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Abstract

We study an intertemporal utility maximization problem where taxpayers 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 audit. 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 avoidance 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.
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 Laer 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 dierent government objectives:
minimizing evasion, minimizing non-compliance (evasion plus avoidance),
or maximizing revenues. Our results suggest that specic policies (e.g.,
tax simplication) need to be implemented to deter avoidance and we
illustrate their impact on evasion.
Keywords
: Tax Avoidance; Tax Evasion; Dynamic Programming; Tax
Simplication
*
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 specic grant from funding agencies in the public, commer-
cial, or not-for-prot 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 eect 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 dierent (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 aected 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 identies three basic principles of these
activities: postponement of taxes, arbitrage across individuals subject to dierent marginal
tax rates, and manipulation of dierent 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 dierent 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
aect 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 aect 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 Laer 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 aected 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 osets any positive
eect 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 eect. In our setting, instead, an increase in the tax rate may have a
non-monotonic eect 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 justied 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
0τ1
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)ytct)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 eectiveness 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 dene as
Ia=e
the indicator function whose value is either
1
if avoidance
is successful, i.e. it is assumed to be dierent 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 eec-
tiveness. If
β= 1
avoidance is riskless (like in Alm, 1988b; Alm et al., 1990),
and deterrence is completely ineective against avoidance. Instead, for
β=η1
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 eort (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= (Aktτ(1 etat)Aktctf(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 (
1etat
),
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+1η λ (1 β)f(at)
atτatτAktctdt,
from which we see that evasion is expedient on average if
Et[dkt]>Et[dkt]et=0 ,
which becomes
1ηλ > 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) <1τ
.
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) = (ctcm)1δ
1δ+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
ctcm
,
(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
(ctcm)1δ
1δeρ(tt0)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=ktH
τηAkt1(λη)1
δ(1 β)a,
(10)
c
t=cm+(ktH)ρ+λ
δ+δ1
δ
1
η+δ1
δ(1 τ)A1
η(λη)1
δ+δ1
δ(τβaf(a)) A,
(11)
in which
(f)1
is the inverse of the rst derivative of the function
f
, and
H:= cm
A(τβaf(a) + (1 τ)).
Proof.
See Appendix A.
Equation (10) shows that optimal evasion is aected 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
cmeA(τ βaf(a)+(1τ))(st)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,
ktH
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 benets 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 aect its net marginal return relative to evasion in case of an
audit. The lack of eect 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 aected 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=ktH
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 eect of increased deterrence on avoidance is
entirely oset 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+δ1
δ
1
η+δ1
δ(1 τ)A1
η(λη)1
δ+δ1
δ(τβaf(a)) A.
Proof.
It is sucient 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 eect 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 aected 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
tcm
. 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 ane 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 conrming the
Yitzhaki puzzle. In our model, the presence of avoidance reinforces the damp-
ening eect already observed when the ne is proportional to evaded taxes. The
same eect 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 aAH2
cm
e
t+ (1 β)a
ktH+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) eect is multi-
plied by
1β
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
a1
1β,
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 scientic literature unanimously reports evidence of substantial tax
motivated prot 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 eectiveness 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
1β,
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
δH1βa
τβaf(a) + (1 τ)+ktH
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 suciently 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 eect (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 specication
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
(τβaf(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: Eect 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
ae
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 modied capital
k
tH
is
γ:= 1
dtEtd(k
tH)
(k
tH)=1
δ(1 τ)A(ρ+λ) + 1
η+ (τβa
tf(a
t)) A1(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
tH
is exponential, and is
always positive if the initial value
k
t0H
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
tH)
(k
tH)=1
dt Etd(k
t)
(k
t).
13
Proposition 3.
The expected dynamics of government revenue
dTt
is
Et[dTt] = τ(1 βa
t)AktktH
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)Aktτ β ∂a
t
∂τ Akt+1
η(1 λη)1(λη)1
δH1βa
(τβaf(a) + (1 τ)),
(15)
1
dt
Et[dTt]
∂τ cm=0
0 βa
tτ∂a
t
∂τ
1
a
t
+ 11.
(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 sucient to substitute
a=κτχ
in (16) with
κ
and
χ
positive.
Corollary 2 shows the condition under which our model entails a Laer curve
behavior. For
τ
suciently low, the revenue increases as
τ
increases because
the rise in the tax rate and the reduction of tax evasion more than oset the
increase in tax avoidance. However, as
τ
increases, the latter eect 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 Laer 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 dierent 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 aect
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 Laer 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 eects 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-
tiable, seem to have limited practical relevance. The costs to en-
gage in avoidance are related to the eort (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 eective 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 simplication, 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; Mccaery, 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 oset 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 eect 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 eorts
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 aected by the preferences of the taxpayer.
The share of avoided income results from a cost-benet analysis: the (certain)
avoidance cost measures the (money equivalent) eort (or hired expertise cost)
needed to engage in avoidance, while the (uncertain) benet 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 dierent 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 Laer 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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