Malas Notches¤
Ben Lockwoody
First version: 4 July 2016
This version: 16 September 2016

Abstract
This paper shows that the su¢cient statistic approach to the welfare properties
of income (and other) taxes does not extend to tax systems with notches, becausewith notches, changes in bunching induced by changes in tax rates have a …rst-ordere¤ect on tax revenues. In an income tax setting, we show that the marginal excess
burden (MEB) and the welfare-maximizing top rate of tax are given by the relevant
formulae for a proportional tax as in Feldstein (1999) plus a correction factor. The
Feldstein formulae always underestimate the MEB and overestimate the revenue
and welfare-maximizing rate of tax. Quantitatively, these mis-estimates can be verylarge; the MEB can be underestimated by an order of magnitude. An application
to VAT is discussed; with a calibration to UK data, the MEB of the VAT is roughly
three times what is would be if VAT was simply a proportional tax.
JEL Classi…cation: H20,H21,H31
Keywords: tax kink, tax notch, excess burden, su¢cient statistic¤

I would like to thank Omiros Kouvavas for outstanding research assistance, and Miguel Almunia, Steve
Bond, Thomas Brosy, Michael Devereux, and Joel Slemrod for helpful comments. I also acknowledge…nancial support from the ESRC under grant ES/L000016/1.
y
CBT, CEPR and Department of Economics, University of Warwick, Coventry CV4 7AL, England;
Email: B.Lockwood@warwick.ac.uk

1

1

Introduction

In a recent survey, Chetty (2009) argues that an important new development in public
economics is the so-called su¢cient statistic approach, which "derives formulas for thewelfare consequences of policies that are functions of high-level elasticities rather thandeep primitives" (Chetty (2009), p 451). In turn, this means that to assess the welfare
properties of these policies, only these elasticities, rather than fully structural models,need to be estimated.1
The su¢cient statistic approach originated in a seminal paper by Feldstein (1999), whoshowed that the marginal excess burden (MEB) of a proportional income tax only depends
on the behavioral responses to the tax via a su¢cient statistic, the personal elasticity oftaxable income (ETI). Feldstein’s paper has given rise to a large literature devoted toobtaining empirical estimates of the ETI (Gruber and Saez (2002), Saez, Slemrod, andGiertz (2012), Kleven and Schultz (2014), Weber (2014)).
Subsequently, Saez (2001) and Saez, Slemrod, and Giertz (2012) showed that theFeldstein formula for the MEB could be extended to the top rate of tax in a progressivepiece-wise linear income tax system, and they also established formulae for the revenue
and welfare-maximizing rate of tax. These formulae also have the su¢cient statisticfeature; speci…cally, they depend only on the elasticity of the ETI, a statistic of theincome distribution, which is constant if the top tail of the income distribution is Pareto2 ,and a possibly a welfare weight.
In this paper, we ask the question as to whether these su¢cient statistic propertiesof key formulae also extend to tax systems with notches. Generally, a tax notch occurs
when there is a discontinuous change in the tax liability as the tax base varies (Slemrod(2013), Kleven (2016)).
In practice, we do see notches in several major kinds of taxes, and these are being
increasingly studied in the empirical literature. Signi…cant notches in the personal income
tax system are quite rare, although they do exist; for example, in Pakistan (Kleven andWaseem (2013)), there are notches where the tax on all income below the notch can riseby as much as 5%, and in Ireland, an emergency income levy after the …nancial crisis
1

Chetty (2009) also argues that this su¢cient statistic approach is also valuable in several othercontexts, such as evaluating the welfare gain from social insurance programs, and the welfare e¤ects ofchanges in taxes with optimization frictions.

2
The formula is that the marginal excess burden equals 1¡¡ where  is the rate of tax,  is the
personal elasticity of taxable income with respect to the net of tax rate 1¡ and  is the Pareto parameter.

2had a notch of up to 4% (Hargaden (2015))3 . There are even small notches in the federal
income tax in the US, and larger notches induced by income-dependent entitlement totax credits (Slemrod (2013)).
Notches also exist in other major taxes. For example, notches are, or were until
recently, present in housing transactions taxes in the UK and the US (Best and Kleven(2014), Kopczuk and Munroe (2014)). They also arise in the corporate income tax in
Costa Rica (Bachas and Mauricio (2015)). Slemrod (2013) notes that there are many
examples of commodity tax notches, where a marginal change in some characteristic canchange the product classi…cation so as to produce a discrete change in the tax liability.4
Finally, as argued by Liu and Lockwood (2015), a VAT threshold can be thought of as
a tax notch; a …rm’s VAT liability changes discontinuously when its sales go over theregistration threshold. Indeed, given the importance and near-ubiquity of VAT, this is infact the most important example of a tax notch.
We …rst study notches in the income tax setting of Saez (2010) and others, where
households di¤er in ability or taste so that the disutility of generating taxable income
varies across households. For simplicity, we assume a two-bracket tax i.e. a tax with alower rate (which could be zero) below a threshold, and a higher rate above. In this setting,our …rst contribution is to derive exact formulae for the marginal excess burden of thehigher rate of tax, and for the welfare-maximizing rate higher rate of tax. These formulae
are the same as in Feldstein (1999) for a proportional income tax, with a correction factor
that captures the e¤ect of the bunching response to an increase in the top rate tax on tax
revenue.
The bunching response measures the change in the number of households bunching at
the threshold to avoid paying the top rate of tax, and is thus distinct from the intensive
margin response of taxable income of given household to the tax rate; the latter has been
the focus of the ETI literature. With a notch, (unlike the case of a kink), the bunching
response a¤ects tax revenue because with a notch, the tax schedule is discontinuous at
the threshold.
Our second key …nding is that the correction factor cannot be expressed as a simple
function of the usual su¢cient statistics i.e. the intensive margin elasticity of the ETI and
the Pareto parameter. It does depend on these variables, but it also depends on the lower
rate of tax and on the size of the bunching interval. So, the su¢cient statistic approach3

From Table 1 of Hargaden (2015), in 2010, earnings of above 26000 Euro incurred a charge of 1040
Euro.
4For example, in the US, the Gas Guzzler Tax, under which high-performance cars are subject upon
initial sale to a per-vehicle tax that is higher, the lower is the fuel economy of the car.

