Tax Evasion, Social Customs and Optimal Auditing

Transcription

1 1 Tax Evasion, Social Customs and Otimal Auditing Gareth D. Myles and Robin A. Naylor University of Exeter University of Warwick May 1995 Abstract: The otimal audit olicy is analysed for an indeendent revenue service when a social custom exists that rewards honest tax-aying. The imlication of the existence of the social custom is that in equilibrium the income level of a taxayer cannot always be inferred exactly from their reort. The structure of the otimal audit olicy is determined both for a fixed (reort-invariant) audit robability and for when the audit robability is a function of the income reort. For the constant robability of audit it is shown that an interior solution exists to the decision roblem of the revenue service and comarative statics results are given. When the audit robability can vary, the audit function is roved to be a decreasing function of the income reort which reaches zero at the highest income reort of a tax evader. ncreases in the fine for evading and in the tax rate raise the otimal audit robability. Keywords: tax evasion, auditing, social customs JEL Classification nos: D82, H26 Acknowledgements: Thanks are due to Ben Lockwood, David de Meza, Jonothan Thomas and articiants in seminars at Essex, Exeter, Kent and the Public Finance Weekend at Warwick. Postal Address: To first author, Deartment of Economics, University of Exeter, Exeter, EX4 4RJ.

2 2 1. ntroduction Tax evasion is a henomenon that affects both develoed and develoing countries. Although estimates of the level of evasion are subject to considerable uncertainty, there can be no doubting that a significant art of national income is unrecorded in official statistics. t is therefore not surrising that tax evasion has become a toic of considerable research activity. n addition to attemts at the measurement of the extent of evasion, research has roduced both models designed to exlain the known facts of evasion and others that aim to rovide guidance for the government in reacting to the existence of evasion. t is to the latter literature that the resent aer aims to contribute. An imortant avenue through which state intervention can affect the level of tax evasion is the olicy of its revenue collection agency. Given that tax rates are set as art of a broader economic olicy and that unishments for evasion are determined by the civil courts in relation to those for other offences 1, the auditing olicy of the revenue service becomes the only dimension of choice. Essentially, the revenue service can be modelled as receiving income reorts from the taxayers and, on the basis of these, determining what roortion of reorts should be audited at each income level 2. The reaction of the taxayers to these robabilities, in terms of the reorts they submit, then determines the level of revenue, net of auditing costs, achieved by the revenue service. By analysing the effect of the auditing scheme on the equilibrium outcome the otimal, revenue maximising, scheme can be characterised. Economic analysis of the determination of the otimal audit olicy 3 has rovided a number of significant results but all such analyses have been based on the standard Allingham and Sandmo (1972) model of tax evasion as an otimal ortfolio decision under uncertainty. Although there is much to say in favour of this model as a reliminary insight into the tax evasion decision, its redictions are not entirely in agreement with the emirical and exerimental evidence. For examle, it fails to redict that aggregate evasion will rise as the tax rate increases (Clotfelter (1983)) or that many taxayers do not evade even when they would accet a gamble equivalent to the tax evasion decision (Baldry (1986)). t also neglects the imortance of social factors as emhasised by the findings of Sicer and Lundstedt (1976) 4. To overcome 1 An alternative view on this issue is given in Pestieau et al (1994). 2 The revenue service may also try to influence ercetions about the chance of being caught by, for examle, rosecuting a small number of "celebrity" cases which are guaranteed ublicity. t may be ossible to model this but the resent aer works on the basis of objective robability assessments. For discussion of evidence on this, see Alm et al (1992). 3 Such as Andreoni (1992), Cremer, Marchand and Pestieau (1990), Reinganum and Wilde (1986), and Sanchez and Sobel (1993). 4 Fuller discussion of these issues can be found in Cowell (1990) and Myles (1995).

