Equality of Opportunity and Freedom from Poverty:
*
Measurement and Optimal Taxation
Paul Hufe, Ravi Kanbur & Andreas Peichl
Preliminary Version: 24.01.2016.
Abstract
In this paper we reconcile the ideals of equal opportunities and absence of poverty from a
perspective of inequality measurement. A particular advantage of our approach is its appeal
to the measurement of unfair inequalities. While standard measures of equality of opportunity are criticized for understating the extent of unfair inequalities due to informational
shortcomings, our approach yields strong upward corrections of measured unfair inequality
by acknowledging that judgments on the fairness of an outcome distribution are informed
by multiple normative principles. Furthermore, we sketch an optimal tax model in which
we ascribe the ideals of opportunity equalization and poverty minimization to the objective
function of the social planner. As in previous works on non-welfarist income taxation, we
nd that marginal tax rates are not bound to be zero at the bottom and at the top of the
ability distribution. Furthermore, the framework allows for negative marginal tax rates at
some points of the ability distribution.
JEL-Codes: D31; D63; D72; H21
Keywords: Equality of Opportunity; Poverty; Optimal Taxation
*
Hufe: ZEW Mannheim and University of Mannheim ([email protected]); Kanbur: Cornell University (sk145@
Peichl (corresponding author): ZEW Mannheim, University of Mannheim, IZA and CESifo.
cornell.edu);
Postal Address: ZEW Mannheim, L7,1, 68161 Mannheim, Germany ([email protected]).
1
Introduction
Why do we care about inequality? Various inuential thinkers on matters of distributive justice
seem to agree that we should not concerned by inequalities per se, but that we should rather focus
on inequalities that are rooted in factors that could not have been inuenced by individual choice
(Arneson, 1989; Cohen, 1989; Dworkin, 1981; Lippert-Rasmussen, 2001; Rawls, 1971; Sen, 1979).
Following the contributions by Roemer (1998), there now is a vivid theoretical and empirical
literature that weaves the idea of personal responsibility into inequality research by drawing
onto the distinction between circumstances and eort (see Ferreira and Peragine, 2016; Roemer
and Trannoy, 2015; Van de gaer and Ramos, 2016, for recent overviews). Circumstance is that
which is outside the control of the individual, eort is the opposite. Prominent examples of the
former are, for example, the biological sex, the place of birth, or the socio-economic outcomes
of parents. It is argued that variations in outcome which are attributable to circumstance are
morally unjustiable and therefore an appropriate target for policy intervention. To the contrary,
variations in outcome attributable to eort are legitimate and thus not appropriate as a target
for policy intervention. If eort was the sole determinant of the observed outcome distribution,
the ideal of equality of opportunity would be realized to its full extent. Yet as corroborated by
a vast literature (e.g. Konow, 2003; Konow and Schwettmann, 2016), it is reasonable to assume
that the fairness preferences of people are not exhaustively explained by the notion of individual
responsibility.
To the contrary, whether an individual holding of any good is perceived to be
fair is inuenced by a multitude of additional factors such as (procedural) desert and overall
eciency.
Another co-determinant, which will be of foremost interest in the following, is the
notion of individual need.
Imagine yourself serving on a soup line. The indigents shue forwards towards you and you
hand out hot soup. But in one case a new piece of information is given to you. You are told that
the outcome of the person in front of you was the result not of circumstance but of eort. Would
you then withdraw your soup holding hand because her outcome is morally justiable according
to the circumstance/eort doctrine?
If not, clearly some other principle is cutting across the
power of the equality of opportunity argument. When an individual holding of the outcome of
interest precludes the owner to make ends meet, questions of how it came about seem secondary
to the moral imperative of addressing the extremity of hunger, of homelessness, of violence
and insecurity.
Bourguignon et al. (2006) recognize the limits of an exclusive focus on equal
opportunities and propose an approach which makes avoidance of severe deprivation a constraint
that must be satised in the process of pursuing the broader objective of equal opportunity.
In this work we are concerned with taking seriously the idea that a given outcome distribution
should reect equal opportunities (EOp) and freedom from poverty (FfP). Therefore, we rst
propose a measure of unfair inequalities that extends current approaches towards quantifying
unequal opportunities by a concern for poverty minimization. In a second step, we specify an
optimal tax model in which the social planner is motivated by both opportunity maximization
and poverty minimization. As such we contribute to the literature on the measurement of unfair
inequalities as well as the literature on optimal labor income taxation.
This paper is organized as follows. In section 2 we clarify the underlying normative principles
of EOp and FfP. We continue by outlining our approach towards reconciling both principles
into one measure of unfair inequalities (Section 3). Furthermore, in this section we provide an
1
illustration of our measurement approach by providing empirical evidence on the extent of unfair
inequalities for a set of 31 European countries. Lastly, in section 4 we specify an optimal tax
model in which the social planner pursues EOp and FfP as co-equal principles.
2
The Principles
Equality of Opportunity
A rapidly expanding literature proposes equality of opportunity
(EOp, or IOp for inequality of opportunity) as an attractive framework for the normative assessment of the distribution of some desirable outcome
y,
such as health status (Rosa Dias, 2009;
Trannoy et al., 2010), education (Ferreira and Gignoux, 2014; Oppedisano and Turati, 2015)
or income (Aaberge et al., 2011; Ferreira and Gignoux, 2011). EOp decomposes the observed
outcome distribution of into a fair and an unfair component.
To do so, the literature on
EOp draws on the concepts of circumstances and eorts the underlying assumption being,
that a set of circumstances
est
y.
Ω
and a scalar
θ
of eort jointly determine the outcome of inter-
Theoretically, EOp is underpinned by two fundamental ideas.
First, people should be
compensated for unequal circumstances. There are two versions of this idea. According to the
principle of ex-ante compensation, outcomes ought to be equalized across people with dierential circumstances. Ex-post compensation, however, entails that outcomes ought to be equalized
across people exerting equal levels of eort. Both principles have been shown to be incompatible
(Fleurbaey and Peragine, 2013). The second idea inherent to the concept of EOp is that people
should be rewarded according to dierential eorts. Again there are dierent formulations of this
idea. One prominent version is the principle of liberal reward which states that income dierences due to dierential eorts ought to be respected without any further interference. This idea
however has provoked criticism as it bars any further redistribution beyond the compensation for
dierential circumstances (Roemer, 2010). Therefore, Van de gaer and Ramos (2016) formalize
an inequality-averse version of the reward principle.
shown to be incompatible with ex-post compensation.
In general, reward principles have been
Therefore, will adhere to the principle
of ex-ante compensation in this paper. A simple way to formulate the requirement of ex-ante
compensation is the following:
µt (y) = µ(y) ∀t ∈ T,
where
µt (y)
(EOp)
is the mean outcome of a specic type, consisting of a set of people with identical
circumstances.
Using the mean as a description of the opportunity set available for types
t,
an equal-opportunity society is characterized by all type means corresponding to the population
mean. To determine which factors are circumstances and therefore provide the ground for ex-ante
compensation is a major normative challenge in the operationalization of the principle.
There is compelling empirical evidence that people disapprove of inequalities that are rooted
beyond individual control. For instance, in a recent work Alesina et al. (2016) use information
treatments to show for a set of industrialized countries that policy preferences with respect to
taxation and spending on opportunity-equalizing policies are robustly correlated with perceptions
on social mobility. Faravelli (2007) demonstrates that perceptions of justice tend to more equal
distributions when income dierences originate from contextual factors that could not have been
inuenced by individuals. The works of Cappelen et al. (2007, 2010); Krawczyk (2010) conrm
that people uphold the EOp ideal even if this adversely aects their own material interests.