3

seems to break down with tax notches, unless the correction factor turns out to be small.
Out third contribution is to investigate this question. Qualitatively, ignoring the correction factor underestimates the marginal excess burden and overestimates the welfaremaximizing rate of tax. Calibrations show that the percentage error from using theFeldstein formulae can be very large. At baseline values, the marginal excess burden isunderestimated by a factor of six, and the revenue-maximizing tax is overestimated byaround half, and the errors can be much larger for some parameter values. So, the conclusion is that at least in the income tax setting, the su¢cient statistic approach is notpractical.
We then turn to apply our approach to the VAT, which is the most empirically important example of a tax notch. We present a simple model of small traders who di¤er
in productivity, and are subject to VAT at rate  above a threshold level of sales. We
show that this model is formally equivalent to our income tax model, in the sense that
registered …rms above the threshold face an e¤ective rate of VAT  on value-added and…rms below the threshold face a lower rate   It may seem counter-intuitive that nonregistered …rms face a positive rate of e¤ective VAT; this is because non-registered …rms
cannot claim back VAT on inputs (so-called "embedded" VAT).
We then show that the MEB of an increase in the statutory rate of VAT is given by theFeldstein formula for a proportional tax plus a correction factor as in the income tax case.
However, the details of the correction factor are more complex, because an increase in thestatutory rate  increases both the e¤ective rates     A calibration of the model showsthat the proportional tax formula for the MEB of the VAT underestimates the true MEB
by a factor of up to three. This framework also allows us to evaluate the e¤ect of increased
compliance costs of VAT in the MEB via its impact on bunching; increased compliancecosts increase bunching and thus increase the MEB, but the e¤ect is quantitatively small.
The remainder of the paper is arranged as follows. After the literature review in
Section 2, in Section 3, we set up the model. Section 4 has the main analytical results forthe income tax, and Section 5 the simulations. Section 6 deals with the extension to the
VAT, and Section 7 concludes.

2

Related LiteratureThis paper speaks to a number of related literatures. First, it is already known that due
to externalities of one kind or another, the su¢cient statistic approach has its limitations.
Saez, Slemrod, and Giertz (2012) give the examples of deductibility from income tax of4charitable giving and mortgage interest payments for residential housing. In these cases,
an increase in the marginal rate of tax will boost charity income and home ownershiprespectively, which may be valuable objectives in themselves. Saez, Slemrod, and Giertz(2012) call these classical externalities5 .
Fiscal externalities, where the actions of the household generate additional revenue forthe government and thus bene…ts other households, can also cause the su¢cient statistic
approach to fail, or at least require adjustment, but in these cases a simple change to theformula is sometimes possible. Chetty’s (2010) analysis of income tax evasion is a casein point6 . As Gillitzer and Slemrod (2016) show, in this case the standard formula forthe marginal e¢ciency cost of funds can be adjusted in the same way it must be adjusted
for any …scal externality, i.e. whenever a change in tax rates induces taxpayers to shift
income to another tax. Our results are rather di¤erent to these cases of both classicaland …scal externality. In our setting, there is no …scal or other externality- rather, thesu¢cient statistic approach fails because the bunching response ha a …rst-order e¤ect on
tax revenue.
A second related literature is on VAT. Here, there are two distinct sets of related
papers. First, there is a growing literature on the e¤ect of VAT thresholds on …rm behavior. Theoretical contributions include Keen and Mintz (2004), Kanbur and Keen (2014)
and Liu and Lockwood (2015), and empirical studies include Liu and Lockwood (2015)
and Harju, Matikka, and Rauhanen (2016). The theoretical work of Kanbur, Keen and
Mintz focusses on the optimal threshold of the VAT, holding the rate of tax …xed, andis thus complementary to this paper, which characterizes the MEB of an increase in therate, holding the threshold …xed. In fact, we e¤ectively ask the question of whether it
is legitimate to ignore the threshold altogether when calculating the MEB of the VAT.
Therefore, our paper relates to as second literature on the marginal excess burden of indirect taxes, including VAT (e.g. Ballard, Shoven, and Whalley (1985), Rutherford, andPaltsev (1999)). In these papers, when the marginal excess burden of VAT is calculated,
it is always assumed that the VAT is a proportional tax i.e. the VAT threshold is ignored.
This paper shows that this simplifying assumption yields seriously biased estimates.
5

See Doerrenberg, Peichl, and Siegloch (2015) for a more formal statement of this argument, andestimates of how deductions respond to tax rate changes for the case of Germany.
6
Chetty shows that when the household can evade the personal income tax at a cost, if that cost
is a pure transfer payment i.e. a …ne times a probability of detection, there is e¤ectively a positive…scal externality of evasion - it generates additional revenue for the government and thus bene…t for all
households. In this case, as we might expect, we see that the elasticity of taxable income over-estimatesthe excess burden of the tax.

5

A third related literature is that on the MEB and welfare-maximizing taxes withkinks in the tax schedule. Here, we make a small contribution as a by-product of ourmain focus, which is on notches. In this case, it generally understood that the marginalexcess burden of the top rate of income tax, and the welfare-maximizing top rate depends
via simple formulae, only on the elasticity of the ETI, and the Pareto statistic of theincome distribution. However, there seems to be some confusion about the conditions
required for this result. Saez, Slemrod, and Giertz (2012) suggest that what is required isthat assumption that "behavioral responses take place only along the intensive margin",or more precisely that the bunching response of an increase in the top rate of tax is ofsecond order relative to the extensive margin response.7 This assumption is very strong,
as even with a kink, there is always a bunching response. Our Proposition 4.1 below showsthat this assumption is not necessary, because no matter what the size of the bunching
response, the response has no e¤ect on tax revenue, to …rst order, as the tax schedule iscontinuous. All that is required is that the distribution of taxpayer types  is continuous,
a standard assumption.
A …nal related literature is the small one on the design of piece-wise linear income
taxes. In any early contribution Slemrod et. al. (1994) consider the design of a twobracket income tax, and they explicitly take into account bunching responses in doing so.
They did not obtain analytical results but their numerical simulations suggest that the
tax schedule should be concave i.e. the higher tax should be below the lower tax. Morerecently, Apps, Long, and Rees (2014) have extended their work. This work is somewhat
related to our …nding that the intensive margin response should be adjusted in the caseof the VAT, as explained below.

7

Speci…cally, they say the following. "The change  could induce a small fraction  of the 
taxpayers to leave (or join if   0) the top bracket. As long as behavioral responses take place only
along the intensive margin, each individual response is proportional to  so that the total revenue e¤ectof such responses is second order ( ) and hence can be ignored in our derivation."

6

3

The Model and Preliminary Results

3.1

Set-Up

We follow Saez (2010) in our set-up. There are individual taxpayers indexed by a skill
or taste parameter  2 [ ] distributed in the population with density (). A type individual has preferences over consumption  and taxable income  of the form
( ; ) =  ¡ (; )
where (; ) is the disutility of earning income  So, in this speci…cation of ( ; )
we assume away income e¤ects for convenience. We also assume:A1.     0     0So, A1 says that a higher  represents are higher skill level (i.e. higher wage), or a lower
taste for leisure. Assumption A1 is satis…ed for example, by the iso-elastic speci…cationof Saez (2010): ³  ´1+ 1(; ) =(1)
1 + 1 
The budget constraint is  =  ¡  () where  () is the tax function. So, a household’sutility over  is (; ) =  ¡  () ¡ (; )
Finally, for future reference, de…ne the optimal taxable income at tax rate  for a type to be;(1 ¡  ) ´ arg maxf(1 ¡ ) ¡ (; )gNote from A1 that 1¡    0 where subscripts denote derivatives. So, 1¡ is the
response of taxable income to the net-of-tax rate. Following Saez, Slemrod and Giertz(2012) we call this the intensive margin response.

