JOURNALOF
PUBLIC ELSEVIER
Journal of Public Economics 60 (1995) 235249
ECONOMICS
Tax evasion and the optimum general income tax Helmuth
C r e m e r a, F i r o u z G a h v a r i b'*
alDEI and GREMAQ, University of Toulouse, Toulouse, France bDepartment of Economics, University of Illinois at UrbanaChampaign, Champaign, IL 61820, USA Received January 1994; revised version received January 1995
Abstract This paper incorporates tax evasion into an optimum general income tax problem with endogenous labor supply. It posits a twogroup model with high and lowwage individuals to investigate the properties of optimal audit and tax structures. The following main results are obtained. First, highwage persons are never audited while lowwage persons are audited with a probability strictly less than one. Secondly, highwage persons should face a zero marginal tax rate. Thirdly, lowwage individuals who have been audited and found innocent should pay a lower tax than those who are not audited. Fourthly, there are two possible tax regimes: lowwage persons face a positive marginal tax rate under one regime and a zero rate under the other. Fifthly, even if honesty cannot be rewarded, the other results of the paper continue to hold.
Keywords: Tax evasion; Optimal taxation J E L classification: D82; H21; H26
I. Introduction T h i s p a p e r a i m s to i n c o r p o r a t e tax e v a s i o n into the o p t i m u m i n c o m e t a x a t i o n p r o b l e m o f M i r r l e e s (1971). It d i s t i n g u i s h e s itself f r o m e a r l i e r s t u d i e s o n t h e s u b j e c t by c o n s i d e r i n g a m o d e l with e n d o g e n o u s l a b o r s u p p l y a n d a l l o w i n g f o r a g e n e r a l i n c o m e tax. It c h a r a c t e r i z e s a s o l u t i o n to such a * Corresponding author. Fax: + 1  217244  7368. 00472727/96/$15.00 O 1996 Elsevier Science S.A. All rights reserved SSD1 00472727(95)015256
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problem in the presence of adverse selection and moral hazard where the agent is risk averse. While the paper focuses on an optimal tax problem, its results can also be applied to certain enforcement or auditing problems that arise in the design of regulatory and environmental policies, or in an accounting context. 1 The central element in the theory of optimal taxation is information. Lack of pubic information on personal characteristics prevents the government from levying optimum lumpsum taxes and forces it to impose taxes on income. If the taxpayer's true income is also unknown to the government, with the possibility of its being observed through costly audits, the whole question of optimal tax design would take a different light. The set of government policy tools would then include the audit strategy as well as the tax rates. This opens up the interesting questions of the optimal audit strategy, the interaction of tax rates and audit strategy and the reformulation of the optimal income tax schedule. While the optimum income tax literature has ignored the problem of the unobservability of income at no cost, the tax evasion literature has to a large extent ignored the labor supply question that lies at the heart of the optimal income taxation problem of Mirrlees. In Mirrlees' model, ability and labor supply of an individual (which determine his income) are unobservable by the tax authority. The 'agency problem' arises because both adverse selection and moral hazard are associated with the taxation of income. The tax evasion literature, however, has traditionally concentrated on pure adverse selection problems where the exogenous income characterizes an individual's type (see, among others, Reinganum and Wilde, 1985; Border and Sobel, 1987; and Chander and Wilde, 1992). In an important exception, Mookherjee and Png (1989) discuss tax evasion in the presence of moral hazard. In their formulation of the problem, all individuals are assumed identical ex ante. Ex post incomes differ because of different drawings from a given probability distribution. The distribution itself is dependent on an action (e.g. labor supply level). However, because all individuals are identical, they will all choose the same action. While taxation is distortionary in this model, it is the same for everyone. This is of course quite different from Mirrlees's original framework. There, each person knows his own type when making a decision, with different individuals responding differently to the tax system depending on their type. As a result, the distortion created by the tax system is different for different individuals. In Mookherjee and Png, once incomes are realized, one is essentially back to the general adverse selection framework considered by Border and Sobel (and the other contributors to this See, Laffontand Tirole (1992), Mookherjee (1992), and Baiman and Demski (1980) for examples and additionalreferences.
