Second-Order Discrimination and Stochastic Dominance
Rafael Salas
Departamento de Análisis Económico
Universidad Complutense de Madrid
Madrid, Spain 28035
John A. Bishop and Lester A. Zeager
Department of Economics
East Carolina University
Greenville, NC 27858
April 2015
Abstract
We offer an alternative formulation of the second-order discrimination curve (SDC) that yields
orderings of distributions equivalent to those from second-order stochastic dominance. In the
original formulation of the SDC, this equivalence is only satisfied for uniform distributions. Our
formulation is a natural refinement of the first-order discrimination curve and the zero-moment
interdistributional Lorenz curve. It also has a clear economic interpretation, as it compares the
probabilities that randomly selected individuals in the reference and comparison distributions
belong to subgroups having the same per capita income. We apply this measure to distributions
of income for seniors (age 50 and older; the baby boomers) and nonseniors in the U.S. and find
that the nonseniors second-order dominate the seniors, whether we compare truncated income
distributions for discrimination (economic advantage) or compare the entire distributions for
stochastic dominance, whereas first-order comparisons of the same groups are inconclusive.
JEL Codes: D31 (personal income, wealth, and their distributions); I3 (welfare, well-being, and
poverty); C1 (econometric and statistical methods and methodology: general)
Keywords: discrimination measurement, economic advantage, generational income comparison,
interdistributional inequality, stochastic dominance
Second-Order Discrimination and Stochastic Dominance
1. INTRODUCTION
When we compare incomes (wages) for males and females, whites and nonwhites, or
younger and older persons, the concern is with inequality between two populations rather than
inequality within one population. The essential data reside in a pair of distributions instead of a
single distribution, as with Lorenz curves. Responding to the rising social concerns about racial
discrimination and gender inequality in recent decades, statisticians and economists have created
various methods for making inter-distributional (vs. intra-distributional) inequality comparisons.
Gastwirth (1975), Dagum (1980), Shorrocks (1982), Ebert (1984), Butler and McDonald (1987),
and others made pioneering contributions to the literature. Efforts have been made since then to
sort out the implications of, and the relationships among, the proposed methods (Dagum 1987,
Gastwirth, et al. 1989, Butler and McDonald 1989, Yitzhaki 1994, and Deutsch and Silber 1999).
More recently, Le Breton, et al. (2012) have focused on dominance approaches to
measuring the inequality between two distributions, and have examined how such measures
relate to stochastic dominance orderings of those distributions. They proposed refinements of
discrimination orderings through the use of integration, which results in first- and second-order
discrimination curves in the manner of stochastic dominance relations. Taking this approach to
the problem has appeal for two reasons. First, it gives the inequality orderings clear foundations
in welfare theory, because we know the properties of the social welfare function associated with
stochastic dominance orderings of distributions. Second, dominance relations are more general
than a single discrimination index (less sensitive to differences in value judgments), and hence,
achieve broader agreement on the inequality orderings between populations.
1 We support the spirit of the Le Breton, et al. (2012) approach, but we propose an
alternative formulation of the second-order discrimination measure with nice features. Like
stochastic dominance, we integrate with respect to the same variable in the first- and secondorder measures. We also demonstrate that our alternative formulation generates orderings that
are equivalent to those obtained from second-order stochastic dominance, and therefore to all Sconvex discrimination indices.
Section 2 presents our alternative formulation and derives its implications. Section 3
illustrates our approach by an application to the incomes of U.S. seniors and nonseniors, using
data from the Current Population Survey. Section 4 offers some concluding remarks.
2. METHODS
We compare the income distributions for two populations with right-continuous and
nondecreasing cumulative distribution functions, 𝐹! (𝑚) =
𝐹! (𝑚) =
!
𝑓
! !
!
𝑓
! !
𝑦 𝑑𝑦 and
𝑦 𝑑𝑦, defined over nonnegative income values, 𝑚 ∈ 0, ∞ where 𝑐 denotes the
comparison population, 𝑟 denotes the reference population. Moreover, let 𝐹! 𝑧! = 𝐹! (𝑧! ) = 1
for some 𝑧! , 𝑧! < ∞ and assume that the lowest income value in each population is positive,
𝑘! , 𝑘! > 0. The left side of Figure 1 illustrates these functions. Le Breton, et al. (2012) define
the “first-order discrimination curve” (FDC), Γ! 𝑡 = 𝐹! [𝐹!!! 𝑡 ], where 𝑡 is the proportion of
the comparison population with incomes less than or equal to 𝑚, and 𝐹!!! 𝑡 stands for the leftcontinuous inverse of 𝐹! 𝑚 , that is, the quantile function. On the right side of Figure 1 we
illustrate the FDC. From inspection of Figure 1, FDC dominance is equivalent to first-order
stochastic dominance (FSD), because Γ! (𝑡) ≤ 𝑡 for all 𝑡 ∈ 0,1 is equivalent to 𝐹! 𝑚 −
𝐹! 𝑚 ≥ 0 for all 𝑚 ∈ 0, 𝑧! .