3 3 this, Gordon (1989) and Myles and Naylor (1995) have modified the Allingham- Sandmo framework by including social customs into the tax evasion decision. With this modification, a decision to evade brings both the ossibility of a fine if caught evading but also the loss of the social custom utility. This secification is sufficiently flexible to lead to results consistent with observed evidence and catures in a simle, stylised, way the role of social interaction between taxayers. The urose of the resent aer is therefore to consider otimal auditing when the level of tax evasion is determined by the social custom model, thus embedding the auditing roblem in a framework with greater exlanatory ower. The most imortant imlication of the existence of the social custom is that in equilibrium the income level of a taxayer cannot always be inferred exactly from their reort since the honesty of the taxayer cannot be observed by the revenue authority. A low reorted income can come from either a low income, but honest, taxayer or from a higher-income taxayer who is engaging in evasion. This seems to concur much more closely with reality than the correct inference that follows from the revelation rincile in the absence of the social custom. n addition, there may be some reorts of high incomes that will only be filed by honest taxayers and this will ut a bias into the audit robabilities. The analysis determines the structure of the otimal audit olicy both for a fixed (reort-invariant) audit robability and for audits which are a function of the reort. The solution rocedure involves first constructing a social equilibrium which catures the interdeendency between taxayers caused by the existence of the social custom. The otimal audit function is then determined through its effects uon the level of evasion in the social equilibrium. For the constant audit robability case, comarative statics analysis of the otimal solution show that, in contrast to the findings of Christiansen (1980) and Kolm (1973), in this framework more auditing is a comlement to an increase in the fine aid on untaxed income rather than a substitute. The analysis of the general case shows that the otimal audit robability is a decreasing function of reorted income, with the audit robability becoming zero at the highest income reort made by a tax evader. The comlementarity of the fine and audit robability is again shown to arise. Section 2 introduces the form of the social custom, analyses individual choice behaviour and secifies the objective of the revenue service. A social equilibrium of verification and reorting is also defined. The characterisation of the otimal constant audit robability and some comarative statics are given in Section 3. Much of the construction in this section also alies in the case of a variable robability of audit

4 4 which is analysed in Section 4. The otimal audit function is characterised and a comarative statics analysis is conducted. Conclusions are resented in Section Taxayers, revenue service and equilibrium The basic structure of the model is that the revenue service chooses an audit function and taxayers choose their level of evasion (which may be zero) or, equivalently, their reorted level of income. The revenue service could calculate the level of evasion that a taxayer of known characteristics would choose but, since the characteristics are not observed, must infer income from the observed reort. The audit robability is chosen to maximise revenue less costs, given the beliefs about true incomes formed on the basis of the signals in the reorts. Reorts are chosen by taxayers to maximise exected utility given the audit robabilities. 2.1 The individual taxayer The model of individual taxayer behaviour combines the standard Allingham and Sandmo (1972) analysis of the evasion decision with a social custom that rovides a return to honesty. The social custom is relevant for determining whether a taxayer chooses to evade. t also introduces an interdeendency between taxayers that features in the determination of equilibrium. Let the utility level for a taxayer of income and facing the tax rate t, who chooses not to evade, be given by ( [ 1 t] ) + br( µ ) U NE U 1, (1) where b 0. The term br ( 1 µ ) is the utility of conforming with the grou of tax ayers (µ is the roortion of oulation evading tax), so that b can be interreted as the imortance the taxayer attaches to the social custom. Each taxayer is fully b, air. The following assumtion characterises the restrictions described by their { } laced uon references. Assumtion 1 U ( ) is smooth with U '( ) > 0, U ''( ) < 0 and lim ψ 0 U ( ψ ). ( ) R '( ) > 0. R is smooth and When the decision to evade tax is taken, the resulting level of utility is U E { } [ ( )] U ( t ) + ( ) U ( t ft[ ]) max 1, (2)

