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Income Tax Avoidance and Evasion: A Narrow

Bracketing Approach

Duccio Gamannossi degl’Innocentiy

duccio.gamannossi@imtlucca.it

Matthew D. Rablenz

m.rablen@she¢ eld.ac.uk

November 21, 2016

Abstract

We characterize optimal individual tax evasion and avoidance when taxpayers “narrow bracket”

the joint avoidance/evasion decision by exhausting all gainful methods for legal avoidance before

choosing whether or not also to evade illegally. We …nd that (i) evasion is an increasing function

of the audit probability when the latter is low enough, yet tax avoidance is always decreasing

in the probability of audit; (ii) an analogous …nding to the so-called Yitzhaki puzzle for evasion

also holds for tax avoidance –an increase in the tax rate decreases the level of avoided income

and the level of avoided tax; and (iii) that, holding constant the expected return to evasion, it

is not always the case that the combined loss of reported income due to avoidance and evasion

can be stemmed by increasing the …ne rate and decreasing the audit probability.

JEL Classi…cation: H26, K42, D82, H21.

Keywords: Tax avoidance, Tax evasion, Narrow bracketing, Financial intermediaries.

Acknowledgements: We thank the Editor, James Alm, and two anonymous referees for helpful comments.

Gamannossi degl’Innocenti gratefully acknowledges …nancial support from the Ministero dell’Istruzione,

dell’Università e della Ricerca (cycle XXVIII) and from the European Commission (Erasmus mobility grant

2015-1-IT02-KA103-013713/5).

yIMT School for Advanced Studies, Piazza S. Francesco 19, 55100, Lucca, Italy.

zDepartment of Economics, University of She¢ eld, She¢ eld, S1 4DT, UK.

1 Introduction

Individuals take a variety of actions to reduce their tax liabilities. The UK tax authority,

for instance, distinguishes three distinct types of action (HM Treasury and HM Revenue and

Customs 2011): those that breach tax law (tax evasion); those that “use the tax law to get

a tax advantage that Parliament never intended”(tax avoidance); and those that “use tax

allowances for the purposes intended by Parliament”(tax planning). By these de…nitions,

both tax evasion and tax avoidance are responsible for signi…cant losses in public revenue:

estimates provided by the UK tax authority put the value of tax avoidance at £ 2.7 bn. and

the value of tax evasion at £ 4.4 bn. (HM Revenue and Customs 2015). Given the …rst order

signi…cance of tax avoidance, it is of note that the …rst economic studies relating to tax

compliance (e.g., Allingham and Sandmo 1972; Srinivasan 1973; Yitzhaki 1974; Christiansen

1980) neglect the possibility of tax avoidance altogether, and the economic literature that

followed has largely retained this bias.

In this paper we introduce tax avoidance into the portfolio model of tax evasion (Yitzhaki

1974). To model the joint tax avoidance/evasion decision we build on insights developed

in psychology and behavioral economics. In particular, we allow for a pervasive propensity

among human decision makers facing multiple-dimension problems –that of narrow brack-

eting. In our context, a decision maker who narrow brackets would decompose sequentially

the joint decision {avoidance,evasion} into narrow brackets, e.g., {avoidance} followed by

{evasion}. A key feature of narrow bracketing is that the decision maker tends to choose

an option in each stage without full regard to the other decisions and circumstances that

he or she faces (Rabin and Weizsäcker 2009). In the present context, therefore, the choice

made …rst is made without the decision maker being assumed to have pre-meditated on

how the second choice will be made. Important in this context is whether the taxpayer is

more likely to make the avoidance or evasion choice …rst. This question – as to the order

in which a complex decision is mentally staged –is thought to depend heavily on mentally

focal qualitative features of the choice set. We argue that a focal feature of the choice set is

that avoidance is ostensibly legal whereas evasion is illegal. Indeed, judiciaries on both sides

of the Atlantic have long upheld the right of a citizen to challenge the proper interpretation

of tax law and to pay only the tax they owe in law. Thus, while tax avoidance can be seen

as the rightful exercise of a basic right by some lights, tax evasion lacks an equivalent in-

terpretation. Accordingly, if taxpayers qualitatively prefer avoidance to evasion, we suppose

they would focus on exhausting opportunities for legal tax avoidance before subsequently

1

focusing on opportunities for illegal evasion.

We are by no means the …rst to propose that taxpayers distinguish qualitatively between

avoidance and evasion, however. This distinction has previously been represented by sup-

posing that a cost owing to social stigma and/or personal guilt is attached to the illegal act

of tax evasion. This cost can be …nancial (e.g., Chetty 2009; Lee 2001) or psychic (e.g., Ben-

jamini and Maital 1985; Gordon 1989; Myles and Naylor 1996; Kim 2003; Dell’Anno 2009).

The concept of narrow bracketing o¤ers an alternative perspective on this distinction: in

our model, illegal evasion is not considered until all gainful avenues for legal avoidance have

been exhausted.1In this way our paper relates to a literature on two-stage decision making

(e.g., Gorman 1959; Strotz 1959; Blackorby et al. 1970). Unlike this literature, however, we

do not seek a sequential approach to the joint avoidance/evasion decision that coincides with

the outcome that would obtain were the joint decision assumed to be made simultaneously.