2 [place Figure 1 here]
In a closely related contribution, Butler and McDonald (1987) had proposed
interdistributional Lorenz curves (ILCs) based upon normalized partial moments of the
distributions, 𝜑! 𝑚; 𝑘 = [
! !
𝑦 𝑓!
!
𝑦 𝑑𝑦] 𝐸! 𝑦 ! and 𝜑! 𝑚; 𝑘 = [
! !
𝑦 𝑓!
!
𝑦 𝑑𝑦] 𝐸! 𝑦 ! .
They proposed two “natural” ILCs. The first ILC sets 𝑘 = 0 and then plots 𝜑! (𝑚; 0) = 𝐹! (𝑚)
and 𝜑! (𝑚; 0) = 𝐹! (𝑚) at corresponding incomes. Here Γ! (𝑡) = 𝜑! 𝜑!!! 𝑡; 0 ; 0 is equivalent
to the FDC above. Another ILC sets 𝑘 = 1 and plots 𝜑! (𝑚; 1) = 𝐿! [𝐹! 𝑚 ] and 𝜑! (𝑚; 1) =
𝐿! [𝐹! 𝑚 ], the Lorenz ordinates for a fixed m, at corresponding incomes in both populations.
However, 𝜑! 𝜑!!! 𝑡; 1 ; 1 does not provide a refinement of FDC comparisons.
To obtain such a refinement, Le Breton, et al. (2012) proposed integrating Γ! (𝑡) as
(1)
Φ! 𝑡 =
! !
Γ
!
𝑢 𝑑𝑢 for all 𝑡 ∈ 0,1 ,
which they call a second-order discrimination curve (SDC). They show that orderings of
discrimination patterns by this measure are equivalent to those generated by a discrimination
index proposed by Gastwirth (1975). One feature of this refinement is a bit unusual. The firstorder curve cumulates according to an income threshold, but the second-order curve cumulates
according to a population proportion. The latter orderings resemble SSD, but they are not
equivalent, except under uniform distributions (Le Breton, et al., 2012, p. 1346).
We propose instead the second-order discrimination curve
(2)
Γ ! : 0,1 → 0,1 , where Γ ! 𝑡 = 𝑇!!! 𝑇! 𝑡
for all 𝑡 ∈ 0,1 ,
and where
(3)
𝑇! : [0,1 → [0, 𝜇! , where 𝑇! 𝑡 =
!(!)
(!!!! ! )!" !
!(!)
!! ! !" !
0
𝑡 ∈ (0,1
𝑡=0
and where 𝑚 𝑡 = 𝐹!!! 𝑡 and 𝜇! is the per capita income of the comparison distribution.
3 Alternatively, we can write
𝑇! 𝑡 =
(4)
where 𝑄! 𝑡 =
! !!
𝐹
! !
𝑝 𝑑𝑝 =
𝑄! 𝑡 /𝑡
0
!(!)
𝑥𝑓! (𝑥)𝑑𝑥 is
!
𝑡 ∈ (0,1
𝑡=0
the cumulative quantile function or the
generalized Lorenz curve, and where 0 ≤ 𝑝 ≤ 1 is a population proportion.
We can interpret 𝑇! 𝑡 as the truncated per capita income of the t poorest proportion of
the comparison population or as the cumulative per capita income up to the quantile m(t) of the
comparison population. Note that 𝑚 1 = 𝑧, 𝑇! 1 = 𝜇! , and 𝑇! 1 = 𝜇! . Notice further that
both 𝑇! 𝑡 and 𝑇! 𝑡 are nondecreasing functions. This follows from equation (4), and
𝑄! 𝑡 =
! !!
𝐹
! !
𝑝 𝑑𝑝, and from the fact that 𝐹!!! 𝑝 is a nondecreasing function. The latter
! !!
𝐹
! !
implies that for any 𝑝 ≤ 𝑡, 𝑄! 𝑡 ≤
𝑡 𝑑𝑝 = 𝐹!!! 𝑡
!
𝑑𝑝
!
relevant areas in Figure 1 confirms this result. Hence, we have
!!!! ! !!!! !
!!
= 𝐹!!! 𝑡 𝑡. Comparing the
!
!! !
!"
!
=
!!! ! !!!! !
!!
=
≥ 0, ∀ 0 < 𝑡 < 1, and likewise for 𝑇! 𝑡 .
