Optimal linear income tax in presence of tax evasion
M
aDolores Navarro Bergas
Universitat Autónoma de Barcelona (UAB) First draft: June 2007
Abstract
This paper studies the optimal linear income tax problem when there are possibilities of tax evasion. The decision about how much to work is made simultaneously with the decision of how much income to report. The following main results are obtained. The government audits according to a decreasing function of taxpayer’s reported income. Second and most important, the tax system is regressive due to the presence of tax evasion and the particular way government controls for it. The audit probability and the penalty rate are jointly set with the tax system by the government to achieve some level of compliance. Moreover the government’s optimal controls will determine the "cut-o¤" level of report that separates the evaders from non evaders in equilibrium. Finally, it is also performed a very simple numerical exercise to show the di¤erent equilibrium scenarios that can be obtained in this particular framework.
JEL classi…cation: H21, H26.
Key words: optimal taxation, tax evasion.
1 Introduction
The so called "hidden economy" is a fact in most countries, and has encouraged a lot of research because of its particular properties that make its study a chal- lenging job. There exists a large literature devoted to the tax evasion problem.1 The aim of this paper is to incorporate tax evasion into the standard model of optimal income tax. It distinguishes itself from earlier studies on the subject by
I am very grateful to my supervisors Amedeo Spadaro and Juan Carlos Conesa. I also thank helpful coments by Joan Maria Esteban. I would like to thank the participants to the UAB macro-micro workshop, and some of my professors and colleagues, especially Eduard Alonso and Luca Paolo Merlino for their interest . I would like to thank as well Fundación Ramón Areces for the …nancial support.
1See Slemrod, J. and Yitzhaki, S.’s chapter in the Handbook of Public Economics for a literature review.
considering a model with endogeneous labor supply, a linear income tax and a probability of detection that will be a function of the reported income.
Some institutions have carried out studies in order to establish the impact of tax evasion in di¤erent countries: IRS2 estimates that about 17% of income tax liability is not paid in US. For the case of Europe the estimated hidden economy, that includes tax evasion activities3, is around one third of GNP for Greece and Italy, around 20% in case of Spain, Portugal and Belgium, 18 to 20%
for the Scandinavian countries and between 13-16% for the central European countries.4 This kind of studies conclude, from the empirical evidence they have, that lower …scal burden and moderate regulatory restrictions reduce the size of the hidden sector in the economy. In the appendix you can …nd in Table 5 the estimations for the hidden economy for …fteen OECD countries.
There may be other important institutions, di¤erent from the tax system, that explain these facts, and in general they are related to social and ethical rules. In any case, we should keep in mind that since income is a private information that individuals may reveal or not and that government may want to infer, then taxpayers have opportunities to reduce their tax payment, although there is a private cost of taking advantage of these opportunities, which may take the form of an altered consumption basket, an increasing probability of detection and a penalty for evasion or a resource cost of concealing evasion. Tax systems are then clearly biased by all those evasion and avoidance activities.5
Optimal taxation theory relies on the fact that since personal characteristics are not observable the government is forced to impose taxes on income, which is assumed to be costless observable, instead of imposing a lump-sum tax system.6 However, if taxpayer’s true income is unknown to the government and only observed through costly audits, then the optimal tax design takes a di¤erent direction. The set of government policy tools is enlarged: the audit system becomes part of the tax system designed by the government. When tax evasion is considered the government has the additional aim that is to ensure some degree of compliance. The purpose of this paper is to analyze how this new objective a¤ects the e¢ ciency and equity properties of the tax system in a particular scenario, and compare the results with the ones we would obtain in a setting where evasion is ignored.
There are few papers that have the same approach to the tax evasion and tax design problem. The …rst one is Sandmo(1981), in his paper the economy is divided a priori into two groups of individuals: the non-evaders and the evaders who have access to the irregular markets where the earned income is hidden
2Internal Revenue Service in US.
3For the case of Europe most of the studies are focused on the measure of the hidden economy; while in the case of US is easier to …nd estimates about tax evasion activities.
Despite of that the hidden economy has a lot to do with tax evasion and for that reason we show those estimates as an approximation to evasion in the European countries.
4Those percentages belong to the study "Informal and underground economy" by Frey, B.
and Schneider, F. , International Encyclopedia of Social and Behavioural Science, 2000.
5The distinction between tax evasion and avoidance is that the second one is legal while the other is not.
6The last system is known to be the …rst best solution since it would not create distortions.
unless the government conducts an audit; both groups are subject to a linear income tax and their size is …xed in the economy. Individuals are risk averse and, identical within groups but di¤er in their corresponding earning ability across groups. Given taxpayer behavior, government chooses tax rates, penalty and a constant probability of detection in order to maximize a utilitarian social welfare function.7 That model can not predict whether the marginal tax rate is going to be lower or higher in case of tax evasion when the number of evaders is huge. Traditionally, it has been claimed that higher tax rates would induce a higher level of evasion, and this could be a reason to lower the tax rates in presence of tax evasion. However Sandmo’s model does not claim this, on the contrary since irregular market labor supply is distorted through the penalty system there could be reasons to increase the taxation instead of decreasing it.
There is another related paper by Cremer and Gahvari (1994) which we follow in some aspects when developing our model; where they introduce tax evasion into the analysis of linear income taxation. They assume that taxpayers can alter the probability of being caught through expenditures on concealment.
They consider purely random audits independent of taxpayer’s report and let the penalty exogeneously …xed as a proportion of the tax rate. Taxpayers choose how much to work, how much to conceal and how much to expend on conceal- ment and their preferences are assumed to be quasi-linear. Government has to determine the optimal linear tax system and the optimal audit probability to maximize a general social welfare function subject to to a budget constraint, considering that there exist a continuum of ability types in the economy. Their results about the progressivity of the tax system in presence of evasion are a bit ambiguous in the sense that they depend on the "concealment technology".
The model developed here intends to do an extension of those papers, us- ing some of their assumptions; the main contribution being that we allow the government to base its decision on whether to audit a particular taxpayer on his reported income, so that the audit probability is no longer constant as it has been assumed in most of the related literature. The cost of auditing is suf-
…ciently high so that auditing always every taxpayer is not optimal; but some number of audits will be performed in order to assure some degree of compliance.
Section 2.3.1 is devoted to explain the reasoning behind the particular strategy we adopt. Just for clarity, it is implicit in the analysis that the audit system is perfect in the sense that once an individual is audited the government observes her true income, with no errors. In this setting the penalty will be endogeneized as in Sandmo (1981). A priori individuals can not be classi…ed in evaders or non evaders, the model and especially the particular audit system will determine in equilibrium those two groups. This is a new feature introduced by the particular modelling strategy that di¤ers from the existing related literature.