5 5 where is reorted income, () the robability of detection given a reort of level, and f, f > 1, the fine rate if caught. The maximiser of (2) is denoted by * and the maximised utility level by U E *. t can be seen from (2) that the return from the social custom is lost when evasion is undertaken. This conforms to the framework adoted in the literature on social customs (see Akerlof (1980), Naylor (1989)) but differs from the model of Gordon (1989) in which the costs of evasion increase roortionately as evasion increases. Define W t and Z t ft[ ] chooses to evade, the otimal level of reorted income, * (, ) the first-order condition 6 [ U ( Z ) UW ( )] [ 1 ] tu '( W ) + [ f 1] tu '( Z ) 0. f a taxayer of income 5, must satisfy '. (3) t is clear from these conditions that for given the otimal choice *, and hence U E *, is indeendent of b. So if an individual deviates from the social custom the extent to which they evade does not deend on the imortance they attach to the social custom. Whether an individual taxayer evades deends uon the value of U NE relative to U E *. Evasion will take lace if U NE < U E * and the taxayer will choose not to evade if U NE U E *. The weak inequality on the latter exression indicates the imlicit assumtion that the taxayer refers to follow the social custom if there is no utility loss to doing so 7. For given, t and (), if there exists a value of µ* satisfying [ 1 ] UW ( ) + U ( Z ) U ( X ) + br( 1 µ *), (4) where X [1-t], then, since the right-hand side is monotonically increasing in µ, b, evades if µ > µ* but does not evade if µ µ*. Hence, given t, if (4) taxayer ( ) holds, there is a critical roortion, µ*, such that the taxayer begins to evade. f [ 1 ] UW ( ) + U ( Z ) > U ( X ) + br( 1 µ ), (5) at µ 0 then the taxayer always chooses to evade. Alternatively, if the reverse inequality to (5) holds at µ 1 then such a taxayer always chooses to ay tax. 2.2 The oulation of taxayers 5 Restrictions on the range of * will be derived below. t should be noted that this ermits income reorts which are less than the lowest actual income level (which is assumed to be ublic knowledge). Although such reorts are known to be false, auditing still requires the use of resources so they may not be audited with robability 1. 6 A corner solution at * cannot occur because the value of U NE is then at least as great since b 0. Allowing the ossibility of negative income reorts removes corner solutions at the other extreme. 7 When comuting integrals below, the choice for this set of measure zero has no consequence.

6 6 Moving to the economy as a whole, the oulation of taxayers is described by a h b,, of b and. This distribution function satisfies the distribution function, ( ) restriction of Assumtion 2. Assumtion 2 h is smooth: (i) ( ) (ii) ( b, ) > 0 ( b, ) 0 h { b, } ( 0, B) (, ) h elsewhere. B (iii) ( b, ) dbd 1 0 h ;, 0 < B <, < < <, > 0, (iv) lim (, ) 0 hbdbd 0, lim (, ) 0 hbdbd 0 Ψ B B. Ψ n (ii) of Assumtion 2, is the minimum level of income in the oulation and the maximum. The valuation laced on the social custom is restricted to be nonnegative, so no-one actively desises it, but with a finite uer limit, B. Part (iii) is a normalisation and (iv) requires that the roortion of the oulation having a given level of income tends to zero as the boundaries of the income range are aroached. Θ Exloiting the monotonicity of R ( ) in µ, (4) can be solved to write µ* ( b,, ( ) ). Given the assumtions on R ( ), it follows that Θ ( ) is continuously differentiable in b and that dµ * db R br' > 0. (6) The strict inequality in (6) shows that µ Θ(,, ( )) (,, ( )) b can be solved to give b β µ, (7) with β ( ) differentiable and strictly increasing in µ. This has the interretation that, given µ and ( ), all taxayers described by airs { b, } that satisfy b < β (, µ, ( )) evade tax. Assuming that for those choosing to evade the second-order condition for the,. Acceting otimisation in (2) is satisfied, (3) can be solved to write ( ( )) that (, ( )) is strictly monotonic in for the resent 8, (, ( )) inverted to give m (, ( )). The function (, ( )) can be m determines the true income of a taxayer who is evading as a function of reorted income and the audit function. 8 Monotonicity will be studied for alternative restrictions on the form of (. ) below.

7 7 Given a value of µ, the roortion of taxayers who choose to evade is b <, µ, G µ, determined by those for whom β ( ( )). Denoting this roortion by ( ) it follows that β ( ) ( ) (, µ, ) h( b ) G µ dbd. (8) 2.3 The revenue service 0, The revenue service is concerned only with maximising tax revenue less the cost of auditing. t takes t and f as fixed 9. Given an income reort, the revenue service infers an exected income level of the taxayer making that reort. Denote the exected M. The cost of enforcing an audit robability of () for level of income by ( ) reorts of level is given by the cost function ( ( )) of Assumtion 3. Assumtion 3 c ( ) is smooth and c ( 0 ) 0, c '( 0) 0, '( ) > 0 R c which satisfies the conditions c for > 0, c' ( ) lim 1. With audit strategy ( ), exected revenue (less costs) is given by ( ( )) [ ( )[ t + ft[ M ( ) ] + [ 1 ( )] t c( ( ))] g ; ( ) ( )d, (9) where g ( ) is the number of reorts of income level received when the audit function is given by ( ). The form of g ( ) will be calculated below. The audit function ( ) g ( ). 2.4 Equilibrium is chosen to maximise (9) given the beliefs ( ) M and the distribution The notion of equilibrium that will be emloyed is defined in two stes. n the first, the audit function is taken as given and a social equilibrium is defined. This equilibrium incororates the social externalities between taxayers arising from the existence of the social custom of conformity. The second ste is to allow for otimisation of the audit function, taking into account the deendence of the social equilibrium uon the audit function. A social equilibrium with otimal audit function is termed a social equilibrium of reorting and verification. A social equilibrium is now defined. Definition 1 A social equilibrium is a level of evasion, ˆ ˆ µ ˆ µ, such that G ( ) µ. 9 This is justified by viewing the revenue service as a distinct body from the courts which set the level of the unishment and the exchequer which determines the tax rate.

