The Optimality of Punishing Only the Innocent: The Case of Tax Evasion

International Tax and Public Finance, 7, 641–664, 2000.c© 2000 Kluwer Academic Publishers. Printed in The Netherlands.

The Optimality of Punishing Only the Innocent: TheCase of Tax Evasion

ROBIN BOADWAY [email protected] of Economics, Queen’s University, Kingston, Ontario K7L3NG Canada

MOTOHIRO SATO [email protected] of Economics, Hitotsubashi University, 2-1 Naka Kunitachi, Tokyo 186 Japan

Abstract

We study the effects on tax enforcement and tax policy of unintentional compliance errors by taxpayers andadministrative errors by tax auditors. The government can impose both penalties for misreporting and rewardsfor honest reporting. Maximal sanctions will not be applied because errors are possible, so evasion cannot beeliminated costlessly. Under optimal policy intentional evasion can be deterred, but innocent tax evaders must bepenalized whether they have unintentionally evaded or have been mistakenly convicted. This deters intentionalevasion, but limits redistribution. Without rewards for honest reporting, the revelation principle need not apply,so intentional evasion can occur.

Keywords: tax evasion, compliance errors, administrative errors

JEL Code: H23, H26

1. Introduction

Errors in administering justice are a fact of life, and can give rise to the conviction ofinnocent people. Such errors may be of two main sorts, both of which arise due to theinevitable complexity of distinguishing guilt from innocence. First, innocent persons maybe mistakenly convicted. Indeed, this is one of the main arguments against the death penalty.In recent years, there have been several cases of persons who were wrongfully convicted andwho spent several years in prison before they were actually found, thanks to DNA evidence,to be innocent. Second, persons might inadvertently commit a crime, either because theydo not know the law well enough or because the boundary between committing a crimeand not committing a crime is itself unclear. For instance, the circumstances in which itis permissible to assault or harass another person might be ambiguous. Unlike in the firstcase, the mistake here occurs on the part of the household committing a crime rather thanthe justice system in deciding on guilt or innocence. Persons who unintentionally commita crime may be found guilty by the justice system and punished. Although this is notwrongful conviction, it might nonetheless be viewed as less deserving of punishment thanpersons who intentionally commit a crime.

In this paper, we explore some of the consequences of errors of these two sorts in thecase of tax evasion. We do so in the context of a tax system that is redistributive in na-ture, a relevant feature of income taxes. Income tax systems are typically very complex,

642 BOADWAY AND SATO

given that the definition of what constitutes income is not clear-cut. For example, someincome comes in imputed forms, such as imputed rent on housing which is taxable in somecountries; some comes in forms that are hard to measure, such as capital gains and self-employment income; some forms are sheltered, such as income on saving for retirement;some comes as in-kind benefits, such as a company vehicle, medical and dental plans, traveland accommodation which may combine business with pleasure, entertainment expenses,frequent flyer points, and so on; and some transactions that occur over the tax year maybe lost or forgotten by taxpayers, such as accepting payment for services performed forfriends and neighbors. In these circumstances, it would not be surprising if unintentionalmistakes are made in reporting one’s own income, or even in filling out the required taxforms. Casual observation would suggest that a significant proportion of income tax returnsthat are filed are found to be incorrect as a result of routine verification by the tax admin-istration. In some instances, the tax authorities may be able to discern to their satisfactionwhether filing mistakes are intentional or unintentional. Thus, tax systems exonerate honestmistakes by imposing light penalties while punishing fraudulent behavior more severely.Our focus is on instances of misreporting where it is not possible to uncover the intent ofthe taxpayer.

By the same token, errors of administration can occur in the process of auditing a tax-payer’s income tax return. An early study of the consequences of administrative error inthe context of optimal redistributive taxation is by Stern (1982). Using numerical exam-ples, he shows that when administrative errors of identifying taxpayers’ types (e.g., theirneeds or abilities) are present, the social gain from implementing differentiated lump-sumtaxes and transfers is lowered, and a simple, but distorting, tax system such as an incometax may be preferred. Accordingly, the presence of administrative errors can have impor-tant implications for the optimal form of taxes and transfers. Our purpose is to considerhow the presence of both administrative and taxpayer errors, combined with the possi-bility of intentional misreporting of one’s income, affects both the government’s optimaltax enforcement policies and its ability to achieve redistributive objectives using the taxsystem.

There is a large literature on the economics of tax evasion. The standard approach, whichoriginated with Allingham and Sandmo (1972) and is amply surveyed in Cowell (1990),treats the decision of how much tax to evade as being akin to a choice of a consumptionstream under uncertainty. Taxpayers, faced with a given penalty and a given probability ofdetection, will choose the amount of evasion which maximizes their expected utilities. Thisapproach has been applied in a wide variety of settings.1 Recent papers by Marhuenda andOrtuno-Ortin (1997) and Chandar and Wilde (1998) study optimal redistributive tax policyin a model in which incomes are exogenously fixed, so there is no standard labor-leisuredistortion, but are unobservable to the government. The government can choose a taxfunction and a probability of audit function, both based on reported income, and a penaltyfunction based on the amount of income evaded for those detected. Given an upper boundon the penalty function, these authors characterize the properties of optimal policies thatinduce honest reporting of incomes. They find, surprisingly, that for what they regard asreasonable limitations on the size of penalties, the tax structure is non-progressive. Cremerand Gahvari (1996) do a similar exercise for the case in which labor supplies are variable

THE CASE OF TAX EVASION 643

and show how the possibility of evasion imposes an additional constraint on the ability ofgovernments to redistribute.

A conceptual difficulty with these papers is that they need to impose arbitrary limitson the penalty function. In their settings, the restrictions on redistribution caused by taxevasion could be avoided by applying maximal sanctions of the Becker (1968), so there isa need to arbitrarily rule that out.2 In our approach, the penalty function is endogenous:maximal sanctions are avoided by the fact that the innocent will be punished. Nonethe-less, our approach does share one feature with these papers. In our model, as long asthe government is not constrained in its choice of enforcement policies, there is noin-tentional tax evasion in the optimum. That is, tax, penalty and audit policies are setso that all persons honestly report their perceived incomes. But, as Cowell (1990) hassuggested, maximal sanctions cannot be applied because of the possibility of errors, sothe Becker solution, in which evasion can be virtually costlessly eliminated, cannot beachieved. The consequence is that the optimal policy is necessarily second-best. The only‘tax evaders’ are innocent ones, and this serves as a limit on the ability of the governmentto redistribute.3

To illustrate our argument, we construct an extremely simple model consisting of a pop-ulation of households whose taxable incomes can take only two possible values, bothexogenously given. The government is allowed to choose the tax function, the audit proba-bility function and the penalty function freely. This means not only that the government canchoose any degree of severity of punishment, including Becker-type maximum sanctions,but can also reward honest reporting. That is, it can provide a reward for those taxpayerswho have been audited and found to have been telling the truth, implying that they arebetter off than honest taxpayers who have not been audited. The optimality of rewardinghonest taxpayers as part of the government’s redistributive tax and enforcement polices hasbeen established by Mookerjee and Png (1989), and is a component of optimal policiesin Cremer and Gahvari (1996) and Chandar and Wilde (1998). In our model, rewardsfor honest reporting are not only a useful component of optimal policy, they are neces-sary if truthful reporting is to be ensured. In their absence, there is no guarantee that therevelation principle will apply in the optimum, in which case there can be intentional taxevasion.