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literature).2 Also, as with these earlier studies, the only way for a person to cheat is to misreport her income. Furthermore, this type of cheating can be discovered by an audit.3 The current paper models a community consisting of two types of individuals: high and lowwage.4 Abilities (types) and labor supply (action) are unobservable and remain private information even if an audit is performed. Income, which is the product of two unobservable variables, is not costlessly observable either but can be discovered through an audit. This implies that more complex incentive constraints must be imposed to elicit truthtelling. Given this information structure, we derive the optimal revelation mechanism and investigate the properties of optimal audit and tax structures. The following main results are obtained. First, highwage persons are never audited while lowwage persons are audited with a probability strictly less than unity. Secondly, highwage persons are assigned an income level equal to what they would choose for themselves (given their taxes) if they are never audited. In its optimization problem, then, the government need not worry about the possibility of these individuals choosing the wrong action. Therefore, the optimal taxcumaudit strategy implies that highwage persons should face a zero marginal tax rate. Thirdly, lowwage individuals who have been audited and found innocent should pay a lower tax than those who are not audited. Fourthly, there are two possible tax regimes: lowwage persons face a positive marginal tax rate under one regime and a zero rate under the other. Fifthly, even if honesty cannot be rewarded, the other results of the paper continue to hold.
2. The model
Consider an economy consisting of two types of individuals: N h individuals of type h and N t individuals of type I. They are risk averse with identical preferences but differ in earning abilities. Individuals of type l earn
2 Except that in Mookherjee and Png (as in our setting) individuals are risk averse while most other authors have assumed risk neutrality. 3 Sandmo (1981), Cremer and Gahvari (1994) and Schroyen (1993) also have attempted to incorporate tax evasion into an optimal income tax problem with endogenous labor supply. The first two papers restrict the income tax schedule to be linear, and consider purely random audits only. Schroyen allows for nonlinear taxation but restricts the penalty to be proportional to the tax evaded. This assumption implies that one can no longer rely on revelation mechanisms for the solution. The outcome is suboptimal as compared with what may be obtained if there" is only an upper bound on penalties (unless the upper bound is very small). In addition, Schroyen assumes that the only way to conceal income is to work in an irregular market to which only highability persons have access. 4 This setup has been popularized by Stiglitz (1987) to discuss Paretoefficient tax structures.
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a lower wage than those of type h. There is a single produced commodity (the numeraire), C. Preferences are separable in C and labor supply, L; they are represented by U = u(C) +
¢(1 L),
(1)
where U is twice continuously differentiable, strictly increasing in C and strictly decreasing in L. To ensure risk aversion, assume that u is strictly concave. A person's consumption must be nonnegative, and u(O)= O. Accordingly, penalties are bounded and cannot exceed an individual's actual income. In other words, there is limited liability and nonmonetary sanctions are excluded. Define (2)
I = w L ,
~bi(I ) = tp I 
,
i = 1, h ,
(3)
where w i is the wage of an individual of type i = l, h with w h > w r Similar notation is used with other variables: subscripts l and h specify the value of a variable for individuals of the corresponding type. An individual's ability and labor supply are not publicly observable. Also, the person's income is not costlessly observable; however, income may be observed through a costly audit. Audit cost, a, is a strictly increasing function of the number of people audited, such that 0 < a' < +o0. Consider a direct mechanism which consists of four functions: I(ff), p ( f f ) , T(ff) and F(ff, la), where ff is reported type and I a is the individual's income as revealed by an audit. A person who reports n3 is assigned 5 to earn an income of I(v~) and is audited with a probability of p(ff). If she is not audited, he pays a tax of T(ff). If he is audited, he will have to pay F(ff, Ia) which includes any applicable fine. Taxes and fines may be negative as well as positive. Nevertheless, they must satisfy the following restrictions: T(ff) ~
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case. 6 Consequently, we can restrict our attention to revelation mechanisms whereby the individual is induced to report her type truthfully and choose the appropriate level of income (labor supply). It is easily seen that, to provide a maximum incentive for truthtelling without affecting the individual's expected utility, the maximum penalty should be applied to any individual who is caught cheating. This implies that, when a person is caught lying, her consumption must be set at its minimum, i.e. at zero:
F(w, l a ) = I a,
ifla~I(r~ ).
