imperfect tax compliance and the optimal provision of public ... - SSRN

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# Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research 2003. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA Bulletin of Economic Research 55:1, 2003, 0307– 3378

IMPERFECT TAX COMPLIANCE AND THE OPTIMAL PROVISION OF PUBLIC GOODS* Alessandro Balestrino and Umberto Galmarini University of Pisa, and ChilD, Turin, Italy and University of Insubria, Como, and Catholic University of Milan, Italy

ABSTRACT

Our aim in this paper is to investigate whether the presence of imperfect income tax compliance affects the optimal provision of public goods within a framework in which public expenditure is financed by a general income tax that also accomplishes redistributive goals. We first derive the income tax structure, and then a generalized Samuelson rule. We argue that, under imperfect income tax compliance, it is desirable to distort public-good supply downwards, in the sense that the sum of marginal rates of substitution between public and private consumption must exceed their marginal rate of transformation.

I.

INTRODUCTION

In this paper we extend the self-selection approach to optimal income taxation proposed by Stiglitz (1982) including imperfect income tax compliance and public-good provision. Tax dodging may be accomplished either by avoiding or by evading taxes. In some cases, the taxpayers may be able to hide part of the tax base from the fiscal authorities, without any risk of being discovered but incurring a cost (e.g. paying a lawyer to learn how to circumvent the law); this is what is usually referred to as tax avoidance (or tax sheltering). In other circumstances, the only way to hide part of the tax base is to expose * We thank a referee and Gianni De Fraja for their constructive remarks. Usual disclaimers apply.

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BULLETIN OF ECONOMIC RESEARCH

themselves to the risk of facing an audit and being fined; then, we talk of tax evasion.1 Irrespective of which form it takes, however, tax dodging is an activity which has great relevance in the real world economies: the common practice of assuming that the tax liability can be ascertained and collected at no cost is a very poor description of reality. For instance, the IRS reports that about 17 per cent of the US income tax liability is not paid and that about 40 per cent of US households underpay their taxes (Andreoni et al., 1998). Tax collection costs are also important, adding up to about 10 per cent of tax revenue (Slemrod and Bakija, 1996). In the present paper, we address the following question: what are the implications of imperfect tax compliance for public expenditure? That is, do avoidance or evasion activities generate any deviation from the level of public expenditure which would arise if these activities were not present? Interestingly enough, it turns out that tax dodging implies a tendency towards a reduction of public expenditure. This result thus could be interpreted as offering an argument in favour of the downsizing of State intervention which is not based on a priori reasons, but on a theoretical analysis of the effects of some relevant features of real economies. We assume that the government has at its disposal a general income tax, but does not observe incomes. Thus, it has to rely on reported income, which may differ from actual income due to tax dodging. Initially, we characterize tax dodging as tax avoidance, since this allows for a relatively straightforward presentation of our main arguments. Later in the paper, however, we show that the model can be reinterpreted, with some qualifications, as one of tax evasion. We study a generalized Samuelson rule which extends the modification proposed by Boadway and Keen (1993). They demonstrated that deviations from the first-best rule for public-good provision are useful as long as they can be used to relax the self-selection constraint, thereby allowing a Paretoimproving change in the income tax schedule. We confirm this result and show how the pattern of the deviations is modified to account for imperfect tax compliance. Specifically, we note a general tendency to induce a ‘downward’ distortion, i.e. what is usually called an ‘underprovision’ (relative to the first-best) of the public good. Optimal public-good provision under non-linear income taxation is studied by Boadway and Keen (1993), Edwards et al. (1994) and Nava et al. (1996) among others, while Boadway et al. (1994) investigate the optimal direct-indirect tax mix in a model with tax avoidance. Tax evasion under general income taxation has been studied by Cremer and Gahvari (1995) and Schroyen (1997), while Falkinger (1991) focuses on 1 We will discuss in more detail the differences and similarities between these two types of tax dodging in Section V.

# Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research 2003.


IMPERFECT TAX COMPLIANCE

39

optimal public-good provision with linear income taxation and tax evasion. Hindricks (1999) deals with tax avoidance and public-good provision at the same time, focusing on the case in which non-linear income and commodity taxes can be avoided. Hindrick’s approach, like ours, builds on Boadway et al. (1994); however, our emphasis is on somewhat different issues. In particular, while Hindricks derives the same rule for public-good provision as we do, he does not go on to identify the factors which differentiate this rule from the one that would hold in a standard model with full tax compliance, and does not investigate the connection between tax avoidance and tax evasion. The paper is structured as follows. Section II introduces the model and characterizes consumer behaviour. Section III sketches the derivation of the optimal general income tax in the presence of tax avoidance. Section IV focuses on how public-good provision is affected by tax avoidance. Section V shows to what extent the model with tax avoidance can be re-interpreted as one with tax evasion. Finally, Section VI summarizes.