Similar to the papers mentioned above, individuals choose simultaneously how much to work and how much to report.8 Given the assumption about
7That is the social welfare function just sums up the individuals utility.
8In Sandmo (1981) individuals choose how much to work in each market, the regular and the irregular, while in Cremer and Gahvari (1994) they decide about labor and about the amount concealed.
quasi-linear preferences our taxpayers will be risk neutral with respect to con- sumption. It is considered that there exists a continuum of earning abilities in the economy and the social welfare function will be a general one restricted to be strictly concave.9 The results show that tax evasion plays an important role when choosing the optimal tax system. The optimal tax rates may be lower or higher depending on the cost associated to persecute evaders and the aversion to inequality of the government. Moreover it is showed theoretically and nu- merically that the system is regressive, in the sense that the expected marginal tax rate is decreasing with the true taxable base. This is another result that distinguish this paper from the ones already mentioned.
It should be stressed that in the papers described the probability of audit has been assumed to be a parameter, sometimes exogeneous like in Sandmo (1981), sometimes endogeneous like in Cremer and Gahvari (1994). In this model it will be a function of the reported income by the individual, that is it will vary across individual’s type, and it will be endogenously determined when solving the governement problem. In the so called optimal auditing literature this is a regular assumption, but the approach to the tax evasion problem is di¤erent;
basically because in those models they look for the best audit strategy given the tax system and some budgetary requirement. In the present setting w the optimal tax and audit system are jointly determined. It will be showed that the key features, when determining the optimal policy, will be the degree of aversion to inequality in governemt’s objective function and the audit cost( which could be interpreted as a measure of the e¢ ciency of the audit technology). Few papers have tried to do the same, and in general they had very restrictive assumptions such as the exogeneity in the labor supply decision10, a discrete number of agents type or a penalty system consisted of zero consumption when and evader is caught.11
The structure of the paper is as follows: section 2 develops the model and explains its main assumptions. Section 2.1 solves for the individuals’problems, the optimality conditions are derived and the progressivity of the tax system is also studied there. Section 2.2 is devoted to the government problem; there the optimal tax rate in an evasion environment is characterized and in Section 2.2.1 the optimal formulas for the rest of government tools are derives. In section 2.2.2 there is an explanation of the results obtained in the numerical computation of the model. The last section concludes and discusses possible extensions of the paper. The tables and graphs cited along the paper are included in the appendix, as well as an explanation of the methodology used to perform the computations.
9Only for the numerical computation we will consider the utilitarian case were the social welfare function is linear in its argument.
1 0This the case in the paper by Cremer, Marchand and Pestieau (1990), were they pointed out a very interesting result about the incompatibility of enforcement and equity considera- tions in many cases; as we will comment on our results as well.
1 1This is the case in Cremer and Gahvari(1995), were they look for the non-evasion equi- librium in an economy with just two types of individuals.
2 The model
2.1 The problem of the individuals
The economy consists of individuals who are identical except for the earning ability, !. Individuals derive utility from consumption and disutility from labor.
Preferences are assumed separable and quasi-linear in consumption for reasons of tractability and to be represented by:
u(c; L) = c B(L) (1)
Where B is a strictly increasing, twice di¤erentiable and strictly convex function.12
The choice of quasilinear preferences has important implications; …rst our individuals are risk neutral, second, it sets aside income e¤ects on labor supply and the marginal utility of income is constant. About the …rst one, it should be noticed that most tax evasion models predict that the more risk averse individ- uals evade less13, then it follows that our taxpayers are more willing to evade than taxpayers characterized by utilities strictly concave in consumption.
Tax schedule is linear with a constant marginal rate of t and a lump sum transfer of a, T(R(!))=tR(!)-a, where R(!) is the reported income of an indi- vidual with earning ability !. A taxpayer’s true pre-tax income I (!)=!L(!)14 is unknown to the government and only observed through a costly audit. Tax- payers may evade taxes by reporting a proportion 2 [0; 1] of the true income.
The taxpayer’s report is hence:
R = I: (2)
Individuals’problem consist of choosing I ( ; !) and ( ; !) in order to max- imize their expected utility subject to a budget constraint, given a government policy which contains all the variables that characterize an optimal tax and audit system as we will show in section 2.3.
In this model individuals can in‡uence the probability of being audited by choosing how much to report. The government will audit taxpayer’s report with certain probability that will be a function of R. In particular given that government pursues some level of compliance in the economy this function has to be decreasing with the reported income, otherwise individuals would be tempted to underreport always and particularly high income taxpayers. This is likely to characterize most tax systems and is a common result in much of the optimal auditing literature.15 In our particular setting we will represent this probability
1 2BL> 0and BLL> 0
1 3That is one of the main conclusions of the …rst tax evasion model by Allingham and Sandmo (1972).
1 4For simplicity reasons we will delete the variable w in the notation, although it is implicit in the formulas for I, R and :
1 5See Border and Sobel (1987) or Scothmer (1987), where p(R) is and endogeneous variable for the tax authority that has to control evasion.
as a decreasing16 function of R given by,
p(R) = R (3)
In section 2.1.3, there will be explained the implications of choosing such an audit strategy for the government and taxpayers.
A person who is audited and found evading is taxed on the true income and additionally pays a proportional penalty17 to the not reported income , so the penalty function (I R) can be written as ,
(I R) = (I R) (4)
with being the constant marginal penalty rate. We are going to impose that this rate has to be bounded from below and above; that is people who evade a few can not be rewarded, so that can not be negative and in the opposite case individuals who evade a lot and are caught can not be left with negative consumption, that is can not exceed certain limit.18 Section 2.3 will discuss in detail those boundaries.
The policy cited above is composed by = (a; t; ; ; ); all of them are endogeneous variables of the model that individuals’ at this stage take as given. Assuming that t; ; ; are positive variables, bounded according some natural restrictions, such as that the tax rate t 2 [0; 1]; that ; are such that p(R)2 [0; 1]; and is a positive and below certain threshold speci…ed in section 2.2.1.
The expected consumption for each individual in this particular framework can be written as,
Ce= (1 p(R))(I tR + a) + p(R)(I tI + a (I R)) (5) and the individual’s expected utility is ,
u = (1 p(R))(I tR + a) + p(R)(I tI + a (I R)) B(I=!) (6) Individuals choose I and to maximize u, formally the individual’s problem can be state in the following way:19
maxI; u=(1 p(R))(I tR + a) + p(R)(I tI + a (I R)) B(I=!) Assuming interior solutions20, and taking into account the relationship es- tablished by (1) we obtain the …rst order condition as follows:
1 6By the SOC of the individuals’problem it will be clear that has to be positive so p(R) is shown to be decreasing.