We consider in sequence each of two cases, that in which evasion is unintentional andthen that in which mistaken conviction occurs. We consider first the case of unintentionalevasion when the government is able to reward honest reporting. Taxpayer error is assumedto occur only for high-income persons. We show that such rewards are necessary for therevelation principle to apply, and that the only taxpayers who are penalized are unintentionalevaders. It is the inability of the government to distinguish unintentional from intentionalevaders that results in the former being punished, and that prevents the government fromachieving its redistributive objectives. We then show that, if it is not feasible to rewardhonest reporting, it is possible that there be intentional alongside unintentional evasion. Inany case, it remains true that unintentional tax evaders must be punished if detected. InSection 3, we consider the case where taxpayer errors occur for low-income taxpayers.Finally, we turn to the case of administrative errors, the analysis of which turns out toparallel the taxpayer error case.


2. The Case of Unintentional Evasion

The economy consists of a large number of persons with identical values of income,Y:adding more income classes of taxpayers would enhance the options for redistribution, butwould not affect the main point of our argument. However, taxable income, which is usedas the basis for tax policy, differs from income. This could reflect the social value that thegovernment places on various components of income. For example, there may be differentcosts associated with earning different types of income, or the same level of true incomemay be earned by taxpayers with different earning abilities. Suppose a proportionα ofthese persons have taxable incomes ofY2 and the remaining 1− α haveY1. We assumeY > Y2 > Y1. Taxable incomes are exogenous, but their values may not be properlyperceived by households when they report them to the tax authorities. The misperceptionmay be due to the complexity or ambiguity of the definition of the tax base, or mistakesin calculating tax liabilities. We suppose in this section that reporting errors only occurfor high-income persons; low-income persons correctly perceive their taxable incomes tobeY1. The presumption is that errors of underestimation are more prevalent than those ofoverestimation, say, because high income earners have more complicated incomes and maybe subject to more complicated forms of preferential treatment.

Let z denote the taxable income that a taxpayer perceives, and suppose thatβ is theprobability that persons with true taxable incomeY2 mistakenly perceives their taxableincome to beY1, where 0< β < 1. Then, denoting true taxable income byy, Pr(z= Y1 |y = Y2) = β and Pr(z = Y1 | y = Y1) = 1. Taxpayers withz = Y2 necessarily have hightaxable income, that is, Pr(y = Y2 | z = Y2) = 1. On the other hand, a proportionµ ofthose withz= Y1 will erroneously perceive themselves to be low-taxable-income persons,whereµ is obtained, using Bayes rule, as follows:

µ ≡ Pr(y = Y2 | z= Y1) = αβ

(1− α)+ αβ (1)

All taxpayers share the same utility function,U (C), which is increasing and strictlyconcave in consumptionC, and satisfiesU (0) = −∞,U ′(0) = ∞and limC→∞U ′(C) = 0.These insure that taxpayers are risk-averse, that the Becker maximal penalty involves drivingconsumption to zero, and that the solution to the government policy problem is always aninterior one.4 A taxpayer’s consumption equals their true incomeY net of taxes and penaltiesor rewards imposed as part of government policy. The government can choose tax rates,audit probabilities, and penalties for misreporting income. We also allow the governmentto reward honest reports, and this turns out to be an important policy instrument.

The amount taxpayers pay to the government in taxes and penalties or rewards dependson the taxable income they report and whether or not they are audited. Letx denote taxableincome reported, wherex = Y1,Y2. If taxable income reported is the same as taxableincome perceived (x = z), we say that the taxpayer has reportedhonestly: this is naturalsince the taxpayer does not know true taxable incomey with certainty so is acting on thebest information available. On the other hand, if the taxpayer reports a different value ofincome than he perceives (x 6= z), he is said to be reportingdishonestly. If it turns out onaudit that reported income differs from true taxable income(x 6= y), that may be either


because the taxpayer whose perceived income differed from true taxable income reportedhonestly(x = z 6= y), or because perceived income and taxable income are the same butthe taxpayer reported dishonestly (x 6= z = y). We refer to the former asunintentionalevasion, and the latterintentional evasion. Since the government has no way of observingperceived taxable income (z), it cannot distinguish intentional from unintentional evasion.This is the source of the possibility that honest tax evaders be prosecuted. It will, however,be able to choose its policies so as to deter intentional evasion.

We assume that the government audits a proportionpi of those reporting incomeYi .For those audited, true taxable incomey is revealed: that is, there are no audit errors, anassumption that is relaxed in a later section. For those who are not audited, the tax liabilitiesare denotedT1 andT2 depending on whether they reported incomeY1 or Y2 respectively.5

Those audited pay a combined tax and penalty/reward ofTi j if they have reportedx = Yi

and have been found to have a true taxable income ofy = Yj . Thus,T12 is the combinedpenalty and tax for a high-taxable-income person who has been audited and found to haveincorrectly reported a low taxable income; a similar interpretation applies toT21. If there isa penalty for misreporting,T12 > T2 andT21 > T1. On the other hand,T11 andT22 are thetax plus rewards for honest reporting. Again, if honest reporting is rewarded,T11 < T1 andT22 < T2. But, we impose no restrictions on either the signs or the magnitudes of penaltiesor rewards. Honest reporting could be rewarded or penalized.

Since taxable income is exogenous, the only decision taken by households is how muchtaxable income to reportx, given their perceived incomez. Consider first the choice of ataxpayer who perceives his taxable income to be high, soz= Y2. This taxpayer necessarilyhas a high taxable income by assumption. If the taxpayer reportsx = Y2 honestly, hispayoff is

E[U | x = z= Y2] = (1− p2)U (Y − T2)+ p2U (Y − T22) (2.1)

On the other hand, if he evades taxes by reportingx = Y1, his expected utility becomes:

E[U | x = Y1, z= Y2] = (1− p1)U (Y − T1)+ p1U (Y − T12) (2.2)

The taxpayer decides whether or not to evade taxes according to the relationship between(2.1) and (2.2). We assume that ifE[U | x = z = Y2] ≥ E[U | x = Y1, z = Y2], thetaxpayer reports his perceived taxable income honestly; otherwise, he under-reports.6

Next consider the taxpayer who perceives a low taxable income,z = Y1. This personcould, in fact, be either a high- or low-taxable-income person. When he reports honestlywhat he perceives his taxable income to be,x = Y1, he will be penalized with probabilityµp1 since his true taxable income isY2 with probabilityµ. Thus, his expected utility isgiven by:

E[U | x = z= Y1] = (1− µ)((1− p1)U (Y − T1)+ p1U (Y − T11))

+ µ((1− p1)U (Y − T1)+ p1U (Y − T12)) (3.1)

Note that, since the government cannot distinguish intentional from unintentional evasion,the same tax and penaltyT12 is imposed here as in (2.2), where evasion is intentional. When


this taxpayer over-reports his taxable income (that is,x = Y2), expected utility becomes:

E[U | x = Y2, z= Y1] = (1− µ)((1− p2)U (Y − T2)+ p2U (Y − T21))

+ µ((1− p2)U (Y − T2)+ p2U (Y − T22)) (3.2)

Again, we assume that the taxpayer reports truthfully ifE[U | x = z= Y1] ≥ E[U | x =Y2, z= Y1]; otherwise he over-reports.

Given that there are two values for perceived taxable incomes, and taxpayers may reporthonestly or dishonestly, there are four potential combinations of reporting strategies thanmight occur in equilibrium, corresponding with the relative values of (2.1) versus (2.2) and(3.1) versus (3.2). The four cases are the following:

Case i) both types of taxpayers report truthfully;

Case ii) both reportx = Y1;

Case iii) both reportx = Y2; and,

Case iv) both report dishonestly.