8 8 The interretation of a social equilibrium is that the roortion evading is selfsuorting. t is straightforward to rove that at least one level of evasion exists that is a social equilibrium. This result is summarised as Lemma 1. Lemma 1 For any audit function ( ), there exists a social equilibrium. Proof β is continuous in µ, it can be seen from the definition in (8) that G defines a As ( ) continuous ma G ( µ ):[ 0,1] [ 0,1]. By Brouwer's Theorem it has at least one fixed oint and each fixed oint satisfies the definition of equilibrium. The definition of a social equilibrium of verification and reorting is now given in which both the audit function and the income reorts are equilibrium strategies. t therefore incororates the otimisation behaviour of the agents and the social interaction between taxayers. The structure of such equilibria will be investigated in the following sections. Definition 2 A social equilibrium of reorting and verification is an array {, M ( ), ( ),µˆ } that: such (i) the income reort,, from each taxayer maximises their utility given the audit and level of evasion µˆ ; olicy ( ) (ii) the beliefs of the revenue service satisfy M ( ) E [ ] taken over the distribution of taxayers making reort ;, where the exectation is (iii) the audit olicy ( ) maximises R ( ( )) given the distribution of reorts g ( ) and beliefs M ( ) ; and ˆ µ ˆ (iv) the level of evasion, µˆ, satisfies G ( ) µ. 3. Restricted audit function This section investigates the otimal audit olicy under the restriction that the audit robability is chosen from the class of constant functions with the intention of characterising the otimal audit strategy and assessing the effects of variations in the underlying structure uon this. This restricted case is considered for its own interest

9 9 and because many of the constructions undertaken in this simlified setting extend to the general case considered in the next section with only minor modification. With attention restricted to the set of constant functions, a choice of audit strategy reduces to the announcement of an audit robability,, which is indeendent of income. The first ste in the characterisation is to determine the distribution of reorts forthcoming from taxayers and the beliefs of the revenue service. To do this, note that any announced audit robability will artition the oulation into those who evade and those who do not. The tyical relation of reorts is illustrated schematically in Figure 1; it is to be exected that the lowest reorts will be rovided by tax evaders and the highest by honest tax ayers. Figure 1: Distribution of income reorts n Figure 1, denotes the minimum reort filed by a non-evader so that reorts below this level are filed only by those evading. Similarly, since denoted the maximum reort from an evader, any reort above this must come from an honest taxayer. f any reort is received between these two levels 10, the revenue service remains uncertain whether it emanates from an honest taxayer with a low income level or from an evader. t is now necessary to verify that (, ( )) (or from (12)* of itzhaki (1974)) it follows that d d t [ ft 1] RA ( Z ) + RA ( W ) [ f 1] R ( Z ) + tr ( W ) A A is monotonic in. From (3), (10) where R A ( M ) is the coefficient of absolute risk aversion at income level M. For the second-order condition for the otimisation in (2) to be satisfied it is necessary that d the denominator of (10) is ositive. For to be ositive it is then necessary that d ft 1 R Z R W. This inequality is assumed to be satisfied 11. t can also be seen [ ] ( ) ( ) A + A from (3) that with ' 0, Assumtion 1 imlies that for a taxayer of income the 10 The ossibility that the set of reorts filed by the evaders may be disjoint from those filed by the non-evaders clearly arises for some audit robabilities. This is taken exlicitly taken into account in the calculations below. 11 As the coefficient of absolute risk aversion is ositive, it is sufficient that either ft > 1 or that absolute risk aversion is non-decreasing in income.