1 7Here the penalty must be interpreted as another tax rate at which not reported income is taxed.
1 8Although for the government could be optimal in order to reduce evasion to set very high and the probability of audit very low since the later is costly and the sooner is not.
1 9It should be noticed that the maximization over I and is equivalent to the one over L and given the relations we set at the beginning.
2 0In the case of assuming interior solution means that we are excluding the full-evasion scheme.
@u
@I = 1 t p(R)((t + )(1 )) p0(R) [(t + )(1 )I] BL=! = 0 (7)
@u
@ = p(R) I (1 p(R))tI p0(R)I[(t + )(1 )I] = 0 (8) The …rst FOC has a clear economic interpretation when rewritten as (9). If individual decides to earn one unit more of income ( or to provide one additional unit of labor) the cost will be the l.h.s of expression (9), that is the disutility from that additional unit of labor divided by the wage per unit of labor, while the bene…t will be the l.h.s; that is the additional unit of income minus the tax over the reported proportion of that unit; both are independent on whether the individual will be audited or not. But in case the individual is audited then he will have to subtract to the previous gain the penalty plus the tax over the not reported income. The last term in the l.h.s represents the gain from increasing the report at the margin, given that the audit probability is decreasing in the report, and that keeping constant earning one unit more of income increases the individual’s report.
1 t p(R)((t + )(1 )) p0(R) [(t + )(1 )I] = BL=! (9) The FOC with respect to has not so clear interpretation as the FOC for I . Dividing all the terms by the pre-tax income level I we can see that an increase in the reported proportion means paying the corresponding additional taxes over the additional report in case of no audit (this is the second term in expression (8), with negative sign); but it reduces the probability of being caught and the corresponding penalty over the additional report (…rst and last term of expression (8), and both have positive sign, remind that p’(R) is negative).
Notice that both …rst order conditions share this last element p’(R)[(t+ )(1 )I]; this is due to the fact that the reported income may increase for two reasons, because I increases or because the reported proportion increases. In both cases the probability of audit will change, actually will decrease by p’(R).
The rest of the expression could be interpreted as the net cost from evasion.
Let v( ; !) be the maximum utility level attained by an individual type !, that is v( ; !)=u(I *( ; !), *( ; !)) where I *( ; !) and *( ; !) are the controls that solve the individual’s problem speci…ed above.
In order to see the properties of the income tax schedule in presence of tax evasion, the second order derivatives and the total derivatives of the FOC will be derived. The aim is to compare those results with the ones that are derived under no evasion scheme. Remind that without evasion, assuming quasi-linear preferences, labor supply and pre-tax income both increase with earning ability
!. In addition linear taxation in such a situation is progressive in the sense that the average tax rate increases with pre-tax income I. Now it is studied whether this is the case when evasion is considered in this particular framework.
2.1.1 Second order conditions , Hessian and total derivatives Using the equation (2) to simplify and express p’(R)=- and p”(R)=0 in the following derivatives we got:
@2u
@I2 = p0(R) (t + )(1 ) p0(R) (t + )(1 ) p00(R) 2[(t + )(1 )I] BLL=!2;
@2u
@I2 = 2 ( + t)(1 ) BLL=!2
@2u
@ 2 = p0(R)(t + )I + p0(R)(t + )I + p00(R)I[(t + )(1 )I] = 2 (t + )I
@u
@I@ = @u
@ @I = p0(R)(t+ )(1 ) p00(R) [(t+ )(1 )I]+p0(R) (t+ ) = (t+ )(1 2 ) Then the Hessian matrix can be written as ;
H=
@2u
@I2
@u
@I@
@u
@ @I
@2u
@ 2
!
= 2 ( + t)(1 ) BLL=!2 (t + )(1 2 )
(t + )(1 2 ) 2 (t + )I
According to SOC for maximization it must be the case that the inequalities below hold;
@2u
@I2 = 2 ( + t)(1 ) BLL=!2< 0
@2u
@ = 2 (t + )I < 0
and@2u
@I2
@2u
@ 2 ( @u
@I@ )2> 0; (2 ( +t)(1 ) BLL=!2)(2 (t+ )I) 2(t+ )2(1 2 )2> 0 Realize that those conditions involve several restrictions among the govern-
ment tools. Then, the optimal controls set by the government {t, ; ; g should satisfy those conditions, otherwise individual would never choose those alloca- tions since she can …nd a better option ( she is not maximizing anymore). This is a bit controversial issue and will be taken into account when solving the gov- ernment problem. Note that the second SOC tells us that the probability of audit will be drecreasing with the reported income, since must be positive in order to full-…ll the condition.21 This will be used and comment later on in the paper.
It is interesting to study the behavior of the pre-tax income I and the re- ported proportion according to the di¤erent earning abilities that we allow
2 1Since the tax rate and the penalty rate are bounded to be positive, and the pre-tax income is by assumption positive as well.
in the economy, the sign of their derivative with respect to ! is analized. In order to get those expressions use the following procedure. First, develop the total derivatives of the FOC and use the mathematical relation that states that the Hessian, H, times the partial derivatives of the controls with respect to ! is equal to minus the derivative of the FOC with respect to !, denoted by A.
That is;
H
@I
@!@
@!
= A
Then applying the Implicit Function Theorem when you have simultaneous -equations to get the expression for the elements in A. It follows that:
A = BL=!2 0
And if using the Cramer’s rule to solve the system to get the analytic ex- pression of the partial derivatives of I and with respect to !, then the sign of those derivatives can be determined using the optimality conditions for the individual as can be seen in Proposition 1 and Proposition 2.
Proposition 1 The higher the earning ability of the individual ,!, the higher the pre-tax income I.
Proof. By the previous calculus we can write the derivative of I with respect to
! as:
@I
@! =[H[H]1] = BL=!2[ 2 (t+ )I]
[H]
the sign of this expression can be easily checked since by SOC the determinant of the Hessian [H] is positive and the expression in the numerator is positive.22
Given this result, the pre-tax income is an increasing function of the earn- ing ability. That is to say that the Spence-Mirrlees condition about agent monotonicity is satis…ed. This condition is important because it guarantees that the tax system has the same properties with respect to the individuals’
earning abilities as to their true earnings.
Proposition 2 The reported proportion has a non monotonic relation with respect to the earning ability,!.
Proof. For the case of the resulting expression for the partial derivative is
@
@! =[H[H]2] = ( BL=!2)[ (t+ )(1 2 )]
[H]
as can be seen the sign of that ratio depends on the magnitude of proportion
; because by SOC the denominator is positive and the rest of the numerator is positive as well.
2 2BLis positive by assumption, -BL=w2is then negative and the term [-2 (t+ )] by SOC is negative.