For ease of notation, define: I i = l(wi), Pi = p ( w i ) , Ti = T(wi) and F i = F(w i, li), with i = l, h. Denote the expected utility of a person of type i who claims to be of type j and earns lj by EUij. We have EU,j  (1  p j ) u ( l j  Tj) +pju(lj  Fj) + O~(Ij),
i and j =l, h .
(4)
T o facilitate a formal statement of the government's problem, we also define
V(w, p, T)= max [ ( X  p ) u ( I  T) + ~p(l  / ) ]
.
(5)
In words, V is the maximum utility that can be attained by an individual with earning ability w who faces a probability p of being audited, and who pays a fine equal to her income if audited and a tax T if she is not. Thus, V~j =V(wi, Pi, Tj) indicates the maximum utility attainable by an i type individual who claims to be of type j but chooses a different income level than the one assigned to type 2'" In addition, denote the value of I which solves the problem in (5) by I(w, p, T). This function shows the level of income an individual would choose for herself if she had to pay a tax equal to all her income if audited and T(ff) if not. The objective of the government (tax administration) is to maximize social welfare given by W = EUtt + pEUhh ,
(6)
where p is a positive constant. The maximization is subject to the revenue and incentive compatibility constraints. In the specification of the selfselection constraints, however, we shall ignore the 'upward' incentive constraints [EU//~ EUlh and EUll ~ Vlh], assuming they are not binding at the optimum. That is, to achieve selfselection, the government need not worry about a lowwage person 'imitating' a highwage person. This is the As in Mookherjee and Png (1989) and Melumad and Mookherjee (1989), risk aversion complicates the proof because one has to deal with the possibility that individuals want to use randomized reporting strategies. Melumad and Mookherjee provide a detailed discussion of this problem; their proof can easily be extended to our setting.
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'normal' case on which most of the literature has concentrated. Intuitively, it means that the tax policy involves redistribution from the rich to the poor. The government's problem is formally stated as follows. Maximize (6) with respect to I t, I h, Pt, Ph, Tl, Th, Ft and Fh, subject to the selfselection constraints EUhh ~ EUh/,
(7a)
EUhh/> Vhh ,
(7b)
EUhh I> Vht ,
(7C)
EU n I> Vtt ,
(7d)
and the revenue constraint
Nt[(l 
pt)T,
+ p,Ft] + Nh[(l  ph)Th
+ phFhl  a(Nzp ' + Nhph) >I I~ ,
(8) w h e r e / ~ denotes the required tax revenue. Selfselection constraint (7a) ensures that a highwage person prefers a truthful statement of her type and a choice of the corresponding income, to mimicking the type and the income of the lowwage. This is the traditional selfselection constraint. However, there remains the possibility that the highwage person reports her type truthfully, but then goes on to earn an income of I ~ I h. Selfselection constraint (7b) ensures that this will not happen. Additionally, she may claim to be of type I while earning an income I ~ I t. Selfselection constraint (7c) ensures that this does not happen either. These latter conditions can be thought of as 'moral hazard' conditions. Finally, inequality (7d) ensures that a lowwage person does not cheat by choosing an income level different from what she is assigned to.
3. Optimal audits We start by discussing two special cases which illustrate how our model can be seen as a natural extension of the traditional optimal tax problem. First, suppose that everyone is audited for certain. Incomes are then observable, and we are back at the traditional optimal tax problem without tax evasion (as in Stiglitz, 1987). It is easily seen that constraints (7b)(7d) are automatically satisfied. As in the traditional setting, we have only the selfselection constraint (7a) to consider. Of course, perfect observability of income now comes at a cost: compared with the traditional setting, the required tax revenue is increased by the full amount of audit costs a(N~ + Nh). It will be seen below that this policy is actually never optimal.