3.2

Kinks and Notches

For simplicity, we focus on a two-bracket tax, although our arguments apply straightforwardly to the case of the highest tax in a piecewise-linear tax system with any numberof brackets. We will assume that the tax system is progressive; that is, the tax rate on
incomes in the higher income bracket is strictly greater than the tax on incomes in thelower income bracket.
So, with a two-bracket tax, for a kink, the tax function is(  · 0 () =(2)
 0 +  ( ¡ 0 )   0
7

for 0  0    ¸ 0; that is, all income below the kink point 0 is taxed at the lower
rate   and all income in excess of the kink is taxed at the higher rate. For a notch, thetax function is(   · 0 () =(3)
    0with    ¸ 0 That is, when taxable income is below 0  a tax at rate  is paid on
all income, but when  is above 0  a tax at rate  is paid on all income.

3.3

Bunching

With either a kink or a notch, all types in an interval  2 [   ] will bunch at taxable
income 0  In both cases, the lowest type who bunches is the one who is just willing toearn taxable income 0 at the lower tax rate i.e. the critical  is de…ned by the condition
(  1 ¡  ) = 0(4)With a kink, the highest type who bunches,   is de…ned by the condition that the
optimal choice of taxable income of the  ¡type at tax  is just 0 i.e.
(1 ¡  ;  ) = 0(5)With a notch,  is de…ned by the condition that the  type must be indi¤erent
between staying at the notch and paying tax  , and choosing  optimally, and paying
 on all income (Kleven and Waseem (2013), Kleven (2016)). To write this indi¤erence
condition, we …rst de…ne the indirect utility function(; ) ´ max f(1 ¡ ) ¡ (; )gNote that the derivative of  with respect to    is ¡(1 ¡ ; ) Then, the condition
de…ning  can be written:(1 ¡  )0 ¡ (0 ;  ) = ( ;  )(6)

The left-hand side of (6) is utility when taxable income is constrained to be at the notch
value 0 . Note that this indi¤erence condition implies (1 ¡    )  0 ; because if(1 ¡    )  0  the  ¡type could choose  optimally and stay below the notch.
Note the di¤erence between indi¤erence condition (6) and the condition (5).

8

3.4

The Bunching Response

Here, we study the e¤ect of a change in  on the mass of individuals who bunch i.e. on
the size of the interval [   ] Note …rst from (4) that  is una¤ected by  for botha kink and a notch. Next, in the kink case, note that

1¡
=
0



(7)

So, we have a bunching response to  : i.e. an increase in the tax rate above the kinkmakes going above the kink less attractive, and so more people bunch below the kink.
In the notch case, note from (6) that  does a¤ect   and in fact, using  = ¡ we
see that(1 ¡    )
=
(8)

 (0 ;  ) ¡  ((1 ¡    );  )Also, as  (; )  0 and (1 ¡    )  0  we see that for any  : ((1 ¡    ); )   (0 ; )and consequently from (8):

0
(9)

So, again we see that the bunching response to a change in  is intuitive; an increase in
the tax rate above the notch makes going above the notch less attractive, and so more
people bunch at the notch.

4

Main Results

4.1

The E¤ect of the Bunching Response on Tax Revenue

Here, we show that the e¤ect of the bunching response on tax revenue with a kink anda notch are qualitatively di¤erent, being zero and negative respectively. With a kink,revenue can be written
µZ 
¶
µZ 
¶
Z 
 = 
(1 ¡  ; )() +0 () + ((1 ¡  ; ) ¡ 0 )()






(10)
Note that all households with  ¸  pay tax at the lower rate on the …rst 0 of earnings.
In the kink case, the bunching e¤ect on tax revenue i.e. the e¤ect of a change in  on via a change in  in  is, from (12):= ¡ ((1 ¡  ;  ) ¡ 0 )( ) = 0

9

(11)

So, overall, with a kink, the e¤ect of the bunching response on tax revenue is zero.
With a notch, revenue is
µZ 
¶
µZ 
¶
Z 
 = 
(1 ¡  ; )() +0 () +(1 ¡  ; )() (12)






Comparing this to (10), we see two di¤erences. Because the higher rate applies to all
income for those earning above 0  the threshold 0 no longer enters into the the tax basefor   and so the upper limit of integration on 0 in the tax base for  falls from  to  re‡ecting the fact that now only individuals below  pay any tax at the lower rate
Note from (12) that;= ( 0 ¡  (1 ¡  ;  ))( )  0


(13)

This is strictly negative as     (1¡ ;  )  0  So, in contrast to the kink case, thebunching e¤ect on tax revenue  from an increase in  is negative, as  0 from (9)This is because a small increase in  has two e¤ects on revenue that are both negative.
First, there is a discontinuity in the tax base; the earnings of these who now locate at thenotch fall discontinuously from (1 ¡  ;  ) to 0  Second, there is a discontinuity in the
tax rate applying to that base; all these earnings are taxed at a lower rate,  rather than So, we conclude:
Proposition 1. The e¤ect of the bunching response on tax revenue is zero for a kink,but strictly negative for a notch.
This result is the key one that drives the rest of the paper. The result that bunching
response on tax revenue is zero for a kink also has a useful implication that helps to clarify
some confusion in the literature. As already noted, Saez, Slemrod, and Giertz (2012) argue
that for su¢cient statistic formulae to apply in the kink case, what is required is thatassumption that "behavioral responses take place only along the intensive margin", ormore precisely that the bunching response of an increase in the top rate of tax is ofsecond order relative to the extensive margin response. Proposition 4.1 shows that this
assumption is not required, because no matter how large   = 0 in the kink case.
 

4.2

The Marginal Excess Burden

Here, we derive a formula for the marginal excess burden (MEB) of  when there is anotch and show that it can be written as the MEB of a proportional tax plus a correction
10

factor. To de…ne the MEB, note that due to quasi-linearity, the natural measure of
welfare is the integral of indirect utilities, say  plus revenue  which is assumed to beredistributed as a lump-sum back to households when calculating the MEB. So,  = ¡( + )


(14)

The minus sign ensures that the marginal excess burden is measured as a positive number.
From (12), we see that the e¤ect of an increase in  on tax revenue is:¯ ¯¯ 

+

¯

 
=  + | {z const}(15)| {z }
intensive-margin

Here

 =

Z 



bunching(16)