2
Freedom from Poverty
"Who is poor and who is not?" (Alkire and Foster, 2011) is one
of the fundamental questions of poverty research.
As in the equal-opportunity literature, the
insight that poverty cannot be construed in a unidimensional fashion say, by focusing on some
denition of household income only resonates in many works on the measurement of poverty
(Aaberge and Brandolini, 2015; Alkire and Foster, 2011; Bourguignon and Chakravarty, 2003).
Regardless of the specic outcome dimension of interest, the measurement of poverty is usually
implemented through a poverty line
yp 1
which provides the threshold of deprivation, and a
poverty index which aggregates the deprivations below the threshold. As such the population
can be decomposed into the groups
P = {i : yi < yp }
R = {i : yi ≥ yp },
and
2
aggregation index is applied to the incomes of the former group.
where the
Akin to the literature on EOp,
the normative concern for poverty operates on a principle of compensation: People should be
compensated to the extent that they fall short of the poverty line. A simple way to formulate
this requirement is the following:
yi ≥ yp ∀i ∈ I.
(FfP)
Naturally, the appropriate setting of the poverty line is a widely debated issue in the literature
that bears consequences on the claim for compensation (among others Decerf, 2015; Foster, 1998;
Lipton and Ravallion, 1995). All else equal, the more lenient the denition of
group
P
yp ,
the larger the
to which compensation is owed.
There is ample evidence that people are willing to compensate not just on the base of exante opportunity dierences but also based on outcome dierences among people with identical
circumstances. For example, in Cappelen et al. (2013) both stakeholder and impartial spectator
tend to equalize outcomes among participants who engaged in risky choices and the dierential
outcomes of which are the result of luck. Furthermore, the fairness preferences of people appear
to be sensitive to individual needs, i.e. the intuition that everybody should have enough to make
ends meet irrespective of how the given outcome came about (Gaertner and Schwettmann, 2007;
Konow, 2001).
3
A Measure for Unfair Inequalities
3.1
Construction
In terms of measurement, there seem to be at least three ways to think about EOp and FfP in a
joint framework. The rst is to simply treat the measurement of EOp and FfP as two separate
exercises conveying dierent normative information. But this is, in eect, to avoid the question
of what happens when the two go in opposite directions, or the deeper question of how the two
concepts are related to each other. Secondly, one could modify standard measures of poverty,
so as to make them sensitive to unequal opportunities. This is the strategy pursued by Brunori
et al. (2013), who propose an opportunity sensitive poverty measure that eectively weights
incomes below the poverty line by the outcomes of the respective types. Another approach is the
construction of opportunity-deprivation proles (Ferreira and Peragine, 2016), where groups
consisting of individuals with comparable opportunity sets are considered opportunity-deprived
1 The
appropriate setting of the poverty line has also attracted strong interest by the research community. See
for example the discussion in Decerf (2015).
2 Widely used indexes are the literature is the family of measures proposed by Foster et al. (1984).
3
3
if their average outcome falls below a pre-specied deprivation threshold.
Third, one can frame
the problem in terms of measuring unfair inequalities, which is precisely our approach in this
paper.
For illustrating our strategy, let us start from the standard approach towards estimating IOp.
In line with ex-ante compensation, the distribution of
by factors that lie within the realm of responsibility
idea, let's specify a function
g : θ × Ω 7→ R+ ,
y would be fair if it was entirely determined
of individuals i ∈ I . To operationalize this
according to which the entire space of outcomes
can be described by the individual combinations of circumstances and exerted eort.
As it
appears reasonable to assume that the distribution of eorts is not orthogonal to circumstances
the relation of interest can be rewritten in the following form:
y = g(Ω, θ(Ω), )
where circumstances
Ω
(1)
are considered as root-causes of unfair inequality beyond individual con-
trol, whereas dierential eort net of circumstance inuence, i.e.
,
yields fair inequality. For
constructing a measure of EOp, one now needs to determine the strength of the inuence of
on the determination of
y.
To do so, one typically partitions the population
i∈I
Ω
into a set of
k
k
types T based on the expressions xi of all circumstances C ∈ Ω. Individuals i and j belong
k
k
k
to the same type t ∈ T if xi = xj ∀C ∈ Ω. To the contrary, individuals i and j belong to
k ∈ Ω : xk 6= xk . Drawing onto this type partition we can condierent types t ∈ T if ∃C
i
j
struct a counterfactual distribution, where each outcome
of the individual's type, here denoted by
µt .
yi
is replaced by the mean realization
This counterfactual (or smoothed) distribution
is indicative of unequal opportunities (and thus unfair inequality) since any inequality can be
traced back to dierential circumstances and thus a non-compliance with the principle of ex-ante
compensation. Using a subgroup decomposable inequality measure (Cowell, 2011) the following
inequality decomposition is instructive:
I(y)
=
X
t
ft I(yit )
Z
+ I( yit f t (y)dy)
y
|
{z
}
(2)
=µt
In line with the idea that circumstances should have no leverage on determining the outcome
distribution (ex-ante compensation), a social planner seeking to realize equal opportunities, would
implement policies that minimize the second component of equation 2.
Employing standard
reward principles, within-type inequality is ethically unproblematic since the constituents of
each type dier by dierential levels of eort exertion, only. However, according to the view put
forward in this paper our judgment as to whether the observed distribution is fair should also
be sensitive to everybody making ends meet. As the principle of FfP operates independent of
3 Eectively, this amounts to applying standard poverty measures to a counterfactual (or smoothed) distribution
where individual incomes have been replaced by the mean incomes of their respective types.
4
individual eort exertion, we have to further decompose within-type inequality:
I(y)
=
X
ft I(yit ) + I(µt )
(3)
t

=
X
t



ft fpt I(yipt ) + frt I(yirt ) + I(µtg ) + I(µt )
| {z } | {z } | {z }
(1)
(2)
(4)
(3)
Equation 4 decomposes inequality within each type into (1) inequality among the poor, (2)
inequality among the rich, and (3) a between-group component. Clearly, FfP deplores (1) and
yp ,
is indierent to (2). If everybody has a justied claim to a minimum holding of
yi < yp .
be no inequality among those for which
the legitimate holding of
y
once
yi ≥ yp .
there shall
To the contrary, FfP provides no guidance on
Yet, keep in mind that according to our conception
the planner pursues both FfP and EOp. The latter is inspired by the idea that circumstances
rt ) is a matter
As such, while I(yi
ought to be compensated while eorts should be rewarded.
rt ) within the rich groups
of indierence from a FfP perspective, EOp tells us not to alter I(yi
of each type in order to comply with the reward principle.
intricate but follows the same logic.
legitimate claim for holding
yp .
The classication of (3) is more
On the one hand, FfP prescribes that everybody has a
µtp < yp ,
Therefore, to the extent that
(3) must be classied
as morally objectionable. On the other hand, FfP is indierent to the distribution of incomes
once
yi ≥ yp ∀i ∈ I .
Therefore, any mean-preserving transfer from
t
with FfP as long as µp
distributions.
guidance.
≥ yp .
R
to
P
would be consistent
The planner would be indierent between any of the ensuing
Yet, again reference to the reward principle inherent in EOp provides further
Among all possible intra-type allocations that respect FfP, our planner has a clear
preference for the allocation that preserves the initial rewards to eort to the greatest extent.
As such she would transfer sucient funds from
R
to
P
to realize
µtp = yp .
With FfP realized
however, she would oppose any further transfer for the purpose of respecting the reward principle.
Therefore, (3) is morally objectionable to the extent that
t
extent that µr
≥ yp .
