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Tax Avoidance and Evasion in a Dynamic

Setting

*

Duccio Gamannossi degl'Innocenti

Rosella Levaggi

Francesco Menoncin

September 20, 2022

Abstract

We study an intertemporal utility maximization problem where tax-

payers can engage in both tax avoidance and tax evasion. Evasion is

costless but is ned if discovered, while avoidance is costly but might be

successful (i.e. deemed legitimate) with a given probability (

β

) upon au-

dit. We nd that traditional deterrence instruments (ne and frequency

of audit) reduce optimal evasion but, in contrast with results in a static

framework, they have no impact on optimal avoidance. In fact, tax avoid-

ance depends negatively on its marginal cost and positively on both its

probability of success (

β

) and the tax rate. We show that non-compliance

behavior may result in a Laer curve for scal revenues and that the

revenue maximizing tax rate is lower the higher

β

. We characterize the

optimal level of

β

by taking into account dierent government objectives:

minimizing evasion, minimizing non-compliance (evasion plus avoidance),

or maximizing revenues. Our results suggest that specic policies (e.g.,

tax simplication) need to be implemented to deter avoidance and we

illustrate their impact on evasion.

Keywords

: Tax Avoidance; Tax Evasion; Dynamic Programming; Tax

Simplication

*

Acknowledgements: We thank Sebastian Blesse, Brandon Pecoraro, David Splinter,

Karsten Staehr and participants at the 7th Shadow conference and XXXIII SIEP confer-

ence for helpful comments.

Declarations of interest: none.

This research did not receive any specic grant from funding agencies in the public, commer-

cial, or not-for-prot sectors.

Corresponding author at: Department of Economics and Finance, Università Cattolica

del Sacro Cuore, Via Necchi 5, 20123, Milan, Italy. E-mail: duccio.gamannossi@unicatt.it.

Department of Economics and Management, University of Brescia, Via San Faustino 74b,

25122, Brescia, Italy.

Department of Economics and Management, University of Brescia Via San Faustino 74b,

25122, Brescia, Italy.

1

1 Introduction

In Europe, income under-reporting is about 20% of GDP, accounting for a po-

tential loss of about 750-900 billion Euros each year (Buehn and Schneider,

2012; Murphy, 2019), i.e. about 13.2% of total revenue (Albarea et al., 2020).

Intentional underreporting of income is about 18-19% of the total reported in-

come in the US, leading to a tax gap (Cebula and Feige, 2012; IRS, 2019) that,

according to some estimates, may have reached 630 billion dollars in 2020 (Sarin

and Summers, 2019) but the latter may be even higher when tax avoidance is

taken into account. Since the revenue loss is only the tip of the iceberg for

what concerns the eect of tax evasion (Slemrod, 2007; Alm, 2012; Dzhumashev

and Gahramanov, 2011; Markellos et al., 2016), reducing non-compliance is a

priority for many governments, both in developed and developing countries.

1

In order to reduce their tax liabilities, taxpayers may take three kinds of

actions: (i) illegal ones (tax evasion), (ii) those that use tax law to gain an

advantage that lawmakers never intended (tax avoidance), and (iii) those that

use tax allowances for the purposes intended by lawmakers (tax planning).

2

Since (iii) is legit, here we focus on (i) and (ii). In contrast to most of the

literature that disentangles the study of avoidance and evasion, we study them

jointly (similar to Gamannossi degl'Innocenti and Rablen, 2017) as the taxpayer

optimally chooses them at the same time.

The rst studies on tax compliance (e.g. Allingham and Sandmo, 1972,

Yitzhaki, 1974) adapted Becker (1968)'s model of crime to tax evasion. Since

then, the economic literature has been mostly focusing on tax evasion (e.g.

Gamannossi degl'Innocenti and Rablen, 2017), either neglecting tax avoidance

or considering it independent of other non-compliance opportunities.

3

This

approach may be misleading: Cross and Shaw (1981, 1982) pointed out the im-

portance of a joint study of evasion and avoidance, since taxpayers may consider

them either substitutes or complements, and tax authorities should take into

account both channels of response to their deterrence activities.

Furthermore, most of this literature relies on a static (timeless) framework

(Dzhumashev and Gahramanov, 2011), even though the properties of the opti-

mal solutions in a static and a dynamic framework may be quite dierent (Wen-

Zhung and Yang, 2001). Actually, the most recent models are cast into a dy-

namic framework (Dzhumashev and Gahramanov, 2011; Levaggi and Menoncin,

2012) and take into account the impact of uncertainty over scal parameters on

evasion and growth (Bernasconi et al., 2015), the relationship between evasion

and investment choices (Levaggi and Menoncin, 2013), and the role of habit in

consumption (Bernasconi et al., 2019).

1

The literature is not conclusive on its exact amount. For Davison (2021) it may also reach

1 trillion dollars, while other estimates are more in line with the tax gap.

2

Despite a marked heterogeneity in the details, in most tax systems (e.g. IRS, 2014;

European Parliament, 2017; HMRC, 2019; UN, 2019) adherence to the letter of the law does

not imply legality. Courts decide whether a given tax liability reduction is admissible based

on its agreement with the purposes of the tax legislation.