The government is assumed to maximize an objective function which incorporates somedesire to redistribute income from those with higher taxable incomes to those with lowertaxable incomes. Recall that the government cares about taxable income rather than trueincome as discussed above. LetW(Yi − Ti ) be the social utility that the governmentattaches to a taxpayer with after-tax taxable incomeYi − Ti , whereW′(·) > 0 > W′′(·)andW′(0) = ∞. We take the government’s objective function to be the sum of these socialutilities over all households, though any quasi-concave objective function would serve aswell.7 The objective function of the government is therefore:

Ä = (1− α)((1− p1)W(Y1− T1)+ p1W(Y1− T11))

+ αβ((1− p1)W(Y2− T1)+ p1W(Y2− T12))

+ α(1− β)(p2W(Y2− T22)+ (1− p2)W(Y2− T2)) (4)

Before proceeding with the analysis of government policy, it is worth summarizing theorder in which events occur and information becomes revealed to the individuals and thegovernment:

1. Individuals observe their perceived incomes (z= Y1,Y2), which may differ from theirunobserved true taxable incomesy.

2. Individuals report their taxable incomes either honestly (x = z) or dishonestly (x 6= z).

3. The government randomly audits a proportionpi of individuals who reportx = Yi ,i = 1,2. This reveals true taxable incomey = Y1,Y2 for those audited.

4. Taxes and penalties are applied.


It turns out that in this section,when rewards for honest reporting are allowed, therevelation principle applies in the sense that the tax/audit/penalty scheme can be designedso that case i) necessarily applies in the optimum. However, this depends critically on thefeasibility of rewards for honest reporting. As we shall see after we establish our mainproposition, if such awards are not allowed, the revelation principle breaks down and it ispossible that allowing some dishonest reporting will be optimal. The following lemma,summarizes this.

Lemma 1 When there are no restrictions on the values of Ti , Ti j , i, j = 1,2, optimalpolicy can be implemented by inducing truthful reporting by all taxpayers.

The proof of this lemma, along with our other main results, can be found in the Appendix.Given Lemma 1, we can impose as self-selection constraints the requirement that taxpay-

ers prefer to report truthfully:

E[U | x = z= Y2] ≥ E[U | x = Y1, z= Y2];E[U | x = z= Y1] ≥ E[U | x = Y2, z= Y1]

where the expressions for expected utility are given by (2.1), (2.2), (3.1) and (3.2). It isimmediate to see that the second constraint is irrelevant in designing policy since it canalways be relaxed by imposing a high penalty on the over-reporting of taxable income bythose withz = Y1, say,T21 = Y. Thus, a taxpayer perceivingz = Y1 can be inducedto reportY1 honestly despite the fact that with probabilityµp1, he will be penalized forunintentionally evading taxes. Moreover, it can be shown that, given that all those whoperceive their taxable incomes to be high are truly high-taxable-income persons, there shouldbe no rewards for honest reporting at the top (T22 = T2).8 The self-selection constraint forthe high-taxable-income taxpayers, therefore, can be reduced to:

U (Y − T2) ≥ (1− p1)U (Y − T1)+ p1U (Y − T12) (5)

Finally, given that there is no reward at the top, it is optimal to make the probability of theauditing at the top as low as possible. Given that we are settingT21 at its maximal value(sayT21 = Y), the self-selection constraint on truthful reporting for those withz= Y1 willbe satisfied ifp2 is indefinitely small. Accordingly, we can letp2 → 0, and ignore it inwhat follows.

To summarize, the government can design an optimal tax/auditing/penalty/reward schemeso that it induces truth-telling. This implies that there is no intentional tax evasion in theoptimum. But because of the presence of taxpayer error, there always exist taxpayers whocommit evasion unintentionally, even though they are honestly reporting their perceivedincomes. Ideally, of course, such unintentional evasion should go unpunished. But, the taxauditor cannot distinguish intentional from unintentional evasion, so a penaltyT12 must beimposed on all taxpayers for whom under-reporting is discovered. In the optimum, this willonly include unintentional evaders. Figure 1 summarizes the payoff structure for taxpayersin the model of this section.

Consider now the policy problem facing the government. It chooses the tax ratesT1 andT2, the audit probabilityp1, the tax-cum-penalty rateT12 and the tax-cum-reward rateT11.


Figure 1.

Let c(p1) be the auditing cost per taxpayer reportingY1, wherec(p1) is strictly convex withc(0) = c′(0) = 0 andc′(1) = ∞. Then, the budget constraint faced by the government is:

(1− α)((1− p1)T1+ p1T11)+ αβ((1− p1)T1+ p1T12)+ α(1− β)T2

−(1− α + αβ)c(p1) = G (6)

whereG is a given revenue requirement. Since we have assumed there are a large numberof taxpayers, the government faces no uncertainty: the probability of taxpayers being in agiven category is the same as the aggregate proportion of them who are. The left-hand siderepresents tax revenues net of auditing costs. The proportion of taxpayers reporting a lowincome is 1− α + αβ, of whomαβ are unintentional tax evaders.

To contrast our argument with other models, it is helpful to consider the situation wherethere are no errors involved (i.e.,β = 0) although such a case is well-known:

Proposition 1 If β = 0, T12→ Y , p1→ 0, T11→ T1 and Y1− T1→ Y2− T2.

This is simply the ‘Becker solution,’ which approaches arbitrarily close to the first-bestallocation with full equality of after-tax income for all households.9 Proposition 1 confirmsthat, in the absence of taxpayer errors, the possibility of tax evasion does not constrainthe ability of the government to redistribute incomes provided it can impose the maximal


sanction. The reason is obvious: imposing a penalty incurs no social costs. Note also thatthere is no need to reward honest reporting in this extreme case.

Let us return to the original case. The government problem is to maximize (4) subjectto the incentive constraint (5) and the budget constraint (6). Making use of the first-orderconditions shown in the Appendix, we can characterize the optimal policy as follows:

Proposition 2 When there are errors in perceiving taxable income by high-taxable-incometaxpayers (Pr(z = Y1 | y = Y2) > 0) and rewarding honest reporting is feasible, optimalpolicy satisfies the following properties:

i) In the optimum, all taxpayers report their perceived taxable incomes truthfully, so thereis no intentional evasion;

ii) the tax is redistributive: T2 > T2;

iii) there is a reward at the bottom and the penalty is not maximum: T11 < T1, Y > Y2 >

T12 > T2;

iv) the full-information egalitarian outcome is not achieved: Y1−T1 < Y1−T11 < Y2−T2;

v) A high-taxable-income taxpayer found to be evading unintentionally is worse off than alow-taxable-income taxpayer who is rewarded for honest reporting: Y2−T12 < Y1−T11.