µtp < yp ,
but morally alright to the
With this reasoning in mind we can further decompose between-group
inequality within each type:
I(y) =
X
h
i
ft fpt I(yipt ) + frt I(yirt ) + I(µtg ) + I(µt )
(5)
t


=
X
t


ft fpt I(yipt ) + frt I(yirt ) + I(µtp , yp ) + I(yp , µtr ) + t  + I(µt ).
| {z } | {z } | {z } | {z } |{z}
(1)
(2)
(3.1)
Equation 6 decomposes between-group inequality among
µtp and the poverty line
error
yp
t .
(3.2)
R
P
yp ,
and
yp , (3.2) inequality between µrp and
(6)
(3.3)
into (3.1) inequality between
and (3.3) an approximation
The approximation error originates from replacing the mean incomes of
R
and
P
with
for the calculation of the morally objectionable share (3.1) and the fair share of between-
group inequality (3.2), respectively. Both replacements create counterfactual distributions, the
t
inequality of which does not add to I(µg ). The approximation error decreases with either of
5
µtr
and
µtr
yp . 4
approaching
Furthermore, it can be shown that
t ≥ 0
conditional on
µtp ≤ yp ≤ µtr
(see Appendix A.1).
Expression 6 now can be written in terms of fair and unfair components:
I(y)
= I(µt ) +
X
ftp I(yipt ) +
X
t
t
{z
|
+
}
Unfair
X
ftr I(yirt ) +
X
ft I(yp , µtr )
t
|t
{z
}
Fair
X
+
ft I(µtp , yp )
ft t
| t {z }
Approx. Error
The approximation error instills a logic of lower and upper bounds. The cautious approach would
allocate
P
t ft t to the fair component of outcome inequality.
For the sake of tractability we will use the following notation for the remainder of this paper:
I(y)
= I(Type) + I(Poor) + I(Group/Unfair)
|
{z
}
Unfair
+ I(Rich) + I(Group/Fair)
|
{z
}
Fair
+
|{z}
.
Approx. Error
3.2
Empirical Application
Data
To illustrate the suggested measure of unfair inequalities we draw onto the 2011 cross-
section of the EU Statistics on Income and Living Conditions (EU-SILC) which covers 31 European countries.
5
In particular, we use the 2011 wave as it provides a module on the inter-
generational transmission of advantages, which allows us to construct types from circumstance
6
variables.
As a default, we focus on disposable household income adjusted by the OECD-
equivalence scale as the outcome of interest. Observations with
yi < 0
are excluded from the
analysis and zero incomes are replaced by 1 in order to foreclose sample size reductions through
logarithmic transformations. To curb the inuence of outliers in the upper part of the income
distribution, we replace all values above the 99th percentile of the country- and year-specic
income distribution by the respective value for the 99th percentile. We hold the poverty rate
4 The
yp
idea is intuitive. If µtr = yp , I(µtp , yp ) = I(µtp , µtr ) = I(µtg ), i.e. all between-group inequality could be
classied as unjust, due the fact that it was only driven by µtp < yp . Reversely, if µtr = yp , all between-group
inequality could be classied as justied, due the fact that it was only driven by µtr ≥ yp .
5 The sample consists of Austria (AT), Belgium (BE), Bulgaria (BG), Switzerland (CH), Cyprus (CY), Czech
Republic (CZ), Germany (DE), Denmark (DK), Estonia (EE), Greece (EL), Spain (ES), Finland (FI), France
(FR), Croatia (HR), Hungary (HU), Ireland (IE), Iceland (IS), Italy (IT), Malta (MT), Lithuania (LT), Luxemburg
(LU), Latvia (LV), Netherlands (NL), Norway (NO), Poland (PL), Portugal (PT), Romania (RO), Sweden (SE),
Slovenia (SI), Slovakia (SK), and Great Britain (UK).
6 The 2005 wave also comprises a module on the intergenerational transmission of advantages for a sample of
26 European countries. Results for the 2005 wave are available on demand.
6
xed at the so called European
At-Risk-Of-Poverty Rate
which is drawn at 60% of the country-
specic median equivalized disposable household income. Furthermore, we restrict the sample to
the age range 18-59. To assure the representativeness of the sample all calculations are performed
considering personal cross-sectional sample weights.
Type partition
For the type partition we use four circumstance variables. First, the biological
sex of the respondent. Second, we proxy the migration status of the respondent indicated by
whether she was born in her current country of residence or not.
on the educational status of the parents.
Third, we use information
Particularly, we use two binary variables indicating
whether either of a respondent's parents attained more than a secondary school degree or not.
Lastly, we proxy the occupational status of both parents by grouping them in either elementary
occupations (ISCO88: 8000-9000), semi-skilled occupations (ISCO88: 5000-7000), or top-rank
positions (ISCO88: 1000-4000). As such each of the considered populations is partitioned into
2 ∗ 2 ∗ 4 ∗ 9 = 144
Results
types.
Figure 1 illustrates the unfair share of inequality as calculated by the decomposition
illustrated in section 3 for the sample of 31 European countries. Recall that inequality between
types gives an indication of IOp.
Drawing onto our partition into 144 types, the share of in-
equality attributed to unequal opportunities hovers around 10% (∅
of countries.
= 12.12%)
for the majority
Only in Luxemburg the share of inequality explained by circumstances exceeds
the 20%-mark (25.29%). In our sample the fairest country from an opportunity perspective is
Slovakia where only 5.89% of inequality can be traced back to inequality between circumstance
types.
According to FfP any inequalities among those below the poverty line is morally objectionable.
Thus, adding outcome inequality below the poverty line to the IOp component yields a strong
upward correction of unfair inequalities in our sample. On average
attributed to inequality below the poverty line
yp .
9.42%
of inequality can be
Lowest inequality among the poor is found in
the Netherlands (1.42%), while inequality below the poverty line has the highest share in Latvia
(24.88%).
The last inequality component of moral concern is the between-group inequality between poor
and rich to the extent that the average poor household has less than the legitimate minimum
holding
yp .
Again we see a sizable upward correction of the unfair share of inequality when
introducing this component. On average 5.56% of observed inequality are due to the poor falling
short of the poverty threshold, with the lowest share being found in the Netherlands (2.31%)
and the highest share in Denmark (13.60%). Taken together, complementing EOp with FfP on
average more than doubles the measured unfair share of inequality in the considered sample.
Note that Figure 1 is constructed based on the most conservative approach towards measuring
FfP as we attribute the approximation error
to the fair share of inequality. Figure 2 illustrates
the impact of relaxing this assumption. Attributing the approximation error to the unfair share
of inequality increases our measure on average by 12.25 percentage points. Even more bluntly
one could attribute the average distance of the rich population to the poverty line to the unfair
share of inequality. Doing so amounts to considering the entire between-group inequality between
P
and
R, I(µtg ),
as unfair. This is reasonable if we assumed that poverty was never a choice, but
always determined by unobserved circumstances. The partial observability of circumstances and
7
Figure 1: Unfair Inequality by Country, in %
Data Source: EU-SILC. Own Calculations. I(Type)
represents I(µt ), i.e. the share of inequality exP
plained by circumstances. I(Poor) is indicative for t ftp I(yipt ), i.e. the share
of inequality explained by
P
within-group inequality below the poverty line. I(Group/Unfair) denotes t ft I(µtp , yp ), i.e. the share
of inequality explained by between-group inequality between the poor and the rich to the extent that the
poor earn less than the required minimum.