3

Notable exceptions considering both avoidance and evasion are Alm (1988a), Alm et al.

(1990), Alm and McCallin (1990), and Gamannossi degl'Innocenti and Rablen (2017).

2

In this paper, we aim at merging most of the cited approaches, by mod-

elling both avoidance and evasion choices (as in Gamannossi degl'Innocenti and

Rablen, 2017) in a dynamic setting (as in Levaggi and Menoncin, 2013).

We assume evasion to be costless since it can be performed quite easily just

by concealing a part of the revenue. Conversely, tax avoidance schemes are

usually sophisticated

4

and require considerable expertise to be devised.

5

Our

model is cast in a dynamic framework and we study the dynamic programming

problem of a representative taxpayer who maximizes the expected utility of

inter-temporal consumption and decides the optimal percentage of evasion and

avoidance. The taxpayer receives utility from the consumption that exceeds a

minimum (subsistence) amount in each period and utility increases with the

consumption of both a private and a public produced good. The taxpayer is

endowed with a linear

Ak

technology and a constant tax rate is levied on the

yield produced.

The model has two sources of uncertainty: the occurrence of an audit and

the success of avoidance.

In particular, the taxpayer knows the frequency of audits but does not know

when they occur. When an audit happens, we assume that evasion is detected

and a ne must be paid on the evaded tax. Conversely, upon audit, there

is some positive probability that avoidance will be successful, because: (i) it

goes undetected, (ii) it is not challenged by the tax authority, or (iii) it is

not recognized as illegitimate by courts. If avoidance is unsuccessful, then in

our model it is considered akin to evasion and ned accordingly.

6

The success

probability of avoidance (

β

) represents, in some way, a broad measure of the

vulnerability of the tax system to avoidance and we assume it to be constant

over time.

Our analysis shows that optimal avoidance is constant over time, it does not

depend on the preference parameters, it is independent of the form of the utility

function and is not directly aected by traditional deterrence instruments, like

the ne and the frequency of audits. Instead, avoidance depends only on (i)

the tax rate (positively), (ii) the avoidance probability of success

β

(positively),

and (iii) the cost of avoidance (negatively). However, avoidance has a negative

impact on tax evasion and, thus, total unreported income crucially depends on

both traditional scal parameters and tax avoidance determinants.

As in Gamannossi degl'Innocenti and Rablen (2017) (i) the taxpayer op-

timally decides both evasion and avoidance, (ii) the audit is performed with

an exogenous frequency and probability, and (iii) when audited, evasion is im-

mediately detected, while avoidance may be successful with a given probabil-

ity. However, in Gamannossi degl'Innocenti and Rablen (2017) the evasion and

4

Stiglitz (1985) provides several examples and identies three basic principles of these

activities: postponement of taxes, arbitrage across individuals subject to dierent marginal

tax rates, and manipulation of dierent types of income that are taxed.

5

Empirical evidence shows that people's understanding of tax law, tax rates, and basic

concepts of taxation is limited, i.e. Blaufus et al. (2015); Gideon (2017); Stantcheva (2021).

6

Notice that in some countries (e.g., the UK), when avoidance is not successful the tax

agency has the legal authority to recover only the tax liabilities (with no additional nes).

3

avoidance decisions are taken in two separate steps, while in our model we as-

sume them to happen simultaneously. As a consequence, it is possible to decide

evasion and avoidance by managing two dierent risks. The risk of avoidance to

be unsuccessful is managed just by choosing an optimal amount of avoidance,

while the risk to be audited is managed inside the optimal decision of evasion.

Instead, if these two decisions are taken separately, the same parameters should

aect both of them, as demonstrated in Gamannossi degl'Innocenti and Rablen

(2017).

We nd that the impact of the tax rate on total unreported income share

is non-monotonic, and depends on the assumptions about the taxpayer's pref-

erences. In particular, we investigate two cases, with and without a subsistence

level of consumption for the taxpayer (notice that in the latter case the taxpayer

has constant relative risk aversion - CRRA). While the presence of a subsistence

level does not aect the optimal avoidance, it makes the optimal evasion (as well

as the optimal consumption) a dynamic variable which converges towards a long

term value.

We show that, for low levels of either avoidance or taxation, an increase

in the tax rate leads to a net improvement in compliance, while the opposite

holds true when the level of either avoidance or tax rate is high. As a result,

in our model non-compliance behavior may result in a Laer curve for scal

revenues, providing a theoretical explanation for a phenomenon documented by

policymakers (Papp and Takáts, 2008; Vogel, 2012). We also show the revenue

maximizing tax rate to be lower the higher the probability

β

, so that the abil-

ity to raise revenues from taxes is directly aected by the vulnerability of the

economic system to avoidance.

Interestingly, an increase in evasion is more likely when minimum consump-

tion is relatively high (which happens, in general, in more developed economies,

see Chetty and Szeidl, 2016; Havranek et al., 2017). Notice, however, that a

reduction in

β

always increases declared income (and expected government rev-

enue) since the reduction induced in avoidance more than osets any positive

eect on evasion.