10 10 U W and K, is continuous in declaration satisfies K(, ) > since lim '( ) 0 lim U '( Z ) > 0. From the structure of the otimisation, ( ) and hence has a minimum over the comact set [ ],. This minimum is denoted K ( ). Thus, for given, the declaration must belong to the comact interval K ( ),. [ ] The distribution of reorts received by the revenue service is constructed from the distinct comonents consisting of the three intervals identified in Figure 1. On these intervals, the distribution of reorts can be calculated as follows. (a) For min{, } (b.i) f β ( m, µ, ), g( ; ) h( bm ( )) <, then for [, ] (b.ii) f >, then for [, ] (c) For max{, } (d) ( ; ) 0 0, db; β ( m, µ, ) B, ( ) h( b, m ( )) db h( b ) g +,, ( ; ) 0 ; β ( m, µ ) g ; > ( ) h( b ) g elsewhere. g, ; B ( m, ), ; β µ 0, db db ; n a similar manner, the beliefs of the revenue service relating reorts to exected income can now be constructed. As already noted in (ii) of Definition 2 these must be correct, in terms of exectations, at the equilibrium. When < the, is simly a weighted average of the exected income following a reort [ ] incomes of evaders and non-evaders making that reort. This is given as (β.i). Conversely, when > no reorts [, ] are actually received. To make the belief function continuous in such a case, the convention is adoted in (β.ii) that the exected income function is linearly extended from the highest income of an evader to the lowest income of a non-evader. This arbitrary convention does not affect any of the conclusions. The belief function is therefore as follows. (α ) For min{, }, M ( ) m (, ) (β.i) f <, for [, ] (β.ii) f >, for [, ] (γ ) For max{, } ; ;, ( ) β ( m, µ, ) B ( ) ( ( )) ( ) ( ) 0 hb, m db + β, µ, hb, β ( m, ) B h( b, m ( )) db + h( b, ) db m M ; µ, 0,, M ( ) >, M ( ) [ ] m ( ) [ ] [ ] [ ] ; + ; ;. β (, µ ) db ;

11 11 Lemma 1 roved that for given any choice of auditing strategy there exists a social equilibrium with resect to that auditing strategy. To show the existence of an equilibrium with reorting and verification, it must be demonstrated that there exists an otimal choice of given the strategy of the taxayers. To roceed with this assume that for all, 1 G '( µ ) 0 at the equilibrium value of µ associated with that so the identity G ( µ ) µ can be solved to write the equilibrium roortion of evasion as µ µ ( ). The analysis below requires that µ ( ) is continuous but this is not restrictive. The only circumstances in which µ ( ) can fail to be continuous are a) when the social equilibrium occurs at µ 0 or µ 1 and (i) '( 0) > 1 G and (ii) there are some values of for which µ 0 or µ 1 is not an equilibrium (see the analysis µ 0,1 and G '( µ ) 1. The latter of these has in Myles and Naylor (1995)), or b) ( ) already been ruled out. Excet for these excetional cases, µ ( ) will be continuous and is henceforth assumed to be so. Emloying (iv) of Assumtion 2, the following lemma is immediate and is stated without roof. Lemma 2 The distribution function g ( ; ) and the belief function ( ) Using the facts that ( ) and M ( ) M ; are continuous. ; for [ max{, }, ] constant robability case, the objective of the revenue service can be written max { } g ( )d max, { } R( ) [ t + ft[ M ( ; ) ] c( ) ] [ t c( ) ] g ;, for the + { } ( max, ; )d. (11) t should be noted that because of the assumtion of Nash behaviour, the revenue service takes the reorts of the taxayers as fixed when it otimises. Effectively, this M ; are treated by the revenue service as constant. imlies that, g ( ; ) and ( ) The existence of a solution to the maximisation decision of the revenue service is demonstrated in the next lemma. Lemma 3 There exists * ( 0,1) Proof The continuity of ( ) such that * maximises R ( ) over [ 0,1]. R and the comactness of the set from which must be chosen ensure that a maximum exists. From Assumtion 2, * < 1. To show that * > 0, note that