Figure 2: Unfair Inequality by Country, in %
Data Source: EU-SILC. Own Calculations. I(Type)
represents I(µt ), i.e. the share of inequality exP
plained by circumstances. I(Poor) is indicative for t ftp I(yipt ), i.e. the share
of inequality explained by
P
within-group inequality below the poverty line. I(Group/Unfair) denotes t ft I(µtp , yp ), i.e. the share
of inequality explained by between-group inequality between the poor and the rich
P to the extent that the
poor earn less than the required minimum. represents
the approximation error t ft t originating from
P
t
the inequality decomposition. I(Group/Fair) denotes t ft I(yp , µr ), i.e. the share of inequality explained
by between-group inequality between the poor and the rich to the extent that the rich earn more than the
poverty line.
8
the ensuing downward bias on IOp measures is a widely discussed issue in the empirical literature
t
(Ferreira and Gignoux, 2011; Niehues and Peichl, 2014). When declaring the entirety of I(µg ) to
be unfair, we presume that if we had perfect information on circumstances, all poor would be
collected in types
t
µt < yp . Then, invoking the EOp criterion all poor types had a claim
level of µ. If we followed this reasoning we would attain another strong
with
to be elevated to the
upward correction of the unfair share of inequality within our sample of 31 European countries
(∅
= 16.84%).
While such an assumption is plausible it remains speculative, which is why we
focus on the more conservative approach in the following.
Naturally, what it means to make ends meet is contentious both philosophically (e.g. Casal,
2007) and empirically (e.g. Meyer and Sullivan, 2012). It is beyond the ambit of this work to
resolve these issues. However, we illustrate the sensitivity of our proposed measure to varying
the level of
yp
for the case of Germany in Figure 3.
In particular, we vary the percentage
Figure 3: Unfair Inequality by Poverty Line, in % (Germany, 2011)
Data Source: EU-SILC. Own Calculations. I(Type)
represents I(µt ), i.e. the share of inequality exP
of inequality explained by
plained by circumstances. I(Poor) is indicative for t ftp I(yipt ), i.e. the share
P
within-group inequality below the poverty line. I(Group/Unfair) denotes t ft I(µtp , yp ), i.e. the share
of inequality explained by between-group inequality between the poor and the rich to the extent that the
poor earn less than the required minimum.
level of the population median income that is required to trespass the poverty threshold in 10
percentage point steps.
yp would equal the population's
P and R, respectively. The 60%-mark
By denition at the the 100%-mark
median income and 50% of the population would be in
replicates the result for Germany from Figure 1.
EOp: Varying the poverty threshold
yp
Note that FfP operates independently of
has no impact on the share of inequality explained by
dierential circumstances. Furthermore, the importance of FfP increases when raising the level
yp . This is intuitive for I(Poor) since raising yp ceteris paribus increases the population share
of P . Thus, the share of total inequality accounted for by inequality among the poor cannot fall
7
when elevating yp . For I(Group/Unfair) this is less clear. Imagine an income distribution that
of
7 At
the limit, if R = ∅, 100% of outcome inequality would be explained by inequality among the poor.
9
completely attens towards its higher percentiles, i.e. there is little inequality among the richer
parts of this population. Then, at some point when moving
yp
along this distribution,
µtp
and
µtr will start to converge. This, in turn leads to lower between-group inequality. Yet as the case
of Germany illustrates, this seems to be a rather theoretical possibility in view of the fact that
income distributions tend to be skewed to the right Cowell (2011). As such, it is reasonable to
assume that an increase of I(Group/Unfair) with rising
yp
is the empirical standard case.
Furthermore, the EU-SILC dataset provides detailed information on dierent income sources.
As a consequence we can evaluate the impact of tax-benet systems on reducing the considered
components of unfair inequality. Figure 3 shows the reduction of unfair inequality from market
Figure 4: Unfair Inequality by Income Concept and Country, MLD
Data Source: EU-SILC. Own Calculations. For each country the upper bar shows the result for equivalized
household market income. The central bar indicates equivalized household gross income, i.e. market
income+government transfers. The lower bar shows the results for equivalized disposable household income,
i.e. market income+government transfers-taxes paid. I(Type)
represents I(µt ), i.e. the share of inequality
P
explained by circumstances. I(Poor) is indicative for t ftp I(yipt ), i.e. the P
share of inequality explained
by within-group inequality below the poverty line. I(Group/Unfair) denotes t ft I(µtp , yp ), i.e. the share
of inequality explained by between-group inequality between the poor and the rich to the extent that the
poor earn less than the required minimum.
income to disposable income as measured by the mean log deviation. Note that in contrast to
the previous graphs we do not present the results in terms of shares of total inequality.
are interested in the reduction of unfair inequalities through tax-benet systems.
We
Looking at
percentage shares, however, conates the eects of tax-benet systems on unfair inequality with
total inequality reduction. Therefore, we focus on an absolute instead of a relative measure of
unfair inequalities. For each country, the upper bar indicates the results for equivalized household
market income, while the central bar shows unfair inequality in equivalized household gross
income. Thus, comparing the upper with the central bar shows the impact of benet payments
on unfair inequality. The lower bar is based on the results for equivalized household disposable
income. Therefore, comparing the central with the lower bar gives an impression of the impact of
the income tax system of the respective country. Clearly, transfer payments have a tremendous
impact on moving the income distribution towards realizing FfP. In particular, benet payments
10
achieve a strong reduction of inequality among the poor, while they also move the poor closer to
the poverty line
our sample.
yp .
In addition we observe a moderate reduction of IOp in 31 countries within
In general, we see that government transfers in all countries are predominantly
directed towards poverty alleviation and thereby contribute strongly to the reduction of unfair
inequalities. The picture is less clear for taxes. In the majority of countries government taxes
exert an adverse impact on FfP, i.e. they unequalize incomes below the poverty line and/or move
the average income of the poor population further from the poverty line. Exceptions to the rule
are Romania, Estonia, Lithuania, Latvia and the UK. To the contrary, income taxes have an
opportunity equalizing impact in all considered countries. Combining the eects on both EOp
and FfP, taxes increase unfair inequalities in nine (CH, PL, DK, IS, SK, CY, NO, FR, EL) out
of the 31 countries of our sample. Looking at the entirety of tax-benet systems, however, we
consistently see a strong negative impact on unfair inequalities, which is mainly driven by the
8
eects of government transfers.
4
Optimal Income Taxation
To the extent that the moral concern about inequality is indeed motivated by EOp and FfP,
redistributional policies ought to be designed to remedy inequalities arising on these grounds.
While it has been convincingly shown that tax-benet systems have not been able to put a halt
on rising inequalities in recent decades (Piketty et al., 2016), they nevertheless remain the main
lever for the ex-post correction of inequalities. Therefore, in this section we will investigate the
design of tax-benet systems that are directed towards correcting inequalities that contradict the
principles of EOp and FfP. For this purpose, we will specify an optimal tax model à la Mirrlees
(1971) in which the social planner follows the objective of maximizing both EOp and FfP. By
doing so we relate to an increasing body of literature on optimal income taxation that diverts
from the standard assumption of a welfarist planner (among others Kanbur et al., 2006; Ooghe
and Peichl, 2015) who seeks the maximization of some transformation of aggregate utility in the
population.
The Household
In particular, we assume a population that can be characterized by means
of a cumulative distribution function
from the set
[a, a].
Individual
i
F (a),
with ability
where
a
a
corresponds to an individual ability level
earns income
z
by working
l
hours, i.e.
z = al.
Furthermore, we assume a quasi-linear utility function:
U (a) = y(a) − ρ
z(a)
a
,
according to which individual utility grows linearly in consumption and decreases convexly in
hours worked.
Households choose their eort levels, i.e.
hours worked, so as to maximize
8 Note that this conclusion is based on a sequential accounting approach. If we followed a factor decomposition
approach (Shorrocks, 1982), that takes account of the interaction between the dierent income components (i.e.
benets are taxable in some countries) the results might dier substantially (see for example Fuest et al. (2010)
for a comparison of both approaches with respect to outcome inequality.).
