Discrimination-Blind versus Discrimination-Sighted
Optimal Income Taxation
Alan Krause
7 April 2008
Abstract
This paper examines optimal nonlinear income taxation in an economy where
some individuals are discriminated against based on an ascriptive characteristic,
e.g., race or gender. We assume that the government is not permitted to tackle
discrimination by basing the tax system on the ascriptive characteristic, but it
may or may not be able to use information on discrimination when designing a
skill-based tax system. In our model, discrimination takes the simple form of
individuals with the same skill level receiving di¤erent wages. We show that if
the government cannot use information on discrimination (discrimination-blind
optimal taxation), it is optimal for high-skill discriminated-against individuals
to pretend to be low-skill, and the government’s budget constraint is violated.
If the government can use information on discrimination (discrimination-sighted
optimal taxation), all individuals self-select the tax contract intended for them
and the government’s budget constraint is satis…ed. We also derive the optimal
marginal tax rates under discrimination-blind and discrimination-sighted optimal
taxation, which in both cases happen to di¤er substantially from the canonical
Mirrlees/Stiglitz results. Our analysis also shows how anti-discrimination laws
can usefully complement discrimination-sighted optimal taxation.
Keywords: Income taxation, wage discrimination, horizontal equity.
JEL classi…cations: D6, H21, J7.
Department of Economics and Related Studies, University of York, Heslington, York, YO10 5DD,
U.K. Email: ak519@york.ac.uk. I thank Peter Simmons and workshop participants at the University of
York for many helpful comments. I also thank Dustin Chambers for his guidance on some of the U.S.
anti-discrimination legislation. Any errors are my responsibility.
1
Introduction
Despite the successes of anti-discrimination laws in reducing the wage gap between
blacks and whites and between men and women, wage discrimination remains pervasive
in most countries.1 The purpose of this paper is to investigate the implications of wage
discrimination for optimal income taxation, given that the government cannot base
the tax system on ‘irrelevant’ characteristics such as race or gender. The prevailing
political climate and social norms embedded in the principle of horizontal equity deem
that individuals who are in all relevant respects identical should be treated equally.
Thus income tax systems tend to be designed to redistribute from the highly-skilled to
the lesser-skilled, with anti-discrimination legislation and related programmes being the
policy instruments used to address discrimination.
However, even if the government cannot base the tax system on ascriptive characteristics such as race or gender, it may or may not be able to use information on wage
discrimination when designing a skill-based tax system. There are at least three reasons why the government may not be able to use information on wage discrimination.
Firstly, such information may become unavailable. For example, a proposed constitutional amendment (known as Proposition 54) that appeared at the California statewide
special election of October 2003 would have prohibited (with some exemptions) state and
local governments from collecting data on a person’s race, ethnicity, colour, or national
origin.2 Although Proposition 54 was clearly defeated,3 its supporters argue that such
measures are a necessary step towards achieving a truly colour-blind society. Secondly,
using any form of race-based or gender-based information for taxation purposes may
be considered immoral. However, the moral objection is not so obvious here, since the
tax system would still discriminate on the basis of skill alone. Individuals of equal skill
would therefore pay the same taxes, and the principal of horizontal equity would be
1
See Altonji and Blank [1999]. Anti-discrimination laws exist in many countries banning discrimination based on factors such as race, gender, age, disability, pregnancy, nationality, and religion. Holzer
and Neumark [2000] provide a comprehensive survey of the e¤ects of anti-discrimination legislation,
a¢ rmative action, and related programmes on labour market outcomes.
2
See, for example, Fryer and Loury [2005] for further discussion.
3
Proposition 54 was defeated with 36 percent of voters in favour and 64 percent of voters against.
2
satis…ed. Thirdly, the government may be reluctant to use information on wage discrimination when designing a tax system because it could be interpreted as the government
admitting that its anti-discrimination policies are ine¤ective. This may have negative
political repercussions for the government.
In this paper, we do not take an ex ante position on whether or not the government
should use information on wage discrimination when designing a skill-based tax system.
Instead, we characterise the optimal tax system under both scenarios and examine how
they compare. We refer to the case when the government cannot use information on
wage discrimination as “discrimination-blind optimal taxation”, and when it can as
“discrimination-sighted optimal taxation”. We do not model the wage discrimination
process, but simply take its existence in the economy as given. Wage discrimination
in our model takes the simple form of individuals with the same skill level receiving
di¤erent wages.
Our analysis reveals some clear-cut factors in favour of discrimination-sighted optimal
taxation. That is, the government should not ignore the problem of wage discrimination when designing a tax system even if the tax system must be based on skill alone.
The reasons are as follows. Firstly, the government’s budget constraint is violated under discrimination-blind optimal taxation.4 This is because it is optimal for high-skill
discriminated-against individuals to pretend to be low-skill and receive a transfer from
the government, while the government has designed the tax system on the expectation
that all high-skill individuals will pay taxes. This also implies, somewhat ironically,
that discrimination-blind optimal taxation violates the principle of horizontal equity ex
post, since high-skill individuals end up receiving di¤erent tax treatments. Secondly,
we show that any reduction in wage discrimination (say through anti-discrimination
laws) increases social welfare under discrimination-sighted optimal taxation, but may
actually decrease social welfare under discrimination-blind optimal taxation. Thirdly,
because it is optimal for high-skill discriminated-against individuals to pretend to be
4
Since our model does not specify how the budget de…cit is …nanced, our analysis can be viewed as
partial equilibrium rather than general equilibrium in nature. We do not seek a resolution to the de…cit
problem here, as our intention is to simply highlight the issues that arise when the government can and
cannot use information on wage discrimination when designing a skill-based tax system.
3
low-skill under discrimination-blind optimal taxation, we argue that this adversely affects their incentives to invest in education and other skill-enhancing activities. We also
derive the implicit marginal tax rate faced by each type under discrimination-blind and
discrimination-sighted optimal taxation, which in both cases happen to di¤er substantially from the canonical Mirrlees/Stiglitz results.5
The issues addressed in this paper have received little attention in the literature.
The paper most closely related to ours is an interesting contribution by Blumkin, et al.
[2007], who show that taxation may be more e¤ective than anti-discrimination laws as
a policy instrument to tackle discrimination. In their model, there are two individuals
with equal skills, but due to discrimination the ‘tall’worker receives a higher wage than
the ‘short’ worker. Because height is observable,6 the government could implement a
system of personalised (height-based) lump-sum taxes. But as Blumkin, et al. [2007]
point out, such a system would clearly violate the principle of horizontal equity. They
therefore propose a ‘height-blind’tax system, in which the government o¤ers two distinct
tax contracts— with one intended for the tall worker and the other intended for the short
worker— such that each worker is willing to self-select the tax contract intended for them.