In addition to the legal distinction between avoidance and evasion, we further assume that

avoidance is costly whereas evasion is costless. Devising avoidance schemes that reduce a tax

liability without ostensibly violating tax law invariably requires a detailed understanding of

tax law, coupled with a degree of ingenuity. A classical form of avoidance scheme, for in-

stance, involves the implementation of a circular sequence of self-cancelling option agreements

that return the seller to his or her original position, but in the process create an allowable

loss. As, however, few taxpayers are equipped to conceive of and implement independently

such avoidance schemes, it is necessary to purchase them on the open market.2Satisfying

this demand for tax avoidance is a substantial industry dedicated to the development and

marketing of avoidance schemes (see, e.g., Sikka 2012; Committee of Public Accounts 2013;

Addison and Mueller 2015). By contrast, many forms of tax evasion require no technical

or legal expertise. Intentionally understating income on the tax return, for instance, may

readily be performed independently.

We …nd that, in two respects, allowing for tax avoidance importantly changes the charac-

teristics of optimal tax evasion. First, under plausible conditions, evasion is an increasing

function of the probability of audit. Second, we re-examine the …nding of Christiansen (1980,

1Strictly speaking, the stigma cost approach and ours and not mutually exclusive. To present our approach

in the simplest possible light, however, we do not allow for a stigma cost.

2People not only have di¢ culties in understanding tax law, but also show poor knowledge about tax rates

and basic concepts of taxation. For evidence that people often signi…cantly under-estimate marginal tax

rates see, e.g., Lewis (1978) and Gideon (in press).

2

391) that “if the …ne is increased, but the e¤orts to detect tax evaders are adjusted so as

to keep the expected gain from tax evasion unaltered, risk averters will always reduce their

tax evasion.”We are again able to prove this result, yet in our model it is the total amount

of lost tax (through both avoidance and evasion) that is economically pertinent. When we

consider both avoidance and evasion, however, it is possible that taxpayers declare more

income if the probability of audit is increased, and the …ne decreased, holding the expected

gain from tax evasion constant.

The paper adds to the small, but growing, economic literature on tax avoidance. The two

closest analyses to ours are Alm and McCallin (1990) and Alm (1988). The former describes

avoidance and evasion as risky assets –each asset has a return characterized by a mean and

variance, and the interdependence between the two returns is characterized by a covariance

– while the latter characterizes avoidance as a riskless, albeit costly, asset. Whereas both

of these analyses consider the simultaneous determination of evasion and avoidance, in our

framework we argue that these are chosen sequentially. Di¤erent from Alm and McCallin,

we model the mean, variance and covariance of evasion and avoidance, rather than taking

these quantities as exogenous. Unlike in Alm (1988), we take avoidance to be risky, owing

to the possibility of e¤ective anti-avoidance measures by the tax authority.

Much of the remaining literature on tax avoidance is, however, concerned with whether

income tax has “real” e¤ects upon labor supply, or simply leads to changes in the “form”

of compensation. Accordingly, in these studies the term “tax avoidance” typically refers to

all form-changing actions that reduce a tax liability.3This de…nition overlaps with ours,

but is broader in the sense that it also includes actions that fall in to our notion of tax

planning. In the context of this broader de…nition of tax avoidance, Slemrod and Kopczuk

(2002), Piketty, Saez, and Stantcheva (2014) and Uribe-Teran (2015) analyze theoretically

the elasticity of taxable income in the presence of avoidance. In the empirical literature,

Slemrod (1995, 1996) …nds pronounced tax avoidance e¤ects in the response of high-earners

to tax changes, while Feldstein (1999) …nds that accounting for tax avoidance signi…cantly

increases estimates of the implied deadweight loss of income taxation. Lang, Nöhrbaß, and

Stahl (1997) estimate that tax avoidance costs the German exchequer an amount equal to

around 34 percent of income taxes paid. Fack and Landais (2010) show that the response

of charitable deductions to tax rates is concentrated primarily along the avoidance margin

3For a detailed discussion of these “form-changing” actions see, e.g., Stiglitz (1985) and Slemrod and

Yitzhaki (2002).

3

(rather than the real contribution margin), while Gruber and Saez (2002) show that the

elasticity of a broad measure of income is notably smaller than the equivalent elasticity

for taxable income, suggesting that much of the response of taxable income comes through

deductions, exemptions, and exclusions.

The plan of the article is as follows: in section 2 we motivate the key behavioral assumptions

behind our analysis, from which section 3 develops a formal model. Section 4 performs the

main analysis and 5 compares our …ndings to the literature. We extend the model in section

6 to allow for risk in the tax authority’s e¤orts to illegalize avoidance schemes, and section

7 concludes. All proofs are in the Appendix.

2 Deciding to Avoid and/or Evade Tax

Two key features of our modelling of the joint decision to avoid and/or evade are that (i) the

taxpayer makes the avoidance and evasion decisions sequentially; and (ii) that the avoidance

decision is made …rst. The …rst feature –that complex decisions are routinely broken down

into smaller ones –is often termed narrow bracketing in the behavioral literature. The second

feature –the choice of how to stage the sub-decisions within the larger composite decision

– is sometimes termed decision staging (Johnson et al. 2012). We discuss each of these

features in turn.

2.1 Narrow Bracketing

A mass of evidence suggests that people narrowly bracket: a decision maker who faces a

multi-dimensional decision tends to break the decision down sequentially, proceeding at each

stage to isolate a single dimension of the problem without full regard to the other dimensions

of the problem. In the context of monetary risk, Tversky and Kahneman (1981) present an

experiment that demonstrates how powerful this propensity is. In their experiment, people

narrowly bracket even when faced with only a pair of independent simple binary decisions

that are presented on the same sheet of paper. Narrow bracketing lies at the heart of

current explanations of phenomena such as the stock market participation puzzle (Barberis,

Huang, and Thaler 2006), the equity premium puzzle (Benartzi and Thaler 1995; Gneezy and

Potters 1997) and choice among lotteries (Camerer 1989; Battalio, Kagel, and Kiranyakul

1990; Langer and Weber 2001).