11
individual utility subject to a household budget constraint:
U (a)
max
z,y
s.t.
y = z − T (z),
(7)
This leads to the optimality condition
a(1 − T 0 (z)) = ρ0
z(a)
a
,
i.e. households choose an eort level at which the marginal return, the net wage rate, corresponds
to the marginal cost of labor.
The Social Planner
resource constraint
λ
The planner in turn, maximizes an objective function
and a set of incentive compatability constraints
∆
subject to the
ψ(a):
Z
max
z(a),y(a)
∆(U (a) + ρ(·), l(a))dF (a)
|
{z
}
(∆)
s.t.
=y(a)


Z


z(a) − (U (a) + ρ(·)) f (a)da = R
|
{z
}
(λ)
=y(a)
−
ρ0 (·)l(a)
a
=0
(ψ(a))
The resource constraint acknowledges that the planner's distributional problem is limited to
the resources available in the economy. The incentive compatibility constraint assures that the
optimal tax schedule instills individuals to reveal their true ability types.
If the tax schedule
incentivized individuals to mimic other ability types, government taxation would create economic
ineciencies the planner seeks to avoid. See also Appendix A.3 for a derivation of
Solution
ψ(a).
The planner's problem can be solved by means of dynamic control optimization. The
Hamiltonian reads as follows:
H = {∆(U (a) + ρ(·), l(a)) + λ [z(a) − U (a) − ρ (·)]} f (a) + ψ(a)
with control variable
l(a), state variable U (a) and co-state variable ψ(a).
ρ0 (·)l(a)
a
,
This yields the following
rst-order conditions:
∂H
= (∆0y − λ)f (a) = −ψ 0 (a)
∂U (a)
∂H
= ∆0y (·)ρ0 (·) + ∆0l (·) + λ(a − ρ0 (·)) f (a) +
∂l(a)
Solving for the marginal tax rate
(8)
ψ(a)
a
ρ00 (·) l(a) + ρ0 (·) = 0
(9)
T 0 (·) gives the following characterization of the optimal marginal
tax schedule (see Appendix A.4 for more details on the solution procedure):
h 0 T 0 (·)
∆y (·)
1
=
−
ρ0 (·) +
0
λ
ρ
(·)
0
1 − T (·)
∆0l (·)
∆0y (·)
i
+
1
(a)
12
+1
1−F (a)
1
af (a) (1−F (a))
Z
a
(1 −
a
∆0y (·)
λ )f (a)da (10)
As shown by Kanbur et al. (2006) the rst term in equation 10 can be understood in terms of the
marginal rates of substitution between labor and consumption of households and the planner,
respectively. While for households MRS
the is dened as MRS
p
=
U0
= − U l0 = ρ0 (·)
h
y
the marginal rate of substitution for
∆0 (·)
− ∆0l (·) . Thus, equation 10 can be rewritten as follows:
y
h 0 i T 0 (·)
∆y (·)
1−F (a)
h
p
1
1
1
=
−
MRS
−
MRS
+
+
1
λ
ρ0 (·)
(a)
af (a) (1−F (a))
1 − T 0 (·)
Z
a
(1 −
a
∆0y (·)
λ )f (a)da
(11)
h
Obviously, if the objective functions of households and the planner coincide MRS
= MRS
p
, the
rst term of equation 10 vanishes and the characterization of the optimal tax schedule reduces
to:
Z a
T 0 (·)
1−F (a)
1
1
(1 −
=
+
1
(a)
af (a) (1−F (a))
1 − T 0 (·)
a
| {z } | {z } |
{z
=A
=B
∆0y (·)
λ )f (a)da
(12)
}
=C
This is ABC-formula prominently put forward by Diamond (1998) for the case of a Utilitarian
planner. The optimal level of marginal taxes varies negatively with the labor supply elasticity
of individuals at ability level
a
(A). The higher the mass of people
f (a),
i.e.
the population
share of the distorted ability type, the lower the optimal level of marginal taxes at this point
of the ability distribution.
To the contrary, increasing marginal tax rates at a specic ability
level also increase average tax rates for all individuals with ability levels greater than
a
without
inducing distortions through behavioral responses. Thus, the greater the mass of people above
a,
the higher the marginal tax rate on
component B. Note that
1 − F (a) = 0
a.
These two countervailing eects are summarized in
for the highest ability individual, resulting in the well-
known result of zero marginal tax rate at the very top of the ability distribution (Sadka, 1976;
Seade, 1977). Lastly, C is indicative for the average marginal social welfare weight of households
above the distorted area. Note that
λ
is the shadow price for loosening the budget constraint.
It is thus evaluated as:
Z
a
(∆y (·))f (a)da
λ=
a
Furthermore, let's rewrite C as follows:
C
=1−
1
(1−F (a))
|
Thus,
g
a
Z
a
∆0y (·)
λ )f (a)da
(
{z
g
is the average marginal utility gain on the interval
(13)
}
[a, a]
relative to
[a, a].
of a welfarist planner and the standard assumption of decreasing marginal utilities,
with
a.
In the case
g
decreases
A Utilitarian planner therefore tends to levy higher marginal tax rates on high ability
individuals. Note that
g
is dened on the interval
the lowest ability individual. Hence marginal tax
[0; 1], where 1 is the welfare weight assigned to
rates are 0 for the lowest ability individuals in
the standard welfarist case. As all components of the ABC-formula are weakly positive, marginal
tax rates cannot be negative at any point of the ability distribution.
While the ability distribution F(a) and labor supply elasticities
13
(a)
are insensitive to the
exact specication of the planner's objective function, the variation of
g
across the ability dis-
tribution remains contingent on the normative objective of the planner.
The same holds for
the rst additive term in equation 10. It only vanishes if the objective functions of households
and planner coincide. To the contrary, if the valuation functions of households and the planner
diverge the rst term in equation 10 persists. In the following we maintain the assumption of
utility-maximizing households but vary the valuation function of the planner for her to respect
the principles of EOp and FfP.
4.1
Minimizing Unfair Inequality
In section 3.1 we developed a measure of unfair inequality that complies with the principles
of EOp and FfP. Thus, a planner who seeks to maximize equal opportunities while minimizing
poverty within the population can be construed as minimizing our measure of unfair inequalities.
∆
Thus, the planner's objective function
Z
max
−
I(µt )
z(a),y(a)
+
reads as follows:
X
ftp I(yipt ) +
X
t
subject to the resource constraint
λ
ft I(µtp , yp )dF (a),
1
(∆ )
t
and the incentive compatibility constraints
ψ(a).
Note,
that the planner's objective is exclusively determined by the consumption levels of households.
Therefore,
∆01
l =0
and equation 10 reduces to
T 0 (·)
∆01
1−F (a)
y (·)
1
1
=
−
+
+
1
λ
(a)
af (a) (1−F (a))
1 − T 0 (·)
Hence, we only need to determine the rst derivative of
a
Z
∆1
(1 −
a
∆01
y (·)
λ )f (a)da
(14)
with respect to consumption
y
to
spell out the optimal marginal tax schedule. Again drawing onto the MLD as the scalar measure
of inequality
∆01
y (·)
reads as follows:
h
 1 −
µ
01
∆y (·) = h t
 1 −
µt
1/T
1/T
1
P
1
P
t
i
+
i
,
µt
t µt
h
1
y(a)
−
1
µtp
i
+
h
1
y(a)
−
1
1/ft [fpt µtp +frt yp ]
i
if
i ∈ P.
(15)
otherwise.
For the entire population it holds that marginal tax rates increase (decrease) the stronger the
mean disposable income of a household's type diverges positively (negatively) from the population
average.