While such a tax system might be suitable for their purposes, it is explicitly based on
the ascriptive characteristic, so one could argue that it violates horizontal equity ex ante.
It also violates horizontal equity ex post, as the two identical individuals (i.e., identical
in all ‘relevant’respects) choose di¤erent tax treatments.7
To a lesser extent, our paper is also related to the literature on taxation and tagging; see for example Akerlof [1978] and more recently Boadway and Pestieau [2006] and
Mankiw and Weinzierl [2007]. This literature is concerned with how redistributive taxation can be improved by using information on some observable characteristic, or ‘tag’,
which is known to be correlated with unobservable skill. Boadway and Pestieau [2006]
note that some may …nd the use of tagging for taxation purposes objectionable since
5
See Mirrlees [1971] and Stiglitz [1982]. The canonical results are that the highest-skill type should
face a zero marginal tax rate, while the lowest-skill type should face a positive marginal tax rate.
6
As is race, gender, etc.
7
If the principle of horizontal equity is interpreted as implying that same-skill individuals should end
up with the same level of utility (as opposed to the same tax treatment), the Blumkin, et al. [2007]
approach still violates horizontal equity since the tall worker obtains more utility than the short worker.
4
it can lead to a violation of horizontal equity. Mankiw and Weinzierl [2007] present
empirical evidence that shows a strong correlation between height and wages. They
therefore argue that the tax system should include a subsidy for short taxpayers and a
surcharge on tall taxpayers. But they also acknowledge that a ‘height tax’would clearly
violate horizontal equity. Our paper di¤ers from the literature on tagging in that there
is no correlation between the ascriptive characteristic and skill, and some individuals are
discriminated against based on this ascriptive characteristic.
The remainder of the paper is organised as follows. Section 2 describes the economic environment we consider, while Section 3 characterises discrimination-blind and
discrimination-sighted optimal taxation. The question arises as to how social and individual welfare compares under discrimination-blind and discrimination-sighted optimal
taxation, but such comparisons generally require additional assumptions regarding the
form of the utility function and the distributions of wages and skills. Therefore, in Section 4 we present a numerical example using a log-linear utility function to illustrate how
social and individual welfare compares as the degree of wage discrimination increases.
Section 5 contains some closing remarks, and all proofs are relegated to an appendix.
2
The Economy
There are four individuals in the economy, who are distinguished by their skill levels in
employment and an ascriptive characteristic, e.g., race or gender. Two individuals are
low-skill with a skill level equal to s1 > 0, and two individuals are high-skill with a skill
level equal to s2 > s1 . The wages of the two low-skill individuals are w1a and w1b with
w1a > w1b > 0, so the b-type is discriminated against. Similarly, the wages of the two
high-skill individuals are w2a and w2b with w2a > w2b . We also assume that w2b > w1a , so
even though the high-skill b-type is discriminated against, he/she still receives a higher
wage than the low-skill individuals.8 The individuals share the common utility function
U : R2++ ! R which for analytical convenience takes the quasi-linear form u(cji )
8
lij ,
We think w2a > w2b > w1a > w1b is representative of the pattern of wage discrimination in most
countries. For example, a male lawyer may earn more than a female lawyer, and a male truck driver
may earn more than a female truck driver, but a female lawyer earns more than a male truck driver.
5
where cji is the wij -type’s consumption and lij is the wij -type’s labour supply. The function
u( ) is increasing and strictly concave, and the pre-tax income of the wij -type is denoted
by yij = wij si lij . Thus if both i-type individuals work the same hours (lia = lib ), they
supply the same amount of e¤ective labour (si lia = si lib ) as their skill levels are the same,
but the b-type earns a lower pre-tax income due to wage discrimination.
The government can design a tax system that discriminates between high-skill and
low-skill individuals, as redistribution from the highly-skilled to the lesser-skilled is considered ethically acceptable (and indeed desirable) in most countries. The government
cannot base the tax system on the ascriptive characteristic, but it may or may not be able
to take wage discrimination into account when designing a skill-based tax system. Under
discrimination-blind optimal taxation, the government must design the tax system using
aggregate data. That is, it knows there are two low-skill and two high-skill individuals
in the economy, who respectively receive the wages w1 and w2 with w1 < w2 . But the
government does not know (or must ignore) the reality that these wages are average
values: w1 = 21 (w1a + w1b ) and w2 = 12 (w2a + w2b ). For the purposes of this paper, one can
think of w1 and w2 as being …xed by a linear production technology, with actual wages
varying around these …xed averages due to discrimination. Under discrimination-sighted
optimal taxation, the tax system must still be based on skill alone, but the government
can use the information that there are two high-skill and two low-skill individuals in the
economy, and due to discrimination their wages are w2a > w2b > w1a > w1b .
3
3.1
Optimal Taxation
Discrimination-Blind Optimal Taxation
If the government cannot use information on wage discrimination when designing the
tax system, the government can be described as choosing a ‘tax contract’ hyi ; ci i for
6
each skill type9 to maximise:
2 u(c1 )
y1
+ 2 u(c2 )
w1 s1
y2
w2 s2
(3.1)
0
(3.2)
subject to:
2 (y1
u(c1 )
u(c2 )
c1 ) + 2 (y2
y1
w1 s1
y2
w2 s2
c2 )
u(c2 )
u(c1 )
y2
w1 s1
y1
w2 s2
(3.3)
(3.4)
The objective function (3.1) is a utilitarian social welfare function, with the individual
utility functions written in terms of the choice variables yi and ci . Equation (3.2) is
the government’s budget constraint, while (3.3) and (3.4) are incentive-compatibility
constraints for the low-skill and high-skill types, respectively. The government knows
the distribution of skills and the form of the common utility function, but an individual’s
skill type is private information. The government must therefore satisfy the incentivecompatibility constraints to try to induce the two high-skill individuals to choose hy2 ;
c2 i and the two low-skill individuals to choose hy1 ; c1 i. Note that under discriminationblind optimal taxation, equations (3.1) – (3.4) are necessarily formulated in terms of
aggregate data regarding the population and wages.
The programme (3.1) – (3.4) is a simple and well-studied problem in the theory of
optimal nonlinear income taxation, whose solution is characterised by:
Lemma 1 Under discrimination-blind optimal taxation, hy2 ; c2 i
hy1 ; c1 i, (3.2) and
(3.4) are binding, while (3.3) is slack.