Cognitive limitations –in perception, attention, memory, and analytical processing – are

4

thought to be an important reason for narrow bracketing (Read, Loewenstein, and Rabin

1999). Accordingly, in the experiment of Read et al. (2001) subjects who were required

to resolve a complex choice problem sequentially actually made better choices than those

subjects who were required to proceed simultaneously. Clearly, however, decision making

outcomes under narrow bracketing can, in other contexts, appear worse than those arrived at

from a wide bracketing perspective. For instance, decision making under narrow bracketing

may violate …rst-order stochastic dominance (Rabin and Weizsäcker 2009) and, in a famous

example, the “one day at a time”bracketing observed among New York cab drivers fails to

maximize earnings per hour across days (Camerer et al. 1997). Thus, while the desirability

of narrow bracketing is still the subject of academic debate (see, e.g., Palacios-Huerta 1999;

K½

oszegi and Rabin 2009), the pervasiveness of the phenomenon is not in doubt. It is thus of

relevance to understand the nature of the joint avoidance and evasion decision under narrow

bracketing.

2.2 Decision Staging

Having established that taxpayers may well mentally separate the joint avoidance and evasion

decision there remains the question as to how this is done. According to Kahneman (2003),

the way individuals will choose to stage or frame a decision is heavily shaped by the features

of the situation at hand that come to mind most easily – to use the technical term, by

the features that are most “accessible.” This notion is supported in the context of tax-

related decision making by McCa¤ery and Baron (2004, 2006). These authors employ a

slightly di¤erent terminology –the isolation e¤ect –which, however, refers to the tendency

of respondents in their study to “decide complex matters –and tax raises a host of complex

matters – by responding to the most salient or obvious aspect of a choice set or decision

problem.”

In the context of the joint avoidance and evasion decision we argue that the most accessible

feature or aspect of the choice set is that tax avoidance and evasion are qualitatively distinct:

one is legal, and the right to practice it has often been defended by the judiciary, whereas

the other is a crime. Courts on both sides of the Atlantic have for many years upheld the

right of citizens to challenge the interpretation of tax law (Barker 2009; Prebble and Prebble

2010). In 1936 Lord Tomlin, ruling in a case involving the Duke of Westminster, surmised

that “[e]very man is entitled, if he can, to order his a¤airs so that the tax attaching under

the appropriate Acts is less than it otherwise would be. If he succeeds in ordering them so

5

as to secure this result, then, however unappreciative the Commissioners of Inland Revenue

or his fellow taxpayers may be of his ingenuity, he cannot be compelled to pay an increased

tax.”This so-called “Duke of Westminster principle”dominated UK tax avoidance law until

the 1980s and remains in‡uential today. Similarly, in the US, Judge Learned Hand stated

in 1947 (in Commissioner vs. Newman) that “[o]ver and over again courts have said that

there is nothing sinister in so arranging one’s a¤airs as to keep taxes as low as possible.

Everybody does so, rich or poor; and all do right, for nobody owes any public duty to pay

more than the law demands: taxes are enforced exactions, not voluntary contributions. To

demand more in the name of morals is mere cant.”

There is evidence that these traditional legal arguments continue to a¤ect public sentiment

towards tax avoidance. In the qualitative study of Kirchler, Maciejovsky, and Schneider

(2003) participants relate tax avoidance to lawful acts enabling tax reduction, to cleverness,

and to costs. Tax evasion, by contrast, is associated with illegal acts such as fraud, crim-

inal prosecution, risk, tax-audits, punishment, penalty, and the risk of detection. In this

sense, we argue that, for many taxpayers, tax avoidance is qualitatively preferred to evasion.

Accordingly, we argue that taxpayers would exhaust the scope for legal avoidance before

subsequently deciding whether or not they additionally wish to evade illegally (rather than

the other way round).

3 Model

Our model is a direct extension of the canonical portfolio model of Yitzhaki (1974). A

taxpayer has an income (wealth) wand faces a tax on income given by tw, where t2(0;1).

Taxpayers behave as if they maximize expected utility, where utility is denoted by U(z) =

log z.4The taxpayer’s true income is not observed by the tax authority, but the taxpayer

must declare an amount x2[0; w]. The taxpayer can choose to avoid paying tax on an

amount of income A2[0; w], and subsequently to evade illegally an amount of income

E2[0; w A], so x=wEA.

Evasion is …nancially costless but avoidance technology must be bought in a market in which

4Thus taxpayers are risk averse and have a constant (unit) co-e¢ cient of relative risk aversion. We adopt

the logarithmic form for reasons of analytic tractability. Other simple speci…cations such as constant absolute

risk aversion or mean-variance utility can instead be used and yield similar results. However, the assumption

of constant relative risk aversion has stronger empirical support (see, e.g., Wakker 2008; Chiappori and

Paiella 2011).

6

“promoters”sell avoidance schemes to “users”.5A common feature of this market is the “no

saving, no fee”arrangement under which the price received by a promoter is linked to the

amount by which their scheme stands to reduce the user’s tax liability. Although system-

atic information regarding the precise contractual terms upon which avoidance schemes are

typically sold is scarce, we understand from a detailed investigation in the UK that, for

the majority of mass-marketed schemes, the fee is related to the reduction in the annual

theoretical tax liability of the user, not the ex-post realization of the tax saved (Committee

of Public Accounts 2013, 11). This implies, in particular, that the monetary risks associated

with the possible subsequent detection and termination of a tax avoidance scheme are borne

by the user.6Accordingly, we assume that the promoter’s fee is a proportion 2(0;1) of

the amount by which the taxpayer’s tax liability stands to be reduced, tA. In this way,

may be interpreted as measuring the degree of competition in the market for tax avoidance

schemes, with lower values of indicating the presence of stronger competitive forces.