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Secondly, suppose no audits are to take place. The only feasible tax instrument is then a uniform lumpsum tax which, from the revenue constraint, is set at T h = T t = R / ( N h + Nt). 7 This type of policy can arise as a corner solution to the above problem if (for instance) audit costs are very high a n d / o r wage differentials very small. While this has to be kept in mind for the proper interpretation of our results, we shall concentrate on the more interesting case of an interior solution. We are now in a position to address the optimal choice of the audit probabilities. Interestingly, it turns out that highwage reports are never audited. Moreover, lowwage reports face stochastic audits: they are audited with a probability that is strictly less than unity. This result is formally stated and proved as: Proposition 1. The optimal audit policy is characterized by Ph = 0 and p~ < 1. Proof. We prove each claim separately_, beginning with Ph = 0. Consider any policy ([h, ~, fib, Pt, Th' Tl, Fh, Ft) which satisfies the selfselection constraints (7a)(7d) as well as the revenue constraint (8). We show that, if the policy entails /~h > 0, then it can be replaced by another policy with Ph = 0 satisfying all the constraints and resulting in a higher level of social welfare. Denote the value of the variables under the alternative policywith a 'hat' over them, and define /~h = 0, Th : (1  ph)'Fh "~phFh, l h = l(Wh, 0 , T h ) ' /~l =/~l, Tt = ~,/¢t = ~ , and It = ~ From these definitions, it follows that the value of EU u under both policies is the same. Turning t o E U h h , strict concavity of u (risk aversion) implies that, even if I h were to remain at [h, the suggested changes for Ph and T h increase EUh h (same expected taxes but no uncertainty). Now, since [h is replaced by I, expected utility can only increase further. 8 Hence, E U h h >EUhh SO that social welfare under the alternative policy will be higher than under the initial policy. It remains to be shown that the new policy is feasible. The above argument shows that the lefthand side of each of the selfselection 7 If the proposed tax is purely redistributive (/~ = 0), then no tax is levied. 8 Formally, from Eqs. (3) and (4) we have:
From the definition of 1, the first bracketed term in the righthand side of the above equation is seen to be nonnegative; while strict concavity of u implies that the second bracketed term is positive.
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constraints (7a)(7c) increases, while the lefthand side of (7d) remains unchanged. Hence, a selfselection constraint can only be violated if its righthand side increases. This is the case only for (7b); but this constraint is automatically satisfied (with equality), since the new policy implies ih = I(W h, O, Th)" Finally, the revenue constraint is satisfied because expected tax revenue is the same, while audit cost is lower under the new policy. To prove the second claim of the proposition, Pt < 1, consider again any policy ([h, ~,Ph,Pl, "Fh, Tt, Fh,Fl) which satisfies selfselection constraints (7a)(7d), as well as the revenue constraint (8). This time we will show that, if the policy entails /~ = 1, then it cannot be optimal. This is proved by showing that p~ can be reduced from its initial value of one in such a way that (i) social welfare does not change while net tax revenue increases, and (ii) selfselection constraints remain satisfied. Consider an alternative policy consisting of/~t  1  e, with ~ 'sufficiently' small, and Tt = ~ with the values of the rest of the variables remaining the same as before. It is easily seen that this policy switch satisfies (i) (it leaves E U h h , EUlt and the expected (gross) tax revenue unchanged, while reducing the audit cost). Turning to (ii), the only constraints that could possibly be violated by the policy switch are (7c) and (7d) ((7a) and (7b) are not affected, because EUhl and Vhh remain unchanged). Now, the original policy involves Pt = 1 so that these conditions are satisfied with strict inequality (cheaters are audited for sure and face a fine equal to their income). Consequently, if e is sufficiently small the constraints continue to be satisfied at the alternative policy. [] Proposition 1 extends an earlier result by Mookherjee and Png (1989). They have derived a similar property in a setting where incentive constraints are much simpler as high and lowwage individuals choose identical levels of labor supply. Some intuition for our result may be obtained from the method of proof. As we have shown, the government can extract the same tax revenues from highwage persons by levying a 'sure' tax on them (no audits) rather than a random tax (random audits). Moreover, it can do this without changing the incentive constraints. The change in policy will then have two distinct benefits. On the one hand, the rich, being risk averse, become better off by the elimination of uncertainty in their consumption. On the other hand, while gross tax revenues remain the same, audit costs are reduced. Turning to the second part of our result, note that, if lowwage persons are audited with probability one, they will pay Ft in taxes for certain. However, they can have the same certain net income with random audits also, provided they pay identical taxes regardless of their being audited or not. Such a shift in policy is welfare improving, because it entails lower audit costs. Regarding feasibility, the crucial point is that, with certain audits,
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cheating by not choosing the assigned income level is a strictly dominated strategy. Consequently, a small decrease in the audit probability will not violate the corresponding selfselection constraint. Proposition 1 may at first glance seem rather paradoxical, because it appears to suggest that only 'poor' persons must be audited. This is not the case. The identities of 'poor' and 'rich' persons are unknown. What is known is who has reported a low or a high income. The government's aim is to prevent rich persons from reporting a low level of income. To do this, it will only need to audit lowincome reports.