12 12 R max {, } ft[ M ( ; ) ] g ( )d 0. (12) ; Since the set of tax evaders will be non-emty when 0, the integral in (12) will be over a set of ositive measure for each of whom M( ; ) > 0. Therefore R 0 > 0 which roves the lemma. Corollary 1 The corollary follows directly. With a constant robability of audit, there exists a social equilibrium of verification and reorting. Proof Lemma 1 roved that a social equilibrium existed for any given. Lemma 3 has now shown there exists an otimal choice of. The aggregate reaction of the taxayers is g ;. The choice of the continuous in and R is continuous in M ( ; ) and ( ) revenue service is therefore continuously deendent on the reaction of the taxayers. 0,1. The lower Furthermore, the otimal robability is drawn from the comact set [ ] limit of declarations, K ( ), is continuous in and, over the comact set [,1] 0, has a minimum K( ). The choice set of the taxayers can then be restricted to the comact K,. Under these conditions, the Nash equilibrium must exist. [ ] set ( ) Given the existence of an equilibrium, it is now ossible to characterise the otimal audit robability and to investigate the effect of changes in the economic environment uon this. Differentiating (11) with resect to, the first-order condition for the otimal choice of the revenue service is given by max {, } ft[ M ( ; ) ] g ( ; ) d c' ( ), (13) where the normalisation assumtion 2.iii has been emloyed to write c ' g ; d c'. The interretation of (13) is that the revenue service ( ) ( ) ( ) chooses the audit robability to balance the marginal return from increased auditing of those already evading tax with the marginal cost. t is imortant to note that, as a consequence of the Nash behaviour, the revenue service does not take account of how a change in audit robability affects the reorting behaviour of those already evading tax nor of how the change in robability affects the decision to evade tax. Since an increase in rovides an incentive for evaders to increase their reorted income and encourages marginal evaders to reort their full income, the left-hand side of (13) understates the true benefit from increased auditing. As a result, the Nash equilibrium

13 13 leads the revenue service to audit less than it would choose to do it were a Stackelberg leader. The effects of arametric changes in the tax and fine rates uon the otimal audit robability are derived in the aendix assuming the otimal robability of audit satisfies the inequality f < 1 so that there is some evasion. Under a set of reasonable assumtions, these suggest that the audit robability will increase if the fine rate increase but will decrease when the tax rate increases. The first of these findings is in direct contrast to the results of Kolm (1973) and Christiansen (1980) that show the two instruments are substitutes with the otimal olicy consisting of an infinite unishment and a infinitesimal robability of detection. The reason for this differing conclusions is that in the Christiansen/Kolm analysis the objective is the minimisation of evasion by an authority that controls and f. n contrast, in the resent analysis is chosen to maximise revenue with f taken as arametric and the revenue service, being concerned only with revenue maximisation, is not interested in minimising evasion as an end in itself. The tax rate result is exlained by the observation that the increase in t raises the declaration of each tax evader (although it may add new tax evaders at the margin) and this is sufficient to induce a reduction in audit robability. 4. The otimal audit function As may be exected, a number of additional difficulties arise once the constancy of the audit function is relaxed. Firstly, the exected utility function of the taxayer need not be concave and continuity of the otimal decision may be lost. Secondly, the conditions guaranteeing an interior solution to the evasion decision need to be strengthened; from (3) it can be seen that the shae of the audit function, as well as its level, will be relevant. Finally, the analysis of the revious section was sufficiently general to be alicable whether the social equilibrium was interior or occurred at a corner and also allowed none or all taxayers to be located at corner solutions. With the extension to a general audit function it will not be ossible to achieve this level of generality. n resonse to similar difficulties, Reinganum and Wilde (1986) assumed linear utility, that the second-order condition for taxayer maximisation was valid and that the equilibrium audit function was such that all taxayers had an interior solution to the choice of evasion level. They did not demonstrate whether this interior equilibrium was dominated by audit functions that lead to corner solutions for all taxayers. Of course, such a discrete comarison cannot be conducted with any

14 14 degree of generality. Although restrictive, this is erhas the best that can be achieved in the circumstances. n consequence, the otimal audit function is now constructed on the basis that the utility function is linear in income and that, in equilibrium, the taxayers objective is concave with those who choose to evade being at an interior solution; that is, (3) holds as an equality for all taxayers declaring <. Given these reliminaries, Lemma 4 follows directly from the assumtion that (3) is an equality. Lemma 4,, '( ) < 0 For ( ). Proof f the evasion choice is to be interior, it follows from concavity that this requires E E U U < 0. When, W Z. Hence from (3) < 0 f < 1. Since this U ' W U ' Z at an interior must hold for all and the linearity of utility imlies that ( ) ( ) solution, (3) can only be satisfied if '( ) < 0. t should be noted that this direct argument is at the heart of the analysis of Reinganum and Wilde (1986) and that the decreasing audit robability follows directly from the assumtion of an interior solution. ts consequences is that the,. revenue service chooses an audit function that is decreasing over the interval ( ) The motivation behind the choice of a decreasing audit robability is the incentive it gives the tax evader to increase their reort in order to benefit from a reduced likelihood of detection. Given Lemma 4, the monotonicity of (, ( )) differentiation of (3) with resect to and. Doing this shows that in can be addressed by d d S U ' ' ft > 0, (14) where S < 0 is the second-order condition for the otimisation. Therefore, the decreasing robability of audit ensures that higher income levels are connected with higher income reorts. This monotonicity also has the imlication that the α γ remain valid constructions given for g (, ( )) in (a)-(d) and M (, ( )) in ( ) ( ) but with the relacement of by the function ( ). The monotonicity result also shows that revenue is still given by (11) but again with the extension that is a function of.