This is consistent with tailoring tax schedules towards the achievement of EOp as
redistribution tends to favor disadvantaged types with
µt < µ.
The sign of the rst component of
equation 15 thus is positive (negative) for disadvantaged types. The second and third component
capture the concern for FfP and apply to the poor population
P
only.
According the second
component, among the poor those with lower (higher) incomes face lower (higher) marginal tax
rates. Thus, the sign of the second component is negative for richer households within
in turn increases their marginal tax rates. Does this imply that some households in
marginal tax rates than comparable households in
component. All
i∈P
R?
yp .
Thus the third component
P . Furthermore, note that the sum of
1/ft [fpt µtp + frt yp ] ≥ µtp . For this to hold, a
of equation 15 is strictly positive for all households in
14
face higher
Clearly, this is not the case due to the third
earn incomes lower than the poverty line
component two and three is always weakly positive if
P
P , which
sucient condition is
µtp ≤ yp
(compare also to Appendix A.1), which holds by denition. Hence,
marginal tax rates for households in
P
cannot exceed the ones for comparable households in
R.
In contrast to the standard welfarist model, marginal tax rates in principle can be zero in our
non-welfarist case. For the purpose of illustration consider the case in which the lowest ability
household is also in the most disadvantaged type. Then, the reformulated ABC-term vanishes
and the marginal tax rate at this point is exclusively determined by the rst additive component
−
∆01
y (·)
λ . As discussed previously, the latter must be negative for poor persons from types that
have an average outcome below the population mean. Reversely, the optimal marginal tax rate
at the top of the ability distribution is not bound to be zero. To see this, note that for the very
top of the distribution the reformulated ABC-term collapses to zero since
1 − F (a) = 0.
As a
consequence the top marginal tax rate is determined by the rst additive term in equation 14
only, which can be both negative and positive depending on the type of the respective household.
4.2
The Loss-Function Approach
Another route for speciying an optimal tax model that is sensitive to both EOp and FfP is to
draw on the loss-function approach put forward by Weinzierl (2014). This approach provides a
general strategy to reconcile multiple normative objectives of the planner within a single optimal
tax model. It consists of four steps. First, one chooses a set of plausible normative principles.
Second, one characterizes ideal distributions that are in accordance with these principles. Third,
one assigns weights to the chosen normative criteria.
Lastly, one minimizes a weighted loss
function to obtain the optimal tax schedule according to the chosen criteria.
In our case the set of normative principles consists of EOp and FfP. We represent the former
by the planner's wish to minimize the the mean-log deviation of the smoothed distribution. As
outlined in section 2 this is consistent with the principle of ex-ante compensation. Thus, the rst
part of the objective function
∆
reads as follows:
Z
−
max
I(µt )dF (a)
z(a),y(a)
(∆
Clearly, the ideal point for an EOp-maximizing planner is any distribution with I(µt )
= 0,
EOp
)
i.e. a
distribution in which there are no dierences in mean outcomes across circumstance types.
In line with Kanbur et al. (1994) we operationalize the FfP principle by means of minimizing
a version of the Foster-Greer-Thorbecke poverty measure (Foster et al., 1984):
Z
max
z(a),y(a)
−
Pα (y(a), yp )dF (a)
(∆
The ideal point for a poverty-minimizing planner is any distribution with Pα (y(a), yp )
FfP
)
= 0.
Hence, a planner that values both EOp and FfP would minimize the following weighted loss
function:
max
βEOp [0 − I(µt )] + βFfP [0 − Pα (y(a), yp )],
max
βEOp [−I(µt )] + βFfP [−Pα (y(a), yp )]
z(a),y(a)
z(a),y(a)
15
or
2
(∆ )
subject to the resource constraint
1 − βFfP
λ
and the incentive compatibility constraints
ψ(a). βEOp =
are the respective weights on the planner's objectives. Note that as in the previous case,
the planner is non-welfarist and thus is only concerned with the levels of disposable income while
disregarding the eort cost of labor. Hence, we again only need to determine the rst derivative
of
∆2
with respect to consumption
y

h
βEOp 1 −
h µt
∆02
y (·) =
βEOp 1 −
µt
to characterize the marginal tax schedule:
1/T
1/T
1
P
1
P
t
t
i
h
i
+ βFfP α N1 ( y1p )α−1
if
i
,
otherwise.
µt
µt
The marginal tax rate on the households in
R
i ∈ P.
(16)
will again be determined by the divergence of
their respective type-mean from the population mean.
Advantaged types will be taxed more
heavily than disadvantaged households. In contrast to the direct minimization of our measure
P will also be co-determined by the
0
specication of the weights βEOp and βFfP . Note that with βEOp → 0, T (·) approaches the peak
9
of the Laer-curve, i.e. the tax revenue maximizing rate, for all households in P .
The poor
again
consistently
face lower tax rates than comparable households in P as both βFfP ≥ 0 and
i
h
1 1 α−1
≥ 0. As in the previous case, marginal tax rates can be negative in principle, for
α N ( yp )
of unfair inequalities, the taxes levied on households in
example if the lowest ability household comes from a household whose type is disadvantaged as
compared to the population average.
On the one hand, the loss function approach aords more exibility on the trade-o of both
FfP and EOp in the planner's objective function. On the other hand, without further guidance
through either normative theory or empirical evidence on equity preferences, the specication of
the weights appears highly arbitrary. Analogously, the well-known result of zero marginal tax
rates at the top does not hold for our non-welfarist case.
5
Conclusion
Summary
In this paper we have shown how to reconcile the ideals of equal opportunities and
absence of poverty from a perspective of inequality measurement. As such we divert from previous
works that have analyzed principles of EOp and FfP by making standard measures of poverty
sensitive to equal-opportunity considerations (Brunori et al., 2013; Ferreira and Peragine, 2016).
A particular advantage of our approach is its appeal to the measurement of unfair inequalities.
Standard measures of EOp can be interpreted as information on the extent of unfair inequalities in
a given outcome distribution. However, concerns have been expressed about them producing very
low estimates of unfair inequality due to the partial observability of circumstances (Kanbur and
Wagsta, 2016). Responses to this have been the call for better datasets and the development of
alternative upper bound measures of IOp (Hufe et al., 2015; Niehues and Peichl, 2014). In this
paper we propose a dierent route by eectively acknowledging that judgments on the fairness
of an outcome distribution are informed by multiple normative principles.
As the empirical
illustration of our approach highlights, complementing the concern for equal opportunities (EOp)
with poverty aversion (FfP) yields a strong upward correction of measured unfair inequality in
9 In this case the model eectively boils down to the one specied in Kanbur et al. (1994), in which the planner
is only motivated by the objective of poverty-alleviation.
16
a given income distribution.
Taking the principles of EOp and FfP seriously, it is reasonable to assume that the design
of redistributive mechanisms should be informed by them. Therefore, we sketch an optimal tax
model in which we ascribe EOp and FfP to the objective function of the social planner. As in
previous works on non-welfarist income taxation, we nd that marginal tax rates are not bound
to be zero at the bottom and at the top of the ability distribution. Furthermore, the framework
allows for negative marginal tax rates at some points of the ability distribution. These results
divert from the standard Utilitarian case.
Yet more detailed descriptions of the optimal tax
schedule aords simulation exercises with precise information on the income distribution, the
ability distribution and the respective allocation of ability types to circumstance types, as well
as labor supply elasticities.
Way forward
This paper is work in progress and will therefore be subject to further develop-
ment and substantial revisions. Specically, the following items shall be addressed as the project
proceeds:
ˆ
Our current approach towards the interplay of EOp and FfP is consistent with the principles
of ex-ante compensation and an poverty-averse reward principle.