In both Gamannossi degl'Innocenti and Rablen (2017) and Levaggi and

Menoncin (2013), an increase in tax rate reduces non-compliance because of

an income eect. In our setting, instead, an increase in the tax rate may have a

non-monotonic eect on government revenues as in Li et al. (ming) who, how-

ever, model the sole avoidance.

Our model is also the rst one to analytically study the impact of the prob-

ability of success of avoidance.

7

The article is structured as follows: Section 2 outlines the model. The

optimization problem for the taxpayer and some possible government goals are

analyzed in Sections 3 and 4. Section 5 discusses the results and their policy

implications, while Section 6 draws some conclusions. The proofs are provided

in the appendices.

7

In Gamannossi degl'Innocenti and Rablen (2017), an analogous investigation was per-

formed only numerically.

4

2 The model

We model the choices of a representative taxpayer who maximizes an inter-

temporal utility function that depends on the consumption of a private good

(

ct

) and a merit good (

gt

) which is nanced by a linear income tax. Our as-

sumption is justied because, starting from the inception of the welfare state,

the supply of goods like health care, education, and other personal services, has

been increasing over time to become one of the biggest shares of public expen-

diture in western countries (OECD, 2021). We assume that the taxpayer does

not perceive the link between public good provision and the income tax (i.e.,

scal illusion exists), and may engage in tax evasion and tax avoidance without

internalizing the consequences of such behavior in the future supply of

gt

.

2.1 Capital accumulation

The taxpayer is endowed with an initial capital

kt0

that is used over the period

[t0,∞[

to produce an income

yt

through the deterministic production function

yt=Akt,

(1)

where

A

measures total factor productivity. Since the stock of capital cannot

produce an aggregate income greater than the capital itself, it is reasonable to

assume that

0<A<1

. Although

A

is exogenous and deterministic, the process

of capital accumulation is endogenous because of the taxpayer's consumption

choice (

ct

), and it is also stochastic due to the choices related to tax evasion (

et)

and tax avoidance (

at)

.

Government levies a linear tax

0≤τ≤1

on income, which is used to nance

the provision of the merit good (

gt)

. With perfect compliance, the net change

in capital is

dkt= ((1 −τ)yt−ct)dt.

(2)

The taxpayer assumes that

gt

does not depend on the income tax paid. For

this reason, the taxpayer may try to reduce their tax burden by either evading

a percentage (

et

) of the yield or by eroding their tax base through avoidance

(

at

), whose eectiveness depends on the vulnerability of the tax system.

Avoidance is always audited together with evasion, and, in the case of an

audit, two scenarios may happen: either avoidance is considered to be legal, or

not. We dene as

Ia=e

the indicator function whose value is either

1

if avoidance

is successful, i.e. it is assumed to be dierent from evasion, or

0

otherwise, i.e.

it is considered to be evasion.

The taxpayer caught reducing their tax base has to pay a ne

η

that is

proportional to the hidden part of the total tax, given by evasion and avoidance

if it is unsuccessful. Accordingly, the total ne that must be paid in case of an

audit is

η(et+ (1 −Ia=e)at)τAkt,

whose expected value at time

t

is

Et[η(et+ (1 −Ia=e)at)τAkt] = η(et+ (1 −Et[Ia=e]) at)τ Akt.

5

We recall that the expected value of the indicator function of an event coin-

cides with the probability of the event. In particular, we set

E[Ia=e] := β,

which measures the probability that avoidance is successful, i.e. is consid-

ered to be legal. This parameter provides a measure of the vulnerability of the

tax system to tax avoidance and is lower in economies in which: (i) tax codes

are simpler and less ambiguous, (ii) tax authorities are endowed with relatively

sizable operational

8

and litigation resources, and (iii) courts have higher eec-

tiveness. If

β= 1

avoidance is riskless (like in Alm, 1988b; Alm et al., 1990),

and deterrence is completely ineective against avoidance. Instead, for

β=η−1

η

the taxpayer's payment conditional on audit is the same as for true income re-

porting. This implies that, for any level above this threshold, the taxpayer gets

an actual discount on their tax bill even in the case of an audit, while if

β

is

below the threshold the taxpayer reduces the nes to be paid if caught.

In line with other works in both the static and the dynamic literature (Wen-

Zhung and Yang, 2001; Levaggi and Menoncin, 2013; Gamannossi degl'Innocenti

and Rablen, 2017), we assume tax evasion to be a costless activity. Conversely,

avoidance is assumed to be expensive since a considerable eort (or expertise)

is needed to reduce the tax burden while not violating the law.

To keep our results as general as possible, the costs of avoidance are rep-

resented through any increasing and convex function

f(at)

.

9

Notably, this

formulation allows accounting for the (likely) occurrence of xed costs

f(0) ≥0

(setup costs, e.g. creation of legal entities) and has the exibility to represent

any mix of avoidance instruments.

In line with Levaggi and Menoncin (2012, 2013); Bernasconi et al. (2015);

Levaggi and Menoncin (2016a,b), we model the audit process through a Poisson

jump process

dΠt

whose frequency is

λdt

which coincides with the two rst

moments of the jump

Et[dΠt] = V[dΠt] = λdt.