The economic intuition behind this last result is not so clear. In the numer- ical computations, included in the appendix, there will be showed that in most of the cases the reported proportion is a decreasing function of the earning ability, which is something one could expect given the decreasing audit function with the reported income and the positive relation between the last one and the earning ability !. So that a higher probability of audit would induce a higher reported proportion , in other words less evasion. However the behavior of the reported proportion is given by the combination of variables that characterize the optimal policy ; and it could be the case that given then is increasing in !. Nevertheless we are mostly interested in the situations where decreases with ! and we will devote more attention to those cases. In the appendix you may …nd examples of both situations depending on the scenarios considered.
Those results show that on the one hand the pre-tax income remains to be increasing with the earning ability even when tax evasion possibilities are considered. And on the other hand the relation between the reported proportion and the earning ability is not always in the same direction, depending on the individual’s attitude towards evasion and the particular policy it may be increasing or decreasing in !.
2.1.2 A …rst look to the progressivity of the system
It is possible to study the progressivity of the system rede…ning the taxes paid by the individuals . Now the amount perceived by the government must be interpreted in expected terms, since there is the possibility of reporting just a fraction of the true income and there is a probability of being discovered. Not having solved the government problem yet, the analysis that follows must be taken as a partial one where the results are based on the individual’s optimality conditions. This analysis will be completed with the section 2.2.2.
Proposition 3 When individuals can evade taxes and government has to con- trol it with a linear tax, a proportional penalty on the unreported income and a probability of audit that decreases with the reported income, the tax system turns out to be regressive.
Proof. The proof is explained using the expected tax rate shown below, so that all the calculus that follow belong to the proof of the proposition.
Firstly de…ne the expected tax payment for an individual of ability ! as:
Te= (1 p(R))tR + p(R)(tI + (I R)) a (10) Derivating this expression23 with respect to I, taking into account (2) and (3), then the expected marginal tax rate teis:
te= @Te
@I = (1 p(R))t + 2 tI + p(R)(t + (1 )) (tI + (1 )I) (11)
2 3Taking into account the fact that R= I; and the fact that according to (2) p’(R)=-
The last expression has positive sign whenever the last term is small enough.
Given the assumptions of the model if all the variables are positive, with t, ; and in the interval (0,1), the …rst and the second terms are positive, and the remainder can be rewritten as :
p(R)(t+ (1 )) (tI + (1 )I) = p(R)(t+ (1 )) I(t+ (1 )) = (p(R) R)(t + (1 ))
By assumption p(R) = R and p(R) is positive in the interval (0,1), however as p(R) < by construction of the probability function we can not guarantee that p(R)- R will be positive as well. We will assume that we are in the situations where this is the case, or in the situations where the last term of (11) is smaller than the rest of expression (11) so that the expected marginal tax rate is positive. In the appendix we show that in most of the cases the expected marginal tax rate was positive24.
Derivating the last expression with respect to pre-tax income I will give us an idea of the progressivity of the tax system,
@te
@I = p0(R) 2t+ 2 t+ p0(R)t+ p0(R) (1 ) t (1 ) = 2 [(1 )(t+ )]
(12) From the SOC it follows that -2 ( + t)I < 0; and given that is a positive proportion by de…nition and that the pre-tax income is also positive, then@t@Ie <
0; in other words the tax schedule turns to be regressive in presence of tax evasion. If there was no evasion te = t and @t@I = 0: This is the same result we get for individuals that full report their income that is = 1: In the appendix you may …nd the plots of the expected tax rate as a function of the pre-tax income and verify that in all the proposed scenarios it was clearly decreasing with respect to this last variable, con…rming the predictions made in proposition 3.(See Graphs: 11 and 12).
In the previous discussion it has been showed that the expected marginal tax rate with evasion is positive but decreasing with income level, since (12) had negative sign indicating some regressivity in the system. This result has a clear intuition behind, when evasion is a fact in the economy the system may not be so progressive in order to deter some evasion especially in the high levels of income. The average expected tax rate was not included since its study did not lead to a clear conclusion, neither did its comparison with the marginal tax rate.25 In the appendix you may …nd graphs that illustrate the regressivity of the tax system ( graphs 11 and 12).
2 4We just obtained negative expected marginal tax rates in one of the scenarios and it was probably due to the extreme case we were considering, see appendix for more details.
2 5The expression for the average tax rate in our model was Tme = (1 p(R))t + p(R)(t + (1 )) a=I;while its derivative with respect to I was
@Tme
@I = p0(R) 2t + p0(R) t + p0(R) (1 ) + a=I2= [(1 )( + t)] + a=I2 We can not guarantee whether the average tax will be decreasing or increasing with I;
analitically.
2.1.3 The audit probability function as a tool to deter evasion As commented in the introduction following the optimal auditing literature, the reasoning behind the audit probability being decreasing with the reported income is that auditing low income reports with higher probability than high- report taxpayers makes it less attractive for high income taxpayers to underre- port income, but it introduces a regressive bias in the e¤ective tax code as we could show in section 2.1. In particular it is possible to see who is going to evade in equilibrium given the following result that is common in much of the related literature.
Proposition 4 If p(R) t+t ; then taxpayers’ reported proportion of income beta is maximum, that is = 1.
Proof. In order to have that truthtelling is the best strategy for individuals it must be the case that the following inequality holds (non-evasion condition):
I tI +a B(I=!) (1 p)[I tR +a]+p[I tI +a (1 )I] B(I=!) (13) then rearranging terms we get that p t+t
Notice that this is just the condition that ensures that the utility of non- evader taxpayer ( l.h.s of expression (13)) is greater than the utility of the evader taxpayer ( r.h.s of expression (13)). The same result could have been obtained by imposing 1 in the FOC with respect to in the individual’s maximization problem. Notice, that in this particular setting, given the separability property of the utility function on its arguments, it is possible to compare this two utilities expressions without loss of generality.
In this particular setting it is possible to reinterpret the audit probability function so that a the cut-o¤ level of report can be calculated. That report separates the evaders from the non-evaders, as a function of the government controls. To see how it can be done we …rst set the equality below using equation (3) and the last proposition:
Rc = t
t + (14)
then Rc = t=(t + )
(15) That is given the optimal values for ; , t and one can establish the cut- o¤ reported income level. People reporting less than Rc are not going to evade taxes, since they are audited with high enough probability, while the ones with reports that exceed that level have incentives to evade since the probability of detection is below the value that ensures truthtelling. It could be the case that Rc is very low so that the degree of compliance achieved by the government is very low as well. Notice that given that result the audit function can be restated as follows:
p(R) = t
t + + (Rc R) (16)
This is a very important result, because introduces a way of endogeneously determine the evader group and the non-evader group in the population, as a direct consequence of including a probability function that is non-constant across individuals’ type as had been normally assumed in the related papers.
