The primal approach
The dual approach
Measurement of Inequality and Social Welfare
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
February 11, 2016
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Ranking income distribution and Lorenz curves: Partial and
complete orderings
(i) Partial orderings: Stochastic and inverse stochastic dominance,
Lorenz dominance
(ii) Complete orderings:
a. Social welfare criteria based on expected utility theory
b. Rank-dependent social welfare criteria
Important issue in both policy work, descriptive analysis and causal
inference:
Statistical oces and gov agencies compare distribution
functions and Lorenz curves across countries, subgroups and
time
Research compares distributions of earnings, income,
consumption and wealth to evaluate economic policies and
social welfare
1
2
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
The cumulative distribution function and its inverse
Let F be a member of the set F of cumulative distribution
functions with mean µF and left inverse dened by
F −1 (t) = inf {x : F (x) ≥ t}
Note that both discrete and continuous distribution functions are
allowed in F , and though the former is what we actually observe,
the latter often allows simpler derivation of theoretical results and
is a valid large sample approximation. Thus, in most cases below F
will be assumed to be a continuous distribution function, but the
assumption of a discrete distribution function will be used where
appropriate. To x ideas, we will refer to F as the income
distribution, although our framework can be applied to any type of
distribution functions.
In order to rank distribution function we introduce the ordering
relation
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Ranking distribution functions: Examples
Suppose we want to rank two distributions, F1 and F0
Assume that the ordering relation satises rst-degree
stochastic dominance, i.e.
F1 (x) ≤ F0 (x) for all x ∈ [0, ∞)⇔F1−1 (t) ≥ F0−1 (t)
for all t ∈ [0, 1].
Can be used as a ranking criterion when distribtion don't
cross. But how do we deal with intersecting distribution
functions (Figures 2 and 3)?
Conventional approach in empirical work: Using summary measures
like the mean, the median and the variance or weighted means.
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Second-degree stochastic and inverse stochastic dominance
Denition
A distribution function F1 is said to second-degree stochastic
dominate a distribution function F0 if and only if
Zy
F1 (x)dx ≤
0
Zy
F0 (x)dx for all y ∈ [0, ∞)
0
and the inequality holds strictly for some y ∈ (0, ∞).
A distribution function F1 is said to second-degree inverse
stochastic dominate a distribution function F0 if and only if
Zu
−1
F1 (t)dt ≥
0
Zu
F0−1 (t)dt for all u ∈ [0, 1]
0
and the inequality holds strictly for some u ∈ (0, 1).
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
As was demonstrated by Atkinson (1970), second-degree stochastic
dominance is equivalent to second-degree inverse stochastic
dominance, which is called generalized Lorenz dominance by
Shorrocks (1983).
Moreover, under the restriction of equal mean incomes second
degree inverse stochastic dominance is equivalent to the criterion of
non-intersecting Lorenz curves.
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Third-degree stochastic dominance
Since situations where second-degree dominance does not provide
unambiguous ranking of distribution functions may arise, it will be
useful to introduce weaker ranking criteria than second-degree
dominance. To this end it appears attractive to consider
third-degree stochastic and inverse stochastic dominance.
Denition
A distribution function F1 is said to third-degree stochastic
dominate a distribution function F0 if and only if
Zz Zy
0
0
Zz
F1 (x)dxdy ≤
Zz Zy
0
F0 (x)dxdy for all z ∈ [0, ∞) ⇔
0
(z − x) (F1 (x) − F0 (x)) dx ≤ 0 for all z ∈ [0, ∞)
0
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Third-degree inverse stochastic dominance
Denition
A distribution function F1 is said to third-degree inverse stochastic
dominate a distribution function F0 if and only if
Zv Zu
0
−1
F1 (t)dtdu ≥
0
0
Zv
Zv Zu
F0−1 (t)dtdu for all v ∈ [0, 1] ⇔
0
(v − t) F1−1 (t) − F0−1 (t) dt ≤ 0, for all v ∈ [0, 1]
0
and the inequality holds strictly for some v ∈ (0, 1).
Note that third-degree stochastic and inverse stochastic dominance
do not coincide.
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Transfer principles associated with second- and third-degree
dominance
Denition
(The Pigou-Dalton principle of transfers). Consider a discrete income
distribution F . A transfer δ > 0 from a person with income x + h (or F −1 (t))
to a person with income x (or F −1 (s)) is said to reduce inequality in F when
h > 0 (s < t) and to raise inequality in F when h < 0 (s > t).
