Fairness and the proportionality principle online appendix
Alexander W. Cappelen and Bertil Tungodden∗
November 12, 2016
Abstract
In this online appendix, we provide some further results on how
our formulations of the egalitarian and liberal ideals relate to other
conditions in the literature.
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Additional analysis
We first consider the relationship between EEE and the most standard formulation of the egalitarian requirement in the literature (Fleurbaey, 2008):
Equal Income for Equal Effort (EIEE). For any a ∈ ΩN , and any k, l ∈
E
N , where aE
l = ak , Fl (a) = Fk (a).
It follows straightforwardly that EEE implies EIEE.
Observation 1: If a redistribution mechanism F satisfies EEE, then it
satisfies EIEE.
Proof. The observation follows directly from the fact that a and ã may
be the same situation in EEE.
∗
Both authors: Norwegian School of Economics, Bergen, Norway. e-mail: [email protected] and [email protected]. We have received extremely useful
comments and suggestions from two anonymous referees. The project was financed by
support from the Research Council of Norway, research grant 236995 and administered by
The Choice Lab.
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It turns out that EIEE and EEE only differ when the distribution of
talent is different at different effort levels. To establish this more formally,
we introduce the domain restriction that talent and effort are independently
distributed, that is, if we divide the population into groups of people with
the same talent level, then the fraction of people exercising any specific level
of effort will be the same within each group.‘1 To do this, let the fraction of
the population with talent level tk exercising effort level ej in situation a be
denoted da (tk , ej ).
Independent Distribution Restricted Domain Richness. The set of possible
characteristics profiles of society is given by Ω̃N = {a ∈ ΩN | da (tk , ej ) =
da (tl , ej ) for all k, l ∈ ΩT and for all j ∈ ΩE }.
This assumption implies that for any given effort level ej and any two
talent levels, tk and tl , we have that da (tk , ej ) = da (tl , ej ).
Further, we introduce the uncontroversial assumption that the redistribution mechanism should be anonymous.
Anonymity (A). For any a, ã ∈ ΩN and permutation function π : N → N ,
if ai = ãπ(i) , ∀i ∈ N, then Fi (a) = Fπ(i) (ã).
We can now establish the following observation.
Observation 2: Given independent distribution restricted domain richness: If a redistribution mechanism F satisfies A, then it satisfies EEE iff
it satisfies EIEE.
Proof. The only-if-part follows from Observation 1. Hence, we only
prove the if-part.
(1) Consider any a, ã ∈ Ω̃N and some l, m ∈ N , where ãT = aT , E(ã) =
E
l
E
E(a), and ãE
l = am = e ∈ Ω .
(2) By the domain assumptionP
and E(ã) = E(a),
it follows that dã (tj , el ) =
P
da (tj , el ) for all j ∈ ΩT and that i∈N f (ãi ) = i∈N f (ai ).
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An alternative approach is suggested by Roemer (1996), who focus on whether people
with different backgrounds are at the same percentile in the distribution of efforts within
their group.
2
ãTr
E
(3) By the domain restriction, there exists r ∈ N such that ãE
r = ãm and
T
= ãl . By EIEE, Fm (ã) = Fr (ã).
(3) By (1), there exists a permutation function π such that ãi = aπ(i) , ∀i 6=
r, l π(r) = l, and aπ ∈ Ω̃N . By A, Fi (ã) =Fπ(i) (aπ ), ∀i ∈ N . Hence, we
have that Fr (ã) =Fπ(r) (aπ ) = Fl (a). Taking into account (2), it follows that
Fm (ã) = Fl (a). By (1), we then have that P Fm (ã)
= P Fl (a)
and the
i∈N f (ãi )
i∈N f (ai )
result follows.
It is interesting to note that we can characterize the generalized proportionality mechanism by EIEE and NEUDT if we strengthen our domain
restriction.
Observation 3: Given independent distribution restricted domain richness, a redistribution mechanism F satisfies EIEE and NEUDT iff F = F P E
Proof. The proof follows immediately from combining Proposition 1 and
Observation 2.
We now turn to a discussion of how NEUDT and NEUT relate to other
liberal conditions in the literature. In the spirit of NEUDT, Boadway et al.
(2002) propose more generally (in a slightly different framework) that there
should never be any redistribution between effort levels.
No Equalization
Between P
Effort Levels (NEBEL). For all a ∈ ΩN and all
P
j ∈ ΩE (a), i∈N j (a) Fi (a) = i∈N j (a) f (ai ).
NEBEL implies NEUDT, but it is not consistent with any reasonable
framework for redistribution. To see this, we introduce the following two
minimal conditions:
E
Minimal Redistribution (MR). There exists a ∈ ΩN where aE
i = aj ,
∀i, j ∈ N and where for some k ∈ N , Fk (a) > f (ak ).
Non Negative Reward (NNR). For any a, ã ∈ ΩN and k ∈ N , if ãE
i =
E
E
E
ai , ∀i 6= k and ãk > ak , then Fk (ã) ≥ Fk (a).
MR imposes a minimal demand for redistribution, which clearly should
be part of the egalitarian ideal. It states that there should exist at least
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one situation where all individuals exercise the same positive effort level and
there is one individual who receives a positive transfer.
NNR introduces the trivial requirement that the post-tax income of an
individual should not decrease if he unilaterally chooses to exercise more
effort.
We can now establish the following impossibility result:
Observation 4: If the pre-tax income function f is continuously increasing in effort, then there does not exist any redistribution mechanism F that
satisfies NEBEL, MR, and NNR.
Proof. (1) Suppose that F satisfies MR. We then know that there exists
E
a ∈ ΩN , where aE
i = aj , ∀i ∈ N and for some k ∈ N , Fk (a) > f (ak ).
E
(2) By the continuity of f , there exists ã ∈ ΩN such that ãE
k > ak , f (ãk ) <
E
E
Fk (a), and ãi = ai , ∀i 6= k. By NEBEL, Fk (ã) = f (ãk ) < Fk (a). However,
this violates NNR and the result follows.
The problem with NEBEL is that it prohibits redistribution between
effort levels even when the talent composition varies across effort levels. Another condition, proposed by Fleurbaey (2008), has similar implications.
Equal Treatment for Equal Class Circumstances (ETTCC). For any a ∈
N
l j
l k
l
T
Ω
:
∀ej , ek ∈ ΩE (a) , then
P if da (t , e )P= da (t , e ), ∀tP∈ Ω (a) and P
j k
i∈N j (a) Fi (a) −
i∈N j (a) f (ai ) =
i∈N k (a) Fi (a) −
i∈N k (a) f (ai )), ∀e , e ∈
ΩE (a).
This principle is logically weaker than NEBEL, but logically stronger
than NEUDT. It captures the idea that the treatment of a particular class
of individuals should be independent of their effort and only related to their
talent. Hence, it may be seen as an application of the standard natural
reward condition to class circumstances.
Equal Treatment for Equal Talent (ETET). For all a ∈ ΩN and j, k ∈ N ,
where aTj = aTk , Fj (a) − f (aj ) = Fk (a) − f (ak ) .
We view both ETET and ETTCC to be too strong interpretations of the
liberal ideal, since they both prohibit redistribution in cases where there are
differences in the talent composition across effort levels. This seems to go
beyond what the liberal ideal of neutrality should imply.
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The following result summarizes the logical structure between the various
liberal principles:
Observation 5: For any redistribution mechanism F, NEBEL =⇒ ET T CC ⇒
N EU DT ⇒ N EU T
Proof. The proofs are straightforward.
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