4. Optimal taxes and fines The finding that Ph = 0 has interesting implications. First, from constraint (7b) and the definition of V (expression (5)), it implies that
I h = I(Wh, O, Th).
(9)
Recall that l(Wh, O, Th) is the level of income a highwage person, who reports her type truthfully, chooses for herself if she is never audited. This property is not surprising: since a highwage individual who truthfully reports her type is not audited, the government has no control over her labor supply. Therefore, the assigned income must equal the level that she would choose anyway. Now, the firstorder condition of problem (5) for a highwage person facing Ph 0 and T h is given by
a u ( l  Th) OC
OqJh(I) OI
+  
=o.
(lo)
From (10) and (9) it follows that the most able individual should face a zero marginal tax rate. Hence, the famous 'no distortion at the top' result of the optimum income tax literature continues to hold in the presence of tax evasion. 9 The following proposition summarizes this result.
Proposition 2. The optimal taxcureaudit strategy implies that highwage persons should face a zero marginal tax rate. Like Proposition 1, this result appears at first somewhat surprising. To see the intuition, note that the benefit of taxing the marginal income of any group is to raise the average tax rate on the higher income brackets. The benefit is to be balanced against the distortionary impact of the tax on the 9 See, among others, Seade (1977) and in the context of the twogroup model, Stiglitz(1987).
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taxed group. When the taxed group is at the top of the distribution there will be no such benefits, only costs. This is why the marginal income of the group should not be taxed. 1° We can now use Ph = 0 and (9) to restate the tax administration's problem in a more tractable form. We have the following Lagrangian expression: A = [(1  p t ) u ( I ,  T,) +ptu(It  Fl) + qJ,(I,)] + p I u ( I h  Th) + O/h(Ih)] + AI[U(Ih  Th) + d/h(lh)


(1  p t ) u ( I ,  Tt)  p t u ( I t  Ft)  ~bh(I,)]
+ Az[Ih  I(Wh, O, Th) ] + A3[u(Ih  Th) + ~Oh(Ih)  V(Wh, p,, T,)] + A4[(1  p t ) u ( I ,  7,) +ptu(I,  Ft) + g/t(I,)  V(wt, p,, Tt) ] + As[Nt((1  P , ) T t + PtFt) + Nh Th  a(Ntp,)  / ~ ] ,
(11)
where the Ai terms denote the Lagrange multipliers. The firstorder conditions are OA / Our + O~bh(Ih) Olh  (p + 3.1 + A3)~O~Oih / + A2 = 0 ,
(12a)
oa 0~h
(lZb)
Ou~ Ol(w h, O, Th) (p + A1 ] A 3 )  ~  A2 OTh + AsNh = 0 ,
OA Oit  ( 1  A
/Out\ I+A4)E~~" )+(1+A4)
oa
Our
O~bt(I1) Oq~h(l,) Olt A 1 Oit O, OVm
OVu
(1 A, +
aT, 
p,)
= O, OA OFt oa Op,
(12d) Ou F (1A 1 +A4)pt'~~+AsNtpt=O, OVm

(12C)
(1  A, + X 4 ) ( u ~  u y )  A 3 ~
= O,
(lEe) OVu
 A4 Op~+
X s N , ( F l  r,
a') (12 0
where u hr  u(l h  Th); u f   u ( l t  Tt); u F =  u ( I t  F t ) ; and E ( O u / O C ) = (1  p l ) ( O u ( I t  Tt)/OC ) + pt(Ou(It  Ft)/OC ). T o interpret these conditions, it is necessary to determine which conlo Distorting one's choices may be beneficialonly if it relaxes a selfselectionconstraint. Now, because at equilibrium nobody wants to mimic the rich, there will be no point in distorting their prices.