Moreover it allows us to calculate numerically the ceiling for the probability and a cut-o¤ level report (see the last rows in the tables shown in the appendix), so that below it people evade taxes and above it they do not.
2.2 The government problem
The government problem is to determine the tax rate, penalty rate and the probability function so as to maximize a social welfare function subject to a resource constraint that considers the fact that government is entitled to raise some critical level of revenue26 and the fact that audits are costly.
The tax authority could be tempted to use very high punishments since pun- ishing is assumed to be costless, while auditing is costly.27 However we are going to restrict our attention to penalty rates that do not exceed certain ceiling, since we consider that for motives of social cohesion or because of social rules people can not be left with negative income if caught evading. Moreover the penalty introduces an additional distortion in the economy and then the government may not impose excessive penalties because of an e¢ ciency reason. When dis- cussing the properties of optimal penalty rate in section 2.2.1, these facts will be taken into account and a ceiling for the penalty rate will be speci…ed.28
Assume that the earning abilities are distributed over [!, ! ] with the cumulative distribution function F(!) and the corresponding density function f(!). We normalize the population size by assuming that F(!) =1. In order to simplify notation, de…ne the expected tax revenue from an individual type ! as:
G(!;t, ; ; ) = I + a B(I=!) v(!; t; a; ; ; ):29
2 6Revenue that may be related to the provision of some public good.
2 7This is a result from Becker (1968) and the optimal monitoring models, that has been reviewed in this kind of literature.
2 8This will be clearer in the numerical exercises and in the appendix example.
2 9u(c,L)=c-B(L); and c=I-T(R), then u(c,L)=I-T(R)-B(L); T(R)=I-B(L)-u(c,L) in the in- dividual’s optimum T(R(w))=I(w)-B(L(w))-v(I(w), (w)), this is what allows us to rewrite government’s budget constraint. Additionally because of quasi-linear preferences G(.) does not depend on a.
G0is going to be the government’s net revenue requirement, and c the con- stant per audit cost; this is an assumption that is made for tractability reasons, it is a bit restrictive but for the present purpose is not very important.30
The government ’s problem can be expressed as follows:
M ax
ft;a; ; ; g
R!
! (v(t; a; ; ; ))f (!)d!
s.t.R!
!fG(!; t; ; ; ) a cp(R)gf(!)d! = G0 ( )
where is an increasing and strictly concave function, that is to say the government has a redistributive objective given the form of the social welfare function.
The Lagrangian expression associated with the government’s problem is:
=R!
! (v(t; a; ; ; )f (!)d! + [R!
!fG(!; t; ; ; ) a cp(R)gf(!)d!
G0]
Assuming interior solution for all the controls in the maximization the FOC are :
@
@t = Z !
!
0(v(:))vtf (!)d! + Z !
!
@G
@t f (!)d! = 0 (17)
@
@a =
Z !
!
0(v(:))vaf (!)d!
Z !
!
f (!)d! = 0 (18)
@
@ =
Z !
!
0(v(:))v f (!)d! + Z !
!
@G
@ f (!)d! = 0 (19)
@
@ =
Z !
!
0(v(:))v f (!)d! + Z !
! f@G
@ cgf(!)d! = 0 (20)
@
@ =
Z !
!
0(v(:))v f (!)d! + Z !
! f@G
@ + cRgf(!)d! = 0 (21) Using the Envelope’s theorem it is possible to obtain the partial derivatives involved in the FOC. Using the de…nition of v(!) and p(R) in equation (2) it follows that:
v(!)=v(I*(!), *(!))=(1-p(R*))(I*-tR*+a)+p(R*)(I*-tI*+a- (I*-R*))-B(I*/!) where R*= I
vt = (1 p(R))R p(R)I (22)
va = 1 (23)
v = p(R)[R I] (24)
v = (R I)(t + ) (25)
v = R(I R)( + t) (26)
3 0In similar papers they assume that this cost is a function of the audit probability, but on the other side they …x that probability.
Equation (18) has to be interpreted as the marginal utility of income, it will be used to characterize the optimal tax and audit system when dealing with the FOC of the government’s problem.
Applying the same procedure to develop the partial derivatives of G function, that is:
@G
@t = (1 BL=!)@I
@t vt (27)
@G
@ = (1 BL=!)@I
@ v (28)
@G
@ = (1 BL=!)@I
@ v (29)
@G
@ = (1 BL=!)@I
@ vd (30)
From equation (23) and the FOC (18) it is the case that:
Z !
!
0(:)f (!)d! = E( 0(:)) = (31)
where is the Lagrange multiplier associated to the revenue constraint.31. Substituting (31) into FOC (17) and using (22) and (27) it can be established that :
Claim 5 The optimal tax schedule satis…es the following relation ship:
cov( 0; (1 p(R))R + p(R)I) = E( 0)E[(1 BL
! )@I
@t] (32)
This last equation re‡ects the equity (covariance term) vs e¢ ciency ( r. h.s) trade o¤ that characterizes the optimal tax rate. That is, a change in the tax t and the corresponding adjustment in a to meet the budgetary requirement, have two di¤erent e¤ects on welfare: the redistributive one illustrated by the covariance term and the impact on e¢ ciency known as the excess burden of the tax represented by the r.h.s of (32). The …rst term captures the e¤ect on social utility of an increase in the tax rate, it is negative since (1 p(R))R + p(R)I is increasing in !32, while 0 is decreasing in ! given the assumptions over the social welfare function. The r.h.s of (32) could be interpreted as the average substitution e¤ect on labor supply due to an increase in the tax rate.
3 1Given the strict concavity of ; 0 is a positive but decreasing function, then E( 0)is positive, so that is positive and then by the Slackness Kuhn-Tucker condition the associated constraint, the budget one is binding.
3 2I(w) by proposition 1 is increasing in w, but R depends on which by proposition 2 has no monotonical relation with w. The hole expression understood as an average of income for the government is expected to be increasing in w.
As can be observed the optimal tax rate will be set in order to balance those two e¤ects at the margin; the gain from redistributing by increasing t must be compensated by the cost of increasing the distortions in the individuals decisions about labor supply and report ( since in our particular setting where evasion is possible we have to take it into account too). This is common in all the optimal taxation literature, however as will seen in detail in the next section, our formulation di¤ers from the traditional one in several aspects and it is because of the evasion possibilities and the way government tries to control it.