Thus, the nal dynamics of capital

kt

is

dkt= (Akt−τ(1 −et−at)Akt−ct−f(at)Akt)dt

(3)

−η(et+ (1 −Ia=e)at)τAktdΠt,

where the tax

τ

is paid only on the income that is not hidden (

1−et−at

),

the avoidance cost

f(at)

is proportional to income, and we assume that the

probability that avoidance is successful is independent of the probability to be

audited.

8

In a recent paper, Guyton et al. (2021) show that more detailed/thorough audits are able

to uncover avoidance activities that are mostly missed by standard random audits.

9

Evidence on the contractual terms upon which avoidance schemes are typically sold is

scarce. However, Committee of Public Accounts (2013) reports that the majority of schemes

entail a fee related to the reduction in the annual theoretical tax liability of the user and

Kantar Public UK (2015) shows that fees vary with the value of the amount of the investment

realized by the scheme.

6

The expected value of

dkt

is

Et[dkt] = (1 −τ)Akt+ (1 −ηλ)etτ Akt+1−η λ (1 −β)−f(at)

atτatτAkt−ctdt,

from which we see that evasion is expedient on average if

Et[dkt]>Et[dkt]et=0 ,

which becomes

1−ηλ > 0,

(4)

and, accordingly, we will assume that the product between the frequency of

audit (

λ

) and the ne (

η

) is lower than

1

. Instead, avoidance is expedient on

average if

Et[dkt]>Et[dkt]at=0 ,

which becomes

f(at)

at

<f(0)

at

+ (1 −ηλ (1 −β)) τ.

Hence, the taxpayer will engage in avoidance if its costs are lower than a thresh-

old dependent on both xed avoidance cost, and scal and enforcement param-

eters.

Since the product

ηλ

is higher than

1

for an expedient evasion, then the

minorant of the right hand side is

βτ

, and so we can impose that

f(at)< f (0) + atβτ,

(5)

in which we further assume that

f(0) <1−τ

.

2.2 Taxpayer's preferences

The representative taxpayer receives utility from consuming both a private pro-

duced good (

ct

) and a public produced good (

gt

), and we assume that such a

utility is additive in these two goods.

The taxpayer's behavior is described by a Hyperbolic Absolute Risk Aversion

utility (see, for instance, Gollier, 2001) written on the instantaneous consump-

tion as

U(ct) = (ct−cm)1−δ

1−δ+v(gt),

(6)

where

cm

is a minimum (subsistence) amount of consumption that the taxpayer

needs to consume, the parameter

δ > 0

measures the risk aversion, and

v(•)

is

an increasing and concave function. The existence of a strictly positive subsis-

tence consumption level allows us to solve some puzzles and reconcile theoretical

ndings with empirical evidence (see, for instance, Sethi et al., 1992; Weinbaum,

2005; Achury et al., 2012 for the role of subsistence consumption in portfolio

7

choice and Strulik, 2010 for its role in modeling economic growth). The Arrow

Pratt absolute risk aversion index is

−

∂2U(ct)

∂c2

t

∂U (ct)

∂ct

=δ

ct−cm

,

(7)

which increases when either

δ

or

cm

increase. In other words, a taxpayer with

a low

δ

but whose consumption is closer to

cm

behaves exactly as a taxpayer

with a higher

δ

but with a consumption level farther from

cm

.

3 The Problem

If the taxpayer discounts future utility at a constant rate

ρ

, the optimization

problem can be written as

max

{ct,et,at}t∈[t0,∞[

Et0"Z∞

t0

(ct−cm)1−δ

1−δe−ρ(t−t0)dt#,

(8)

under the capital dynamics (3).

Proposition 1.

The optimal solution to Problem (8), given the capital dynam-

ics (3), is

a∗= (f′)−1(τβ),

(9)

e∗

t=kt−H

τηAkt1−(λη)1

δ−(1 −β)a∗,

(10)

c∗

t=cm+(kt−H)ρ+λ

δ+δ−1

δ

1

η+δ−1

δ(1 −τ)A−1

η(λη)1

δ+δ−1

δ(τβa∗−f(a∗)) A,

(11)

in which

(f′)−1

is the inverse of the rst derivative of the function

f

, and

H:= cm

A(τβa∗−f(a∗) + (1 −τ)).

Proof.

See Appendix A.

Equation (10) shows that optimal evasion is aected by all the model pa-

rameters. The same is not true for (9), even if both variables are subject to the

same risk of being audited.

In the proposition above,

H

is a constant whose value coincides with the

present value of a perpetuity. In fact, we can write

H=Z∞

t

cme−A(τ βa∗−f(a∗)+(1−τ))(s−t)ds,

which is always positive because of condition (5). Thus, we can conclude that

H

represents the total present value of the future subsistence consumption

cm

,

8

discounted at a rate given by the total factor productivity corrected by both the

tax rate and a function of avoidance. Accordingly,

kt−H

can be considered as

the disposable capital that remains after saving enough for nancing the future

streams of subsistence consumption.