(i) If µF1 = µF0 , the condition of second-degree inverse stochastic dominance is
identical to the Pigou-Dalton transfer principle.
Denition
(The principle of diminishing transfers, Kolm,1976). Consider a discrete income
distribution F . A transfer δ > 0 from a person with income x + h1 to a person
with income x is said to reduce inequality in F more than a transfer δ from a
person with income x + h1 + h2 to a person with income x + h2 .
(ii) If µF1 = µF0 , the condition of third-degree inverse stochastic dominance is
identical to the principle of diminishing transfers.
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Rank-preserving transfers
Denition
(The principle of positional transfer sensitivity, Mehran, 1976).
Consider a discrete income distribution F . A rank-preserving
transfer δ > 0 from a person with income F −1 (s + h) to a person
with income F −1 (s) is said to have a stronger equalizing eect on
F than a transfer δ > 0 from a person with income F −1 (t + h) to a
person with income F −1 (t) when s < t .
(iii) If µF1 = µF0 , the condition of third-degree inverse stochastic dominance is
identical to the principle of rst-degree downside positional transfer sensitivity.
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Lorenz dominance
Denition
A Lorenz curve L1 is said to rst-degree dominate a Lorenz curve
L0 if
L1 (u) ≥ L0 (u) for all u ∈ [0, 1]
and the inequality holds strictly for some u ∈ [0, 1].
First-degree Lorenz dominance is identical to the Pigou-Dalton
transfer principle. A social planner who prefers the dominating one
of non-intersecting Lorenz curves favors transfers of incomes which
reduce the dierences between the income shares of the donor and
the recipient, and is therefore said to be inequality averse.
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Second-degree Lorenz dominance
To deal with situations where Lorenz curves intersect a weaker
principle than rst-degree Lorenz dominance is called for. To this
end it is normal to employ second-degree upward Lorenz dominance
dened by
Denition
A Lorenz curve L1 is said to second-degree upward dominate a
Lorenz
curve RL0 if
Ru
u
0 L1 (t)dt ≥ 0 L0 (t)dt for all u ∈ [0, 1]
and the inequality holds strictly for some u ∈ [0, 1].
Second-degree upward Lorenz dominance is identical to the principle of rst-degree
downside positional transfer sensitivity. Under the restriction of equal mean incomes
third-degree (upward) inverse stochastic dominance is equivalent to the criterion of
seond-degree upward Lorenz dominance.
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Higher degrees of inverse stochastic dominance and Lorenz
dominance
Since situations where second-degree (upward or downward) inverse
stochastic and Lorenz dominance do not provide unambiguous
ranking of distribution functions and Lorenz curves may arise, it is
useful to introduce weaker dominance criteria than third-degree
inverse stochastic dominance and second-degree Lorenz dominance.
To this end two hierarchical sequences of nested inverse stochastic
(Lorenz dominance) criteria might be introduced; one departs from
third-degree upward inverse stochastic dominance (second-degree
upward Lorenz dominance) and the other from third-degree
downward inverse stochastic dominance (downward Lorenz
dominance). More on this in Aaberge (2009, SCW).
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Complete orderings: Social welfare criteria based on
expected utility theory
The problem of ranking income distributions formally corresponds
to the problem of choosing between uncertain prospects. This
relationship has been utilized by e.g. Kolm (1969) and Atkinson
(1970) to characterize the criterion of second order (upward)
stochastic dominance. Atkinson reinterpreted the standard theory
of choice under uncertainty and demonstrated that inequality
aversion can in fact be viewed as being equivalent to risk aversion.
This was motivated by the fact that in cases of equal mean incomes
the criterion of non-intersecting Lorenz curves is equivalent to
second-degree stochastic dominance, which means that the
Pigou-Dalton transfer principle is identical to the principle of mean
preserving spread introduced by Rothschild and Stiglitz (1970). To
choose between F0 and F1 we can then use the following criterion
R∞
0
u(x)dF1 (x) ≥
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
R∞
0
u(x)dF0 (x)
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Axiomatic justication of the primal approach
Assume that the preference relation of the social planner satises the
following axioms:
(Order). is a transitive and complete ordering on F .