(1 ¡  ; )()

is the base in which the higher rate of tax is levied.
So, (15) is composed of three terms, the mechanical e¤ect   and two behaviorale¤ects on tax revenue, the intensive-margin and bunching e¤ects. The intensive-margine¤ect on tax revenue is standard; it describes how tax revenue changes because of changes
in earnings, conditional on the taxpayer staying the same tax bracket. The bunchinge¤ect on tax revenue and its impact on the marginal excess burden is the focus of ourinvestigation.
To compute   note …rst that the integral of indirect utilities is
Z 
Z 
Z 
 =
(1 ¡  ; )() +(0 (1 ¡  ) ¡ (0 ; ))() +(1 ¡  ; )()






(17)
Note that by de…nition, a small change in  has no e¤ect on welfare, because  isde…ned by (6) above. So, using  = ¡(1 ¡  ) we see that
Z 

=¡
(1 ¡  ; )() = ¡
(18)



So, plugging (15), (18) back into (14), dividing through by   and multiplying by
1 ¡   and noting that holding  constant, (1¡= ¡ 
 we see that)  =

  + 
1 ¡    =¡
1 ¡  ¡   +   
11

(19)

Here,¯Z
1 ¡   ¯¯
1 ¡   (1 ¡  ; )==() (1 ¡  ) ¯ const(1 ¡  )


(20)

is the intensive-margin elasticity of the tax base  with respect to the net of tax rate
1 ¡   and  is a correction factor, which captures the e¤ect of a changing   thebunching response, on the MEB, via its e¤ect on revenue. Of course, given the speci…cation(1),  is a constant independent of   This formula is standard, except that it includes
the e¤ect of the bunching response on tax revenue in both numerator and denominatorvia .
We then have;
Proposition 2. Assume A1, and that the distribution of ability (and pretax-income) is
Pareto, with shape and scale parameters  . Then, the MEB with a notch is  =where=  + 
1 ¡  ¡   ¡ (1 ¡  )( (1 ¡  ) ¡  0  )( ¡ 1)(1 + ) 0
³ ´(1+)
0
1+
(1 ¡  )
¡ 

(21)

(22)

Moreover, in (22),  is de…ned by (6).

Some comments are appropriate at this point. First, the MEB (21) is the formula forthe marginal excess burden of a proportional income tax, as shown by Feldstein (1999),
plus the correction factor  This is intuitive; all households above  are paying tax at
rate  on all their income, so for  …xed,  is indeed a proportional tax. So, as alreadyremarked, the correction factor  just captures the e¤ect of a changing   the bunching
response, on the MEB, via its e¤ect on revenue.
Second, we can ask how the MEB compares to the MEB in a kinked tax system. As
shown for example, by Saez (2001), the latter is  = 
1 ¡  ¡  

Clearly,   depends only on simple su¢cient statistics; other than the tax rate  
it depends only on  the intensive-margin elasticity of taxable income, and  the shapeparameter of the income distribution.
By contrast, from (22) that  is a complex object. It depends not only on su¢cient
statistics   and the top rate of tax,   but also on other parameters of the tax system12  0  and …nally, it also depends on   In turn, generally,  cannot be solved for inclosed form from (6). Given this, one question is whether we can get a good approximationto  by setting  = 0 This is a question addressed in Section 5 below, where we will
see that the approximation is generally very inaccurate.

4.3

The Welfare-Maximizing Rate of Tax

In his well-known article, Saez (2001) derived a formula for the welfare-maximizing rate oftax for a one-bracket tax system.8 He showed that this tax depended only on  and  plus
a parameter he called  which is the "ratio of of social marginal utility for top bracket
taxpayers to the marginal value of public funds for the government." In the special casewhere  = 0 this gives the revenue-maximizing rate of tax.
Here, we develop a similar formula for the optimal   We will show that it is equal
to the formula for the welfare-maximizing proportional tax, plus a term in the correction factor  above. To do this, we assume now, following Saez, that the government’s
objective is not the integral of indirect utilities as in (17), but the integral of a strictlyincreasing, concave transformation () of utilities. The function  captures social aversion to inequality in the usual way. Also, we suppose that the government has a revenue
requirement  Also, we assume following Saez (2001) that the welfare weight  = 0 isconstant above  at some ; if this is not the case, the optimal tax has an additional termin the covariance of  and  Finally, normalize the Lagrange multiplier on the government
revenue constraint  ·  to unity. Then we can show:
Proposition 3. The welfare-maximizing level of  is;
¤ =

1¡¡
1¡+

(23)

with  de…ned in (22) above.
To interpret this, note …rst that there is a direct connection of (23) to the formuladerived by Saez (2001) for the optimal linear tax on top earners, which is equation (9) in
his paper. He allowed for an income e¤ect in labour supply in his setting, so setting thisequal to zero, his equation, in our notation, reduces to
¤ =
8

1¡
1 ¡  + This is given in equation (9) of Saez (2001).

13

(24)

Moreover, it is easily checked that if there is no exempt income in the one-bracket tax
system, so it becomes a proportional tax,  = 1 in the above formula, so
¤ =

1¡
1¡+

(25)

Comparing (23) to (25), we see that the former is equal to the optimal proportional tax
minus a correction factor 1¡+ This factor re‡ects the fact that with a notch, there isan additional cost to taxation because of the bunching response.
As already remarked,  is a complex object; it depends not only on su¢cient statistics  and the top rate of tax,   but also on other parameters of the tax system   0 
and …nally, it also depends on   In turn, generally,  cannot be solved for in closed
form from (6). Given this, one question is whether we can get a good approximation to¤ by setting  = 0 This is a question addressed in Section 5 below.

5

Simulations

We have seen that the MEB of an increase in  and the optimal  are given by thecorresponding formulae for a proportional tax  plus a correction factor. Moreover,the formulae for a proportional tax are very simple, depending only on the intensivemargin elasticity  and thus can easily be calculated This raises the question of whether
the MEB and optimal tax, calculated assuming that  is a proportional tax, are good
approximations to the true MEB and optimal tax. To investigate this, we calibrate themodel.
We require values for       and 0  First, we assume that the intensive-margin
elasticity (20) is constant as in (1), so we only need to calibrate the parameter  Our
baseline parameter values are chosen as follows. Following Piketty and Saez (2003), we set
 = 15, and following Saez, Slemrod, and Giertz (2012) and Kleven and Schultz (2014),
we set  = 025 Regarding the tax rates, we …rst set  = 02 which is broadly in linewith the average income and payroll tax paid by US households9 . It is also the basic rateof income tax in the UK. For the notch, we use the fact that notches in personal income
tax, where they exist, are small. For example, Kleven and Waseem (2013) show that in
the Pakistani income tax, the notch ranges between 2 and 5 percentage points. So, wewill take our baseline notch  ¡  = ¢ = 003
9"Overview Of The Federal Tax System As In E¤ect For 2015", Joint Committe on Taxation, Congtressof the United States.