Although traditional arguments around the morality of tax avoidance continue to a¤ect im-

portantly public sentiment, there has nonetheless been a discernible shift in the attitudes

of the judiciary, beginning in the 1980s (Stevens 2013). Increasingly, as our opening de…-

nition of tax avoidance suggests, courts apply a purposive interpretation of the law. This

interpretation is summarized by Permanent Judge Ribeiro (in Collector of Stamp Revenue

vs. Arrowtown Assets Ltd.) who states that “the ultimate question is whether the rele-

vant statutory provisions, construed purposively, were intended to apply to the transaction,

viewed realistically.” Armed with this purposive interpretation of the law, tax authorities

now routinely seek to have particular avoidance schemes ruled illegal. Yet taxpayers may le-

gitimately continue to use an avoidance scheme while the (often lengthy) process of shutting

it down is ongoing. Moreover, if the scheme is eventually declared illegal the tax authority

can only seek the amount of tax that was properly due. That is, it cannot levy …nes for

tax evasion retrospectively once a scheme has been outlawed. Inherent in our de…nition of

tax avoidance (as distinct from tax planning) is that –should the tax authority learn of the

scheme –it will consider it illegal.

The taxpayer’s income declaration is audited with probability p2(0;1). If audited, Eand A

5For analyses of the market for tax advice see, e.g., Reinganum and Wilde (1991) and Damjanovic and

Ulph (2010).

6It is apparent that such arrangements give promoters incentives to mis-represent the level of risk involved

in particular schemes. Consistent with this point, Committee of Public Accounts (2013, 11) indeed …nds

evidence of such mis-selling.

7

are observed and the taxpayer has to pay [1 + f]tE on account of the amount of evaded tax,

where f > 0is the …ne rate. The tax authority mounts a legal challenge to the avoidance

scheme, which is successful with probability pL. In the event that the legal challenge is

successful, the tax authority obtains the right to reclaim the tax owed (but cannot levy a

…ne). In this case, instead of paying tx in tax, the taxpayer must instead pay t[x+A].

The taxpayer’s expected utility is therefore given by

EU(A; E) = [1 p]U(wn) + ppLU(was) + p[1 pL]U(wau);(1)

where wnis the taxpayer’s wealth in the state in which they are not audited, wasis the

taxpayer’s wealth in the state in which they are audited and the tax authority’s legal challenge

is successful, and wauis the taxpayer’s wealth in the state in which they are audited and the

tax authority’s legal challenge is unsuccessful:

was(A; E) = wt[wE][1 + f]tE tA;(2)

wau(A; E) = wt[wAE][1 + f]tE tA;(3)

wn(A; E) = wt[wAE]tA: (4)

A key distinguishing factor between evasion and avoidance in this context is that avoidance

entails a cost tA in all states of the world. Thus, if avoidance is detected and the scheme

closed down a taxpayer is worse-o¤ for having chosen to avoid, even though they are not

…ned on avoided income. To ensure that the amount of taxes, …nes and fees never exceeds a

taxpayer’s wealth for any A+E2[0; w]we must assume [1 t]=t > max f; f g.

We suppose that taxpayers choose their preferred level of avoidance and evasion sequentially:

gainful opportunities for tax avoidance are exhausted before the taxpayer decides whether

to engage additionally in evasion. Thus, taxpayers …rst choose avoided income as

A= arg maxAEU(A; 0) ; (5)

and then evaded income as

E= arg maxEEU(A; E). (6)

8

4 Analysis

We now present an analysis of the model of the previous section. For analytic tractability,

we shall consider the special case of the model with pL= 1, such that legal challenges by

the tax authority are always successful. In a later section we shall demonstrate numerically

how the results with pL= 1 relate to the results for the more general case with pL<1.

To begin, it is helpful to de…ne the function R(z) = [1 z]=z, such that, e.g., R(p)is the

classical odds ratio found in decision theory. We may then state our …rst Proposition:

Proposition 1 An interior optimum for avoidance and evasion satis…es

A=pR (t)

1[R(p)R()1] w

and

E=pR(t)

1

[1 p] [1 f R()]

fw

where

R(p)R()>1> fR () ; pR(t)

1

[1 p] [1 f R()] + f[R(p)R()1]

f<1:

Proposition 1 gives closed-form expressions for optimal avoidance and evasion when both

are at an interior maximum, and the conditions needed for a such an interior maximum to

arise. The …rst sequence of inequalities at the bottom of the Proposition guarantee that

A; E>0. The left-side inequality, R(p)R()>1, is the condition that the avoidance

gamble be better than fair. As a necessary condition for both inequalities to hold it must be

that R(p)R()> f R (), which implies R(p)> f. This is the standard restriction in the

portfolio model of tax evasion that the evasion gamble be better than fair. The right side

inequality at the bottom of the Proposition ensures that A+E< w. To gain insight into

how Aand Eare related, note that we may write one as a (linear) function of the other:

E(A) = p[wR (t)A] [R(p)f]

fpA:(7)

From (7) we note that

@E

@A=f R() + R(p)

f[1 + R(p)] [1 + R()] <0;(8)

9

so the amount of evaded income Eis negatively related with the amount of avoided income

A. This …nding matches that of Alm, Bahl, and Murray (1990) using data from Jamaica,

who …nd that evasion and avoidance are substitutes. We now consider the comparative

statics of optimal avoidance and evasion:

Proposition 2 At an interior optimum for avoidance and evasion it holds for Athat

@A

@w =A

w>0;

@A

@t =A

t[1 t]<0;

@A

@f = 0;

@A

@ =wR(t) [p+ (1 p)R()2] [1 + R()]2

R()2<0;

@A

@p =R(t)w

[1 ]<0;

and for Ethat

@E

@w =E

w>0;

@E

@t =E

t[1 t]<0;

@E

@f =E

[1 fR()] f<0;

@E

@ =f[R()]2+ 1

[1 ] [1 f R()]E>0;

@E

@p =R(p)1

1pE?0() R(p)?1:

Proposition 2 is derived via straightforward di¤erentiation of the expressions for Aand E

in Proposition 1 so we omit the proof. Beginning with the comparative statics of A, we

see that wealthier people are predicted to avoid more income than less wealthy people. The

second result is an extension of the well-known Yitzhaki paradox for evasion to the case of

avoidance –avoided income falls as the tax rate is increased. The intuition for this result is

analogous to that for evasion: a higher marginal tax rate makes the taxpayer feel poorer, and

thereby more risk averse. As would be expected, an increase in the competitiveness of the

market for avoidance schemes (a decrease in ) increases avoided income, and an increase

in the probability of audit decreases avoided income. Of course, knowing @A=@t < 0does

10

not warrant that the total tax avoided, tA, also falls. It is straightforward to show, however,

that

@tA

@t =1

t[1 t]1A<0:(9)

Turning to evasion, the logic of the chain rule implies that, for an arbitrary exogenous

variable z, it must hold that

@E

@z =@E

@z A=cons:

+@E

@A

@A

@z ;(10)

where the …rst term on the right side is the direct e¤ect of zon evasion, and the second

term captures the indirect e¤ect on evasion arising from the e¤ects of zupon avoidance.

Intuitively, the indirect e¤ect is the income e¤ect imparted upon the evasion choice by

movements in avoidance. Noting from equation (8) that @E=@A<0it follows that if

@A=@z and @E=@zjA=cons: are of the same sign –as turns out to be the case for each of

the variables fw; t; f; pg–then the direct and indirect e¤ects in equation (10) oppose each

other. This observation notwithstanding, the …rst three results in Proposition 2 –those for

fw; t; f g–are each unambiguous and consistent with Yitzhaki (1974). Unambiguity in this

context arises as the direct e¤ect can be shown to always dominate the indirect e¤ect. To

take the e¤ect of the tax rate on evasion as an example, we …nd the direct e¤ect –using (7)

–as

@E

@t A=cons:

=w[1 + R(t)]2[R(p)f]

f[1 + R(p)] <0;(11)

Combining equation (11) with @A=@t in Proposition 1 and @E=@Awe can rewrite the

direct e¤ect in terms of the indirect e¤ect,

@E

@t A=cons:

=[R(p)f] [1 + R(p)] R()

[R(p) + fR()] [R(p)R()1]

@E

@A

@A

@t , (12)

such that, by equation (10), we obtain an alternative form for @E=@t to that given in

Proposition 2:

@E

@t =[R(p)f] [1 + R(p)] R()

[R(p) + fR()] [R(p)R()1]

@E

@A

@A

@t +@E

@A

@A

@t (13)

=R(p) [1 + R()] [1 f R()]

[R(p) + fR()] [R(p)R()1]

@E

@A

@A

@t <0:(14)

11

As well as evaded income being decreasing in the tax rate, it is straightforward to show that

evaded tax, tE, is decreasing in the tax rate too. As it has no direct e¤ect on evasion,

the e¤ect of competition in the market for avoidance (as captured by ) is given by (10) as

simply @E=@ = [@E=@A] [@A=@]>0. That is, we …nd that a decrease in competition

in the market for avoidance increases evasion by making it more attractive relative to the

alternative of avoidance.

The …nal …nding is that tax evasion is increasing in the probability of audit if R(p)>1

(equivalently, p < 0:5) and decreasing otherwise. In this case there are again competing

direct and indirect e¤ects upon evasion, but now the direct e¤ect does not always dominate

the indirect e¤ect. From (7) we obtain the direct e¤ect as

@E

@p A=cons:

=wR(t) [1 + R()] [1 + f R(p)R()]

f[1 + R(p)] R()<0;(15)

which is the pure income e¤ect …rst observed by Yitzhaki (1974). This may be rewritten in

the form @E

@p A=cons:

=[1 + fR(p)R()]

R(p) + fR()

@E

@A

@A

@p ;(16)

such that

@E

@p =[1 + f R(p)R()]

R(p) + fR()

@E

@A

@A

@p +@E

@A

@A

@p (17)

=[R(p)1] [1 f R()]

R(p) + fR()

@E

@A

@A

@p . (18)

From (18) it is immediate that the direct e¤ect dominates when R(p)<1and the indirect

dominates when R(p)>1.

How plausible is the condition p < 0:5required for evasion to be increasing in the probability

of audit? A-priori it appears highly plausible given that the IRS audits only around 0.96

percent of individual tax returns …led in calendar year 2012 were examined (IRS 2014). Even

if we suppose that these audits were concentrated on the 20 percent or so of people in the

US who are self-employed (and thus not subject to third-party reporting) this probability

rises to 4.8 percent, still well below the 50 percent level.