(Continuity). For each F ∈ F the sets {F ? ∈ F : F F ? }and
{F ? ∈ F : F ? F } are closed (w.r.t. L1 -norm).
(Dominance). Let F0 , F1 ∈ F . If F1−1 (t) ≥ F0−1 (y ) for all t ∈ [0, 1] and the
inequality holds strictly for some t∈ (0, 1) then F1 F0 .
(Independence). Let F0 , F1 and F2 be members of F and let α ∈ [0, 1]. Then
F1 F0 implies (αF1 (x) + (1 − α)F2 (x)) (αF0 (x) + (1 − α)F2 (x)).
Von Neuman and Morgenstern (1936) proved that a preference relation that
satises Axioms 1-4 can be represented by the following family of social welfare
functions
Z
Eu(X ) =
u(x)dF (x)
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Measures of inequality based on the primal approach
Atkinson (1970) proposed to use
Iu (F ) = 1 −
u −1 (Eu(X ))
µ
as a measure of inequality, where u −1 (Eu(X )) is denoted the equally
distributed equivalent income
and
µIu (F ) is measure of the loss in social welfare due to inequality in the
distribution F .
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
The relationship between dominance criteria and primal
social welfare functions
Theorem
Let F1 and F0 be members of F . Then the following statements are
equivalent.
(i) F1 second-degree upward inverse stochastic dominates F0
(ii) EF u(X ) > EF u(X ) for all increasing concave u
1
0
Theorem
Let L1 and L0 be members of L. Then the following statements are
equivalent.
(i) L1 rst-degree dominates L0
(ii) Iu (F1 ) < Iu (F0 ) for all increasing concave u
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Complete orderings: The family of rank-dependent social
welfare functions
The general family of rank-dependent measures of social welfare
introduced by Yaari (1987,1988) is dened by
Z
WP (F ) =
1
P 0 (t)F −1 (t)dt,
0
and can be interpreted as the equally distributed equivalent income.
The weighting function P 0 is the derivative of a preference function
that is a member of the following the set of preference functions:
P = {P : P 0 (t) > 0 and P 00 (t) < 0
for all t ∈ (0, 1), P(0) = P 0 (1) = 0, P(1) = 1}
WP preserves 1st-degree dom, since P 0 (t) > 0, and
WP preserves
P 00 (t) < 0
2nd-degree dom (and Pigou-Dalton), since
WP ≤ µF , and WP = µF i F is the egalitarian distribution
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Complete orderings: The family of rank-dependent social
welfare functions
The general family of rank-dependent measures of social welfare
introduced by Yaari (1987,1988) is dened by
Z
WP (F ) =
1
P 0 (t)F −1 (t)dt,
0
and can be interpreted as the equally distributed equivalent income.
The weighting function P 0 is the derivative of a preference function
that is a member of the following the set of preference functions:
P = {P : P 0 (t) > 0 and P 00 (t) < 0
for all t ∈ (0, 1), P(0) = P 0 (1) = 0, P(1) = 1}
WP preserves 1st-degree dom, since P 0 (t) > 0, and
WP preserves
P 00 (t) < 0
2nd-degree dom (and Pigou-Dalton), since
WP ≤ µF , and WP = µF i F is the egalitarian distribution
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Dual measures of inequality
Since WP (F ) can be interpreted as the equally distributed
equivalent income the dual family of inequality measures is dened
by
JP (F ) = 1 −
WP (F )
,
µ
where µ = EX = xdF (X ).
Note that µJP (F ) is a measure of the loss in social welfare due to
inequality in the distribution F .
R
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Axiomatic justication of the dual approach
Assume that the preference relation of the social planner satises the
following axioms:
(Order). is a transitive and complete ordering on F .
(Continuity). For each F ∈ F the sets {F ? ∈ F : F F ? }and
{F ? ∈ F : F ? F } are closed (w.r.t. L1 -norm).
(Dominance). Let F0 , F1 ∈ F . If F1−1 (t) ≥ F0−1 (y ) for all t ∈ [0, 1] and the
inequality holds strictly for some t∈ (0, 1) then F1 F0 .
(Independence). Let F0 , F1 and F2 be members of F and let α ∈ [0, 1]. Then
F1 F0 implies
−1 −1
αF1−1 (t) + (1 − α)F2−1 (t)
αF0−1 (t) + (1 − α)F2−1 (t)
.