14

To choose  0 we assume that only the top 20% of the population pay a higher rate ofincome tax, roughly the proportion in the UK. De…ne 0 to be the skill level corresponding
to taxable income just at the notch i.e. 0 (1 ¡ ³) =´0  This requires that 80% of thepopulation have skills below 0 i.e.  (0 ) = 1 ¡ 0
= 08 or 0 = (02)115 = 0342
Given that only the ratio 0 is determined, we set  = 1 so 0 = 2924 But then
0 = 2924(08)025 = 2168
Finally, from (22), we need a value for   Under the assumption (1), the indi¤erence
condition (6) reduces to1( )¡1 (0 )1+  +  (1 ¡  )1+ ¡ (1 ¡  )0 (1 + ) = 0(26)

Equation (26) has two roots, and we take the larger root to ensure that  (1 ¡  ) 
0  Finally, parameter values are chosen so that the denominator in (21) is positive, whichis equivalent to   0 i.e. that the tax rate is on the right side of the La¤er curve.
This requires simply that the notch is greater than 0001510
Figures 1(a)-(c) show both the true MEB, as given by (21), and the approximation,
treating  as a proportional tax i.e. setting  = 0 in (21). The former is denoted by  in the Figures, and the latter by   - Figure 1 in here The error in using   at the baseline values can be read o¤ from Figure 1(a),setting  = 025 It can be seen that true MEB is about 0.6, whereas the approximation isabout 0.1. So, the error in using the proportional formula is about a factor of six. Figure1(a) also shows that   is increasing in  at a faster rate than    so when  = 04
for example, the error in using   is almost an order of magnitude.
Figure 1 (b) shows that  is also increasing in  the Pareto parameter which measures (inversely) the size of the tail of the income distribution. As  is independentof  this means that the the error in using   is increasing in 
Finally, Figure 1(c) shows     as the size of the tax notch varies. We
can see that as the notch becomes very small, the true MEB becomes very large. Thisis because  ! 1 as  !   While this cannot be proved analytically, the intuitionis clear from (22). As  !   then  !  = 0 (1 ¡  )¡  or (1 ¡  ) ! 0  So, both numerator and denominator in (22) tends to zero, but the denominator does sofaster10

For the denominator in (21) to be positive, we require 1 ¡  (1 + )   which is satis…ed for ¡   00015

15

Figures 2 and 3 show the optimal tax ¤ for the cases where …rst  = 0 (revenuemaximization), and  = 025 (welfare-maximization). Again, we show ¤ as de…ned in(23) along with the approximation setting  = 0 which we denote by ¤  In each …gure,
we show both ¤  ¤ as both  and  vary.
- Figures 2 and 3 in here Both of these taxes are decreasing in  as we might expect. Also, both taxes are
decreasing in  The error in using ¤ as an approximation for ¤ is generally smaller thanfor the MEB. For example, at baseline parameter values, the true revenue-maximizing tax
is about 0.55, whereas the approximation is 0.8.

6

An Application to VAT

6.1

The Set-Up

As remarked in the introduction, perhaps the most important example of a tax notch isthe value-added tax. Then In this section, we present a simple model of value-added tax,
based on Liu and Lockwood (2015), which mathematically, is equivalent to the model
developed above. We then calibrate the model using UK data from Liu and Lockwood(2015), to estimate the MEB from the VAT, taking into account the welfare e¤ects ofbunching at the threshold.
Consider a single industry with a …xed, large number of small traders  2 [ ] producing
a homogenous good. Small trader  combines his own labor input  with an intermediate
input  to produce output  via a …xed coe¢cients technologyn o = min 
(27)
where  measures the the input requirement per unit of output. In particular, for all
traders, one unit of output requires  units of input
We assume that trader has an iso-elastic disutility of labor(; ) =1 + 1

µ ¶1+ 1

(28)

So, traders di¤er in their disutility of labour. This assumption is not essential, butfacilitates comparison to the income tax case.11
11

For example,  could enter into the production function, (27) instead, as in Liu and Lockwood (2016).

16

For simplicity, it is assumed that traders only sell to …nal consumers, who have perfectly elastic demand for the good at price  = 1 This is analogous to the assumption
made in the taxable income literature that the wage is …xed, i.e. labor demand is perfectly elastic at a …xed wage. Finally, the intermediate input is produced only from labour
supplied by non-trader households via a …xed-coe¢cients technology where one unit oflabour are needed to produce one unit of the intermediate input. So, the tax-exclusive
price of the output is  the wage
The traders face and the producers of the intermediate inputs face a VAT system. If
the trader is registered, he must charge VAT on sales  at rate  but can claim back any
VAT paid on inputs. The trader must register for VAT if the value of sales  exceeds thethreshold 0 , but can register voluntarily even if   0 
Note that when not registered, the price of the input is (1 + ) So, the pro…t for thenon-registered trader is = (1 ¡ (1 + )) = (1 ¡ (1 + ))  = (29)

where  is the cost of inputs relative to revenue per unit sold. For the registered trader,
we reason as follows. This trader must charge VAT on his output. None of the output
VAT can be passed on to the buyer, as he has perfectly elastic demand. So, revenue perunit sold is (1 + ) But, if the trader is registered, he can claim back VAT on the input
use  so the price of the input is  So, overall, the pro…t for the registered trader is
µ
¶
µ
¶
1
1
 =
¡   =¡  
(30)
1+
1+

We now assume, to make the analysis interesting, that 1  (1 + ) From (29), thisensures that non-registered …rms make a positive pro…t. Also, it ensures that for a given
value of sales      so there is no voluntary registration. This is important because
then the VAT threshold functions exactly like a tax notch.

6.2

E¤ective VAT Rates

Now de…ne  = (1 ¡ ) Then, after some rearrangement, we can show that the utilityof trader  can be written as a function of value-added  = (1 ¡ ) and the VAT system
as follows; ³  ´1+ 1(; ) =  ¡  () ¡(31)(1 ¡ ) 1 + 1 
where(   · 0
 () =
  =
  =
(32)(1 + )(1 ¡ )1¡    017

As  is a free parameter, we set it equal to 1 ¡  Then, (32) is mathematicallyequivalent to (3).
Here,    are the e¤ective tax rates faced by nonregistered and registered traders
respectively on the value-added they generate. Obviously, both e¤ective rates are increasing in the statutory rate, . Also, note that both rates are increasing in input intensity
 Moreover, from our assumption 1  (1 + ),    .
So, faced with the tax schedule (32), all traders in the interval  2 [   ] will bunchat the VAT threshold 0 . Moreover,  = 0 (1 ¡  )  and  solves (26) with   
replaced by    
Finally, letting (1 ¡ ; ) be the value-added chosen by an unconstrained …rm facingtax  it can be shown that the revenue from the VAT is as in (12), with    replacedby    i.e.
µZ 
¶
µZ 
¶
Z 
 = 
(1 ¡  ; )() +0 () +(1 ¡  ; )() (33)






In (33), the base on which  is levied is the value-added of non-registered traders, andthe base of  is the value-added of registered traders.