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straints are or are not binding. The following lemma summarizes our results in this regard. Lemma
1. W e have:
(i)
A2 = 0 ,
(13)
(ii)
A3 = A4 = 0 is n o t p o s s i b l e ,
(14)
(iii)
A1 = 0 i m p l i e s A4 = 0 .
(15)
P r o o f . The first claim follows from Eqs. (12a) using (10) and (9). Turning to (ii), from Eqs. (12d) and (12e), it follows that if h 3 = A4 = 0, then T l = F t. This in turn implies, from (12f), that a ' = 0  a contradiction. The last part of the lemma is established as follows. If h 1 = 0, then it follows from (12c) that I t is set to maximize E U n (if the first selfselection constraint is not binding, there is no reason to distort the lowwage individuals' labor supply). Given the definition of Vt~, this implies that as long as Pt is positive constraint (7d) is satisfied as a strict inequality. []
We next examine the intuition behind L e m m a 1 and its implications. First, Az = 0 implies that its associated equality constraint (9) (which with Ph = 0 replaces the inequality constraint (7b)) does not involve any welfare loss (it is redundant). In other words, the solution would satisfy (9) even if it were not imposed as a constraint. To understand this result, recall that, in the traditional optimum taxation problem, the labor supply decision of highability persons is not distorted. In that model, income is observable and can be taxed at the margin. It is not desirable to do so, however. In our setting, given Ph = 0 , the highability person's labor supply c a n n o t be distorted. H o w e v e r , just as in the traditional model, it would not be desirable to do so anyway. With only downward incentive constraints binding, a distortion at the top can only reduce welfare without relaxing any of the selfselection constraints. Turning to (14), it says that at least one of the moral hazard conditions must be binding. This is quite intuitive; if neither of these constraints is binding, it will be possible to reduce p~ and increase welfare. The major role of auditing is indeed to ensure compliance with these constraints. That at least one of Lagrange multipliers h 3 o r / ~ 4 must be positive helps determine the relative magnitudes of T l and F~H. This question has been widely discussed in the auditing literature. Border and Sobel (1987), and M o o k h e r j e e and Png (1989) have shown that honesty must be rewarded, so that F t < Tv The following proposition shows that this result also holds in our model. 11With ,oh =
0,
the size of Fh is irrelevant.
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Proposition 3. The optimal taxcumaudit strategy implies Ft < T t, i.e. honest reporting should be awarded. Proof. Dividing (12d) by (12e) and simplifying yields Our/OC asNt  [~3/( 1 P/)] OVhl/OT,  [a4/(1  P l ) ] OVu/OT, OuF/ Oc = AsN,
(16)
From (14) and the fact that OV/OT < 0, the righthand side of (16) must be strictly greater than one. The concavity of u then immediately implies that
F, < T,.