In words, when errors exist in the perception of taxable income by high-income persons,the tax, audit and penalty structure should be set such that there is no intentional evasion.Nonetheless, there will be unintentional evasion by those who misperceive their true taxableincomes. These taxpayers must be penalized if detected (i.e.,T12 > T2) as a means ofdeterring intentional evasion. Of course, the penalty will not be maximal and the probabilityof detection will not be minimal either (T12 < Y, p1 > 0), contrary to the Becker solution:this would entail too great a cost in terms of social welfare, given that the innocent arepunished. Thus, inducing truthful reporting requires either devoting sufficiently a largeamount of resources to auditing or making tax evasion less attractive. The former raises theaudit probabilityp1, while the latter involves increasingT1 and reducingT2. The rewardfor the verification of honest reporting is an essential feature of optimal tax scheme in thepresent context:T11 < T1. Through auditing, the government can find truly low-taxable-income persons, and, given its redistributive objective, it would prefer to lower their taxliability by rewarding honest reporting. WhileT1 is paid by some high-taxable-incomepersons who misperceive their taxable incomes and luckily are not audited, there is nopossibility thatT11 accrues to high-taxable-income persons. Nor does the reward affect theincentive constraint.10

Overall, the tax system is less progressive than it would be if taxable incomes wereperfectly observable, sinceY1 − T1 < Y2 − T2: the after-tax taxable income distributiondeviates from the first-best utilitarian outcome. Note, however, that the relationship betweenY1 − T1 and Y2 − T12 is ambiguous: unintentional evaders who are convicted may bebetter or worse off than low-taxable-income taxpayers who are not audited. The ambiguitypartly arises from the fact thatT1 applies not only to low-taxable-income persons, but


also to a proportionαβ(1− p1) of the high-taxable-income persons, which makes incomeredistribution through the tax system less valuable. In addition,T1 andT12 influence theincentive constraint (5) differently. On the other hand, we can obtain clear-cut relationshipbetweenY1− T11 andY2− T12 according to result v) in the Proposition. The high-taxable-income persons who are caught are worse off than the low-taxable-income ones who areaudited and rewarded.

Finally, note that an increase in the audit probabilityp1 induces both gains and costs, thelatter of which would not occur in the presence of taxpayer error. The costs involve notonly the cost of auditing (which would be insignificant in a model without errors becausep1 would be set as low as possible) but also the reduction in social welfare resulting fromthat fact that innocent taxpayers must be penalized because of their unintentional mistakes.On the other hand, the gain includes the fine revenue from the penalties and utility gain oftruly low-taxable-income taxpayers. These offsetting factors imply thatp1 is chosen so asto balance the gains with the costs at the margin.

For this solution to be implemented, the tax authorities must be able to commit to pun-ishing innocent tax evaders. Since reporting is honest, all tax evasion in the optimum isunintentional. An obvious credibility problem arises. Having found tax evaders and know-ing them to be honest, the tax authorities may not be able to avoid being lenient. In theextreme, if the government will not impose a penalty on innocent tax evaders, avoidingintentional evasion becomes impossible, and the government will be unable to implementany redistribution.11 More generally, there will be a trade-off between the severity of pun-ishment imposed on the innocent and the extent of redistribution that is achieved in theoptimum.

The above analysis establishes that when there are no constraints on enforcement poli-cies, the optimal solution requires that there should be rewards for honest reporting (as inMookerjee and Png, 1989). Given that such rewards are often ignored in the tax evasionliterature and are not used in practice, it is useful to consider the case where they are ruledout. To investigate the implications of there being no rewards for honest reporting, weimpose the conditionTii = Ti on the problem of the previous section. We assume thatTi j

for i 6= j can take on any value. That is, penalties can be positive or negative, and they canapply for either under- or over-reporting. Otherwise, the same assumptions apply.

When rewards for honest reporting are no longer feasible, the optimal policy can changedramatically. We can no longer ensure that the revelation principle applies. As the followingProposition shows, there are now three possible outcomes, only one of which involvestruthful reporting and the other two of which do not.

Proposition 3 When Tii = Ti is imposed, there are three possible cases in the optimum:

Case i) all taxpayers report honestly (x= Yi if z = Yi );

Case ii) all taxpayers report x= Y2;

Case iii) all taxpayers report dishonestly (x= Yi if z = Yj , i 6= j ).

The payoff structures for these three cases are shown in the three panels of Figure 2. Case i)is the standard case in which all taxpayers report truthfully, so the only tax evasion in the


Figure 2.

optimum is unintentional. In this case, the tax/penalty/auditing policy satisfies the followingproperties:T1 < T2 < T12 < Y2, Y2− T2 > Y1− T1 andp1 > 0. The difference is that theinability to reward honest reporting imposes a further constraint on redistribution. Case ii)suggests that it may be optimal to induce taxpayers who observe a low taxable income(z= Y1) to over-report their income to the tax authorities (x = Y2). It is apparent thatT21

must be less thatT2 because of the government’s redistributive objective. Redistributiontakes place entirely through the favorable value ofT21. There now needs to be auditingat the top (p2 > 0). In this case, obviously, the optimal tax/auditing scheme does notcause self-screening by taxpayers. The screening occurs only through auditing. Case iii)goes further than this. In addition to inducing those withz = Y1 to over-report, it inducesthose withz= Y2 to under-report. It does this by making the tax payment for low-taxable-


income persons,T1, sufficiently low relative the tax-sum-penalty if caught,T12, but not lowenough relative toT21 to induce those who observe low taxable incomes to report truthfully.Now, the penalty implicit inT12 (> T2) becomes part of the redistributive policy alongsideT21 (< T1). The former redistributes from high-taxable-income persons, while the latterredistributes to low-taxable-income persons.

Any one of these three cases can be globally optimal depending on the parameter values.12

And, each will be inferior to the general case which imposed no restrictions on the penaltiesor rewards. The three cases, however, may not be equally palatable in the following sense.Though case iii) does not reward honest behavior, it does reward over-reporting by dishonesttaxpayers withz = Y1 who turn out to truly low-income. Societal norms may precluderewarding dishonest behavior. IfT21 < T1 is ruled out, so is case iii). Furthermore, case ii)could also be interpreted as rewarding dishonest over-reporting. Taxpayers withz = Y1

who over-report will incur a lower tax-cum-penalty if they are audited than if they are not(T21 < T2). (Recall thatT1 is not relevant in case ii).) If this is ruled out, we are left withcase i), and the revelation principle is again obtained.

4. Errors by Low-Income Taxpayers

In the previous model, only high-taxable-income earners were subject to taxpayer errors.In this section, we suppose that it is low-taxable-income earners who are subject to errorsin perceiving their true taxable incomes. Letβ again designate the probability of taxpayersmaking mistakes, here the probability that those low-taxable-income persons mistakenlyobservez= Y2: β = Pr(z= Y2 | y = Y1). For simplicity, we assume that no errors occurfor high-taxable-income persons (Pr(y = Y1 | z = Y1) = 1). The proportion of personsobservingz= Y2 who are actually low-taxable-income persons is given by:

µ ≡ Pr(y = Y1 | z= Y2) = (1− α)β(1− α)β + α

Consider the choice of taxpayers perceiving their taxable incomes to be highz = Y2.They can report truthfully, or under-report their taxable incomes. Contrary to our previousmodel, the present case does not rule out the necessity of auditing taxpayers reportingY2

for the following reason. When the government attempts to induce truthful reporting oftaxpayers (that is, inducex = Y2 for those withz = Y2), auditing at the top will serve asa device to discover taxpayer mistakes. This mitigates the risk of declaringY2 when thetaxpayer is not sure about the true value of his taxable income. Once the error is discovered(so the taxpayer reportingY2 is found to be a low-taxable-income person),T21 is imposedwhen over-reporting is verified:T21 can be either greater or less thanT1. As before, weallow the government to award the honest reporting, soT22 is levied instead ofT2 if theaudited taxpayer’s true taxable income is found to be high as reported. Note again that thegovernment cannot distinguish intentional from unintentional tax evasion. So, the penaltyis imposed according to discovered true taxable income irrespective of the intention of thetaxpayers.