Using the individual’s FOC ((7) and (8)), the equation that characterizes the optimal tax rate (32) can be rewritten and manipulated in order to obtain the following expression that allows us to make comparisons with the non evasion scenario:
t = covf 0; (1 p(R))R + p(R)Ig + E( 0)Ef(p(R) )@I@tg
E( 0)Efp(R)( @I@t)g (33)
When there is no tax evasion, Dixit and Sandmo(1977) applying the tech- niques used in commodity taxation simpli…ed the formulation of the optimal linear tax rate and showed that it could be written as (using the notation in the model):
t = cov( 0; I)
E( 0)Ef @I@tg (34)
Once those two particular results are obtained it is possible to make pre- dictions about the optimal tax rate in presence of evasion. As can be seen the numerator in the expression (33) has a new term E( 0)Ef(p(R) )@I@tg; that in- cludes the tools we have introduced in the government menu so as to control evaders, and p(R); this term is going to have negative sign given that with quasilinear preferences the derivative of the pretax income with respect to t is a total substitution e¤ect and thus is negative; while the remainder is positive.
Then, according to the numerator the optimal tax rate is expected to be smaller in presence of tax evasion. The problem is when comparing the denominators from both expressions since this does not lead to a clear conclusion, because in (33) it is expected to be smaller since the same expression is multiplied by p(R) which is a number between zero and one, thus the tax rate would be bigger in evasion presence.
There is an additional fact that makes comparisons under both cases even tougher; in our model labor supply is distorted when evasion is considered, this is di¤erent from previous works as Cremer and Gahvari (1994) where the labor supply is the same with and without evasion under particular assumptions. This can be deduced just looking at the FOC for the individual. It is natural to think that the presence of evasion activities creates new incentives in the economy and thus individuals can behave in a di¤erent way when choosing their optimal
allocations. 33
Given the di¢ culties of making comparisons between this framework and the nonevasion one at the theoretical level, a numerical computation of the model was performed, in order to better explode its policy implications. The reader may …nd in the appendix the details of the computation methodology, the results in tables and graphs, and the main conclusions in section 2.2.2.
2.2.1 Optimal penalty rate and optimal audit policy
Following the same procedure as before it can be established a similar result for the penalty rate, substituting (24) and (28) into the FOC (19) and with (31);
Claim 6 The optimal penalty rate satis…es the following relationship ;
cov( 0; p(R)(I R)) = E( 0)E[(1 BL
! )@I
@ ] (35)
It should be realized that equation (35) illustrates that is set in the same way as if it were a tax rate, just by balancing at the margin the equity e¤ect( co- variance term) and the the e¢ ciency e¤ect ( r.h.s). It follows that as mentioned few lines above, the penalty rate depresses the labor supply and introduces an additional distortion in the economy. This could be a reason not to use very high punishments in this setting.
In particular the penalty rate can not exceed certain limit:
(1 t)
(1 min)+ a
(1 min) Im ax (36)
Where min represents the minimum reported proportion and Im ax the maximum pre-tax income level observed in the economy.
And since it was assumed that honest taxpayers or even low-type evaders can not be rewarded there is another lower bound so that:
0 (1 t)
(1 min)+ a
(1 min) Im ax (37)
The last inequality follows from the fact that the consumption, in case the evader is audited, can not be negative then,
I tI + a (1 )I 0 (38)
Rearranging terms, the inequality above is obtained. This result is related to the discussion opened at the very beginning of this section. Since punishing
3 3Remind that expression (33) is not an explicit formula for t, it has terms on the r.h.s.
that depend on t as well, so far we have solved the model implicitly, for tractability reasons.
is costless government could be tempted to use a very high punishment, but for motives of political ethics she can not leave people with zero or even negative consumption.34 Then the government strategy is to use penalty as well as audit probability as a tools to prevent evasion. Notice that in the expression (38) there are two terms that are not constant, I and are di¤erent for every individual.
From (36), observe that increasing increases while increasing I decreases : If the ceiling for has to satisfy (38) and by extension it should guarantee non- negative consumption for any individual type, then it should be considered the minimum that could exist in the economy and the maximum income. Notice that the lower bound the on the penalty is due to the omission of cases where
"compliants" could be rewarded
In the numerical computations it will be showed di¤erent scenarios where the optimal penalty could be very low or very high depending on the cost per audit and the redistributive aim government may have. (See appendix and section 2.3.1).
It is possible to derive the optimal values for the variables and from the government problem following the same procedure as before, substituting (25), (29) and (31) into (20) and (26), (30) and (31) into (21).
Claim 7 The variables that characterize the optimal audit function satisfy the following two equations:
cov( 0; (I R)(t + )) + E( 0)E[(1 BL
! )@I
@ ] E( 0)c = 0(39) cov( 0; R(I R)(t + )) + E( 0)E[(1 BL
! )@I
@ ] + E( 0)E(R)c = 0(40) From the previous expression it can be commented that the optimal values for and are set following a somewhat similar reasoning to the tax and penalty rate, trading o¤ the bene…ts and costs of increasing that particular government tool . However it is di¢ cult to make predictions about them because expressions (39) and (40) involve some partial derivatives, @@I and @I@ ; whose sign is not very clear. By the assumptions of the model it seems to be the case that the higher probability of being audited the less evasion, but whether the labor supply will be more or less distorted is undetermined, it could be the case that individuals do not evade so much but they decide to work less.
Notice that from (15) a higher value for would mean a higher value for the Rc and thus a greater number of individuals reports being audited with high enough probability in order to induce them not to evade taxes. While increasing or the rate at which the probability of detection decreases with
3 4In some papers like Cremer and Gahvari (1995) they let the penalty to be such that in case of auditing an evader his consumption is set to zero; we do feel that this is a very strong punishment that could not be implemented in reality, although it simpli…es the structure of the theoretical model. Other authors agree that with a less strict punishment some degree of compliance can be achieved, see Schroyen (1993).
the reported income, decreases the value of Rc; this is because the probability function would fall to low values faster then in case of smaller ; being below the critical level t/t+ for very low reports, that is only very low reports would be the ones corresponding to the non-evaders. Then in order to induce a higher degree of compliance government must increase and reduce . This will be clearer in the results of the numerical computations in the appendix and their corresponding comments in the section that follows.
Up to this point what can be learnt from the analytical results are basically two main features: …rstly that the tax system in expected terms shows regres- sivity, in the sense that the expected marginal tax rate was decreasing with the true taxable base.35 Secondly, introducing a probability of audit that di¤ers across individuals we were able to calculate a cut-o¤ level that separates evaders from non-evaders, and it can be endogenously determined the size of each group depending on the value at which the policy variables are set in the optimum. To better explore the properties of the model, in particular its behavior according to changes in the parameter values, the analysis will be completed with some numerical computations that are explained in the following subsection and in the appendix.
