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) 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) 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) 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) 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) 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)=-ε yi- 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) 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) 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: Iε is 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) 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) 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

C B

Lorenz dominance

Multidimensionality 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 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

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)