Due to the feasibility of rewards, it can be shown that Lemma 1 is relevant in the presentcontext, so we can restrict our attention to the case of truth telling.13 Note, however, that


when all taxpayers report their perceived incomes honestly, the tax plus penalty for under-reporting,T12, is not paid in the optimum. This in turn implies that in contrast to the casewhere the perception errors occur at the top, there is no social cost associated with raisingthe penalty rate. Thus, the Becker solution applies:T12 can be arbitrarily high for any valueof p1. The self-selection constraint for the person perceiving a high taxable income is noteffective: only the self-selection constraint for a truly low-taxable-income person is a con-cern. Also, the concavity of the utility function along with the utilitarian objective impliesthat there should be no reward at the bottom:T11 = T1.14 Since we can letT12 = Y andT11 = T1, p1 can be set arbitrarily low:p1→ 0. Therefore, the effective policy instrumentsareT1, T2, T22, T21 and p2. The following proposition characterizes the optimal policy.

Proposition 4 When there are errors in perceiving taxable income by low income taxpayers(Pr(z = Y2 | y = Y1) > 0) and rewarding honest reporting is feasible, optimal policysatisfies:

i) the self-selection constraint for a low income person is not binding;

ii) the tax is redistributive: T2 > T2;

iii) there is no penalty for over-reporting (T21 = T1), and honest reporting by a truly highincome person is penalized (T22 > T2).

The presence of possible tax evasion does not limit the ability of the government to re-distribute income because the incentive constraint is not binding. The problem is, however,that sinceT2 accrues to some truly low-taxable-income persons reporting a high tax lia-bility, the government is constrained from setting it too high. In addition, the governmentconducts auditing at the top not to deter intentional tax evasion, but to detect unintentionalover-reporting of income by low-taxable-income taxpayers. It might seem surprising thathonest reporting by high-taxable-income persons should be penalized. The reason is thatauditing at the top enables the government to identify truly high-taxable-income persons. Asmentioned,T2 is paid by both low- and high-taxable-income persons, soT22 needs to be in-creased above that to make the tax structure progressive. This, of course, might be seen to beparticularly unfair and may be ruled out on those grounds. If so, the revelation principle willagain fail so that both intentional and unintentional tax evasion can occur in the optimum.

5. The Case of Auditing Errors

So far we have considered errors by the taxpayers. As noted by Stern (1982) and Cowell(1990), there may also be errors by the tax authorities. In this section, we abstract fromreporting errors by the taxpayers, and assume that the only errors are those made (uninten-tionally) by tax auditors. To make matter simple, we now assume that taxpayers correctlyobserve their taxable incomes. It is now the government that may misperceive incomes.The auditing of persons reporting low incomes may result in type I errors, where these per-sons are actually low income but are mistakenly identified as high income, or type II errors,where tax evaders with high incomes are not identified as such. In this section, we restrict


ourselves to the former case, since this is the case which gives rise to the prosecution of aninnocent person. It might plausibly occurs in situations where the measurement of taxableincome is difficult, or where the tax auditor does not have enough resources to conductcareful audits.

Let y (= Y1,Y2) now denote true taxable income,x be income reported by the taxpayer,andzbe the income observed by an auditor. As before, the probability of auditing taxpayerswho reportYi is pi , which carries a cost per person reportingYi of c(pi ). The probabilityof a mistake in auditing isβ, which here is the probability that the auditor observesY2 froma low-income taxpayer, soβ = Pr(z = Y2 | x = y = Y1). The tax payments for personswho are not audited areT1 andT2, while the taxes net of penalties (or rewards) for thoseaudited areTi j when a taxpayer reportingYi is audited and the tax collector perceives hisincome asYj .

Again, Lemma 1 applies, so our attention can be restricted to the truth-telling scheme.The self-selection constraints become:

p1([βU (Y − T12)+ (1− β)U (Y − T11)] + (1− p1)U (Y − T1)

≥ p2[βU (Y − T22)+ (1− β)U (Y − T21)] + (1− p2)U (Y − T2) (7)

p2U (Y − T22)+ (1− p2)U (Y − T2) ≥ p1U (Y − T12)+ (1− p1)U (Y − T1) (8)

where as beforeY is the taxpayers actual income, which can differ from taxable income.15

The left-hand side of (9.1) reflects the fact that, when a low-income taxpayer reports honestly,he is audited with probabilityp1 and penalized with probabilityp1β because the tax auditormistakenly observes a high income. If there is no auditing error, the taxpayer obtainsT11,which can differ fromT1: honest reporting can be rewarded. The right-hand side states thatif the low-income person over-reports his tax liability, he is audited with probabilityp2,and, with probabilityp2β, he is considered to be a high-income taxpayer and is mistakenlyrewarded (i.e.,T22 is imposed). As shown in (9.2), since there are no errors in auditinghigh-income persons (that is, no type I errors),β does not appear in the incentive constraintof a high-income taxpayer.

As in Section 2, condition (9.1) can be always relaxed by raisingT21 sufficiently high (sayT21 = Y). Thus, in the optimum, low-income taxpayers will be induced be report truthfullydespite the possibility of being convicted of tax evasion through auditing error. In practice,the taxpayer might be able to appeal the auditor’s decision and, with some probability,be found innocent. If so, he has a chance of avoiding the penalty, although at some costof litigation. Our model could be extended to such a circumstance. Also note that theconcavity of the utility function along with the redistributive objective of the governmentimplies that there should be no reward at the top:T22 = T2. (The proof proceeds the sameway as our basic model.) So, we can letp2 be arbitrarily low, which allows us to omit it asa policy instrument.

Given that both low- and high-income taxpayers report their incomes truthfully, the gov-ernment’s problem becomes:

max{T1,T2,T12,T11,p1}

(1− α)[(1− p1)W(Y1− T1)

+ p1βW(Y1− T12)+ p1(1− β)W(Y1− T11)] + αW(Y2− T2)


subject to the incentive and revenue constraints:

U (Y − T2) ≥ (1− p1)U (Y − T1)+ p1U (Y − T12) (γ )

(1− α)[(1− p1)T1+ p1βT12+ p1(1− β)T11] + αT2− (1− α)c(p) = G (λ)

From the first-order conditions shown in the Appendix, optimal policy can be characterizedas follows.

Proposition 5 If there are type I errors in auditing taxpayers reporting low incomes (Pr(z=Y2 | x = y = Y1) > 0), then when tax system is redistributive in the optimum (T1 < T2),optimal policy satisfies the following properties:

i) there is no intentional evasion;

ii) there is a reward at the bottom and the penalty is not maximum: T11 < T1, T2 < T12 <

Y1 < Y ;

iii) the full-information egalitarian outcome is not achieved: Y1−T1 < Y1−T11 < Y2−T2.

Thus, as in Section 2, any evasion detected reflects an honest mistake, this time by theauditor. Despite this, the innocent (low-income) taxpayers who are erroneously thoughtto be evading must be punished when the income tax is redistributive (T1 < T2). This isnecessary in order to deter evasion by high-income persons. Since the penalty cannot bemaximal, the ability of the government to tax and transfer is limited. Again, a credibilityproblem will arise: the government may not be able to commit itself to punishing theinnocent, or there may be an objection against this idea among the public because ofsocial norms. In the limit, if innocent taxpayers cannot be punished at all (T12 = T2), noredistribution can be achieved by the tax system (T1 = T2).

It should be noted that the results of this section rely on the government adopting autilitarian objective function. By Proposition 5,T12 > T2 > T1, so the social utility of thelow-income taxpayers convicted in error,W(Y1− T12), is less than what they would obtainin the no-redistribution case (T12 = T2 = T1). This implies that objective functions withhigh enough aversions to utility inequality would prefer no redistribution. For example, ifthe objective function were maxi-min, no redistribution would be tolerated in the presenceof auditing errors.