A potential problem the government faces in our model is that the individuals might
not choose the tax contracts intended for them as their wages are, in reality, di¤erent
from w1 and w2 . And the government knows of this problem only after o¤ering hy2 ; c2 i
9
A tax contract consists of pre-tax income and post-tax income (which is equal to consumption) for
each skill type. The di¤erence between pre-tax income and consumption is total taxes paid (or transfers
received). While we do not observe such an income tax system in practice, the ‘Revelation Principle’
implies that any tax system (or any mechanism) can be replicated by an incentive-compatible direct
mechanism.
7
and hy1 ; c1 i. Indeed, we obtain the following result:
Proposition 1 Under discrimination-blind optimal taxation, the w2a -type chooses hy2 ;
c2 i, while the w2b -, w1a -, and w1b -types choose hy1 ; c1 i.
Thus the w2b -type is better-o¤ pretending to be low-skill rather than choosing the tax
contract intended for high-skill individuals. The government o¤ers hy2 ; c2 i on the basis
of a high-skill wage of w2 , but the w2b -type’s wage is actually less than w2 . Therefore,
the w2b -type would have to work longer to earn y2 than the government intends high-skill
individuals to work, and it so happens that the w2b -type is better-o¤ working less and
earning y1 even though the consumption associated with y1 is less than that associated
with y2 . The preference the w2b -type has for hy1 ; c1 i follows from the redistributive goals
of the government (as re‡ected in the social welfare function, which is strictly concave in
the choice variables). The intended redistribution from high-skill to low-skill individuals
creates an incentive for the hypothetical w2 -type to mimic the hypothetical w1 -type, but
not vice versa. As a result, (3.4) is binding while (3.3) is slack. The hypothetical w2 -type
is therefore only indi¤erent between choosing hy2 ; c2 i and hy1 ; c1 i, rather than having
a strict preference for hy2 ; c2 i. And anyone with a wage lower than w2 strictly prefers
hy1 ; c1 i. Note that an ironic implication of Proposition 1 is that discrimination-blind
optimal taxation violates the principle of horizontal equity ex post, as the two high-skill
individuals end up with di¤erent tax treatments.
Our second proposition states the pattern of implicit marginal tax rates:
Proposition 2 Let M T Rij denote the marginal tax rate faced by the wij -type. Under
discrimination-blind optimal taxation: (i) M T R2a > 0, M T R2b > 0, M T R1a > 0, and
M T R1b cannot be signed, and (ii) M T R2b > M T R1a > M T R1b .
The positive marginal tax rate faced by the w2a -type follows from w2a > w2 , so when
the w2a -type chooses hy2 ; c2 i he/she works less than the government intends high-skill
individuals to work. The w2a -type’s labour supply is therefore implicitly distorted downwards through a positive marginal tax rate. Similarly, the w2b - and w1a -types choose hy1 ;
c1 i, and because s2 w2b > s1 w1a > s1 w1 they work less than the government intends, implying that they also face positive marginal tax rates. Since s2 w2b > s1 w1a , the w2b -type
works less than the w1a -type to earn y1 , so the w2b -type faces a higher marginal tax rate
8
than the w1a -type. The w1b -type also chooses hy1 ; c1 i and has to work longer than both
the w2b - and w1a -types to earn y1 , since s2 w2b > s1 w1a > s1 w1b . The marginal tax rate
faced by the w1b -type is therefore lower than those faced by the w2b - and w1a -types. Moreover, it is not necessarily positive because w1 > w1b . Under discrimination-blind optimal
taxation, the government imposes a positive marginal tax rate on the hypothetical w1 type in order to relax the hypothetical w2 -type’s incentive-compatibility constraint.10
Thus the hypothetical w1 -type’s labour supply is distorted downwards. However, since
w1 > w1b , the w1b -type may have to work longer to earn y1 than the government intends.
The w1b -type may therefore face a negative marginal tax rate. The w1b -type could also
possibly face a zero marginal tax rate, if it just so happens that his/her labour supply
need not be distorted to earn y1 . For these reasons, the marginal tax rate faced by the
w1b -type cannot be signed.
A further problem with discrimination-blind optimal taxation is that it results in a
budget de…cit:
Proposition 3 Under discrimination-blind optimal taxation, the government’s budget
de…cit is equal to 2(c1
y1 ) > 0.
The government o¤ers hy1 ; c1 i and hy2 ; c2 i on the expectation that two individuals will
choose hy1 ; c1 i and two individuals will choose hy2 ; c2 i, such that the government’s budget
constraint (3.2) is satis…ed. But from Proposition 1 we know that three individuals
will choose hy1 ; c1 i and only one will choose hy2 ; c2 i. And because the tax system is
redistributive (see the discussion of Proposition 1), y1 < c1 and y2 > c2 . Accordingly,
three individuals receive a subsidy and only one individual is taxed, which results in a
budget de…cit.
3.2
Discrimination-Sighted Optimal Taxation
If the government can use information on wage discrimination when designing a skillbased tax system, it chooses hy1 ; c1 i and hy2 ; c2 i to maximise:
u(c1 )
y1
+ u(c1 )
w1a s1
y1
+ u(c2 )
w1b s1
10
y2
+ u(c2 )
w2a s2
y2
w2b s2
(3.5)
This is the well-known Mirrlees/Stiglitz result that the lowest-skill type should face a positive
marginal tax rate.
9
subject to:
2 (y1
c1 ) + 2 (y2
c2 )
0
(3.6)
u(c1 )
y1
w1a s1
u(c2 )
y2
w1a s1
(3.7)
u(c1 )
y1
w1b s1
u(c2 )
y2
w1b s1
(3.8)
u(c2 )
y2
w2a s2
u(c1 )
y1
w2a s2
(3.9)
u(c2 )
y2
w2b s2
u(c1 )
y1
w2b s2
(3.10)
The restriction that the tax system be based on skill alone implies that the two highskill individuals are ‘pooled’into receiving hy2 ; c2 i, and the two low-skill individuals are
pooled into receiving hy1 ; c1 i. However, because the government can use information on
wage discrimination, equations (3.5) – (3.10) incorporate the data that there are four
individuals in the economy who receive distinct wages. Equation (3.5) is the utilitarian
social welfare function, while (3.6) is the budget constraint. The incentive-compatibility
constraints (3.7) and (3.8) require that both low-skill individuals be willing to choose
hy1 ; c1 i, while the incentive-compatibility constraints (3.9) and (3.10) require that both
high-skill individuals be willing to choose hy2 ; c2 i. Thus under discrimination-sighted
optimal taxation, each individual will choose the tax contract intended for them and the
government’s budget constraint will be satis…ed. It follows that discrimination-sighted
optimal taxation satis…es the principle of horizontal equity, ex ante and ex post.
The solution to the programme (3.5) –(3.10) is characterised by:
Lemma 2 Under discrimination-sighted optimal taxation, hy2 ; c2 i
hy1 ; c1 i, (3.6) and
(3.10) are binding, while (3.7), (3.8) and (3.9) are slack.