6.3

The Marginal Excess Burden of the VAT

With the VAT, a change in the statutory rate  of VAT will change both e¤ective tax
rates    unless  = 0 i.e. no intermediate inputs are used. This is of course, analogous
to a reform that changes both  and  in the income tax model. So, for the VAT,
the formula for the MEB becomes somewhat more complex. To present the formulafor the MEB in this case, we need a few more de…nitions. First, note from (33), using
(1 ¡ ); ) = (1 ¡ )  the e¤ective bases of  and  are
Z 
Z 

 =
(1 ¡  ) () + 0 (( ) ¡ ( ))  =(1 ¡  ) () (34)Then, from (34), the intensive-margin elasticities of    with respect to the net-of-tax
rate are¯¯
1 ¡   ¯¯
1 ¡  ¯¯
= 
= 
(35)
  ¯ const (1 ¡  ) ¯ const
where

R 


(1 ¡  ; )()

1
(36)

The term  captures a new e¤ect of bunching; with bunching, the mass ( ) ¡ ( )of the non-registered …rms that are bunching are unresponsive to a change in the rate=18

of VAT, which lowers the aggregate intensive-margin elasticity of the tax base  withrespect to  .12
Moreover, recall that an increase in  causes both  and  to increase, so= 
1¡   
+ 1¡
1¡ 
 (37)measures the importance of a change in  on revenue relative to  . Armed with thesenew de…nitions, we can state our result.
Proposition 4. Assume A1, and that the distribution of sales (and pretax-income) is
Pareto, with shape and scale parameters  . Then, the MEB of the VAT is +   =(38)
1 ¡  (1 + ) ¡ where = (1 ¡ ) +    =and …nally the correction factor is
=¡




³

(1 ¡ )  + (1 ¡ ) +  +   
  
+ 1¡
1¡ 
 

´

(39)

(40)

So, we note now that bunching impacts the calculation of the MEB in two ways. First,
as before, there is a correction factor  in (38). The correction factor is more complex
than in the income tax case. The reason for the additional complexity is clear from (40);an increase in  now increases both    and in turn, both of these e¤ective taxes a¤ect  the top of the bunching interval, and thus revenue. An explicit formula for  in termsof parameters can be derived as in (22) above; this is done in the Appendix.
In addition, there is a second, new e¤ect of bunching in (39). Bunching dampens theintensive-margin response to a change in  because at a …xed     …rms in this interval
will not adjust their sales in response to a change in  This is captured by the term 
An interesting special case is where the small traders do not use any intermediate
input, so i.e.  = 0 Then  = 0  = 1+
=  so (38) simpli…es to  =

+
1+

1 ¡ 1+ (1 + ) ¡ 

12

(41)

A similar point has been noted before by Slemrod et. al. (1994) and Apps, Long, and Rees (2014)
who consider the design of a two-bracket income tax. Because the tax system studied is kinked, notnotched, the formula for the optimal lower rate of tax depends only on the intensive margin elasticity,
but this elasticity is dampened by the fact that taxpayers at the kink do not adjust their behavior in
response to the tax.

19

It can be checked that in this case,  is given by the explicit formula (22), replacing   by   0 respectively.13
Finally, how realistic is it that the distribution of sales  (or value-added ) is Pareto?
As already remarked, in the US, there is evidence that the size distribution of …rms as
measured by sales is Pareto (Luttmer (2007)). TO BE COMPLETED

6.4

Simulations

Here calibrate the VAT model, and plot the true  in (38) and an approximation to
the MEB as parameters vary. The approximation is the one treating VAT as a proportionaltax i.e. setting  = 0 in (41), which gives
1+  =
1 ¡ 1+ (1 + )

The parameters are calibrated as follows. In the UK, the statutory rate of VAT is20%, so  = 02 Liu and Lockwood (2016) calculate that for the universe of …rms in the
UK that …le a corporate tax return,  = 045. This gives  = 016  = 030 As alreadyremarked, there is evidence that the size distribution of …rms as measured by sales is
Pareto; Luttmer (2007) has a value for the US of  = 106 [TO BE UPDATED]
Next, de…ne 0 to be the productivity level corresponding to turnover just at thethreshold i.e. 0 (1 ¡  ) = 0  From Liu and Lockwood (2016), 62.5% of …rms are below³ ´106
the threshold. So, 0 must satisfy  (0 ) = 1 ¡ 0= 0625 or 0 = (0375)1106 =0396 Given that only the ratio 0 is determined, we set  = 1 so 0 = 253 But then
0 = 253(084)025 = 2422
Our results are given in Figures 4 and 5. Figure 4 shows the simpler case with nointermediate inputs i.e.  = 0 in which case we know that formula (38) reduces to theformula with a notched income tax.
- Figure 4 in here We can see that at the baseline …gures for the parameters e.g.  = 025 in Figure4(a), the true MEB is about 50% higher than the approximation. This di¤erence is muchsmaller than in the income tax case, and is driven partly by the lower value of  in the VAT
case. Indeed, we can see in Figure 4(b) that the accuracy of the approximation  13

If there is no bunching i.e. if  =   then  = 1  = 0 so   = 1¡(1+) But this requiresthat  = 1 so in this case,  ! 0 Also, before this point, 1 ¡  (1 + )  0 so this case is not interesting.

20

falls rapidly as  rises, because   is increasing in  whereas  is independentof 
Figure 5 shows the more realistic case with  = 045 Here, we see that the di¤erence
between the true MEB and the approximation is somewhat higher; the true MEB is about
3 times higher than the approximation. As in the case with no inputs, the true MEB is
increasing in both  and - Figure 5 in here -6.5

The Marginal Excess Burden and the Cost of VAT Compliance

In practice, there are signi…cant compliance costs to being VAT-registered i.e. preparing
and …ling a tax return, and paying any tax owed. In the UK, these costs are relatively
low as a proportion of turnover, even for …rms at the threshold. For example, a recent
literature review found that for the UK, at the registration threshold, these costs werearound 1.5% of turnover, declining to 0.1% or less for large companies (Federation of Small
Businesses (2010)). However, compliance costs can be much higher in other countries.
For example, a report by PwC found that for a …ctional small …rm, the hours taken forcompliance with VAT vary by region from an average 73 hours within the EU to 192 forLatin America, and even within the EU, there are substantial di¤erences, with 22 hours
in required in Finland to 288 in Bulgaria (PwC (2009)). So, it is de…nitely of interest to
ask how the marginal excess burden varies with compliance costs.
We can model compliance costs as follows. Let  be the cost of compliance as a fractionof sales at the threshold. We assume a …xed cost 0  or 1¡0 of compliance if registered,so that net utility of the trader with registration is  (; ) ¡ 1¡0 
The MEB can then be calculated exactly as before, except that  now solves
1(1 ¡  )0 (1 + ) ¡ ( )¡1 (0 )1+  ¡  (1 ¡  )1+ +(1 + )
0 = 01¡The results of variation in registration costs  on the MEB are shown in Figure 6. We
allow  to vary between 0% and 5% of sales. All other parameters are at their baseline
values, with  = 045. We expect that an increase in  will increase bunching and thusincrease the correction factor and the MEB, and this is exactly what happens.
- Figure 6 in here -21

Figure 6 shows that the MEB of the VAT does increase with  but the e¤ect is verysmall. An increase of compliance costs of zero to 5% of turnover at the threshold only
increases the MEB from 0.334 to 0.341, an increase of about 2%.

