1. Measurement 12
Inequality

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

3. 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)

4. 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

5. 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

6. 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.

7. 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

8. 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)

9. 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

10. 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]

11. 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

12. C B

13. Lorenz dominance

14. 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

15. 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

16. 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)