To the best of our
knowledge the latter has not been formally characterized in the literature. We therefore
will provide a more elaborate denition in the future.
ˆ
We provide a new measure for unfair inequalities that is informed by EOp and FfP. An
axiomatic characterization of this measure may be an interesting exercise for strengthening
the theoretical basis of our measure.
ˆ
Naturally, the optimal tax models presented in this paper lend themselves to further extension.
First, poverty is highly correlated with non-participation in the labor market.
Therefore, one avenue for further research would be the inclusion of extensive labor supply
responses. Second, the partition of population into types based on circumstances implicitly
assumes the observability of these characteristics for the planner. Thus, a natural extension
would be to include a tagging model into our analysis.
ˆ
A further approach to optimal taxation with multiple normative objectives is suggested by
Saez and Stantcheva (2016). We will draw on their work to provide another alternative
approach to model optimal tax schedules with a planner motivated by both EOp and FfP.
ˆ
It is a well-known fact in the optimal tax literature, that it is inherently dicult to make
statements on optimal tax schedules beyond some general results, based on the theoretical
model alone. Thus, the optimal tax schedules derived in this paper shall be simulated using
suitable data sets (i.e. the German Taxpayer Panel).
17
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20
A
Appendix
A.1
Decomposing I(µtg )
For the sake of illustration we use the mean-log deviation as our inequality index of choice. Then,
t
I(µg )
t
I(µp , yp )
t
I(yp , µr )
= ln
= ln
= ln
Np t
N µp
Np t
N µp
Np
N yp
+
+
+
−
1
N
(Np ln µp + Nr ln µr ) ,
Nr
N yp
−
1
N
(Np ln µp + Nr ln yp ) ,
Nr t
N µr
−
1
N
(Np ln yp + Nr ln µr ) .
Nr t
N µr
The bias is dened as
t = I(µtg ) − [I(µtp , yp ) + I(yp , µtr )]
N
N
N
= ln Np µtp + NNr µtr − ln Np µtp + NNr yp − ln Np yp + NNr µtr + ln yp
N
N
N
N
= ln Np µtp + NNr µtr − ln Np µtp + NNr yp − ln Np yp + NNr µtr + ln Np yp + NNr yp
h i h i
N
N
N
N
= ln Np µtp + NNr µtr − ln Np µtp + NNr yp − ln Np yp + NNr µtr − ln Np yp + NNr yp .
|
{z
} |
{z
}
(a)
(b)
(a) ≥ 0 and (b) ≥ 0 if µtr ≥ yp . Since the logarithmic function is concave it must
t
t
t
that |(a)| ≥ |(b)| if µp ≤ yp . Taken together this implies that t ≥ 0 if µp ≤ yp ≤ µr .
Clearly,
hold
21
also
A.2
Poverty as a circumstance
Our decomposition can also be framed in terms of a standard regression-based approach towards
estimating IOp in which poverty is categorized as a circumstance variable. However, as outlined
in the following this has important drawbacks.
Table 1 shows a detailed decomposition of unfair inequality into its components for equivalized
disposable household income in Germany in 2011 (compare to Figures 1 and 2).
Table 1: Inequality Decomposition
Component
MLD
Ineq. in %
Decomposition Group/Type
I(Type)
I(Poor)
I(Rich)
I(Group)
0.010
0.009
0.056
0.050
8
7
45
40
Decomposition I(Group)
I(Group/Fair)
I(Group/Unfair)
0.027
0.005
0.018
22
4
14
Data Source: EU-SILC. Own Calculations.
I(Type) represents I(µt ), i.e. the share of inequality
explained
by circumstances. I(Poor) is indicative for
P
pt
f I(yi ), i.e. the share of inequality explained by
t tp
within-group inequality
below the poverty line. I(Rich)
P
is indicative for t ftr I(yirt ), i.e. the share of inequality explained by within-group P
inequality above the
poverty line. I(Group) denotes t ft I(µtp , µtr ), i.e.
the share of inequality explained by between-group inequality
between poor and rich. I(Group/Fair) denotes
P
t
t ft I(yp , µr ), i.e. the share of inequality explained
by between-group inequality between poor and rich to
the extent that the rich earn
P more than the poverty line.
I(Group/Unfair) denotes t ft I(µtp , yp ), i.e. the share
of inequality explained by between-group inequality between the poor and the rich to the extent that the poor
earn less than the required
minimum. represents the
P
approximation error t ft t originating from the inequality decomposition.
Note:
8% of total inequality can be explained by between-type inequality, 7% by inequality among
the poor, and 40% by between-group inequality among
P
and
R.
The latter component can
now again be decomposed into its (un-)fair components and the approximation error
conservative measure of unfair inequality thus equals to 19%.
Panel 1 of Table 1 gives a more detailed picture of inequality patterns in Germany.
22
.
The
Table 2: Poverty as a Circumstance: Smoothed Distributions (Germany)
Group Component
Types
Pop. in %
Mean
MLD
Ineq. in %
P0
P1
0
0
0
0
100
15
85
100
21706
8541
24034
21706
0.125
0.066
0.072
0.054
100
8
49
43
0.150
1.000
0.000
0.150
0.038
0.250
0.000
0.038
139
139
139
139
100
15
85
100
21706
20518
21916
21706
0.010
0.015
0.009
0.000
8
2
6
0
0.002
0.015
0.000
0.000
0.000
0.003
0.000
0.000
234
234
234
234
100
15
85
100
21706
8541
24034
21706
0.060
0.005
0.007
0.054
48
1
4
43
0.150
1.000
0.000
0.150
0.038
0.250
0.000
0.038
1023
1023
1023
1023
100
15
85
100
21706
8541
24034
21706
0.069
0.066
0.007
0.054
55
8
4
43
0.150
1.000
0.000
0.150
0.038
0.250
0.000
0.038
F(y)
All
Poor
Rich
Between
Φ1
All
Poor
Rich
Between
Φ2
All
Poor
Rich
Between
Φ3
All
Poor
Rich
Between
Data Source: EU-SILC. Own Calculations. The rst panel shows the outcome distribution F(y). The second panel
shows a smoothed distribution without treating poverty as a circumstance. In the third panel we complement the regular circumstance set with a binary poverty indicator. In the fourth panel we replace the binary poverty indicator with a
variable that is constant for all individuals in the rich group and equal to their outcome for all individuals in the poor
group. Additionally all circumstances are interacted with a binary poverty indicator. In each panel the rst line yields
the statistics for the entire distribution. The second and the third line show the statistics for the truncated distributions
of poor and rich, respectively. For the fourth line we have replaced the incomes of the poor and the rich by their respective group means. Column MLD shows the mean log deviation of the respective distribution. Column Ineq. in % gives
the percentage of the respective MLD-value in terms of total outcome inequality. P0 and P1 show the Foster et al. (1984)
poverty measure with α = 0 (poverty headcount) and α = 1 (poverty gap), respectively.
Note:
While the mean income
µ
amounted to
e21,706
in 2011, inequality attained a level of 0.125
as measured by the MLD. The share of poor people in the population
Foster-Greer-Thorbecke poverty measure (Foster et al., 1984) when
NP
N is a version of the
α = 0:
1 X yp − yi α
Pα =
.
N
yp
(17)
i∈P
P0 ,
or the poverty headcount, amounted to 15.0% in the Germany in 2011.
amounted to
3.8%.
P1 ,
the poverty gap,
Thus, while the number of poor appears quite sizable (15.0%), their average
shortfall from the poverty threshold seems rather small (3.8%). Zooming into the distributions of
P
and
R
respectively shows that the average income of
above the poverty line earned on average
yp
amounts to
100%,
while it is
0%
e24,034.