7

Conclusions

This paper has shown that the su¢cient statistic approach to the welfare properties ofincome (and other) taxes does not extend to tax systems with notches, because with
notches, changes in bunching induced by changes in tax rates have a …rst-order e¤ect
on tax revenues. In an income tax setting, we showed that the marginal excess burden(MEB) and the welfare-maximizing top rate of tax are given by the relevant formulae fora proportional tax as in Feldstein (1999) plus a correction factor. The Feldstein formulaealways underestimate the MEB and overestimate the revenue and welfare-maximizing rateof tax. Quantitatively, these mis-estimates can be very large; the MEB can be underestimated by an order of magnitude, but the errors in calculating the welfare-maximizing
tax are somewhat smaller.
An application to VAT was also studied. A simple model of small traders who di¤er inproductivity, and are subject to VAT at rate  above a threshold level of sales was shown
to be formally equivalent to the income tax model. We then show that the MEB of anincrease in the statutory rate of VAT is given by the Feldstein formula for a proportional
tax plus a correction factor as in the income tax case. A calibration of the model showsthat the proportional tax formula for the MEB of the VAT underestimates the true MEB
by a factor of up to three. This framework also allows us to evaluate the e¤ect of increased
compliance costs of VAT in the MEB via its impact on bunching; increased compliancecosts increase bunching and thus increase the MEB, but the e¤ect is quantitatively small.

22

8

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23

Kleven, H. J., & Schultz, E. A. (2014). Estimating taxable income responses using
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24

A

Appendix

Proof of Proposition 1. It remains to derive a formula for  From (8), noting that
1 ¡(1+1)
1 ³  ´1+1
1+ 1
 = ¡
()
=
¡

(42)
 1 + 1
1+ 
and (1 ¡ ; ) = (1 ¡ )  we have

=


(1 ¡  )  (1 + )
³ ´1+1(1 ¡  )1+ ¡ 0(43)

Next, from (13) and (16), using the fact that (1 ¡ ; ) = (1 ¡ )  we have1 ( 0 ¡  (1 ¡  ;  ))( )=R 
(1 ¡  ; )()

(44)



=

( 0 ¡  (1 ¡  )  )( )R
(1 ¡  )  ()

So, plugging (43),(44) into (??), we have:=(1 ¡  )( (1 ¡  )  ¡  0 )( ) (1 ¡  )  (1 + )R
³ ´1+1(1 ¡  )  ()
(1 ¡  )1+ ¡ 0

=

(1 ¡  )( (1 ¡  )  ¡  0 )( )
(1 ¡  ) [ j ¸  ](1 ¡ ( ))

(45)



 (1 + )
³ ´1+1(1 ¡  )1+ ¡ 0R
where in the second line we have used  () = [ j ¸  ] (1 ¡ ( )) Now, given that  follows a Pareto distribution with shape and scale parameters  
we also know that

()

[ j ¸  ] ==(46)
 ¡ 1 1 ¡ ()Plugging (46) into (45), we get:
=

(1 ¡  )( (1 ¡  ) ¡  0  )( ¡ 1)(1 + )µ³ ´1+1 ¶1+
(1 ¡  ) ¡ 0(47)

as required. ¤Proof of Proposition 4.3. The government objective, written as a Lagrangean including
the constraint  =  is
Z 
Z 
 =
((1 ¡  ; ))() +((0 (1 ¡  ) ¡ (0 ; )))()


+

Z 





((1 ¡  ; ))() + ( ¡ )
25

So, the welfare-maximizing top rate of tax is de…ned by
Z 


=¡
()(1 ¡  ; )() + 
=0




(48)

where  = 0  So, plugging (15), in (48), and rearranging, we get;"#¯ ¯¯ 
¡( )(1 ¡ ( )) ¡  +   + +=0 ¯ const  

(49)

where

( ) =

Z 



¤

¤

( ¡ ) ()  = (1 ¡ ( )  =Z ¤ ()



Dividing though (49) by  and  and using (??), we get¡ + 1 ¡ 
1  
+=01 ¡    

(50)

with

( )(1 ¡ ( )) + 
Multiplying (50) through by 1 ¡  and using the de…nition of  in (??), we get=(1 ¡  )(1 ¡ ) ¡   ¡  = 0
Rearranging this last expression, assuming ( ) = 0 and normalizing  to 1 gives(23). ¤
Derivation of (31), (32), (33). We …rst derive (31), (32). Trader utility is pro…t minus
the disutility of labour. So, combining (42), (29), (30) and using  = (1 ¡ )  =  get:µ¶1+ 1
(1 ¡ ) = (1 ¡ (1 + )) ¡1 ¡  1 + 1µ¶µ¶1+ 11
(1 ¡ )
 =
¡ ¡
1+
1 ¡  1 + 1

(51)Now, using  = (1 ¡ ) in (51), we get

1 ¡ (1 + )
 ³  ´1+ 1
 =
¡
1¡
1 ¡  1 + 1 
µ
¶
1


 ³  ´1+ 1
 =
¡
¡
(1 + )(1 ¡ ) 1 ¡ 
1 ¡  1 + 1 
26

(52)

Finally, we note that for (52) to imply (32), we require1 ¡  =1 ¡ (1 + )1
 1 ¡  =¡1¡
(1 + )(1 ¡ ) 1 ¡ (53)But, solving (53) for     we get (32) as required.
Now we derive (33). Let () be the sales of an  ¡  trader. Then, revenue fromthe from the VAT is
Z 
Z 

=
()() + 
()()
(54)
1 +  
The …rst term is revenue from VAT levied on the value of sales of registered …rms, because
the sale price is 1(1 + ) and the second term is revenue from inputs sold by the intermediate input producer to …rms that do not register for VAT. Using () = ()(1 ¡ )
we can write this as
Z 
Z 


=
()() +()()
(55)(1 + )(1 ¡ ) 1¡ Finally, replacing () by (1 ¡  ; ) 0  or (1 ¡  ; ) where appropriate, we get (33)
as required. ¤Proof of Proposition 6.3. Let    be the bases of the e¤ective taxes    de…nedin (34) Then from (17),(33), and remembering that a change in the statutory rate of VAT
 changes    via (32), we have:
µ
¶



=¡
 +

(56)



Ã
!
Ã
!
¯
¯


 ¯¯ ¯¯= + 
+ + 
¡ 0
(57)
¯
¯


  const  const
where

µ
¶
     =¡+   
So, plugging (56),(57) into (14), we have, after rearrangement
0(58)

( + )