Given that 𝑇! (𝑡) is nondecreasing, there is a well-defined left-continuous inverse,
𝑇!!! (𝜇), such that
(5)
𝑇!!! : [0, 𝜇𝑟
→ [0,1 , where
𝑇!!!
𝜇 = 𝑖𝑛𝑓 𝑡 = 𝐹! 𝑠 :
!
!!! ! !" !
!
! !! ! !" ≥ 𝜇 for all 𝜇 ∈ [0, 𝜇! .
It is the proportion of the (ascending-ordered) reference population with truncated per capita
income lower than or equal to 𝜇. Therefore, Γ ! 𝑡 is the plot of (𝑇!!! 𝜇 , 𝑇!!! 𝜇 ), or the plot of
the inverse of the truncated per capita income curves. The left side of Figure 2 illustrates the
inverse truncated per capita income functions in equation (5), while the right side shows the
corresponding SDC in equation (2).
4 Under our proposed SDC, second-order discrimination exists if and only if
𝑇!!! 𝜇 ≤ 𝑇!!! 𝜇 , 𝜇 ∈ 0, 𝜇! , or
(6)
(7)
𝑇! 𝑡 ≤ 𝑇! 𝑡 , 𝑡 ∈ [0,1 ,
or equivalently, if Γ ! (𝑡) is below the diagonal in Figure 2, which can be written as
(8)
Γ ! 𝑡 ≤ 𝑡 for all 𝑡 ∈ [0,1 .
The interpretation of condition (8) is as follows: the probability that a randomly selected person
in the reference population belongs to the subgroup with cumulated per capita income 𝜇 is lower
than or equal to the probability that a randomly selected person in the comparison population
belongs to the subgroup with the same cumulated per capita income 𝜇, for all 𝜇. Notice that the
𝑠 value in expression (5) may not be the same for both populations, but the 𝜇 value (cumulated
per capita income) is the same in both populations.
[place Figure 2 here]
To see that this SDC ordering is consistent with the SSD ordering, notice that the
conditions above are equivalent to SSD, because
(9)
𝑇! 𝑡 ≤ 𝑇! 𝑡 𝑡 ∈ (0,1 ,
is equivalent to
(10)
!! !
!
≤
!! !
!
𝑡 ∈ (0,1 ,
and thus to
(11)
𝑄! 𝑡 ≤ 𝑄! 𝑡 𝑡 ∈ (0,1 ,
and 𝑇! 0 = 𝑇! 0 = 𝑄! 0 = 𝑄! 0 = 0 for 𝑡 = 0. Given that SDC and SSD orderings are
equivalent, it follows that all S-convex (Chackravarty, 1988, p. 147) inequality indices can be
used to measure discrimination consistently.
5 3. APPLICATION
For an application of our interdistributional inequality measure, we take up one of the
most challenging problems confronting the United States in the coming decades: baby boomers
entering retirement (Kotlikoff and Burns, 2004). The baby boomers were born between 1946
and 1964, so the youngest boomers turned 50 years old in 2014. At age 50, individuals also
qualify for AARP membership. Thus, we divide the U.S. population into seniors (age 50 or
older) and nonseniors (under age 50) and compare the income distributions for these two
populations.
Our income data come from the Current Population Survey in 2006, 2009, and 2012,
expressed in 2012 dollars. We treat the household as both the income sharing unit and the unit
of analysis, and adjust for household size by the square-root rule. Given that income for seniors
depends heavily on government transfers (e.g., Social Security), we use comprehensive incomes
[cash income plus in-kind transfers (except government medical benefits, which are problematic
to measure) minus taxes, but ignoring unrealized capital gains (also problematic to measure)] as
the income concept. We select quantiles in the pooled distribution of income as the thresholds
for our comparisons.
We plot the cumulative income distributions for the two populations in Figure 3a, and
find that the distribution functions cross. Therefore, the FDC fails to yield an ordering of the two
populations. Next, we consider a refinement by applying our SDC measure. We plot the inverse
truncated per capita income functions in Figure 3b, and find that they do not intersect. Thus, the
SDC lies below the diagonal in Figure 4, so U.S. nonseniors (the reference population) SDC and
SSD dominate the seniors (the comparison population) at present. It will be interesting to track
whether the inequality between the two populations grows or diminishes in the future as more
6 baby boomers shift into retirement, resulting in dramatic changes in government taxes and
transfers.
[place Figures 3a, 3b, and 4 here]
4. CONCLUSIONS
Our proposed SDC is a natural refinement of the first ILC of Butler and McDonald
(1987) and the FDC of Le Breton, et al. (2011). We demonstrate equivalence between SDC
orderings of distributions truncated at any cumulative per capita income and SSD orderings of
the entire distributions. This contribution is along the lines of Foster and Shorrocks (1988), who
relate orderings by the Foster-Greer-Thorbecke (FGT) class of poverty measures for distributions
truncated at alternative poverty lines to stochastic dominance orderings of the complete income
distributions.