We now wish to characterize the total level of undeclared income, A+E:

12

Proposition 3 An interior optimum for avoidance and evasion it holds for A+Ethat

@[A+E]

@w =A+E

w>0;

@[A+E]

@t =A+E

t[1 t]<0;

@[A+E]

@f =@E

@f <0;

@[A+E]

@ =p2wR(t)R(p)f1ff R (p) [R()]2g f

[1 ]2f?0

() R(p)1ffR (p) [R()]2f?0;

@[A+E]

@p =wR(t) [1 + R()] fR(p) [1 f2f R()] 1fg

f[1 + R(p)] R()?0

() R(p) [1 f2f R()] [1 + f]?0:

The …rst three results of Proposition 3 follow immediately from Proposition 2, for the com-

parative static e¤ects for both Aand Ego in the same direction. Both of the remaining

two e¤ects –those for and p–may go in both directions, however.7Thus, for instance, an

increase in audit probability can cause the total amount of hidden income to increase (albeit

avoided income must fall). Reassuringly, however, unlike the condition for Eto increase

in p, the condition needed for this …nding seems far from being satis…ed empirically. In

particular, it requires setting an especially low f, which, in turn, forces the tax rate to be

implausibly high. Similar remarks apply to the condition needed for A+Eto be increasing

in . The results of Proposition 3 allow us to characterize readily the comparative statics of

declared income x(wAE). For all exogenous variables except wwe obtain that the

e¤ect for declared income will take the opposite sign to the e¤ect for total undeclared income,

i.e., @x=@ () = @[A+E]=@ (). For w, however, we obtain @x=@w =x=w > 0.

As a …nal perspective on the properties of optimal avoidance and evasion, we may characterize

the properties of the proportion (or share) of unreported income that is avoided: sA

A= [A+E]. This shall be instructive when we come to compare our results with the existing

literature.

7To demonstrate this by example, setting t= 0:85; p = 0:40; = 0:51; f = 0:085 and w= 10 we

obtain fA; Eg=f0:64;9:36gand @[A+E]=@p = 0:74. Alternatively, setting t= 0:70; p = 0:40;

= 0:51; f = 0:02 and w= 10 we obtain fA; Eg=f1:54;8:46gand @[A+E]=@p =10:10. Turning

to , setting t= 0:80; p = 0:40; = 0:51; f = 0:12 and w= 10 we obtain fA; E g=f0:90;9:10gand

@[A+E]=@ = 13:35. Alternatively, setting t= 0:80; p = 0:40; = 0:25; f = 0:10 and w= 10 we obtain

fA; Eg=f4:67;5:33gand @[A+E]=@ =5:87.

13

Proposition 4 At an interior optimum for avoidance and evasion it holds that

@sA

@t =@sA

@w = 0;

@sA

@f =p2[1 p]R(p)# > 0;

@sA

@ =p2[1 pf p]R(p)# < 0;

@sA

@p =f[p]2[1 f R()] 1 + R(p)2R()# < 0;

where #nA

f[A+E][1p]o2>0.

Proposition 4 clari…es that the share of undeclared income that is avoided is independent

of the tax rate (t) and the taxpayer’s wealth (w). This follows from the observation in

Proposition 1 that wand R(t)enter both avoidance and evasion as multiplicative factors.

We …nd that the probability of audit (p) unambiguously reduces the share of undeclared

income that is avoided, even though the e¤ect of pon evasion can be of either sign. The

results for the e¤ects of the …ne rate (f) and the cost of avoidance () follow immediately

from Proposition 2 (as these variables a¤ect avoidance and evasion in opposing ways).

5 Comparison with the literature

We now compare the …ndings of the previous section to the existing literature. First, where

our results overlap with the empirical …ndings on the interrelationship between evasion and

avoidance of Alm, Bahl, and Murray (1990), we observe agreement. These authors …nd that

the quantity [1 ft]1is negatively related to evaded income, implying that an increase in

either for treduces evasion (consistent with Proposition 2). They also …nd, again consistent

with Proposition 2, that avoided income is decreasing in the cost of avoidance (as measured

in our model by the parameter ). A caveat to these agreement in …ndings, however, is

that Alm, Bahl, and Murray identify avoidance as a riskless asset, somewhat di¤erent from

the de…nition of avoidance as a risky asset we employ here. Second, we may compare our

…ndings for optimal evasion to those of Yitzhaki’s (1974) canonical model of tax evasion. Our

…ndings for the e¤ect of wealth, the tax rate and the …ne rate on evasion are consistent with

Yitzhaki, but the …nding that evasion may increase in the probability of audit is di¤erent

from that in Yitzhaki (where evasion is always decreasing in the probability of audit).

14

Third, we may compare our …ndings to those of Alm and McCallin (1990), who report

comparative statics results for reported income (x) and for the share of undeclared income

that is avoided (sA). Like these authors, we …nd that higher …nes for evasion increase reported

income and increase the share of undeclared income that is avoided. Di¤erent from these

authors, however, we retain the well-known result of Yitzhaki (1974) that a tax rate rise will

increase reported income (whereas Alm and McCallin report the opposite relationship) and,

whereas Alm and McCallin …nd that a tax rate rise increases the share of undeclared income

that is avoided, we …nd that this share is independent of the tax rate. It can be shown that

this independence is robust to allowing for pL<1, and allowing for a coe¢ cient of constant

relative risk aversion that is di¤erent from unity. We are unable, however, to follow Alm

and McCallin (1990) in examining the comparative statics e¤ects of quantities such as the

mean and variance of the return to evasion and avoidance, however, as in our model these

quantities are determined endogenously. Last, we may compare our …ndings to those of the

theoretical model of Alm (1988), albeit an important di¤erence between his model and ours

is that he models avoidance as a riskless asset. Alm presents comparative statics results

for the quantities wEand the share sx[wAE]=[wE] = x= [wE], i.e., the

fraction of income net of evasion that is avoided. In his very general framework, Alm …nds

all comparative statics for the share sxto be ambiguous in sign. We similarly …nd that the

e¤ects of and pon sxare ambiguous, but we …nd that the e¤ects of fand ton sxare

unambiguous –in both cases proportional to a positive constant:

@sx

@f /2p3[1 t]2R(p) [R(p)R()1] >0;

@sx

@t /pf 23R() [R(p)R()1] >0:

We …nd that sxis independent of a taxpayer’s wealth: @sx=@w = 0. The only other two

clear-cut results in Alm (1988) are that evasion is decreasing in the …ne rate and in the audit

probability. In our model the …rst of these e¤ects is preserved, but we …nd that evasion can

be increasing in the probability of audit.