We note that when avoidance is not expedient (i.e.

a∗= 0

), the discount

rate is given by the total factor productivity net of tax and xed avoidance

costs:

A((1 −τ)−f(0))

. We can immediately check that, under condition (5),

the presence of avoidance (i.e

a∗>0

) increases optimal consumption.

Optimal tax avoidance is constant across time and does not depend on the

audit parameters

η

(the ne) and

λ

(the frequency of controls), while it simply

depends on its cost (the shape of the function

f(•)

), the vulnerability of the

tax system to avoidance

β

, and the tax rate

τ

. In particular, from Eq. (9), we

see that the representative taxpayer balances the marginal costs of avoidance

(the derivative

f′(•)

) with the marginal benets from avoidance (

τβ

).

Intuitively, avoidance exploits a loophole in the law to attain a lower expected

penalty relative to evasion. Accordingly, optimal avoidance depends only on the

parameters that aect its net marginal return relative to evasion in case of an

audit. The lack of eect of classic deterrence instruments on avoidance is in

sharp contrast with the rest of the static literature (Gamannossi degl'Innocenti

and Rablen, 2017; Alm, 1988b; Alm et al., 1990).

10

Interestingly, we show that

not only is avoidance not aected by the preference parameters (taxpayer's risk

aversion), but it is even independent of the functional form chosen for utility (see

Appendix A, Remark 1, where rst order conditions are computed). Given that

avoidance does not depend on preference parameters, our model implies that all

the taxpayers will avoid the same share of income if the system is vulnerable to

avoidance (

β > 0

).

Tax evasion, on the contrary, is used as a top up to tax avoidance even

if the substitution rate is not one. The share of evaded income depends on

the scal parameters and is similar to the optimal tax evasion of other dynamic

models (e.g., Levaggi and Menoncin, 2013), but it is lower in magnitude given

that taxpayers substitute it in part with avoidance.

In our setting, we show that tax avoidance, while reducing evasion, increases

total unreported income share, i.e., the sum of optimal evasion and avoidance:

E∗

t=e∗

t+a∗=kt−H

τηAkt1−(λη)1

δ+β(f′)−1(τβ).

(12)

The existence of a subsistence level of consumption implies that optimal

evasion is time dependent as shown in Proposition 1. In the following corollary,

we show that with

cm= 0

evasion is constant over time and so is consumption

share.

Corollary 1.

The optimal solution to Problem (8) for a CRRA taxpayer (i.e.

cm= 0

), given the capital dynamics (3), is

10

One exception is Li et al. (ming), where the eect of increased deterrence on avoidance is

entirely oset by the endogenous adjustment of price by suppliers.

9

a∗= (f′)−1(τβ),

e∗=1

τηA 1−(λη)1

δ−(1 −β)a∗,

c∗

t

kt

=ρ+λ

δ+δ−1

δ

1

η+δ−1

δ(1 −τ)A−1

η(λη)1

δ+δ−1

δ(τβa∗−f(a∗)) A.

Proof.

It is sucient to set

cm= 0

in Proposition 1.

Corollary 1 highlights that, with

cm= 0

, all the optimal strategies are time

independent. Instead, with

cm>0

, the dynamics of optimal solutions arise

because of a habit eect that leads to a gradual convergence towards long term

levels coinciding with the case

cm= 0

.

The results in Proposition 1 and Corollary 1 allow drawing some interesting

conclusions on the dynamic path of the choice variables of the taxpayer. While

the optimal share of avoided income is xed, the dynamics of consumption,

capital, and tax evasion are more nuanced. In Figure 1,

11

black lines show the

case with

β > 0

, so that the optimal avoidance share is positive, while gray

lines show the case with

β= 0

which implies zero optimal avoidance. Thus,

the gure makes it easier to compare our result with the previous literature (i.e.

Levaggi and Menoncin, 2013). When

cm>0

, these variables are aected by the

(random) audits, so we perform

N= 1000

replications and report the average

(solid line) along with the zero and one quantile (dashed lines).

Panel a) shows that, for

cm= 0

(dot-dashed line), the evaded share of in-

come is xed in time due to the constant relative risk aversion. When

cm>0

,

the evasion is increasing in time and in the long run it tends to its optimal level

with

cm= 0

. This dynamics of evasion is driven by the growth of taxpayer's

consumption

c∗

t

, that reduces risk aversion

δ

c∗

t−cm

. Conversely, when an audit

occurs, we observe a sharp drop in quantile lines driven by the fall in the tax-

payer's consumption. While the level of optimal evasion when

a∗>0

(in black)

is lower than the one when

a∗= 0

(in gray), the graph highlights the identical

behavior of the two, given that optimal avoidance is constant.

Panel b) illustrates the evolution of consumption as a ratio of capital. The

dynamics of

c∗

t/kt

when

cm= 0

is shown to be constant over time (Corollary 1)

and lower than the case with

cm>0

. When

cm>0

(Proposition 1),

c∗

t/kt

is

decreasing over time due to a more sustained growth of the denominator. Given

that optimal consumption is an ane transformation of capital, quantile lines

in this panel experience a jump upon audit (since

c∗

t

decreases less than

kt

).