[]
At first glance, Proposition 3 appears counterintuitive because differentiating F t from T t entails a cost. Given risk aversion, individuals prefer F~ to be as close to Tt as possible (to have less risky consumption). However, raising tax revenue through T l has a benefit that is not shared by F t. This is the impact of an increase in T t on incentive constraints (7c) and (7d). Specifically, increasing Tt makes it less attractive for individuals to report I t while earning some other level of income. (Recall that, when individuals are caught cheating, they lose all their income.) Hence, if at least one of these constraints is binding, then an increase in Tt can relax it. This positive effect would then have to be balanced against the earlier negative effect. To obtain an insight into the tax treatment of the lowwage individuals, one must differentiate between two possible regimes implied by Lemma 1. The regimes are as follows. (i) /~1 ~> 0, /~2 = 0, and either ;t3 > 0 or /~4 > 0 (or both): the 'traditional' selfselection constraint plus (at least) one of the moral hazard constraints are binding. (ii) ~1 = 0, A2 = 0, ~3 ~> 0 and /~4 ~0: the traditional selfselection constraint is not binding; redistribution is constrained by the fact that highwage persons must be prevented from reporting to be of lowwage type while choosing I ~/~. By analogy to the concept of marginal tax rate for highwage persons, based on Eq. (10), one may define the 'marginal tax rate' (MTR) faced by a lowwage person as MTR, = 1
Ol, e(Ou,/OC) "
(17)
We can now state and prove the following result.
Proposition 4. The optimal taxcumaudit strategy implies that the marginal tax rate faced by lowwage persons must be positive under regime (i) and zero under regime (ii).
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Proof. Manipulating Eq. (12c) and making use of the definition of MTR t
yields IOu,\ (1  ~1 ~ A4)E~~~)MTRt =
A (O~Ohh((lt) O~t(It)~ 1\
OIl
(18)
OI1 ] "
Now, from the definition of ~O,(.) and the concavity of q~(.), we have OOh(It Oit )
1 ~p' ( 1 __w_.~h)< 1 Wh It
(i  ~t) OIt "' It =  O~bt(It))
while Eq. (12e) implies that (1 /~1 + /~4) > 0. Given these two properties, it follows from Eq. (18) that MTRt is positive if A~ > 0 (regimes (i)) and zero if ;q = 0 (regime (ii)). [] This result is quite intuitive. When At > 0, constraint (7a) is binding. The distortionary tax on lowwage persons is useful here, because it weakens this selfselection constraint. On the other hand, when this constraint is not binding, there is no fear of highwage persons wanting to mimic lowwage persons. Consequently, there is no need to distort the behavior of lowwage persons. Under this circumstance, taxation has only a cost with no corresponding benefit. As a final observation, note that, in practice, it may be politically objectionable to tax those who have been audited and found innocent at less than their original tax liability. Indeed, this violates the principle of horizontal equity. To address this question, one can reexamine the whole problem with the added constraint that F t = Tt. The revelation principle continues to hold with this additional constraint. One can then demonstrate that the constraint will be binding at the optimum and that the level of social welfare will be reduced. However, the other results of the paper continue to hold. This result is summarized as follows. Proposition 5. A s s u m e F t = T t. The optimal taxcureaudit policy is characterized by Ph = 0 , Pt < 1, zero marginal tax rate at the top and positive or zero marginal tax rate for lowwage persons. The constraint is binding; it results in a lower level o f social welfare.
5. Concluding remarks This paper has attempted to incorporate tax evasion into an optimum general income tax problem with endogenous labor supply. The costliness of obtaining information on incomes compounds the problem of lack of information on individuals' types. Thus, taxpayers are able to choose the
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wrong action as well as to misrepresent their income. The twolayer cheating makes it quite a formidable task to discover the general properties of an optimal taxcumaudit policy when there are many individuals with differing abilities. Our twogroup model is a first attempt to understand this complex issue. The results suggest that a good number of the earlier insights of the literature continue to hold, even with the endogeneity of labor supply. Of particular interest is the result that highwage persons must never be audited, while lowwage persons must be audited with a probability strictly less than one. The same is true regarding the result that highwage persons should face a zero marginal tax rate, while lowwage individuals who have been audited and found innocent should pay a lower tax than those who are not audited. Whether these results carry over to more complicated situations is an open question well worth investigating.
Acknowledgements We thank two anonymous referees for helpful comments.
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