The pattern of implicit marginal tax rates is given by:
Proposition 4 Under discrimination-sighted optimal taxation: (i) M T R2a > 0, M T R2b <
0, M T R1a > 0, and M T R1b cannot be signed, and (ii) M T R1a > M T R1b .
The intuition behind Proposition 4 is similar to that behind Proposition 2, the only
di¤erence being that under discrimination-sighted optimal taxation the w2b -type faces a
negative marginal tax rate. This follows from the fact that the w2b -type now chooses hy2 ;
10
c2 i, while the government o¤ers hy2 ; c2 i on the basis of a ‘weighted average’11 high-skill
wage that is greater than w2b . The w2b -type therefore has his/her labour supply distorted
upwards to earn y2 . The pattern of marginal tax rates for the other types is the same
as under discrimination-blind optimal taxation, and for similar reasons. That is, the
w2a -type’s labour supply is distorted downwards since w2a is greater than the ‘weighted
average’high-skill wage. The w1a -type’s labour supply is distorted downwards because w1a
is greater than the ‘weighted average’low-skill wage (and to relax the binding incentivecompatibility constraint). And the w1b -type’s marginal tax rate cannot be signed because
w1b is less than the ‘weighted average’low-skill wage.
3.3
Further Discussion
The main distinction between discrimination-blind and discrimination-sighted optimal
taxation is that the former creates a disincentive for high-skill discriminated-against
individuals to work and earn higher incomes. That is, under discrimination-blind optimal taxation, high-skill discriminated-against individuals are better-o¤ working less
and earning less. This can be undesirable for a number of reasons. For example, in the
real world where innate skill is augmented by education and/or work experience (e.g.,
through learning-by-doing), high-skill discriminated-against individuals have less incentive to invest in education, and working less may lead to their skills deteriorating over
time. Moreover, unless wage discrimination can be completely eradicated (i.e., unless
anti-discrimination laws are fully e¤ective), the incentive high-skill discriminated-against
individuals have to mimic low-skill individuals under discrimination-blind optimal taxation remains (see the discussion following Proposition 1).
As a result of this mimicking behaviour, social welfare under discrimination-blind
optimal taxation may be increasing or decreasing in the degree of wage discrimination:
Proposition 5 Let w1a = w1 + "1 , w1b = w1
"1 , w2a = w2 + "2 , and w2b = w2
"2 where
"1 > 0 and "2 > 0. Under discrimination-blind optimal taxation, the e¤ect on social
welfare of increasing "1 and "2 is ambiguous.
11
Under discrimination-sighted optimal taxation, the government uses (wia si ) 1 + (wib si ) 1 for each
skill type when choosing the tax contracts, rather than 2(wi si ) 1 as it does under discrimination-blind
optimal taxation (see appendix). It can be shown that (wia si ) 1 + (wib si ) 1 > 2(wi si ) 1 , hence the
term ‘weighted average’.
11
On the other hand, under discrimination-sighted optimal taxation society always
bene…ts from a reduction in wage discrimination:
Proposition 6 Under discrimination-sighted optimal taxation, social welfare is decreasing in "1 and "2 .
To interpret Proposition 5, …rst note that an increase in wage discrimination under
discrimination-blind optimal taxation does not change the government’s optimal choices
of hy1 ; c1 i and hy2 ; c2 i because average wages are …xed. Now consider an increase in w1a
and a matching decrease in w1b . This makes the w1a -type better o¤ and the w1b -type worse
o¤. However, because the w1b -type works longer to earn y1 than does the w1a -type, the
w1b -type’s utility falls by more than the w1a -type’s utility increases, the net e¤ect being
a decrease in social welfare. The same logic would apply to the w2a - and w2b -types for
an increase in w2a and a matching decrease in w2b if the w2b -type were to choose to earn
y2 (as the w2b -type would then work longer than the w2a -type). But because the w2b -type
chooses to earn y1 , he/she will typically work less than the w2a -type. As a result, the
w2a -type’s utility increases by more than the w2b -type’s utility falls, making it possible for
an increase in wage discrimination to increase social welfare under discrimination-blind
optimal taxation.
Under discrimination-sighted optimal taxation, an increase in wage discrimination
always reduces social welfare (Proposition 6) because there is no mimicking. The w1b type therefore works longer than the w1a -type, and the w2b -type works longer than the
w2a -type. As a result, decreases in w1b and w2b reduce the utilities of the w1b - and w2b -types
by more than increases in w1a and w2a raise the utilities of the w1a - and w2a -types. The net
e¤ect is a fall in social welfare. Finally, we note that an increase in wage discrimination
under discrimination-sighted optimal taxation does change the government’s optimal
choices of hy1 ; c1 i and hy2 ; c2 i. However, by the Envelope Theorem the ‘indirect e¤ect’
of changes in hy1 ; c1 i and hy2 ; c2 i on social welfare can be ignored when considering
small increases in wage discrimination.
12
4
A Numerical Example
Deciding which of discrimination-blind or discrimination-sighted optimal taxation is better from a social and individual welfare point of view requires a comparison of the solutions to each approach. In general, such comparisons depend upon the exact form
of the utility function and the distributions of wages and skills. Therefore, we present
a numerical example using the often-employed log-linear form of the utility function,
and we examine how social and individual welfare compares as the degree of wage discrimination increases. Speci…cally, we assume that the utility function takes the form
p
lncji lij , and that s1 = w1 = 2 and s2 = w2 = 2. These wages and skills are chosen
so that the average e¤ective wage of the high-skill individuals (w2 s2 ) is double that of
the low-skill individuals (w1 s1 ). We then examine the e¤ects of increasing the di¤erence
between wib and wia , holding skills and average wages constant.
While the numerical example enables us to make social and individual welfare comparisons, we note at the outset that an equal-footing comparison between discriminationblind and discrimination-sighted optimal taxation is impossible since the former violates the government’s budget constraint. It should therefore be kept in mind that
discrimination-blind optimal taxation has an ‘unfair advantage’in this respect.
Table 1 presents the solution to the optimal tax problem if there were no discrimination in the economy,12 while Tables 2 – 4 compare discrimination-blind and
discrimination-sighted optimal taxation as the degree of discrimination increases. Table
2 presents the results when the degree of discrimination is relatively small, in that the
w1a - and w2a -type’s wages are 2.5 percent above the respective averages, while the w1b and w2b -type’s wages are 2.5 percent below the respective averages. Table 3 presents the
results for the 5 percent case and Table 4 presents the results for the 10 percent case.