10. Conclusions

We have shown that taxpayer errors and auditing errors in an income tax system have severalimplications for the ability of the tax system to achieve the government’s redistributiveobjectives and for the policies used to enforce tax compliance. Even if there are noa prioriconstraints on the audit and penalty structure, maximal penalties of the Becker (1968)sort will not be optimal. The optimal tax and enforcement policy will be one in whichall taxpayers report honestly. This will involve both penalizing unintentional tax evasion,whether it results from taxpayer error or auditing error, and rewarding honest reporting.


Moreover, redistribution will be significantly limited relative to the no-error case. In thelatter, a first best can be achieved by imposing maximal sanctions. If rewards for honestreporting are ruled out, it may be optimal to induce dishonest reporting in the optimum,either by those perceiving low incomes or by all taxpayers. But, even in this case, it is stillnecessary to punish unintentional tax evasion.

The model we have used is an extremely simple one, and avoids a number of issues that inthe real world may be relevant. The most obvious one is the credibility problem that wouldinevitably arise if the government tried to impose the optimal truth-telling scheme. If it isknown that the only tax evaders are innocent ones, enforcing a penalty on them might bedifficult. And, if such penalties cannot be credibly imposed, the ability to avoid dishonestreporting and therefore to redistribute will be jeopardized.

We have also avoided considering other possible remedies to the existence of errors. Onepractical response to adminstrative errors might be to institute a system of appeals, as isthe case in actual tax administrations. Persons who learn that they have been wrongfullyconvicted might, by way of appeal, be able to direct more resources to verifying the resultsof audits. Though costly, this may serve to mitigate the adverse consequences of wrongfulconviction. By the same token, if penalizing honest mistakes is unavoidable, taxpayersmight take measures to reduce the possibility of unintentional reporting errors. For example,they might hire tax accountants or lawyers to advise them on filing their tax returns. Giventhat taxpayer error is most likely to arise as a result of complications in defining andmeasuring taxable income, perhaps the best response is to design the tax system to be assimple and transparent as possible.

Appendix

Proof of Lemma 1

In what follows, we show that for any policy(Ti , pi , Ti j ) (i, j = 1,2) that does not inducetruth-telling, there exists an equivalent policy that does. There are three possible cases.

i. Both report x= Y1. We have

E[U | x = z= Y2] < E[U | x = Y1, z= Y2];E[U | x = z= Y1] ≥ E[U | x = Y2, z= Y1]

where these expressions are given by (2.1), (2.2), (3.1) and (3.2) in the text. Note that(1− p1)T1 + p1T12 is the average tax revenue from a person perceiving a high taxableincome, who here reports a low taxable income (x = Y1, z = Y2). Since all taxpayersreportx = Y1, the audit cost per taxpayer becomesc(p1). Define an alternative policy by(T ′i , p′i , T

′i j ), (i, j = 1,2) such that:

T ′1 = T ′2 = T1; T ′11 = T ′21 = T11; T ′22 = T ′12 = T12; p′1 = p′2 = p1


Under the new policy,E[U | x = z = Y1] = E[U | x = Y2, z = Y1] and E[U | x = z =Y2] = E[U | x = Y1, z= Y2], so truth-telling occurs without changing the expected utilityof either type of taxpayer, tax revenues or the auditing cost.

ii. Both report x= Y2. Here we have

E[U | x = z= Y2] ≥ E[U | x = Y1, z= Y2];E[U | x = z= Y1] < E[U | x = Y2, z= Y1]

The average tax revenue from a person who perceives a low taxable income but reportsa high one (x = Y2, z = Y1) is (1− α)((1− p2)T2 + p2T21) + α((1− p2)T2 + p2T22).Since all taxpayers reportx = Y2, the audit cost per taxpayer becomesc(p2). Define analternative policy(T ′i , p′i , T

′i j ), (i, j = 1,2), by:

T ′1 = T ′2 = T2; T ′11 = T ′21 = T21; T ′22 = T ′12 = T22; p′1 = p′2 = p2

Again, the new policy induces truth-telling without changing the expected utility of anytaxpayer, tax revenues or auditing costs.

iii. Both report dishonestly.In this case, we have:

E[U | x = z= Y2] < E[U | x = Y1, z= Y2];E[U | x = z= Y1] < E[U | x = Y2, z= Y1]

The average tax revenue from a person perceiving high taxable income is(1−p1)T1+p1T12,while that from a person perceivingY1 is (1− α)((1− p2)T2+ p2T21)+ α((1− p2)T2+p2T22). In this case, the audit cost per taxpayer is((1− α)+ αβ)c(p2)+ α(1− β)c(p1).Let the alternative policy beT ′1 = T2, T ′2 = T1, T ′12 = T22, T ′21 = T11, T ′11 = T21,T ′22 = T12, p′1 = p2 and p′2 = p1. The new policy replacesE[U | x = Yj , z = Yi ]by E[U | x = z = Yi ], (i = 1,2, i 6= j ). Thus, the new policy induces the truth-telling without changing the expected utility of any taxpayer, tax revenues or the auditingcosts.

Proof of Proposition 2

The Lagrangian expression is

L = (1− α)((1− p1)W(Y1− T1)+ p1W(Y1− T11))

+ αβ((1− p1)W(Y2− T1)+ p1W(Y2− T12))

+ α(1− β)W(Y2− T2)+ γ [U (Y − T2)− (1− p1)U (Y − T1)− p1U (Y − T12)]

+ λ[(1− α)((1− p1)T1+ p1T11)

+ αβ((1− p1)T1+ p1T12)+ α(1− β)T2− (1− α + αβ)c(p1)− G]


The first-order conditions are as follows:

−(1− α)W′(Y1− T1)− αβW′(Y2− T1)+ γU ′(Y − T1)+ λ((1− α)+ αβ) = 0 (T1)

−α(1− β)W′(Y2− T2)− γU ′(Y − T2)+ λα(1− β) = 0 (T2)

−αβW′(Y2− T12)+ γU ′(Y − T12)+ λαβ = 0 (T12)

−W′(Y1− T11)+ λ = 0 (T11)

(1− α)(W(Y1− T11)−W(Y1− T11))− αβ(W(Y2− T1)−W(Y2− T12))

+γ (U (Y − T1)−U (Y − T12))

+λ[(1− α)(T11− T1)+ αβ(T12− T1)− (1− α + αβ)c′(p1)] = 0 (p1)

Result i) follows from Lemma 1. To establish the other results, note first the followinglemma:

LEMMA A.1 T12 > T2 if γ > 0, and T12 = T2 if γ = 0.

Proof: Rearranging(T2) and(T12) gives:

α(W′(Y2− T12)−W′(Y2− T2)) = γU ′(Y − T2)

1− β + γU ′(Y − T12)

β≥ 0

The strict inequality holds iffγ > 0. The desired result is immediate.

Proof of Result ii): SupposeT1 ≥ T2. First, it can be shown thatγ = 0. If not, then byLemma A.1,T12 > T2. SinceT1 ≥ T2 andT12 > T2, U (Y − T2) > (1− p1)U (Y − T1)+p1U (Y − T12), implying γ = 0, which is a contradiction. Becauseγ = 0, T2 = T12 andW′(Y2− T2) = λ. Substituting this into the first-order condition forT1 yields:

(1− α)(W′(Y2− T2)−W′(Y1− T1))+ αβ(W′(Y2− T2)−W′(Y2− T1)) < 0

which is a contradiction.