Finally, the plot shows that allowing for avoidance leads to a higher

c∗

t/kt

since

the last term in (11) is positive when

a∗>0

by condition (5).

Insert Figure 1 about here

11

The code to reproduce all gures in the paper is openly available at

https://github.com/dgdi/Tax_Avoidance_and_Evasion_in_a_Dynamic_Setting.

10

3.1 Comparative statics

Here, we compute how the three choice variables

a∗

,

e∗

t

, and

E∗

t

respond to a

change in the model parameters.

Optimal avoidance

a∗

increases with respect to both

β

and

τ

as expected.

This result matches evidence in the empirical literature

12

and shows that the

Yitzhaki puzzle (Yitzhaki, 1974) does not hold for tax avoidance in a dynamic

setting. The positive relationship between

a∗

and

τ

is in contrast with Gaman-

nossi degl'Innocenti and Rablen (2017) and follows from optimal avoidance be-

ing driven only by its marginal net return relative to evasion in the case of an

audit. In the static framework, a similar result is observed only in Alm and Mc-

Callin (1990), where the joint avoidance and evasion decision is analyzed using

a portfolio approach.

From (10), optimal evasion decreases if

τ

increases, thus conrming the

Yitzhaki puzzle. In our model, the presence of avoidance reinforces the damp-

ening eect already observed when the ne is proportional to evaded taxes. The

same eect can be observed for an increase in

λ

and

η

. Instead, the reaction of

e∗

t

to changes in

β

, measured by

∂e∗

t

∂β =τ a∗AH2

cm

e∗

t+ (1 −β)a∗

kt−H+a∗−(1 −β)∂a∗

∂β ,

is not trivial to compute. In fact, the elements of its latter term,

(1 −β)∂a∗

∂β ,

are impacted in opposite directions by a change in

β

. Indeed, an increase in

β

increases avoidance (hence reducing evasion) but this (negative) eect is multi-

plied by

1−β

which is decreasing in

β

.

For

cm= 0

(i.e.

H= 0

), the optimal tax evasion may be either increasing

or decreasing w.r.t.

β

:

∂e∗

t

∂β cm=0

=a∗−(1 −β)∂a∗

∂β ⋛0⇐⇒ ∂a∗

∂β

1

a∗⋚1

1−β,

i.e. evasion is increasing in

β

only if the elasticity of

a∗

w.r.t.

β

is lower

than a given threshold. This result is interesting from a policy point of view

because of the twofold interpretation of the parameter

β

: it is the probability

of avoidance to be successful, but it can also be interpreted as the vulnerability

of the scal system to avoidance. Thus, decreasing the tax system vulnerability

to avoidance (lowering

β

) reduces (increases) tax evasion if the probability of

success in avoidance (

β

) is low (high). Hence, for a system rather vulnerable

to tax avoidance, a marginal decrease in the probability of success of avoidance

may worsen tax evasion statistics. Other things being equal,

∂e∗

t

∂β

is higher when

12

Long and Gwartney (1987), Alm et al. (1990), and Lang et al. (1997) show that tax

avoidance increases with the tax rate for US, Jamaican, and German households. As reviewed

in Riedel (2018), the scientic literature unanimously reports evidence of substantial tax

motivated prot shifting. Also Beer et al. (2020) perform a comprehensive meta-analysis of

existing studies suggesting an elasticity of before tax income to corporate tax rate of minus

one.

11

cm>0

, meaning that the eectiveness of

β

in preventing evasion is higher if

there is a positive subsistence consumption level.

It is interesting to observe that if

cm= 0

, the value of

β

which minimizes

the evasion must satisfy the condition

∂a∗

∂β

β

a∗=β

1−β,

that coincides with the odds of the event that avoidance is successful.

The total unreported income share (12) reacts to changes in

τ

in an am-

biguous way because of the reduction in tax evasion and the increase in tax

avoidance. The derivative can be written as:

∂E∗

t

∂τ =−1

τηAkt1−(λη)1

δH1−βa∗

τβa∗−f(a∗) + (1 −τ)+kt−H

τ

| {z }

<0

+β∂a∗

∂τ

|{z}

>0

,

and even for the simpler case with

cm= 0

the sign remains ambiguous:

∂E∗

t

∂τ cm=0

=−1

τ2ηA 1−(λη)1

δ

| {z }

<0

+β∂a∗

∂τ

|{z}

>0

.

Notably, when

τ

is relatively high, the negative term is smaller (in abso-

lute value) and the positive one is bigger. This implies that an increase in

tax rate reduces total reported income share when taxation is suciently high,

i.e., increasing scal pressure improves compliance in economic systems with

a relatively high tax burden and vice versa. Our result is in contrast with

both Gamannossi degl'Innocenti and Rablen (2017) and Levaggi and Menoncin

(2013), where a tax raise only induces an income eect (on both avoidance and

evasion or evasion only) that lowers non-compliance.

A result analogous to ours is reported by Alm (1988b) in a static framework,

but, in that case, the ambiguity is made possible by the very general specication

of the ne, tax and avoidance cost functions.