5.1 Audit probability vs. …ne rate

As a …nal comparison to the literature, we consider the …nding of Christiansen (1980) that,

for a constant expected return to evasion, the amount evaded is always reduced by increasing

the …ne rate and by decreasing the audit probability. Following Christiansen (1980), we …rst

15

restrict analysis solely to evasion. For a given level of avoidance the expected return to

evasion is given by Ep[1 + f]1. Holding this constant by appropriate variation of f,

and di¤erentiating Ewith respect to p, we obtain:

Proposition 5 An interior optimum for avoidance and evasion it holds that

@E

@p E=cons:

=wR(t) ([1 R(p)] f2R() + [1 + 2f]R(p)f)

f2[1 ] [1 + R(p)] >0:

According to Proposition 5 we are able to replicate Christiansen’s …nding: it always worsens

evasion to raise the audit probability and lower the …ne rate, holding the expected return

to evasion …xed. In the context of a model containing both avoidance and evasion, however,

what will be relevant to a tax authority seeking to maximize tax revenue is the e¤ect of

varying pand fon the total level of income that does not get taxed. On this question we

have that:

@[A+E]

@p E=cons:

=wR(t)f[1 p]f1 + f[3 2f R ()]g f[1 + f]g

[1 ]f2?0:(19)

As the right side of (19) can take either sign, depending upon parameter values, we now

no longer …nd that raising …nes is always superior to raising audit probability. Intuitively,

this …nding stems from the observation that increasing the …ne rate only a¤ects the evasion

decision, whereas increasing the audit probability a¤ects both avoidance and evasion. The

range of parameters for which audit probability can dominate the …ne rate is limited by

the fact that evasion is increasing in pbelow p= 0:5. We know of numerical examples,

however, that con…rm that audit probability can dominate the …ne rate even in the region

where @E=@p > 0.8

6 Probabilistic Anti-Avoidance Outcomes

Up until this point, the analysis has been undertaken with the simplifying assumption that,

if the tax authority mounts a legal challenge to the avoidance scheme, its challenge is always

successful. While important in securing a tractable model, clearly tax authorities are not

8To demonstrate this by example, setting t= 0:50; p = 0:40; = 0:53; f = 0:05 and w= 10 we obtain

fA; Eg=f2:93;5:60gand @[A+E]=@pjE=cons: = 40:65. Alternatively, setting t= 0:30; p = 0:055;

= 0:94; f = 2:5and w= 10 we obtain fA; Eg=f2:07;6:80gand @[A+E]=@pjE=cons: =91:56.

16

always successful in their attempts to shut-down avoidance schemes, so it is of interest to

understand how this consideration a¤ects our …ndings.

Solving for Ausing the de…nition in equation (5) and the full expression for expected utility

given in (1) we obtain

A=ppLR(t)

1[R(ppL)R()1] w; (20)

which can be obtained from the solution for Agiven in Proposition 1 (for the case of pL= 1)

simply by replacing pwith ppL.9It follows that the comparative statics results for Agiven

in Proposition 2 continue to hold, and that the e¤ects of pLupon avoidance are analogous

to those of p. The solution for Ecoming from (6) is complex, however. We note, though,

that the the taxpayer’s wealth and the tax rate both still enter the solution multiplicatively

–as they do also for Ain equation (20) –so these two variables stay independent of the

share of undeclared income that is avoided, sA.

To make further progress we assess the properties of optimal evasion via a numerical opti-

mization procedure that locates optimal avoidance and evasion for a speci…ed set of para-

meter values. Figure 1 depicts optimal avoidance and evasion as pLis allowed to vary on

the unit interval.10 For very low values of pLin the interval denoted [0;^pLjE=0]avoidance is

seen to be maximal, and the taxpayer does not evade. In a second interval, denoted in the

…gure by [^pLjE=0;^pLjA+E=w], the taxpayer both avoids and evades, and reports no income

(A+Ewis binding). In a third interval, denoted [^pLjA+E=w;1], the taxpayer again

both avoids and evades, but now A+E< w (this is the case to which our comparative

statics analysis applies). Within this interval we observe that optimal evasion increases as

the probability of a successful legal challenge increases. This is as expected, for an increase

9Whereas the expression for Agiven in Proposition 1 is the unique solution to a …rst order condition

linear in A, the expression for Ain (20) is one of a pair of solutions to a …rst order condition quadratic in

A. The other solution to this …rst order condition is A=wR (t)=[1 ]<0, which however, may be

dismissed as, by de…nition, A0.

10 The parameter values that produce Figure 1 are: w= 10; p = 0:5; t = 0:8; f = 0:1; = 0:22. We note

that these values are chosen purely to illustrate cleanly the full range of possible outcomes of the model. As

is well-known, models such as ours, which implicitly assume taxpayers know the true probability of audit,

signi…cantly over-predict non-compliance if calibrated realistically (see, e.g., Alm, McClelland, and Schulze

(1992), footnote 3). This di¢ culty does not appear especially consequential in this context, however, for

insights such as probability weighting (Kahneman and Tversky 1979) have been shown to dramatically reduce

predicted levels of non-compliance, while not importantly a¤ecting its comparative static properties (these

being our interest in this paper).