15 15 To construct the form of the otimal audit function, evaluate revenue at the δ in the otimal function. The otimal ( ) and consider a small variation ( ) assumed otimality of ( ) imlies that this variation should have no effect uon the level of revenue. Calculating the variation, under the assumtion of Nash behaviour, shows that the latter statement is equivalent to ft, for (,max{, } ] [ [ M ] c' ] g 0, (15) and c ' g 0, for > max{, }. (16) The form of (15) shows that the extra flexibility admitted by the allowing the robability of audit to vary is emloyed to equate the marginal return from auditing to the marginal cost at every reort level. n contrast, condition (13) equates the marginal benefit averaged across income reorts to the marginal cost. Theorem 1 From (15) and (16), the following theorem can be roved. For >, ( ) 0 and lim ( ) 0. Proof > max, the first statement of the theorem can be seen directly from For { } Assumtion 3 and (16). f >, since ( ( ) ), ( ) 0 trivially satisfies (15) and any other choice of ( ) must lead to lower revenue. To rove the g, 0 for {, } second art note that Assumtion 2iv imlies that the distribution satisfies β (, µ, ( ) ) lim 0 h( b, ) db 0 and that at, M for all b > β (, µ, ( ) ). Hence Ψ lim ftm g, giving c'g 0. The second art of the theorem then follows Ψ [ ] 0 from Assumtion 3. Combining Lemma 5 and Theorem 1 shows that the audit function is a, and is then 0 for. ositive but decreasing function over the range ( ) This audit function is efficient in the sense that no resources are sent auditing income reorts that are known to originate from truthful revelation. However, if the revenue service could act as a Stackelberg leader, the reactions of the taxayers (i.e. increased reorts) would tend to raise the robability at each oint and consequently the cut-off oint. The equilibrium does ensure that all those evading tax stand a nonzero robability of being audited in contrast to the cut-off rule of Scotchmer (1987) where dishonest reorts above the cut-off are not audited.

16 16 Turning now to the comarative statics of the otimal audit function, over the range where the function is ositive, effects of variations in the underlying arameters can be found. These are stated as Lemma 5. Lemma 5 The effect of an increase in the fine rate uon the detection robability is given by d df [ M ] t ftm > 0, (17) c'' and the effect of an increase in the tax rate is Proof d dt [ M ] f ftm Directly from (15). > 0. (18) c' ' These results suort the contentions discussed following (13). An increase in the fine rate raises the robability of audit as does an increase in the tax rate. n this model where the audit robability is determined by an indeendent revenue service, an increase in auditing is a comlement to an increase in the fine rather than being a substitute. 5. Conclusions The aer has considered the otimal audit olicy for an indeendent revenue service in a social custom model of tax evasion. The fact that the revenue service could not always distinguish between evaders and non-evaders resulted in a signalling game of incomlete information. For the constant robability of audit it was shown that an interior solution existed to the decision roblem of the revenue service and some comarative statics results were given. These results, which were confirmed for the case of a general audit function, showed that the audit robability was a comlementary instrument to the fine rate and the tax rate. This is in marked contrast to the outcome that arises when a central government agency controls all three instruments. When the audit robability could vary the audit function was roved to be a decreasing function of the income reort, reaching zero at the highest income reort of a tax evader. Given the additional comlication caused by the existence of the social custom, it is surrising that this characterisation is stronger than that obtained without it.