Finally, the derivative of

E∗

t

with respect to

β

can be written as:

∂Et

∂β =1

τηAkt1−(λη)1

δHτa∗

(τβa∗−f(a∗) + (1 −τ)) +a∗+β∂a∗

∂β >0,

so that reducing the system's vulnerability to avoidance, despite possibly wors-

ening tax evasion, always increases declared revenue. The comparative statics

results derived in this section, along with the results on government revenue in

the next section, are summarized in Table 1.

12

Table 1: Eect of enforcement/scal parameters on avoidance, evasion, and

tax revenue. The table presents the sign of derivatives (null, positive, negative,

or undetermined) of the function in the column with respect to the parameter

in the row: sign

∂

Col

∂

Row

a∗e∗

tE∗

t=a∗+e∗

tEt[dTt]

λ0− − +

η0− − +

β+

und.

+−

τ+−

und. und.

4 The optimal capital dynamics and government

revenue

We rst consider the optimal capital dynamics and its relation with

β

.

Proposition 2.

The expected growth rate of optimal modied capital

k∗

t−H

is

γ∗:= 1

dtEtd(k∗

t−H)

(k∗

t−H)=1

δ(1 −τ)A−(ρ+λ) + 1

η+ (τβa∗

t−f(a∗

t)) A−1−(λη)1

δλ,

(13)

whose rst derivative with respect to

β

is

∂γ∗

∂β =1

δ

τ

ηa∗

tA > 0.

(14)

Proof.

See Appendix B.

Equation (13) follows from Equation (11) and describes the dynamics of the

capital in excess of

H

,

13

which measures the discounted present value of the

future subsistence consumption levels

cm

.

We stress that the solution to the process

k∗

t−H

is exponential, and is

always positive if the initial value

k∗

t0−H

is positive. Thus, we can conclude

that the optimal capital will never fall below the value

H

if

k∗

t0> H

, i.e., the

taxpayer behaves in such a way to guarantee that their capital is always able to

nance the future ow of subsistence consumption.

Given condition (14), the capital growth is maximized by choosing the high-

est value for

β

(i.e.

β∗= 1

). This result is due to the nature of the good

produced by the government: by assuming a merit good that does not increase

private capital productivity, growth is maximized with minimum tax revenue.

We now characterize government revenue and its relationship with the suc-

cess probability of avoidance, the ne, and the tax rate.

13

Since

H

is constant, we get

1

dt Etd(k∗

t−H)

(k∗

t−H)=1

dt Etd(k∗

t)

(k∗

t).

13

Proposition 3.

The expected dynamics of government revenue

dTt

is

Et[dTt] = τ(1 −βa∗

t)Akt−kt−H

η(1 −λη)1−(λη)1

δdt,

whose derivatives are

∂

∂β 1

dtEt[dTt]>0,

∂

∂η 1

dtEt[dTt]>0,

1

dt

∂Et[dTt]

∂τ = (1 −βa∗

t)Akt−τ β ∂a∗

t

∂τ Akt+1

η(1 −λη)1−(λη)1

δH1−βa∗

(τβa∗−f(a∗) + (1 −τ)),

(15)

1

dt

∂Et[dTt]

∂τ cm=0

⋛0⇐⇒ βa∗

tτ∂a∗

t

∂τ

1

a∗

t

+ 1⋚1.

(16)

Proof.

See Appendix C.

Proposition 3 shows that increases in

β

lead to higher expected revenues

through a reduction in total unreported income share. A positive relationship

also holds between

η

and expected revenues despite a reduction in expected nes

caused by the lower evasion.

Equation (16) in Proposition 3 reveals that there is an ambiguity about how

government tax revenue reacts to changes in the tax rate. A relevant result is

obtained for a particular case.

Corollary 2.

If the elasticity of the optimal avoidance w.r.t the tax rate is

constant (i.e.

∃κ, χ > 0 : a∗=κτ χ

), then

1

dt

∂Et[dTt]

∂τ cm=0

⋛0⇐⇒ τ⋚((γ+ 1) βκ)−1

χ.

Proof.

It is sucient to substitute

a∗=κτχ

in (16) with

κ

and

χ

positive.

Corollary 2 shows the condition under which our model entails a Laer curve

behavior. For

τ

suciently low, the revenue increases as

τ

increases because

the rise in the tax rate and the reduction of tax evasion more than oset the

increase in tax avoidance. However, as

τ

increases, the latter eect becomes

prevalent and the revenue starts decreasing. Notably, the level of the revenue

maximizing tax rate is inversely related to

β

. The analogous result in Li et al.

(ming) arises due to a non-monotonic impact of the tax rate on the minimum

income to engage in avoidance.

Figure 2 shows a graphical representation of the Laer curve in our setting.

On the horizontal axis, we measure the tax rate while on the vertical axis we

14

measure the variation in expected revenues relative to capital. In the gure,

we consider three dierent constants of variation

ω

in a power cost function of

the form

f(a) = ωaγ

. The plot shows the relationship between tax rate and

expected revenues becoming negative at lower tax rates when

ω

is smaller. This

behaviour is expected, as the increase in avoidance following a rise in the tax

rate is higher when the marginal cost

f′

is lower.