When discrimination is present (Tables 2 – 4), social welfare under discriminationblind optimal taxation is measured higher than under discrimination-sighted optimal
taxation, but this outcome needs to be quali…ed since discrimination-blind optimal tax12
If there is no discrimination in the economy, discrimination-blind and discrimination-sighted optimal
taxation yield the same results.
13
ation results in a budget de…cit. Corresponding to the outcomes for social welfare,
individual utility is higher under discrimination-blind optimal taxation for the w2b -, w1a -,
and w1b -types, but interestingly it is lower for the w2a -type. We conjecture that this result
is quite general, since w2 > w2b and it is the w2b -type’s incentive-compatibility constraint
that binds under discrimination-sighted optimal taxation. This means the government
has to o¤er high-skill individuals a more attractive tax contract to satisfy the incentivecompatibility constraint under discrimination-sighted optimal taxation than it does under discrimination-blind optimal taxation. And since the w2a -type chooses the high-skill
tax contract under both tax scenarios, he/she is better o¤ under discrimination-sighted
optimal taxation.
As our theoretical results suggest, the outcomes for the w2b -type are the most di¤erent
of any type under the two tax scenarios. Under discrimination-blind optimal taxation,
the w2b -type chooses the tax contract intended for low-skill individuals. This results in
the w2b -type working less than any other type, and facing the highest marginal tax rate.
Under discrimination-sighted optimal taxation, the w2b -type chooses the tax contract
intended for high-skill individuals. He/she works longer than any other type, and faces
a negative marginal tax rate.
As the degree of discrimination increases, social welfare increases under discriminationblind optimal taxation. As our theoretical results suggest, this is a possibility although
not necessarily the case. The utilities of the w1a - and w2a -types are increasing (as their
wages are increasing), while the utilities of the w1b - and w2b -types are decreasing (as their
wages are decreasing). However, because the w2b -type’s labour supply is very small (due
to mimicking), his/her utility loss is marginal. The net e¤ect, therefore, is an increase
in social welfare.
Under discrimination-sighted optimal taxation, social welfare is decreasing in the
degree of discrimination. Our theory suggests this is necessarily the case. The utility of
the w2a -type is increasing (as his/her wage is increasing), while the utilities of the w1b and w2b -types are decreasing (as their wages are decreasing). Interestingly, the utility of
the w1a -type is also decreasing even though his/her wage is increasing. This re‡ects the
fact that the e¤ect of changes in the optimal tax contracts cannot be ignored when non-
14
marginal changes in the degree of discrimination are considered. Recall that it is the w2b type’s incentive-compatibility constraint that binds under discrimination-sighted optimal
taxation. Therefore, as w2b decreases the government has to o¤er a more attractive highskill tax contract to ensure that the w2b -type is still willing to choose the tax contract
intended for high-skill individuals. But o¤ering a more attractive high-skill tax contract
requires the government to o¤er a less attractive low-skill tax contract in order to satisfy
the budget constraint. So despite an increase in w1a , the w1a -type can be made worse o¤
since they also receive a less favourable tax treatment.
Finally, we note that our theoretical results suggest that the sign of the marginal tax
rate faced by the w1b -type is ambiguous under both discrimination-blind and discriminationsighted optimal taxation, but in our example it is always positive. This re‡ects the ‘small’
degree of discrimination in our example, with the w1b -type’s wage being no more than 10
percent below the average low-skill wage.13 Recall the canonical Mirrlees/Stiglitz result
that the lowest-skill type should face a positive marginal tax rate. In our model, however, it is possible for the w1b -type to face a non-positive marginal tax rate since w1b < w1 .
But because in our example w1b is never substantially lower than w1 , the w1b -type faces
a positive marginal tax rate as in the Mirrlees/Stiglitz model.
5
Concluding Comments
This paper has examined optimal nonlinear income taxation in the presence of wage
discrimination, and in an environment where the government cannot o¤set discrimination by basing the tax system on ascriptive characteristics such as race or gender. Social
norms embedded in the principle of horizontal equity prohibit race-based or gender-based
taxation, and it is unlikely that this will change in the foreseeable future. Instead, the
government is constrained to use non-tax policy instruments, such as anti-discrimination
laws, to tackle discrimination. However, even if the government cannot base the tax
13
After reviewing the large empirical literature on wage discrimination, Holzer and Neumark [2000]
conclude that the extent of both white/black and male/female wage di¤erentials attributable to discrimination is probably no greater than 10 percent. Altonji and Blank [1999] produce similar estimates.
This degree of discrimination corresponds (approximately) to the 5 percent case that we consider.
15
system on ascriptive characteristics, the main conclusion of this paper is that the government should at least use information on wage discrimination when designing a skillbased tax system. Our analysis also suggests that anti-discrimination laws can usefully
complement such a tax system.
Given that our interest in this paper is optimal taxation, we have interpreted the
principle of horizontal equity as implying that individuals who are in all relevant respects
identical should be subjected to the same tax treatment.14 There are, however, other
ways in which the principle of horizontal equity can be interpreted.15 For example, one
could interpret the principle as implying that same-skill individuals should obtain the
same level of utility. Another possible interpretation is that an individual’s tax liability
should be a function of his/her ability to pay, which in turn depends upon the individual’s
skill level and wage rate. While we think these alternative interpretations of horizontal
equity are entirely reasonable, implementing them would require the government to
base the tax system on the ascriptive characteristic, which would be di¢ cult in the
prevailing political climate. Nevertheless, these issues remain interesting avenues for
future research.
6
Appendix
Proof of Lemma 1
Since the objective function (3.1) is strictly concave in the choice variables, we solve
the discrimination-blind optimal taxation problem under the hypothesis that (3.4) is
binding, and we then show that the remaining parts of Lemma 1 are true. The …rstorder conditions can be written as:
2
1
+2 +
w1 s1
2u0 (c1 )
2
2
1
=0
w2 s2
(A.1)
0
(A.2)
2u
14
(c1 ) = 0
Jordahl and Micheletto [2005], for example, also adopt this interpretation in an optimal tax context.
For a recent discussion of possible interpretations of the principle of horizontal equity, see Galbiati
and Vertova [2008].
15
16
2
1
+2
w2 s2
2u0 (c2 )
2 (y1
u(c2 )
where
2
2
2 +
1
=0
w2 s2
(A.3)
0
(c2 ) = 0
(A.4)
c2 ) = 0
(A.5)
2u
c1 ) + 2 (y2
y2
w2 s2
u(c1 ) +
y1
=0
w2 s2
(A.6)
> 0 is the multiplier on constraint (3.4), and
constraint (3.2). It can be seen from (A.4) that
0 is the multiplier on
> 0, and therefore (3.2) is binding.