Proof of Results iii) and iv): Supposeγ = 0. By Lemma A.1,T12 = T2. SinceT1 < T2,

U (Y − T2) < (1− p1)U (Y − T1)+ p1U (Y − T2)

Thus (4) is violated, which is a contradiction. Therefore,γ > 0, which leads toT12 > T2.Also, T12 < Y2, so the penalty cannot be maximal because the expected utility of taxpayerswho are penalized is included in social welfare: there is a social cost associated withpenalizing evaders. By rearranging the first-order condition forT1 and usingγ > 0,

(1− α)W′(Y1− T1)+ αβW′(Y2− T1)

(1− α)+ αβ > λ


The left-hand side is less thanW′(Y1 − T1) and thusW′(Y1 − T1) > λ = W′(Y1 − T11),which yieldsT1 > T11. On the other hand,W′(Y2−T2) < λ = W′(Y1−T11). These implyY1− T1 < Y1− T11 < Y2− T2.

Proof of Result v): Sinceγ > 0, W′(Y2 − T12) > λ = W′(Y1 − T11) from the first orderconditions. Then the desired result is immediate.

Proof of Proposition 3: We can eliminate the case where both reportx = Y1. Supposeboth reportx = Y1. Given this policy,T2 is not effective. DefineT ′2 so thatU (Y −T ′2) = (1− p1)U (Y − T1)+ p1U (Y − T12). The concavity of the utility function impliesT ′2 > (1 − p1)T1 + p1T12 where the right-hand side represents per capita tax revenuefrom a truly high-taxable-income person. Thus, inducing the taxpayers withz= Y2 reporttruthfully raises the tax revenue without affecting their utility and without violating anyincentive constraint.

Proof that Both Cannot Report Dishonestly if Penalty is Non-Negative:Suppose bothreport dishonestly. Given such a policy, the following should be satisfied:

(1− µ)((1− p2)U (Y − T2))+ p2U (Y − T21))+ µU (Y − T2)

≥ (1− µ)U (Y − T1)+ µ((1− p1)U (Y − T1)+ p1U (Y − T12)) (A1)

and

(1− p1)U (Y − T1)+ p1U (Y − T12) ≥ U (Y − T2) (A2)

If the second equation is binding, as shown above, it is Pareto-improving to induce thosewho perceive high taxable incomes to reportY2, so this case cannot be optimal. Supposethat it is not binding. From the first equation, this implies:

(1− p2)U (Y − T2))+ p2U (Y − T21) > U (Y − T1) (A3)

SinceT21 ≥ T1, the above in turn impliesT2 < T1. BecauseT12 ≥ T2, then we obtain

(1− p1)U (Y − T1)+ p1U (Y − T12) < U (Y − T2)

which is a contradiction. Thus, this case is either non-optimal or non-feasible.

Proof of Proposition 4: The Lagrangian expression for the government problem is:

L = (1− α)[(1− β)W(Y1− T1)+ β((1− p2)W(Y1− T2)+ p2W(Y1− T21))]

+ α((1− p2)W(Y2− T2)+ p2W(Y2− T22))

+ γ [U (Y − T1)− (1− p2)U (Y − T2)− p2U (Y − T21)]

+ λ[(1− α)((1− β)T1+ p2βT21)+ (1− p2)((1− α)β + α)T2+ αp2T22− G]


First, we ignore the incentive constraint of truly low-taxable-income persons, and show thatit is not binding in the optimum. The first-order conditions are:

W′(Y1− T1) = W′(Y1− T21) = W′(Y2− T22) = λ(1− α)βW′(Y1− T2)+ αW′(Y2− T2) = λ((1− α)β + α)

From the first equation,T1 = T21, so there is no penalty for over-reporting. The secondequation leads toW′(Y1−T2) > λ, which impliesT2 > T1. Also, using the same equation,W′(Y2 − T2) < λ = W′(Y2 − T22). Thus, we obtainT2 < T22. Finally, sinceT1 < T2 andT21 = T1,

U ′(Y − T1) > (1− p2)U (Y − T2)+ p2U (Y − T21),

so the self-selection constraint is not binding.

Proof of Proposition 5: The first order conditions forT1, T2, T12, T11 and p1 are:

−(1− α)W′(Y1− T1)+ γU ′(Y − T1)+ λ(1− α) = 0 (T1)

−αW′(Y2− T2)− γU ′(Y − T2)+ λα = 0 (T2)

−(1− α)βW′(Y1− T12)+ γU ′(Y − T12)+ λ(1− α)β = 0 (T12)

−W′(Y1− T11)+ λ = 0 (T11)

(1− α)[−W(Y1− T1)+ βW(Y1− T12)+ (1− β)W(Y1− T11)]

+γ (U (Y − T1)−U (Y − T12))

+λ(1− α)(βT12+ (1− β)T11− T1− c′(p1)) = 0 (p1)

First note that the self-selection constraint must be binding in the optimum (γ > 0). Supposenot. Then the utilitarian first-best outcome is achieved:Y1− T1 = Y1− T11 = Y1− T12 =Y2 − T2, and p1 = 0 (see the first-order conditions). AsT2 > T1 = T12, this violates theself-selection constraint,U (Y − T2) < U (Y − T1), which is a contradiction.

Using γ > 0, it can be shown that either i)T1 < T2 < T12, or ii) T12 < T2 < T1 ispossible. First, supposeT12 > T1. Then the binding incentive constraint impliesT1 < T2.Also, if T2 > T12 the constraint is violated, so case i) results. Second, supposeT12 < T1.Again, the binding incentive constraint rules out the situation such thatT2 > T1 ≥ T12 andT1 ≥ T12 > T2, so case ii) results.

We still need to eliminate the case whereT1 = T2 = T12. It can be established thatstarting fromT = T1 = T2 = T12 and p1 > 0, an increase inT12 followed by an increasein T2 and a reduction inT1 raises the welfare. Any changes inT1, T2, andT12 must satisfythe incentive and budget constraints:

U (Y − T2) = (1− p1)U (Y − T1)+ p1U (Y − T12) (A4)

(1− α)[(1− p1)T1+ p1βT12+ p1(1− β)T11] + αT1− (1− α)c(p1) = G (A5)


Differentiating these two constraints with respect to tax changes, solving fordT1 anddT2,and evaluating atT = T1 = T2 = T12 yields:

dT1

dT12= − p1

1− p1(α + β(1− α)) < 0; dT2

dT12= p1(1− α)(1− β) > 0 (A6)

Let T21 = Y1, so we can ignore the incentive constraint for a low-income person. Differen-tiating the objective functionÄ(·) with respect to tax changes yields:

dÄ

dT12= α(1− α)p1(1− β)(W′(Y1− T)−W′(Y2− T)) (A7)

An increase inT12 starting at the pooling scheme improves welfare, which leads to thedesired result.

Thus, the optimum must be one of two cases: i)T1 < T2 < T12, or ii) T12 < T2 < T1. Theabove proof shows that starting from the pooling scheme, the policy change in the directionof case i) (that is,T1 < T2 < T12) is welfare-improving although we cannot rule out case ii)in general. In Proposition 5, we focus on the case that tax system is redistributive, so case i).

Proof of Result ii): T12 < Y1 (< Y2) is obvious because the welfare of the taxpayerswho are penalized due to the auditing error,W(Y1 − T12), is included in the government’sobjective (See equation (T12)). From equations(T1) and(T11),

W′(Y1− T1) = λ+ γ

1− αU ′(Y − T1) > λ = W′(Y1− T11), (A9)

which leads toT1 > T11. As our focus is on case i), we haveT11 < T1 < T2 < Y2.