(59)
µ
¶
µ
¶
¯
¯
 
1¡ ¯
 1¡ ¯
+ 1¡    (1¡ ) ¯+ 01¡   (1¡ ) ¯ const
 const
µ
¶
µ
¶
=
¯
¯
 
1¡  ¯ 
1¡  ¯
1 ¡  ¡    ¯+ 1¡  1 ¡  ¡    ¯¡ 01¡   = ¡ const=

  
 + 1¡  +  01¡    
(1 ¡  (1 + )) + 1¡(1 ¡  (1 + )) ¡  0
1¡ 
 

27

 const 
where in the last line, we have used (35).So, dividing top and bottom of (59) by 1¡+  
and using the de…nition of  from (37), and the de…nition of  from (40), we get1¡   =(1 ¡ )  +   + 
1 ¡ (1 ¡ ) (1 + ) ¡  (1 + ) ¡ (60)

 +
Finally, using the de…nitions of  = (1 ¡ ) +    = (1¡)
 (60) can be(1¡) +rearranged to (39), as required. ¤28

Figure 1: The Marginal Excess Burden

1
MEB
MEBA

0.9
0.8
0.7

MEB

0.6
0.5
0.4
0.3
0.2
0.1
0
0.1

0.15

0.2

0.25
0.3
0.35
e
(a) Changes in e, 4t = 0.2, a = 1.5

0.4

2
MEB
MEBA

1.8
1.6
1.4

MEB

1.2
1
0.8
0.6
0.4
0.2
0

1.1

1.2

1.3

1.4

1.5
a

1.6

1.7

1.8

1.9

2

(b) Changes in a, 4t = 0.2, e = 0.25

18
16

MEB
MEBA

14

MEB

12
10
8
6
4
2
0

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.052 t(c) Changes in 4t, e = 0.25, a = 1.5

Figure 2: The Revenue-Maximising Top Rate of Tax

0.95
t*
t*A

0.9
0.85
0.8

t*

0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.1

0.15

0.2

0.25
e

0.3

0.35

0.4

(a) Changes in e, , a = 1.5, ḡ = 0.0

t*
t*A

0.85

t*

0.8

0.75

0.7

0.65

0.6

1.02

1.04

1.06
a

1.08

(b) Changes in a, , e = 0.25, ḡ = 0.0

3

1.1

Figure 3: The Wefare-Maximising Top Rate of Tax

1
t*
t*A

0.9

t*

0.8

0.7

0.6

0.5

0.4
0.1

0.15

0.2

0.25

0.3

0.35

0.4

e
(a) Changes in e, , a = 1.5, ḡ = 0.25

0.9
t*
t*A

0.85

t*

0.8

0.75

0.7

0.65

0.6
1.02

1.04

1.06

1.08

a
(b) Changes in a, e = 0.25, ḡ = 0.25

4

1.1

Figure 4: The Marginal Excess Burden of the VAT

0.12
MEB
MEBA

0.11
0.1
0.09

MEB

0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.1

0.15

0.2

0.25
e

0.3

0.35

0.4

(a) Changes in e, a = 1.06, s = 0.0

0.095
MEB
MEBA

0.09
0.085

MEB

0.08
0.075
0.07
0.065
0.06
0.055
0.05

1.02

1.04

1.06

1.08
a

(b) Changes in a, e = 0.25,s = 0.0

5

1.1

1.12

Figure 5: The Marginal Excess Burden of the VAT

0.25
MEB
MEBA
0.2

MEB

0.15

0.1

0.05

0
0.1

0.15

0.2

0.25
e

0.3

0.35

0.4

(a) Changes in e, a = 1.06, s = 0.45

0.16
MEB
MEBA
0.14

MEB

0.12

0.1

0.08

0.06

0.04

1.02

1.04

1.06

1.08
a

(b) Changes in a, e = 0.25,s = 0.45

6

1.1

1.12

Figure 6: The Marginal Excess Burden of the VAT and Registration Costs

0.1343MEB
0.13420.13410.134

MEB

0.13390.13380.13370.13360.13350.13340.1333
0

0.01

0.02

0.03

0.04

k
(a) Changes in k , e = 0.25,a = 1.06, s = 0.45

7

0.05

Malas Notches
Ben Lockwood
Not-For Publication Appendix
Details of MEB Simulation for the VAT Case. We need to express all the relevant
elements of the   in terms of the parameters,   0 and     In turn, we knowthat  = 0 (1 ¡  ) and that  is determined by1(1 ¡  )0 (1 + ) ¡ ( )¡1 (0 )1+  ¡  (1 ¡  )1+ = 0(1)Assume that the distribution of …rms is Pareto with shape and scale parameters  .
Without loss of generality, we assume  = 1; so, the distribution and density of  is () =
1 ¡ ¡  () = +1
 So, using these formulae and (1 ¡ ; ) = (1 ¡ )  we have by
routine calculation;
Z  = (1 ¡  )() = (1 ¡  )
( )1¡
(2)
¡1

Z 

 = (1 ¡  )() + 0 (( ) ¡ ( ))1= (1 ¡  )



¡
¢

(1 ¡ ( )1¡ ) + 0 ( )¡ ¡ ( )¡¡1Moreover, from the formulae for    in the paper, we have:


1


=

=
2

(1 ¡ )(1 + )
(1 ¡ )(3)

So, plugging (3) into the formula for  in the paper, we can write
=


1¡


+ 1¡
(1 + )2
1¡


(4)

Plugging (2) into (5) allows us to compute  as a function of   0 and    .
Next, using (1 ¡ ; ) = (1 ¡ )  and the properties of the Pareto distribution, we
have;
=

R 


(1 ¡  ; )()
(1 ¡  ) ¡1(1 ¡ ( )1¡ )=

So, using (2), (5),  can be computed as a function of   0 and    .
1

(5)

Finally, recalling the de…nition of  in the paper, we have:³´  +  
=¡  
+ 1¡ 1¡ 
³
´



1

+


(1+)2 
=¡


1
 + 1¡ (1+)
2
1¡

(6)

where in the second line, we use (3).
 
It remains to calculate    From (1), we have:  

=


=


(1 ¡  )  (1 + )
³ ´1+1
1+(1 ¡  ) ¡ 0(7)

¡0 (1 + )
³ ´1+1
1+(1 ¡  ) ¡ 0Moreover, from the formula for in the paper, and the iso-elastic form of (1 ¡  )

we get= ( 0 ¡  (1 ¡  )  )( )(8)Plugging (7),(8) into (6), and using the formula for the density of the Pareto density to
substitute out ( ), we eventually get:³´1( (1 ¡  )  ¡  0 ) (1 ¡  )  (1+)(1 + )2 ¡ 0 µ¶=³ ´1+1 ³´( )+1
1(1 ¡  )1+ ¡ 0+
1¡ (1+)21¡This expression for  is computable knowing   0 and    . Thus, all the componentsof   in the paper can be calculated. ¤2