Using (A.2) and (A.4) we obtain:
2
2+
u0 (c2 )
= 0
u (c1 )
2
2
(A.7)
which implies that u0 (c1 ) > u0 (c2 ). Therefore, c2 > c1 as u( ) is strictly concave. Using
(3.3) we obtain c2 > c1 =) y2 > y1 , and therefore hy2 ; c2 i
is slack, suppose to the contrary that u(c1 )
w1 s1 u(c1 )
y1
y1
w 1 s1
hy1 ; c1 i. To see that (3.3)
u(c2 ) +
w1 s1 u(c2 ) + y2
y2
w 1 s1
0. Then:
(A.8)
0
Solving (A.5) and (A.6) for y1 and y2 , we obtain:
y1 =
1
[c1 + c2
2
w2 s2 u(c2 ) + w2 s2 u(c1 )]
(A.9)
y2 =
1
[c1 + c2
2
w2 s2 u(c1 ) + w2 s2 u(c2 )]
(A.10)
Substituting (A.9) and (A.10) into (A.8) and simplifying, we obtain:
w2 s2
which is a contradiction. Hence u(c1 )
(A.11)
w1 s1
y1
w 1 s1
u(c2 ) +
y2
w 1 s1
> 0 and (3.3) is slack.
Proof of Proposition 1
To show that u(c2 )
y2
w2a s2
> u(c1 )
y1
w2a s2
and therefore the w2a -type chooses hy2 ; c2 i, let
17
w2a = w2 + "2 where "2 > 0. Since (3.4) binds at an optimum:
u(c2 )
y2
"2 )s2
(w2a
u(c1 ) +
y1
=0
"2 )s2
(w2a
(A.12)
Equation (A.12) can be simpli…ed to:
u(c2 )
y2
w2a s2
u(c1 ) +
y1
"2
= a [u(c2 )
a
w2 s2
w2
u(c1 )] > 0
(A.13)
which shows that the w2a -type is strictly better-o¤ choosing hy2 ; c2 i.
To show that u(c2 )
note that w2b = w2
y2
w2b s2
y1
w2b s2
< u(c1 )
and therefore the w2b -type chooses hy1 ; c1 i,
"2 . Since (3.4) binds at an optimum:
u(c2 )
(w2b
y2
+ "2 )s2
u(c1 ) +
(w2b
y1
=0
+ "2 )s2
(A.14)
Equation (A.14) can be simpli…ed to:
u(c2 )
y2
w2b s2
u(c1 ) +
y1
"2
= b [u(c1 )
b
w2 s2
w2
u(c2 )] < 0
(A.15)
which shows that the w2b -type is strictly better-o¤ choosing hy1 ; c1 i.
To show that u(c1 )
y1
w1a s1
y2
w1a s1
> u(c2 )
y1
w1a s1
suppose to the contrary that u(c1 )
w1a s1 u(c1 )
y1
and therefore the w1a -type chooses hy1 ; c1 i,
u(c2 )
y2
.
w1a s1
This is true if and only if:
w1a s1 u(c2 ) + y2
0
(A.16)
Substituting (A.9) and (A.10) into (A.16) and simplifying, we obtain:
[w1a s1
w2 s2 ] [u(c1 )
u(c2 )]
0
(A.17)
which is a contradiction. Hence the w1a -type is strictly better-o¤ choosing hy1 ; c1 i.
To show that u(c1 )
y1
w1b s1
> u(c2 )
y2
w1b s1
18
and therefore the w1b -type chooses hy1 ; c1 i,
let w1b = w1
"1 where "1 > 0. Since (3.3) is slack at an optimum:
u(c1 )
(w1b
y1
+ "1 )s1
u(c2 ) +
(w1b
y2
>0
+ "1 )s1
(A.18)
Equation (A.18) can be simpli…ed to:
u(c1 )
y1
w1b s1
u(c2 ) +
y2
"1
> b [u(c2 )
b
w1 s1
w1
(A.19)
u(c1 )] > 0
which shows that the w1b -type is strictly better-o¤ choosing hy1 ; c1 i.
Proof of Proposition 2
To obtain an expression for the implicit marginal tax rate faced by each individual,
suppose the individuals faced a smooth nonlinear income tax schedule T (yij ). Each
individual would choose cji and lij to maximise u(cji )
lij subject to cji
yij
T (yij ). The
…rst-order conditions can be written as:
u0 (cji )
1 + wij si
where
(A.20)
=0
T 0 (yij )wij si = 0
(A.21)
> 0 is the multiplier on the individual’s budget constraint. Equations (A.20)
and (A.21) can be manipulated to yield:
M T Rij := T 0 (yij ) = 1
1
(A.22)
u0 (cji )wij si
Therefore, for the w2a -type:
M T R2a = 1
where
a
2
u0 (c
1
=1
a
2 )w2 s2
u0 (c
1
2 )w2 s2
a
2
(A.23)
> 0. Using (A.3) and (A.4) we obtain:
(2 +
(2 +
2
1
2 ) w 2 s2
)u0 (c
2)
=
2
1
=) 0
=1
2
u (c2 )w2 s2
19
(A.24)
Hence (A.23) simpli…es to:
M T R2a =
a
2
(A.25)
>0
For the w2b -type:
M T R2b = 1
where
b
2
u0 (c
1
=1
b
1 )w2 s2
u0 (c
1
1 )w1 s1
b
2
(A.26)
(c2 )
<1
1)
(A.27)
> 0. Using (A.1) and (A.2) we obtain:
2 w11s1
2u0 (c
1)
=
1
2 w 2 s2
2
2 +
2
u0 (c
1)
=)
1
2 +
=
0
u (c1 )w1 s1
2 +
2u
2
0
u0 (c
where use has been made of (A.24) and the fact that u0 (c1 ) > u0 (c2 ). Hence:
M T R2b = 1
u0 (c
1
+
1 )w1 s1
b
2
(A.28)
>0
For the w1a -type:
M T R1a = 1
where
a
1
u0 (c
1
=1
a
1 )w1 s1
u0 (c
1
1 )w1 s1
a
1
(A.29)
> 0. Using (A.27) we obtain:
M T R1a = 1
u0 (c
1
+
1 )w1 s1
a
1
(A.30)
>0
For the w1b -type:
M T R1b = 1
where
b
1
1
=1
u0 (c1 )w1b s1
1
u0 (c1 )w1 s1
b
1
(A.31)
< 0. Using (A.27) we obtain:
M T R1b = 1
u0 (c
1
+
1 )w1 s1
b
1
(A.32)
which cannot be signed.
The result that M T R2b > M T R1a > M T R1b follows from (A.26), (A.29), (A.31), and
20
s2 w2b > s1 w1a > s1 w1b .