References
Butler, R. and McDonald, J., “Using Incomplete Moments to Measure Inequality,” Journal of
Econometrics, 42, 109-119, 1989.
—, “Interdistributional Income Inequality,” Journal of Business and Economic Statistics, 5, 1318, 1987.
Chackravarty, S., “Extended Gini Indices of Inequality,” International Economic Review, 29,
147-156, 1988.
Dagum, C., “Inequality Measures between Income Distributions with Applications,”
Econometrica, 48, 1791-1803, 1980.
7 —, “Measuring the Economic Affluence between Populations of Income Receivers,” Journal of
Business and Economic Statistics, 5, 5-12, 1987.
Deutsch, J. and Silber, J., “Inequality Decomposition by Population Subgroup and the Analysis
of Interdistributional Inequality,” in Silber, J. (ed.), Handbook of Income Inequality
Measurement, Dordrecht: Kluwer, 363-397, 1999.
Ebert, U., “Measures of Distance between Income Distributions,” Journal of Economic Theory,
32, 266-274, 1984.
Foster, J. E., and Shorrocks, A. F., “Poverty Orderings,” Econometrica, 56, 173-177, 1988.
Gastwirth, J. L., “Statistical Measures of Earnings Differentials,” The American Statistician, 29,
32-35, 1975.
Gastwirth, J. L., Nayak, T. K., and Wang, J-L., “Statistical Properties of Measures of BetweenGroup Income Differentials,” Journal of Econometrics, 42, 5-19, 1989.
Kotlikoff, L .J. and Burns, S., The Coming Generational Storm, Cambridge: MIT Press, 2004.
Le Breton, M., Michelangeli, A. and Peluso, E., “A Stochastic Dominance Approach to the
Measurement of Discrimination,” Journal of Economic Theory, 147, 1342-1350, 2012.
Shorrocks, A. F., “On the Distance between Income Distributions,” Econometrica, 50, 13371339, 1982.
Yitzhaki, S., “Economic Distance and Overlapping Distributions,” Journal of Econometrics, 61,
147-159, 1994.
8 Figure 1. Construction of the first-order discrimination curve (FDC). The FDC combines the cumulative distribution functions for the
comparison and reference populations into a single function, 𝛤 1 (𝑡) = 𝐹𝑟 [𝐹𝑐−1 (𝑡)]. If the FDC lies below the line of interdistributional
equality, 𝛤 1 (𝑡) = 𝑡, the reference population FDC dominates the comparison population, which also implies first-order stochastic
dominance.
Figure 2. Construction of the second-order discrimination curve (SDC). The SDC combines the inverse truncated per capita income
functions for the comparison and reference populations into a single function, 𝛤 2 (𝑡) = 𝑇𝑟−1 [𝑇𝑐 (𝑡)], and is a refinement of the firstorder discrimination curve. If the SDC lies below the line of interdistributional equality, 𝛤 2 (𝑡) = 𝑡, the reference population SDC
dominates the comparison population, which also implies second-order stochastic dominance.
Fc(m) Fr(m)
Cumulative distribution functions
1
0.9
0.8
Population share
0.7
0.6
Fc(m)
0.5
Nonseniors
Seniors
0.4
Fr(m)
0.3
0.2
0.1
F-1r(𝛤2(t))
F-1c(t)
m
0
0
20000
40000
60000
80000
100000
120000
140000
Income
Figure 3a. Cumulative income distribution functions for U.S. seniors (age 50 and over, the comparison population) and nonseniors
(the reference population). These functions cross, so the first-order discrimination comparison is inconclusive.
Tc–1(μ) Tr–1(μ)
Inverse truncated per capita income functions
1
0.9
Tc–1
0.8
Tr–1
0.7
Population share
t
0.6
𝛤2(t)
0.5
Nonseniors
Seniors
0.4
0.3
0.2
0.1
Tc(t)
μ
0
0
10000
20000
30000
40000
50000
60000
70000
Truncated per capita Income
Figure 3b. Inverse truncated per capita income functions for U.S. seniors (age 50 and over, the comparison population) and nonseniors
(the reference population). These functions do not cross, so a second-order discrimination ordering is possible.
Tr–1(μ)
Second-order discrimination curve
Population share in the reference distribution below a fixed μ
1
0.9
0.8
0.7
0.6
𝛤2(t)
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
t
0.7
0.8
Population share in the comparison ditribution below a fixed μ
0.9
1
Tc–1(μ)
Figure 4. Second-order discrimination curve (SDC) for U.S. seniors (age 50 and over, the comparison population) and nonseniors (the
reference population). The SDC lies below the line of interdistributional equality, so the seniors SDC dominate the nonseniors.