17

in pLleaves the returns to evasion una¤ected, but reduces the returns to avoidance, making

evasion more attractive relative to avoidance.

In Figure 2 we explore the e¤ect of varying pLon our earlier …nding that evasion can be

increasing in the probability of audit.11 On the interval of Figure 2 where both optimal

evasion and avoidance are interior, we see that reducing pLbelow unity reduces to a value

below one-half the threshold audit probability above which evasion is decreasing in p. Thus

lower values of pLimply a smaller set of parameter values for which evasion is observed to

be increasing in audit probability.

Other numerically generated results we have analyzed – which we do not report here for

brevity –indicate that the qualitative nature of the results given in propositions 2-4 continue

to hold. In particular, the taxpayer’s wealth and the tax rate continue to act as multipliers in

the expressions for optimal avoidance and evasion, and A+Emay be either an increasing

or decreasing function in and p.

7 Conclusion

Although the economic literature has largely limited itself to the study of tax evasion, tax

avoidance is empirically observed alongside tax evasion. We therefore examine the choice of

a taxpayer of how much tax to avoid and how much to evade, under the assumptions that

(i) a taxpayer narrow brackets the joint avoidance/evasion decision –breaking the decision

down into separate avoidance and evasion sub-decisions, and taking the …rst of these two sub-

decisions in isolation from the second; and (ii) that a taxpayer will decide …rst on whether

and how much tax to avoid legally before deciding whether and how much tax to evade

illegally. Among our results are, …rst, that an analogous …nding to the so-called Yitzhaki

puzzle for evasion also holds for tax avoidance –an increase in the tax rate decreases the level

of avoided income and the level of avoided tax. Second, for a small enough audit probability,

evasion is an increasing function of the audit probability. Although tax avoidance is always

decreasing in the probability of audit, in some circumstances even the total amount of income

lost to evasion and avoidance can be increasing in the probability of audit. Last, holding

constant the expected return to evasion, it is not always the case that combined loss of

11 The parameter values that produce Figure 2 are: w= 10; t = 0:85; f = 0:11; = 0:17; pL2 f0:7;1g.

The same qualitative conclusions obtain if the parameter values used to draw Figure 1 are instead used, but

these alternative parameter values yield improved visual clarity.

18

reported income due to avoidance and evasion can be stemmed by increasing the …ne rate

and decreasing the audit probability.

We …nish with some possible avenues for future research. First, it would be of interest to

allow for imperfect audit e¤ectiveness, as in Rablen (2014) and Snow and Warren (2005a,b),

for it might be that evasion and avoidance di¤er in the amount of tax inspector time required

to detect them. Second, it might also be of interest to model more carefully the market for

avoidance. In practice there are a range of providers of tax advice, ranging from those

that o¤er solely tax planning, to those that are willing to o¤er aggressive (or even criminal)

methods, making it important to understand the separate supply- and demand-side e¤ects.

A last suggestion is to embed the model within a general equilibrium framework (see, e.g.,

Alm and Finlay 2013; Neck, Wächter, and Schneider 2012), for the partial equilibrium setting

explored here may miss some important wider interactions between avoidance and evasion

that should properly be accounted for.

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24

Appendix

Proof of Proposition 1: From (1) we have that

@EU(A; 0)

@A =[1 p][1 ]

A[1 ] + wR(t)p

wR(t)A (A.1)

=p R(p)R()

A[1 ] + wR(t)1

wR(t)A . (A.2)

Solving for the point @EU(A; 0) =@A = 0 gives

A= arg maxAEU(A; 0) = pR (t)

1[R(p)R()1] w. (A.3)

Rewriting Ain (A.3) as A=wR(t) [1 p]1[1 ]1, substituting this expression

for Ainto (1), and di¤erentiating with respect to Egives

@EU(A; E)

@E =[1 p]

[1 p]wR(t) + E pf [1 ]

pwR(t)Ef [1 ](A.4)

=p R(p)

[1 p]wR(t) + E f R()

pwR(t)Ef [1 ]:(A.5)

Evaluating at @EU(A; E)=@E = 0 and solving for Egives

E= arg maxEEU(A; E) = pR(t)

1[1 p] [1 f R()]

fw. (A.6)

From (A.3) and (A.6) we see that

A>0() R(p)R()>1; E>0() fR()<1:(A.7)

From (A.3) and (A.6) we compute

A+E=pR(t)

1[1 p] [1 f R()] + f[R(p)R()1]

fw, (A.8)

so

A+E< w () pR(t)

1R(p)f+fR(p) [R(p)R()1]

f<1. (A.9)

Proof of Proposition 5: Noting that

@2E

@p@f E=cons:

=wR(t)f2 [1 + f]R(p)fg

f3[1 ] [1 + R(p)] <wR(t) [1 + 2f]

f2[1 ] [1 + f]<0;(A.10)

it follows that if @E=@pjE=cons: >0when fapproaches its maximum possible value, i.e.,

f!R()1, then @E=@pjE=cons: >0for all f < R ()1. At f!R()1we indeed have

@E

@p E=cons:

=wR(p)R(t)R() [1 + R()]

[1 ] [1 + R(p)] >0:(A.11)

25

Figures

Figure 1: Optimal avoidance and evasion for pL2[0;1].

Figure 2: Optimal avoidance and evasion for pL<1and pL= 1.

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