17 17 Aendix To develo the comarative statics of the otimal audit robability, it is necessary to b β, µ,, t, f m,, t, f, emloy (4) to analyse ( ), (3) to study the relation ( ) (8) to determine the equilibrium function µ (, t, f ) to construct g ( ;, f, t) and M (, f, t) to comute the total derivative of (13). From (4), total differentiation gives db dµ db d db df db dt β µ µ and then (a)-(d) and (α )-(γ ) ;. Given this information, it is then ossible br' > 0, (A1) R ( Z ) UW ( ) U β < 0, (A2) R ( Z )[ ] tu ' β f < 0, (A3) R [ U '( X ) fu '( Z )] β t, (A4) R and db d [ 1 t] [ fu '( Z ) U '( X )] β. (A5) R Under the assumtions made to this oint, neither β t or β is signed. However, the analysis of Myles and Naylor (1992) shows that the model redicts results in accord with emirical evidence when fu '( Z ) U '( X ) > 0. Adoting this restriction, β t < 0 and β >0. Emloying the first-order condition (3) shows that d d S m > 0, (A6) [ 1 U ] '( W )[ ft 1] R ( Z ) + R ( W )] where S is the second-order condition for the otimisation, d df d d [ f 1] U ''( Z )[ 1] [ 1 U ] '( W )[ ft 1] R ( Z ) + R ( W )] A mf < 0, (A7) [ f 1] U '( Z ) + U '( W )] [ 1 U ] '( W )[ ft 1] R ( Z ) + R ( W )] A m < 0, (A8) A A A A

18 18 and d dt m t > 0. (A9) The equilibrium identity (8) then rovides the results that dµ µ d h 1 ( β, ) h( β, ) β d β µ d < 0, (A10) dµ h µ t dt 1 ( β, ) h( β, ) βd t β µ d < 0, (A11) and dµ µ f df h 1 ( β, ) h( β, ) β d f β µ d < 0, (A12) where the signs follow from (A1) to (A4) and the stability requirement h( β, ) β d > 0. 1 µ n the comarative statics analysis of (13) two cases can arise deending on whether is less than, or greater than,. Only the case of < will be treated here; the analysis of the other is a simle extension. Since <, g ( ;, f, t) 0 for ( ], and for (, ], it follows that M (, f, t) m ( ;, f, t) otimality condition (13) can then be written as (, f, t ) ft[ m ( ;, f, t) ] g ( ;, f, t ) d c' ( ) 0 ;. The. (A13) Since a reort of must arise from a tax evader with an income of, it follows from g ;, f, t. Emloying these restrictions, total Assumtion 2.iv that ( ) 0 differentiation of (A13) gives and d dt [ m ] gtd + [ ] f m + ft[ m ] g d c' ' ftmd t + ft gd, (A14) ftm d d df ftm d + ft f [ m ] gfd + t[ m ] ft[ m ] g d c' ' ftm d + gd. (A15)

19 19 To sign these, note from (a) that g g t B (, m)[ β m + β µ + β ] + h m db h β m µ 0 m, (A16) B (, m)[ β m + β µ + β ] + h mdb h β m t µ t t 0 m t, (A17) and g f B (, m)[ β m + β µ + β ] + h m db h β µ. (A18) m f f f 0 m f These exressions cature the two effects of a change: it changes the uer limit of the integral and changes the true level of income associated to each reort. Although there is an exectation for an income distribution that h m < 0, this need not necessarily hold in the resent setting since the derivatives here are taken with b constant. To allow clear results to be derived, assume that h m 0. This imlies from (A1)-(A5) and (A10)-(A12) that g < 0 and g f < 0 but still leaves g t unsigned. Returning to (A14) and (A15) it can be seen that the denominator of both exressions is negative. n the denominator of (A15) the first two terms are negative, since an increased fine raises comliance, whilst the third is ositive as it reresents the increased income from the raising of the fine rate. Although the net effect cannot be restricted, there is a strong ossibility that an increase in the fine rate will raise the robability of detection. n the denominator of (A14) the first and third terms are ositive, with the sign of the second being uncertain. The exectation is therefore that an increase in the tax rate will reduce the otimal audit robability. References Akerlof, G.A. (1980) "A theory of social custom of which unemloyment may be one consequence", Quarterly Journal of Economics, 95, Allingham, M.G. and A. Sandmo (1972) "ncome tax evasion: a theoretical analysis", Journal of Public Economics, 1, Alm, J., G.H. McClelland and W.D. Schulze (1992) "Why do eole ay taxes?", Journal of Public Economics, 48, Andreoni, J. (1992) "RS as a loan shark - tax comliance with borrowing constraints", Journal of Public Economics, 49, Baldry, J.C. (1986) "Tax evasion is not a gamble", Economics Letters, 22,

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