Proof of Proposition 3
Since three individuals choose hy1 ; c1 i and one individual chooses hy2 ; c2 i, the government’s budget surplus (BS ) is equal to:
BS = 3(y1
c1 ) + y2
(A.33)
c2
which can be rewritten as:
2BS = 2(y1
c1 ) + 2(y2
c2 ) + 4(y1
c1 )
(A.34)
Using (A.5), we obtain:
BS = 2(y1
(A.35)
c1 )
To see that c1 > y1 , consider the free market equilibrium where c1 = y1 and c2 = y2 .
Because w2 s2 > w1 s1 , the hypothetical w2 -type’s utility is greater than the hypothetical
w1 -type’s utility. And because the social welfare function (3.1) is strictly concave, the
government can increase social welfare by imposing a small tax on the hypothetical w2 type and transferring the proceeds to the hypothetical w1 -type. These transfers continue
until the hypothetical w2 -type’s incentive-compatibility constraint binds. Hence c1 > y1
and c2 < y2 .
Proof of Lemma 2
Since the objective function (3.5) is strictly concave in the choice variables, we solve
the discrimination-sighted optimal taxation problem under the hypothesis that (3.10) is
binding, and we then show that the remaining parts of Lemma 2 are true. The …rst-order
conditions can be written as:
1
1
w1a s1
w1b s1
2u0 (c1 )
+2 +
1
1
w2b s2
1
w2b s2
b 0
2 u (c1 )
2
w2a s2
b
2
+2
21
b
2
=0
(A.37)
=0
1
w2b s2
(A.36)
=0
(A.38)
2u0 (c2 )
2 (y1
u(c2 )
where
b
2
2 +
b 0
2 u (c2 )
c1 ) + 2 (y2
y2
w2b s2
=0
(A.39)
c2 ) = 0
(A.40)
u(c1 ) +
y1
=0
w2b s2
(A.41)
> 0 is the multiplier on constraint (3.10), and
constraint (3.6). It can be seen from (A.38) that
0 is the multiplier on
> 0, and therefore (3.6) is binding.
Using (A.37) and (A.39) we obtain:
2
2+
b
2
b
2
=
u0 (c2 )
u0 (c1 )
(A.42)
which implies that u0 (c1 ) > u0 (c2 ). Therefore, c2 > c1 as u( ) is strictly concave. Using
(3.7) we obtain c2 > c1 =) y2 > y1 , and therefore hy2 ; c2 i
is slack, suppose to the contrary that u(c1 )
w1a s1 u(c1 )
y1
w1a s1
u(c2 ) +
w1a s1 u(c2 ) + y2
y1
hy1 ; c1 i. To see that (3.7)
y2
w1a s1
0. Then:
(A.43)
0
Solving (A.40) and (A.41) for y1 and y2 , we obtain:
y1 =
1
c1 + c2
2
w2b s2 u(c2 ) + w2b s2 u(c1 )
(A.44)
y2 =
1
c1 + c2
2
w2b s2 u(c1 ) + w2b s2 u(c2 )
(A.45)
Substituting (A.44) and (A.45) into (A.43) and simplifying, we obtain:
w2b s2
w1a s1
y1
w1a s1
which is a contradiction. Hence u(c1 )
u(c2 ) +
(A.46)
y2
w1a s1
> 0 and (3.7) is slack.
To see that (3.8) is slack, suppose to the contrary that u(c1 )
y1
w1b s1
u(c2 ) + wyb2s1
0.
1
Then:
w1b s1 u(c1 )
y1
w1b s1 u(c2 ) + y2
22
0
(A.47)
Substituting (A.44) and (A.45) into (A.47) and simplifying, we obtain:
w2b s2
w1b s1
y1
w1b s1
which is a contradiction. Hence u(c1 )
(A.48)
u(c2 ) +
y2
w1b s1
> 0 and (3.8) is slack.
To see that (3.9) is slack, suppose to the contrary that u(c2 )
y2
w2a s2
u(c1 )+ wya1s2
2
0.
Then:
w2a s2 u(c2 )
w2a s2 u(c1 ) + y1
y2
0
(A.49)
Substituting (A.44) and (A.45) into (A.49) and simplifying, we obtain:
w2a s2
which is a contradiction. Hence u(c2 )
w2b s2
y2
w2a s2
(A.50)
u(c1 ) +
y1
w2a s2
> 0 and (3.9) is slack.
Proof of Proposition 4
Recall from the proof of Proposition 2 that:
1
M T Rij = 1
(A.51)
u0 (cji )wij si
Therefore, for the w2a -type:
M T R2a = 1
u0 (c
1
a
2 )w2 s2
(A.52)
Using (A.38) and (A.39) we obtain:
1
2
=
a
0
u (c2 )w2 s2
where
2
=
1
w2b s2
1
w2a s2
2 (1
+
b
2)
2
<1
(A.53)
> 0. Hence:
M T R2a =
2 (1
+
2
23
b
2)
>0
(A.54)
For the w2b -type:
M T R2b = 1
u0 (c
1
b
2 )w2 s2
(A.55)
Using (A.38) and (A.39) we obtain:
u0 (c
1
2 +
=
b
2
2 )w2 s2
2
(A.56)
>1
Hence:
2
M T R2b =
<0
(A.57)
1
a
1 )w1 s1
(A.58)
2
For the w1a -type:
M T R1a = 1
u0 (c
Using (A.36) and (A.37) we obtain:
2 + b2 wb1s2
1
2
=
b 0
u0 (c1 )w1a s1
2 + 2 u (c1 )
where
1
=
1
w1b s1
1
w1a s1
1
(A.59)
<1
> 0. To show that the expression in (A.59) is indeed less than one,
it is su¢ cient to show that
1
w2b s2
< u0 (c1 ). Suppose to the contrary that
1
w2b s2
u0 (c1 ).
Then:
b
2)
(2
1
b
2.
where from (A.37) we know that 2 >
(2
b
2)
1
(A.60)
Using (A.37) and (A.36) we obtain:
2 =
w2b s2
b
0
2 )u (c1 )
(2
w2b s2
1
w1a s1
+
1
w1b s1
b
2
1
w2b s2
(A.61)
which simpli…es to:
2
1
1
w2b s2
w1a s1
+
1
w1b s1
(A.62)
which is a contradiction. Hence from (A.58) and (A.59) we obtain M T R1a > 0.
For the w1b -type:
M T R1b = 1
u0 (c
24
1
b
1 )w1 s1
(A.63)
Using (A.36) and (A.37) we obtain:
2 + b2 wb1s2 +
1
2
=
b
b 0
0
u (c1 )w1 s1
2 + 2 u (c1 )
1
(A.64)
which may be greater than, less than, or equal to one. Hence M T R1b cannot be signed.
