aversion under the veil of ignorance

axiomatics 4
Axiomatics (4)

Inequality index I(y1, …, yN):

A1: Anonymity

A2: Pigou-Dalton principle: transfers from rich to poor decrease inequality

A3: Relative: I(λy1, …, λyN) = I(y1, …, yN)

A3’: Absolute: I(y1+δ,…,yN+δ) = I(y1,…,yN)

A1+A2+A3: Lorenz dominance and usual indexes (Gini, coefficient of variation, Theil, Atkinson)

axiomatics 5
Axiomatics (5)

Wε = μ [1- Iε]

With Iε : Atkinson-Kolm inequality index:

Iε=1 - [(1/N).Σi(yi/μ)1- ε/(1-ε)] 1/(1- ε) for ε≠1

I1=1-exp[(1/N).Σilog(yi/μ)] for ε=1

with μ is mean income

 Wε measures the « equivalent-income » of an equal distribution: Iεis the share of total income I am ready to loose to reach an equal distribution with the same welfare as with ŷ and prevailing inequalities

theil indexes 1
Theil indexes (1)

A4: Additive decomposability

Define mutually inclusive groupings (social classes, etc.) When can I write?:

I = I[between group means]+I[within groups]

Only with “generalized entropy” of the form:

GE(β) = [1/β(β-1)] Σi yi/μ [(yi/μ)β-1 -1]

theil indexes 2
Theil indexes (2)

GE(β) = [1/β(β-1)] Σi yi/μ [(yi/μ)β-1 -1]

β0 : Theil-L (linked to Atkinson’s I1), also named mean logarithmic deviation

(weights = simple population weights)

β1 : Theil-T or simply Theil index

(weights= income weights)

β=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


2 variables x and y (ex. income & health)

Dominance on x and dominance on y

  • No problem, A > 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

measurement errors
Measurement errors

Inequality indexes most sensitive to low and high incomes

Especially low incomes for GE(β) with β<0

Especially high incomes for GE(β) with β>1

For Gini and 0≤β≤1

- Theil-T (β=1) more sensitive to high incomes

(see example in homework)

- Theil-L (β=0) more to low incomes but nos as much

Simulations suggest that Gini or Theil-L could be preferred on those grounds


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)