Yaari (1936) proved that a preference relation that satises Axioms 1-4 can
be represented by the following family of social welfare functions
Z
WP (F ) =
1
P 0 (t)F −1 (t)dt,
0
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Normative justication of the general family
The normative justication of WP can be made in terms of a
(a) Theory for ranking distribution functions:
With basic ordering and continuity assumptions, the dual
independence axiom characterizes WP (Yaari, 1988)
(b) Value judgement of the trade-o between the mean and
(in)equality in the distributions (Ebert, 1987; Aaberge, 2001)
WP
= µF [1 − JP (F )]
where µF is the mean of F and
JP (F ) is the family of rank-dependent measures of inequality
aggregating the P 0 -weighted Lorenz curve of F
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
The Gini subfamily
If we choose
P1k (t) = 1 − (1 − t)k−1 , k > 2
then WP is equal to the extended Gini family of social welfare
functions (Donaldson and Weymark, 1980)
WG k
= µ [1 − Gk (F )] =,
k >2
where
Gk (F ) is the extended Gini family of inequality measures
G3 (F ) is the Gini coecient and WG = µ
2
Note that {µ, WGi (F ) : i = 3, 4, ...} uniquely determines the
distribution function F (Aaberge, 2000)
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
The Lorenz subfamily
If we instead choose
P2k (t) =
(k − 1) t − t k−1
,k >2
k −2
then WP is the Lorenz family of social welfare functions (Aaberge,
2000)
WDk
= µ [1 − Dk (F )] ,
k >2
where
Dk (F ) is the Lorenz family of inequality measures
D3 (F ) is the Gini coecient
Note that {µ, WDi (F ) : i = 3, 4, ...} uniquely determines the
distribution function F (Aaberge, 2000)
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
The relationship between dominance criteria and dual
criteria of social welfare and inequality
Theorem
Let F1 and F0 be members of F . Then the following statements are
equivalent.
(i) FR1 second-degree upward
inverse stochastic dominates F0
R
(ii) 01 P 0 (t)F1−1 (t)dt > 01 P 0 (t)F0−1 (t)dt for all increasing concave
P (P 00 (t) < 0)
Theorem
Let L1 and L0 be members of L. Then the following statements are
equivalent.
(i) L1 rst-degree dominates L0
(ii) JP (F1 ) < J(F0 ) for all increasing concave P
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Partial dual ordering - Third degree upward dominance
Note that second degree inverse stochastic dominance is dened by
2
ΛF (u) ≡
Z u
F −1 (t)dt,
0
u ∈ [0, 1]
To dene third degree upward inverse stochastic dominance, we use
the notation
Λ3F (u) ≡
Z u
0
Λ2F (t)dt =
Z u
(u − t)F −1 (t)dt,
0
u ∈ [0, 1]
Denition
A distribution F1 is said to third degree upward inverse stochastic
dominate a distribution F0 if and only if
Λ3F (u) ≥ Λ3F (u) for all u ∈ [0, 1]
1
0
and the inequality holds strictly for some u ∈ (0, 1).
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Transfer principle
∆s WP (δ , h): change in WP of a fixed progressive transfer δ from an
individual with rank s + h to an individual with rank s.
∆1st WP (δ , h) ≡ ∆s WP (δ , h) − ∆t WP (δ , h).
Denition
(Zoli, 1999; Aaberge, 2000, 2009) WP satises the principle of rst
degree downside positional transfer sensitivity (DPTS) if and only if
∆1st WP (δ , h) > 0,
when s < t.
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare
The primal approach
The dual approach
Equivalence result
Let P3 be the family of preference functions dened by
n
o
000
P3 = P ∈ P : P (t) > 0,
Theorem
Let F1 and F0 be members of F . Then the following statements are
equivalent.
(i) F1 third-degree upward inverse stochastic dominates F0
(ii) WP (F1 ) > WP (F0 ) for all P ∈ P3
(iii) WP (F1 ) > WP (F0 ) for all P ∈ P where WP satises
rst-degree DPTS
⇒ (i) and (ii): least-restrictive set of social welfare functions that
unambiguously rank in accordance with 3-UID
⇒ (i) and (iii): normative justication for 3-UID
Rolf Aaberge, Research Department,Statistics Norway
E-mail address: [email protected]
Measurement of Inequality and Social Welfare