The result that M T R1a > M T R1b follows from (A.58), (A.63), and w1a > w1b .
Proof of Proposition 5
Social welfare under discrimination-blind optimal taxation is equal to:
W = u(c1 )
y1
+ u(c1 )
w1a s1
y1
+ u(c2 )
w1b s1
y2
+ u(c1 )
w2a s2
y1
w2b s2
(A.65)
where hy1 ; c1 i and hy2 ; c2 i are the solution to programme (3.1) – (3.4). By assumption, an increase in wage discrimination does not change average wages. Therefore,
under discrimination-blind optimal taxation, an increase in wage discrimination does
not change the government’s optimal choices of hy1 ; c1 i and hy2 ; c2 i. Thus:
W =
Let
w1a =
y1
a a
w1 w1 s1
w1a +
w1b and
w2a =
y1
b b
w1 w1 s1
w1b +
y2
a a
w2 w2 s2
w2b , so that
w2a +
w1b > 0 and
y1
b b
w2 w2 s2
w2b
(A.66)
w2b > 0 corresponds to a
reduction in wage discrimination. Then (A.66) can be simpli…ed to:
W =
y1
b b
w1 w1 s1
y1
a a
w1 w1 s1
w1b +
y1
b b
w2 w2 s2
y2
a a
w2 w2 s2
w2b
(A.67)
The …rst bracketed term in (A.67) is positive, but the second bracketed term in (A.67)
cannot be signed since y1 < y2 and w2b w2b s2 < w2a w2a s2 . Hence the e¤ect on social
welfare of a decrease in wage discrimination under discrimination-blind optimal taxation
is ambiguous.
Proof of Proposition 6
The Lagrangian corresponding to the discrimination-sighted optimal taxation problem
25
can be written as:
y1
+ u(c1 )
w1a s1
L = u(c1 )
+
[2 (y1
c1 ) + 2 (y2
c2 )] +
y1
+ u(c2 )
w1b s1
b
2
u(c2 )
y2
+ u(c2 )
w2a s2
y2
w2b s2
u(c1 ) +
y2
w2b s2
y1
w2b s2
(A.68)
Let W (w1a ; w1b ; w2a ; w2b ; s1 ; s2 ) be the associated value function. By the Envelope Theorem
we obtain:
W =
b
y1
y1
y2
y2
2
a
b
a
b
w
+
w
+
w
+
w
+
(y2
1
1
2
2
a a
b b
b b
b b
s
w
w1a w1a s1
w
w1 w1 s1
w2 w2 s2
w2 w2 s2
2 2 2
y1 ) w2b
(A.69)
Let
w1a =
w1b and
w2a =
w2b , so that
w1b > 0 and
w2b > 0 corresponds to a
reduction in wage discrimination. Then (A.69) can be simpli…ed to:
W =
y1
b b
w1 w1 s1
y1
a a
w1 w1 s1
w1b +
y2
b b
w2 w2 s2
b
y2
2
+
(y2
w2a w2a s2 w2b w2b s2
y1 )
w2b
(A.70)
which is positive if
w1b > 0 and
w2b > 0.
26
TABLE 1
No Discrimination: w1b = 2 , w1a = 2 , w2b = 2 , w2a = 2
Optimal Taxation
Individual Type
w2b
w1a
4.000
1.333
4.864
0.469
1.216
0.235
0.000
0.333
0.170
0.053
0.447
0.000
a
2
Consumption
Pre-tax income
Labour supply
Marginal tax rate
Utility
Social welfare
Budget balance
w
4.000
4.864
1.216
0.000
0.170
27
w1b
1.333
0.469
0.235
0.333
0.053
TABLE 2
2.5% Discrimination: w1b = 1.38 , w1a = 1.45 , w2b = 1.95 , w2a = 2.05
Discrimination-Blind Optimal Taxation
Individual Type
w2b
w1a
1.333
1.333
0.469
0.469
0.120
0.229
0.658
0.350
0.167
0.059
0.473
−1.728
a
2
Consumption
Pre-tax income
Labour supply
Marginal tax rate
Utility
Social welfare
Budget balance
w
4.000
4.864
1.186
0.024
0.200
w1b
1.333
0.469
0.241
0.316
0.047
Discrimination-Sighted Optimal Taxation
Individual Type
w2b
w1a
3.965
1.365
4.744
0.586
1.217
0.286
0.334
−0.017
0.161
0.025
0.418
0.000
a
2
Consumption
Pre-tax income
Labour supply
Marginal tax rate
Utility
Social welfare
Budget balance
w
3.965
4.744
1.157
0.033
0.220
28
w1b
1.365
0.586
0.300
0.300
0.011
TABLE 3
5% Discrimination: w1b = 1.34 , w1a = 1.49 , w2b = 1.90 , w2a = 2.10
Discrimination-Blind Optimal Taxation
Individual Type
w2b
w1a
1.333
1.333
0.469
0.469
0.124
0.224
0.649
0.365
0.164
0.064
0.497
−1.728
a
2
Consumption
Pre-tax income
Labour supply
Marginal tax rate
Utility
Social welfare
Budget balance
w
4.000
4.864
1.158
0.048
0.228
w1b
1.333
0.469
0.247
0.298
0.041
Discrimination-Sighted Optimal Taxation
Individual Type
w2b
w1a
3.927
1.393
4.629
0.691
1.218
0.329
0.337
−0.033
0.150
0.002
0.386
0.000
a
2
Consumption
Pre-tax income
Labour supply
Marginal tax rate
Utility
Social welfare
Budget balance
w
3.927
4.629
1.102
0.065
0.266
29
w1b
1.393
0.691
0.364
0.267
−0.032
TABLE 4
10% Discrimination: w1b = 1.27 , w1a = 1.56 , w2b = 1.80 , w2a = 2.20
Discrimination-Blind Optimal Taxation
Individual Type
w2b
w1a
1.333
1.333
0.469
0.469
0.130
0.213
0.630
0.394
0.157
0.074
0.539
−1.728
a
2
Consumption
Pre-tax income
Labour supply
Marginal tax rate
Utility
Social welfare
Budget balance
w
4.000
4.864
1.105
0.091
0.281
w1b
1.333
0.469
0.261
0.259
0.027
Discrimination-Sighted Optimal Taxation
Individual Type
w2b
w1a
3.840
1.440
4.405
0.875
1.224
0.398
0.345
−0.067
0.122
−0.033
0.312
0.000
a
2
Consumption
Pre-tax income
Labour supply
Marginal tax rate
Utility
Social welfare
Budget balance
w
3.840
4.405
1.001
0.127
0.344
30
w1b
1.440
0.875
0.486
0.200
−0.121
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