2.2.2 Optimal policy in presence of tax evasion: numerical results The aim of this section is to show under very simple functional forms and stan- dard distributional assumptions the way the model works in di¤erent scenarios;
that will allow us to perform a comparative statistics exercise and to highlight some properties about the progressivity of the tax system that were not very clear in previous sections
For the numerical computations several scenarios were considered, accord- ing to the revenue requirement, the cost per audit, the degree of aversion to inequality in the government’s objective function and it was also considered that the elasticity of labor supply could be di¤erent36(a or b in the tables de- notes di¤erent elasticity of labor supply, for a more detailed discussion see the appendix). Additionally two cases concerning the lump-sum transfer a were computed, since it was realized that without restrictions it was naturally opti- mal to set it to zero value in most of the scenarios. Then there are two cases:
one in which the government can freely choose any level of positive a (Tables numbered 1 and 2), and another where he is entitled to provide some minimum level of income, called a037 (Tables numbered 3 and 4). With the numerical results, some comments about the properties of the tax and audit system can be made, and the results derived analytically through out the discussion can be now showed.
3 5According to the de…nition of tax progressivity that we used in section 2.1.2.
3 6In the numerical exercises about optimal taxation the elasticity of labor supply is in general a key element, that may alter the results, this is the reason why it is included in the analysis, see Tuomala(1984) or Tuomala(1994).
3 7Whenever the lump-sum transfer is …xed, the resulting optimal policy is a restricted one and so the welfare there is lowere compared to the scenarios were this variable can take any positive value.
First about the optimal tax rates. For the particular cases that were com- puted tax rates are in general low (below the 20% of the reported income), especially when the lump-sum transfer a can be any positive number38. When- ever the auditing cost increases they increase as well, as one could expect (see the change in variable t through the columns of any of the tables). Another reason that can explain a tax increase is a higher degree of inequality aversion in the government’s objective function. Comparing the utilitarian scenarios (with no aversion to inequality) with the ones where there is some aversion to in- equality (non-utilitarian scenarios), it can be clearly observed that tax rates are higher in this last setting (compare Tables 1a.1 and 2a.1 or 1b.1 and 2b.1 for instance). This is due to the inequality aversion , since a higher tax rate may induce more redistribution. Additionally whenever there is an increase in the revenue requirement (in the computations from G=0 to G=0.100, see Table 1a.1 and 1a.2) or when some a0 was imposed ( in the computations a0= 0:001); tax rates go up (compare Table 2a.2 and 4a.2). The computations for the optimal tax rate in the standard model where evasion is ignored, show lower tax rates in most of the cases (see Tables 6 and 7). This could be explained by means of the revenue needed to perform audits and persecute evasion in the present model.
What makes the di¤erence are the e¤ects on the optimal audit system con- sisted of the probability variables and and the proportional penalty : About
; with a being any positive number and high elasticity of labor supply (Tables 1a.1, 1a.2 and 2a.1 and 2a.2) it remains to be low and near the lower bound we used in the numerical computations, but decreases as the cost per audit does, so that the probability of audit decreases faster when auditing is very costly as could be expected. When there is a …xed lump-sum transfer the result about the choice variable is no longer true. The government now plays around with its value and it may be far from the previous result depending on the particular scenario (see Table 4a.2 for an example). This is also the case when the compu- tations for the optimal policy are carried out in case of lower elasticity of labor supply; although the changes in are not so evident (see Tables 3b.1, 3b.2 and 4b.1, 4b.2). An explanation for this phenomenon is that, when …xing the lump sum transfer, we are restricting the set of choice variables and so the government may alter its optimal decision with respect to the other variables, in particular to this . Remind that this variable represents the maximum probability of au- dit, since the probability function is decreasing with the reported income, and so it has clear e¤ects on the taxpayer’s evasion decision (see formula 14).
The penalty rate is much higher than the tax rate in all the scenarios, but it does not always go to the maximum value we allowed in the computations.
Whenever the cost per audit is low enough, the optimal penalty rate decreases a lot (see Table 1a.1 and 2a.1). As the tax rate, the penalty rate is a¤ected by the degree of aversion to inequality and the revenue requirement. It shows, in general, higher values in the non-utilitarian scenarios (compare Table 1a.1 and 2a.1 or 1a.1 and 1a.2 for instance), and if we impose some larger revenue
3 8And so the optimal policy implies a zero lump-sum transfer as can be observed in tables 1.1-1.8.
requirement (compare Tables 1a.1 and 1a.2). The pattern is very similar when we …xed the lump-sum transfer, being the values for the penalty rate even higher than in the previous cases (see Tables 3a.1, 3a.2 and 4a.1, 4a.2). In the scenarios with lower elasticity of labor supply, the behavior of the penalty rate is very similar (Tables with subindex b). It is then proved that for the government it may not be always optimal to set the penalty rate at its maximum level, the reason being the distortions introduced, that is the e¢ ciency concern.
Whenever the elasticity of labor supply is set at a lower value the results are a bit di¤erent. For instance tax rates are higher in most of the cases, compared to the cases of higher elasticity (see tables 1a.1 and 1b.1 for instance), as one could expect given that now individuals do not react that much to changes in the parameters that a¤ect their net earning. Although …xing the lump-sum transfer a or a bad audit technology (high cost per audit) may alter this general result (see tables 2a.1 and 1b.1). But one should not forget that there exists another tax rate in this economy, the penalty rate, and in the lower elasticity scenarios it is set to higher values than in the high elasticity ones ( compare for instance Table 2a.2 and 2b.2). What makes that the e¤ective marginal tax rate39 at which not reported income is taxed would be higher in the low elasticity scenarios. Concerning this last comment, in this particular framework all the policy variables determine the so de…ned expected marginal tax rate (equation (11)), and this is the variable that really a¤ects individuals decisions about how much to earn and how much to report. But since all the policy variables are being jointly and simultaneously set, it is di¢ cult to asses the impact of the shift of one of them in the expected marginal tax rate.
An interesting feature of that model, is that once we have the optimal policy we can determine who evades and who does not in this economy, that is to show the evader and non evader group in equilibrium. Several situations can happen according to the optimal policy. In graph 1 and 2 we have the case where the optimal policy completely deters evasion in equilibrium; everybody fully reports his pre-tax income. The audit probability is above the threshold level that ensures truthtelling for all the individuals in the population (see graph 3 and last column of Table 3a.1 for the numerical values of the choice variables).