Insert Figure 2 about here

5 Discussion and policy implications

The results and the comparative statics presented in the previous sections high-

light the importance of studying tax evasion and tax avoidance as a joint de-

cision. The results of our model indeed show that several interesting policy

implications can be derived from this analysis and that the institutional setting

(especially

β

) may change the outcome of policies aimed at reducing non com-

pliance. In what follows we summarize and discuss the most important results

of our model.

1. Tax avoidance depends neither on taxpayer's risk attitude, nor their utility

function, nor on traditional audit parameters (frequency and nes). This

implies that government cannot alter the avoidance decision using ordinary

tax enforcement tools. Instead, this result can be obtained through an

increase in the quality (litigation resources and thoroughness) of the audits

or scal/legal reforms (which in our model would aect

β,

the vulnerability

of the tax system to avoidance). However, avoidance deterrence might

entail unintended consequences:

(a) Even if evasion is decreasing in the tax rate, there are limits to the

use of the tax rate as an instrument to improve compliance due to

the presence of a Laer curve on total government revenue, which

provides a theoretical explanation for a phenomenon documented by

policymakers (Papp and Takáts, 2008; Vogel, 2012). This nding

follows from the three eects induced by a rise in the tax rate: (i) a

mechanical increase of revenues due to the higher marginal tax rate,

(ii) a reduction in evasion, and (iii) an increase in avoidance.

(b) Policies aimed at increasing avoidance costs, while theoretically iden-

tiable, seem to have limited practical relevance. The costs to en-

gage in avoidance are related to the eort (or expertise) required to

have a deep understanding of the loopholes in the tax law. An

increase in these costs entails a trade-o, as these costs also apply

to intended economic activities. A more eective way to reduce

tax avoidance is to lower the probability of a successful avoidance

(i.e. reduce

β

), through a simplication of the tax system.

14

Invest-

14

On specic anti avoidance reforms of the tax system, see Gravelle (2009).

15

ing in tax simplication, intended as the reduction of the extent of

variation in possible tax treatments of economic activities (number

of deductions, exemptions and instances of preferential treatment of

income), has also been recommended in the literature (e.g., Skinner

and Slemrod, 1985; Mccaery, 1990; Kopczuk, 2006) for its several

desirable outcomes.

2. Our analysis shows that tax avoidance deterrence performed by lowering

either the tax rate or the probability that the audited avoidance is success-

ful, might entail an unwanted increase in tax evasion, which can however

be oset by raising either the frequency of audit or the ne.

3. The opposite impacts of the tax rate on avoidance and evasion may pro-

vide an alternative interpretation for the Yitzhaki puzzle. While, from a

theoretical point of view, it is possible to disentangle evasion from avoid-

ance, the distinction is much more complex in an empirical setting. An

imperfect measure of tax evasion (which may also include a part of tax

avoidance) would lead to a spurious estimation, as the recent estimates on

the tax gap show (Sarin and Summers, 2019).

Over the last few decades, the most striking worldwide trend in tax policy has

been the decline in corporate income tax rates. Some argue (e.g. Tørsløv et al.,

2020) that this is an eect of the tax reduction performed in many countries

to face the competition of tax havens. We show that a similar mechanism

might also be at work for individual income tax: when avoidance is more prof-

itable (higher

β

), the tax rate that maximizes government revenue and the

revenues themselves are lower. Our results suggest that anti-avoidance eorts

of tax authorities/governments/international organizations should be extended

to personal income to prevent a deterioration of government revenue.

6 Conclusion

In this paper we developed what can be considered, to the best of our knowledge,

the rst dynamic model studying taxpayer's avoidance and evasion. Evasion is

costless, but entails the payment of a ne if detected. Instead, avoidance is

costly, but there is a probability that it will be considered legitimate upon

audit.

Contrary to previous studies in a static framework, our results show that

optimal avoidance does not depend on audit parameters (frequency of the audits

and ne to be paid when caught evading) in an intertemporal setting. Tax

avoidance, unlike evasion, is also not aected by the preferences of the taxpayer.

The share of avoided income results from a cost-benet analysis: the (certain)

avoidance cost measures the (money equivalent) eort (or hired expertise cost)

needed to engage in avoidance, while the (uncertain) benet is the potential

reducion of the probability of being ned when audited.

From a policy point of view, our model shows that reducing tax evasion may

be a government objective that is rather dierent from maximizing revenue,

16

especially in the presence of tax avoidance. Given the opposite impact of the

tax rate on avoidance and evasion, we nd that a Laer curve exists between

the tax rate and scal revenue. Our analysis also shows the importance of

the probability that the audited avoidance is successful

β

and highlights its

possible detrimental impact on evasion. In particular, a reduction in

β

leads

to an increase of collected revenues but might entail a rise of tax evasion for

economies more vulnerable to avoidance.

Future research may seek to model more closely the standard tax admin-

istrative procedures by considering an audit technology where the concealing

activities are audited separately but a crossover between audits might occur

and extend the model to account for institutional settings where unsuccessful

avoidance only entails the payment of the tax due (e.g, the UK).

17

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