Proof of Result iii): Rearranging the first-order condition forT2 yields:

W′(Y2− T2) = λ− γα

U ′(Y − T2) < λ (A10)

Along with (A9), we can establish the desired result.

Acknowledgments

Financial support of the Social Sciences and Humanities Research Council of Canada isgratefully acknowledged. Comments of two anonymous referees were very valuable.

Notes

1. A representative set of applications include the choice of audit and enforcement strategy (Reinganum and Wilde,1985; Cremer, Marchand and Pestieau, 1990), the effect of tax evasion on the marginal cost of public funds(Usher, 1986), evasion of commodity taxes (Cremer and Gahvari, 1993), the implications of social normson evasive activity (Gordon, 1989; Bordignon, 1993), and the use of evasion to mitigate time consistencyproblems (Boadway and Keen, 1998; Renström, 1998).


2. Becker argues that crime can be deterred by various combinations of punishment and probabilities: if theseverity of punishment is decreased (increased), the probability of detection must be increased (decreased) tomaintain a given level of deterrence. Since it is costly to increase the probability of detection, increasing thelevel of punishment indefinitely while letting the probability of detection fall to zero is the least costly wayof deterring crime. Becker’s maximal sanctions refer to a level of punishment sufficient to deter all crimewhen the probability of detection approaches zero. Note that in many applications, maximal sanctions includenon-monetary ones, like incarceration or physical punishment.

3. The relevance of the Becker solution has been widely discussed in the economics of crime literature: seeGaroupa (1997) for a survey. It might be presumed that when there are law enforcement or compliance errors,the case for maximal sanctions lessens. But Kaplow and Shavell (1994) have argued that the sanctions maywell be maximal even when there are type I and II enforcement errors, so that the innocent may mistakenly besubject to severe penalties. Their result rests, however, on the assumption that individuals are risk-neutral, sothere are no costs of risk-bearing associated with the possibility of being mistakenly convicted. On the otherhand, Polinsky and Shavell (1979) showed that with risk aversion, maximal sanctions may not be optimaleven if enforcement errors are not present. In their case, crime was not a zero-sum gain: the private benefitsit creates for the perpetrators might exceed the external cost imposed on society. In the case where the crimeis tax evasion, the consequences are purely distributive (the tax evader’s gain is the government’s loss). Whenoptimal income taxes are in place, sanctions are maximal so tax evasion is eliminated provided there are noconviction errors.

4. Note that our assumption thatU (0) = −∞ implies that we need not resort to non-monetary sanctions toimplement the Becker solution. If, for example, we had assumedU (0) = 0, as in Cremer and Gahvari (1996)and Chandar and Wilde (1998), allowing only monetary sanctions will not be sufficient to achieve the Beckersolution. It is for that reason that the possibility of tax evasion imposes a limit on redistribution in their models.If they had allowed for non-monetary sanctions, the first best could have been achieved merely be the threatof severe penalties with minimal auditing.

5. We place no restrictions on the sign of tax liabilities: they can be either positive or negative. If both arenegative, the model can be interpreted as applying to the part of the population who are transfer recipientsrather than taxpayers.

6. Technically speaking, whenE[U | x = z= Y2] = E[U | x = Y1, z= Y2], the taxpayer is indifferent betweenreporting honestly and dishonestly. To ensure truth-telling, therefore, the utility from honest reporting shouldalways be greater than that from dishonest reporting at least by a sufficiently small amount, sayε > 0.Following convention, we assume thatε can be taken sufficiently close to zero and thus, when indifferent,individuals behave honestly. Otherwise, we would have to allow for the possibility that individuals randomizetheir choice of reported income, which would complicate our analysis unnecessarily. This is a commonproblem in the literature.

7. The utilitarian form presumes zero aversion to inequality of social utilities. Alternatively, we could havetaken the other extreme case of a maxi-min objective function which exhibits infinite aversion to inequality.The same qualitative results would result. Only if the government redistributes from the less well-off to thebetter-off would our results not apply.

8. To see this, supposeT22 6= T2. Thus, expected revenue from a taxpayer withz = Y2 is T2 + p2(T22− T2).Now, change the tax payment imposed on a high-taxable-income person toT ′2 = T2 + p2(T22− T2). Strictconcavity of the utility function impliesU (Y2 − T ′2) > (1− p2)U (Y2 − T2) + pU(Y2 − T22). This changeimproves the utility of the high-taxable-income taxpayers and thus relaxes the self-selection constraint whilekeeping the expected revenue from them the same. Moreover, the value of the government’s objective functionincreases sinceW(·) is strictly concave. Since those withz = Y1 are not affected, this is a Pareto-improvingchange.

9. As above, the solution to this problem does not exist, since probability of detection approaches zero withouta lower bound. This is a well-known problem with the Becker solution.

10. The result thatT11 < T1 depends on government’s objective function. If the latter were max-min instead, thegovernment would have no incentive to raise the welfare of low-taxable-income persons who are audited atthe expense of low-taxable-income persons who are not audited: we would obtainT11 = T1 in the optimum.

11. The problem of time consistency has been recognized in the literature in other contexts. For example, Boadway,Marceau and Marchand (1996) argue that punishing criminals after they have been found guilty is itself time-inconsistent since after the crime is committed a penalty serves as no deterrent and merely reduces the welfare ofthe criminal. And, it has been argued that reducing the intensity of auditing capital tax evasion may be a rationalepolicy response to tax policy itself being time-inconsistent (Boadway and Keen, 1998; and Renström, 1998).


12. To see this, consider the case ofβ = 1, so all taxpayers perceive a low taxable income. In case i), theyall report x = Y1. It can be shown that the optimal policy will be such thatT∗1 < T∗12 and p∗1 > 0

(T2 is redundant). DefineT2, T21 and p2 by W(Y2 − T2) = (1 − p∗1)W(Y2 − T∗1 ) + p∗1W(Y2 − T∗12),

p2W(Y1− T21)+ (1− p2)W(Y1− T2) = W(Y1−T∗1 ), and p2 = p∗1. Then the switch from case i) to case ii),in which all taxpayers reportx = Y2, is desirable if

α[T2 − ((1− p∗1)T∗1 + p∗1T∗12)] > (1− α)[T∗1 − ((1− p∗1)T2 + p∗1T21)]

This will hold whenW(·) is more concave aty = Y2 than aty = Y1.13. The incentive constraints are:

µ((1− p2)U (Y − T2)+ p2U (Y − T21))+ (1− µ)((1− p2)U (Y − T2)+ p2U (Y − T22))

≥ µ((1− p1)U (Y − T1)+ p1U (Y − T11))+ (1− µ)((1− p1)U (Y − T1)+ p1U (Y − T12))

(1− p1)U (Y − T1)+ p1U (Y − T11) ≥ (1− p2)U (Y − T2)+ p2U (Y − T21).

14. The proof is as follows. LetT11 6= T1. The government revenue from a taxpayer who reportsY1 is (1−p1)T1 + p1T11. Let T ′1 = (1− p1)T1 + p1T11. ReplacingT1 andT11 by T ′1 does not change the governmentrevenue, but raises the utility of low income persons:U (Y − T ′1) > (1− p1)U (Y − T1) + p1U (Y − T11).Social welfare also improves sinceW(·) is strictly concave.

15. In the context of administrative errors, it is not necessary to distinguish between taxable and actual income.We do so to make the analysis parallel to the case of taxpayer error.

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The Optimality of Punishing Only the Innocent: The Case of Tax Evasion