In graph 4 and 5, there is another optimal policy that only ensures partial compliance: the low earning ability taxpayers fully report their income, while the others report just a fraction that decreases as the earning ability increases. The probability of audit is above the threshold level just for some of the individuals ( see graph 6 and middle column of Table 1a.2 for the numerical values of the choice variables). Finally it may be the case that the optimal policy can not ensure compliance, given the restrictions and the requirements it may be optimal to have a population of evaders, like it is shown in graphs 7 and 8.40 Notice that the reported proportion never reaches one, but it increases as the earning ability does. It is also possible to …nd cases were the optimal policy can
3 9That is t+ :
4 0This last case corresponds to one not reported in the Table 1.4, where the cost per audit was set to 0,003. The tax rate was 0,0247, the alpha variable was 0,05, gamma was 1,0507 and the penalty rate was 0,3440.
not ensure compliance at all, but in which the variable is decreasing with the earning ability (and this was the case in most of the low-elasticity scenarios)
Those cases illustrate some facts we had already mention: …rst, the behavior of the reported proportion with respect to the earning ability !; which can be monotonically increasing or decreasing and secondly the possibility of evasion in equilibrium. In graphs 10, 11 and 12, it is reported the behavior of the marginal tax rate in this three cases, just to show the regressivity of the tax system found in Proposition 3. The marginal tax rates are decreasing with the pre-tax income in presence of tax evasion.
It is hard to establish a general statement about the properties of the optimal policy mix of variables that lead us to one or another case (with respect to the behavior of the reported proportion ). In general, lower tax rates, high penalties or high probabilities of audit (because of low values for instance) generate situations of no evasion or partial evasion (a group of compliants and another of non-compliants) in this particular economy. In the rest of the cases there is no-compliance at all, that is all the individuals in the sample underreport part of their true income. Then it may be optimal for the government to allow people evade in some situations.
In general the government must trade-o¤ redistribution versus compliance and e¢ ciency. A higher cost per audit (or a higher revenue requirement) means that more resources have to be devoted to the audit system to reach some compliance level. On the other side higher tax rates may induce more evasion but also more redistribution. The joint e¤ect is an increase in the optimal tax rate when there is aversion to inequality or higher cost per audit. In fact the conclusion is that evasion in general increases the optimal tax rate due to the more resources that the government has to devote to control it( to perform audits). Although a minimum income level requirement and the elasticity of labor supply may alter this general result, whenever the cost of auditing is low (see Tables 1b.2 and 2b.2).
As mentioned in the introduction the cost per audit can be interpreted as a measure of the e¢ ciency of the audit process. Then a high cost per audit means that the audit process is not very e¢ cient. As showed in the numerical computations, in this situation if government has to reach some level of com- pliance and has some redistributive goal, then the government should be really concerned about announcing "tax cuts", for instance. Remind that the optimal policy in such an environment is expected to have really high tax rates com- pared to another scenario where the audit technology is more e¢ cient ( lower cost per audit). That is to say that when evasion is introduced into the problem the audit technology becomes a key ingredient in determining the optimal tax policy.
3 Conclusions, comments and further extensions
This paper shows with a very simple model that introducing tax evasion into the optimal linear income tax model has important implications for the progressivity of the tax system and the way the government may control it or part of it. It is worth to recover some of the results obtained through out the discussion concerning the properties of the optimal tax and audit system.
In the section 2.1.2, considering just the optimality conditions for the in- dividual, it was concluded that the tax system was regressive in presence of tax evasion, since it was showed that the expected marginal tax rate was posi- tive but decreasing with income level. This is what distinguish this work from the previous ones; in most of them, like Cremer and Gahvari (1994) the pro- gressivity of the system in presence of tax evasion could not be clearly stated.
When solving the government problem, considering both optimality conditions for the individual and for the government and helped by some numerical com- putations, the tax rates in presence of evasion were showed to be higher than the ones derived in an environment without evasion possibilities (especially in those situations where auditing was costly).
The optimal penalty rate behaves as another tax rate, trading o¤ e¢ ciency versus equity considerations. The values obtained in the numerical computa- tions seem to indicate that its value could be lower than some upper bound;
especially when audits are not very expensive or there is no aversion to inequal- ity. According to those comments it is clear that this variable is as sensitive as the tax rate to the changes in the evasion scenarios and inequality preferences of the government; and that the upper bound calculated in section 2.3 could be binding just in some cases.
The variables that characterize the audit function, and , are changing as well with the cost per audit, as expected, in general they decrease as the cost increases, so that the probability of audit is decreased and more evasion possibili- ties emerge in the economy. Moreover the joint system {t, ; ; g will determine the "cut-o¤ level" for the report that divides the taxpayers into evaders and non-evaders. It is important to remark that in this particular setting there is room for evasion in equilibrium and the degree of it will be depending on the level at which the government variables are set.41
In summary, tax evasion introduces distortions in the way economy works:
it distorts the labor supply decision, it a¤ects the redistributive properties of the tax system and makes it tougher for the government to allocate resources e¢ ciently since she must take into account an additional cost of monitoring taxpayers and so devote resources to do it as well. It may be the case that it is welfare improving for the society to allow for some degree of evasion, instead of trying to establish a system that does eliminate evasion in equilibrium. It was not mentioned, but we do feel that this is the case and that the later could not be implementable in practice.42
4 1See equation (38) for Rc:
4 2Because it would be too expensive or because it would violate some social, ethical rules.
Some comments have to be made about the assumptions of the model.
Firstly, as pointed in section 2.1, with quasilinear preferences we are imposing risk neutrality of individuals and thus a source of evasion activities compared with other scenarios where individuals could be risk averse. Remind that the evasion activity is a risky decision, where attitude toward risk can play a decisive role. Secondly, the functional form imposed to the audit probability function could be seen as too restrictive, but it was made for tractability reasons and in order to illustrate how such an audit strategy would a¤ect the hole tax system.
The assumption about having a constant cost per audit is also restrictive, in the sense that a more general setting could be considered. For instance one could think about a system where the audit cost depended on individual’s character- istics , such as the main income source. But not to complicate the model it was assumed away.
Finally, dealing with linear income taxation, which turns to simplify the government problem since we do not have to consider the so called self-selection
43constraint, it also simpli…es the issue of taxation itself by allowing a unique tax rate in the economy for any income level. A natural extension of the model would be to introduce a general tax function into the analysis. Another interesting and challenging issue could be the development of a theoretical model where we do impose no evasion in equilibrium, so we would be looking for a tax and audit system that do deter evasion activities; but as mentioned before it could be the case that it does not exist in a very general setting or that it can not be implemented in practice.44 Those possible extensions of the model showed here are left for future research.
4 3This is an incentive compatible constraint that ensures that type w individuals will choose the corresponding pair C(w) and I(w) and not the pair associated to another type w’. In the linear case it can be interpreted as being already included into the problem through the indirect utility function that we computed for the government ob jective function.
4 4Cremer and Gahvari (1995) introduced tax evasion into the general income tax problem and seeked for the tax and audit system that was incentive compatible in such a situation.
The point is that they just allowed for two individual’s type in the economy and they set the penalty to zero consumption in case an evader was discovered. Our analysis tries to be more
‡exible and general.