P
amounted to
e8,541
while households
By denition the poverty headcount below
for the remainder of the population. Inequality in Germany
is to a large extent explained by inequality among the rich (49%), while inequality among the
poor is only a minor contributor (8%). The between-group component, where the incomes of
poor and rich are replaced by the respective mean income of their group, accounts for another
43% of total inequality in Germany.
In this paper we construct the smoothed distributions in a non-parametric fashion (Checchi
and Peragine, 2010).
This is equivalent to estimating a fully saturated model in a regression
23
framework:
yi =
T
X
βt
Y
t=1
Cik + i
(18)
k
µt = β̂t .
(19)
µt ∈ Φ denotes the type-specic mean realization of outcome y for type t, whereas the distribution
Φ yields the outcome distribution as predicted by the interaction of all considered circumstances
C k ∈ Ω.
In Panel 2 of Table 1 we construct the smoothed distribution Φ1 in such a fashion by drawing
onto the set of circumstances outlined in section 3.2. Clearly, Φ1 is a mean-preserving spread
of F (y) as µ(Φ1 ) remains around e21,706. Remarkably, however, the mean-incomes of the
poor and the rich group move closely together when replacing individual income with typespecic average income.
This suggests, that poverty and opportunity-disadvantages are only
moderately correlated, i.e. based on the employed type partition, ending up in poverty seems
to be dependent on own eort instead of aliation with a specic type. This can also be seen
when looking at the prevalence of poverty in
Φ1 .
The value for
type earns an average income below the poverty threshold.
than I(F (y))
= 0.125.
More specically, only
P0
drops to 0.002 as almost no
Naturally, I(Φ1 )
= 0.010
is lower
8% (0.010/0.125) of the observed inequality can be
attributed to to inequality between types, whereas the remaining variation is due du dierential
eort. This is the classic Roemer measure of IOp, which also corresponds to I(Type) in Table 1.
Panel 3 shows analyses the smoothed distribution
Φ2 ,
for the construction of which the set
of circumstances is complemented with a binary measure of poverty.
If
yi < y p
this measure
10
assumes value one and zero otherwise. As a consequence the type partition almost doubles.
Φ2
When introducing being poor as a circumstance,
is again a mean preserving spread of
F (y).
Φ2 at yp yields truncated distributions that are mean-preserving spreads
of their analogues in F (y). As a consequence, the between-group inequality component, and the
measures of Pα are exactly as in F (y). Treating membership in P as a circumstance increases the
relative measure of IOp to 48%. The dierence between Φ1 (8%) and Φ2 (48%) is also equivalent
to I(Group) (40%) in Table 1. Thus, the overall increase of inequality from Φ1 to Φ2 is exclusively
In addition, separating
driven by assuming that all inequality between rich and poor is ethically problematic.
The last type partition we consider introduces a circumstance, that assumes the following
values:

y ,
i
yc =
yp ,
if
i ∈ P.
otherwise.
Remember that we estimate a fully saturated model. Hence,
poverty indicator. As a consequence,
fact that
yc
is a perfect predictor of
people and therefore
yi
µt = β̂t = yi
yc
is also interacted with the binary
for every poor person. This follows from the
in the poor group. To the contrary,
yc
is a constant for rich
Φ in this group is exclusively determined by the group-specic mean income
as determined by the initial circumstances.
of the rich group is the same for
Φ2
and
More succinctly: While the censored distribution
Φ3 , Φ3
10 Theoretically
does not replace
yi
by
µt
for the poor group.
it should double, however not all type cells are populated. For instance, there may be very
advantaged types, in which no individual falls below the poverty line yp .
24
Therefore, each
1023 (Table 2).
yi < yp
denes a separate type leading to an increase of the type partition to
As one would expect,
Φ3
again is a mean-preserving spread of
F (y)
for the
Φ2 , the between-group
component and the measures of Pα are exactly equal to their analogues in F (y). Additionally, the
MLD in the rich group remains unaltered at 0.007. Thus, the increase in IOp from 48% to 55%,
overall distribution and both truncated distributions.
Therefore, as in
is exclusively driven by attributing all variation in the poor group to the ethically objectionable
part of inequality. The dierence between
Φ2
(48%) and
Φ2
(55%) is also equivalent to I(Poor)
(7%) in Table 1.
In conclusion, when treating poverty as a circumstance the decomposition outlined in the
main section of this paper can be partly implemented. This requires augmenting the set of regular
circumstances by a circumstance variable that captures all incomes below the poverty line and
remains constant otherwise, as well as a binary poverty-indicator that is interacted with the full
set of circumstances. A drawback of the regression approach towards constructing the smoothed
distribution is that it does not allow for a more nuanced treatment of inequalities among
P.
Total unfair inequality equals
55%,
R
and
which is the result of adding I(Type)+I(Group)+I(Poor)
(see Table 1). By default, I(Group) is allocated to the morally objectionable share of inequality,
whereas we cannot make a dierentiation between its fair and unfair components.
25
A.3
Incentive Compatibility
The incentive compatability constraint aims to discourage the mimicking of types. Therefore the
following must hold:
0 U y(a0 ), z(aa )
≤ U y(a), z(a)
∀a ∈ [a, a].
a
Setting
∂U (a,a0 )
∂a0
= 0,
one gets:
−a(Uy (·)(1 − T 0 )(·) +
Uz (·) 0
a )l (a)
From individual utility maximization we know that
= Uy (·)(1 − T 0 (·))l(a)
Uy (·)(1 − T 0 )(·) = (1 − T 0 ) =
(20)
Uz (·)
a
= − ρ(·)
a .
Hence equation 20 reduces to
Uy (·)(1 − T 0 (·))l(a) = −
26
ρ0 (·) l(a)
= 0.
a
(ψ(a))
A.4
Solution Procedure Optimal Taxation Model
It is a standard assumption that
ψ(a)
is not binding for
a
and
a,
ψ(a) = ψ(a) = 0
i.e.
(these are
the so called transversality conditions). Hence integrating the rst FOC yields the following:
Z
a
ψ(a) =
ψ 0 (a)da
a
Z
=
a
(λ − ∆0y (·))f (a)da
a
Recall that
ρ0 (·)
a
= 1 − T 0 (·).
Thus we can reformulate the second FOC:






∂H
ρ0 (·)
0
0
0
= ∆y (·)ρ (·) + ∆l (·) + λa (1 − a ) f (a) +
∂l(a) 
| {z }


ψ(a)
a
ρ00 (·) l(a) + ρ0 (·) = 0
T 0 (·)
The elasticity of labor supply is dened as
that
(1 − T 0 (·))a = w = ρ0 (·).
(a) =
w ∂l(a)
w
l(a) ∂w . We know that l(a)
=
(1−T 0 (·))a
and
l(a)
Hence:
(a) =
1−T 0 (·)a 1
l(a)
ρ00 (·)
⇐⇒ l(a)ρ00 (·) =
ρ0 (·)
(a)
Thus we can reformulate the second FOC:
∂H
= ∆0y (·)ρ0 (·) + ∆0l (·) + λa(T 0 (·)) f (a) +
∂l(a)
ψ(a)
a
00
ρ (·) l(a) + ρ0 (·) = 0
|
{z
}
ρ0 (·)
0
(a) +ρ (·)
Using again
ρ0 (·)
a
= 1 − T 0 (·),
we can solve for
T 0 (·)
h 0 T 0 (·)
∆y (·)
1
=
−
ρ0 (·) +
λ
ρ0 (·)
1 − T 0 (·)
Now, substituting
ψ
and divide by
∆0l (·)
∆0y (·)
i
+
1
(a)
1 − T 0 (·)
+1
by the reformulated rst FOC yields equation 10.
27
to get:
ψ(a)
aλf (a)