II.

THE MODEL

The economy is inhabited by two types of individuals, distinguished by their ability. There are n h individuals of type h, h ¼ 1; 2, with w 2 > w 1 , where w h is the wage rate (taken, as usual, to represent ability).2 The utility function is concave, identical across types and defined over a composite consumption good, x, labour supply, l, and a public good, G; it is written u ¼ u(x; l; G). Production is linear and uses labour as the only input; production prices are normalized to unity for both x and G. The government has at its disposal a general income tax, but it does not observe gross income, wl. Rather, it has to rely on reported income, which may differ from wl. Misreporting income is lucrative because it reduces the tax liability without taking risks, but is done at a cost. Let C(H; wl) denote the cost of avoidance,3 where H, 0 ˘ H ˘ wl, is the amount of income which is not reported to the tax authorities. Note that we assume that negative reports (H > wl) are not feasible, since true income can never be negative. We also rule out overreporting (H < 0), which can be shown never to occur in equilibrium. We make several assumptions on C. Firstly, we posit CH > 0 and CHH > 0 for H > 0, that is the marginal cost of tax avoidance is positive and increasing in the level of concealed income; C(0; wl) ¼ CH (0; wl) ¼ 0, 2 Superscripts denote households, while subscripts denote partial derivatives; however, to simplify the notation, we will drop superscripts whenever possible. 3 We follow the common practice of not describing the tax avoidance activity in a specific way. For an example in which tax avoidance is modeled explicitly as a form of arbitrage between taxed and untaxed assets, see Agell and Persson (2000).



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that is full reporting is costless and the marginal cost of concealing the first unit of income is zero; and CH (wl; wl) > 1, that is the marginal cost of concealing the last unit of income is greater than one. Secondly, we assume that C(wl ) ˘ 0, which implies that, for given H, an increase in true income does not increase the cost of avoidance; this is what Slemrod (2001) refers to as the ‘avoidance-facilitating’ value of true income. Finally, we assume that C(:) is homogeneous of degree one in H and wl, i.e. C(H; wl) ¼ C(H; wl) for all > 0; in words, the tax avoidance technology exhibits constant returns to scale. Although this latter assumption is not essential to reach our conclusions concerning the optimal provision of public goods, it allows us to redefine the cost of tax avoidance in a form which is very convenient to solve the model and interpret the results. Let H ¼ (1 s)wl, where 0 ˘ s ˘ 1 is the fraction of actual income which is reported to the tax authorities. Then define the per-unit-of-true-income tax-avoidance-cost as c(s) C[(1 s)wl; wl]= wl ¼ C(1 s; 1). The function c(:) depends on how large is the fraction of reported income; it is independent of wl. It is immediate to see that the assumptions made on C(:) imply the following Lemma 1: cs < 0, css > 0 for s < 1, c(1) ¼ cs (1) ¼ 0, cs (0) < 1. The households choose x h , l h and s h optimally, thereby positioning themselves on a given point of the tax schedule. Following Boadway et al. (1994), we can linearize the after-tax budget constraint at that point. Each linearized budget will be defined by a household-specific marginal tax rate, t h ˘ 1, and lump-sum tax, T h . We can then define the so-called virtual budget constraint as: x h ¼ [1 t h s h c(s h )]w h l h T h :

(1)

Each household maximizes its utility function subject to (1) by choice of x h , l h and s h . The first-order conditions immediately lead to:

u hl u hx

¼ w h (1 t h s h c h );

t h ¼ c hs ;

(2)

where 1 t h s h c h > 0 for all t h ˘ 1, since the second part of (2) implies that t h > t h s h þ c h .4 Notice that c hs has the opposite sign of the marginal tax rate; then, given our assumptions on the cost-of-avoidance function, we have that t h 2 (0; 1] implies s h < 1 and t h ¼ 0 implies s h ¼ 1. In other words, the household underreports its income only when income is taxed 4 To see this, consider the cs curve, which is decreasing in the [0; 1] interval. The optimal s is given by the intersection between that line representing the value of Ð 1curve and a straight Ð1 Ðs t. Now, we have that ts þ c 0 tdz s cs (z)dz < t 0 tdz, since cs (z) < t for z 2 (s; 1) from the first order condition for s. Clearly, if ts þ c < t, then ts þ c < 1 for t ˘ 1.




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at the margin. From the maximization problem, we can derive an indirect utility function in the usual way, denoted v(t h ; T h ; G). In the self-selection models of income taxation, the social planner can observe gross income and, of course, also net income. Thus, the selfselection constraint says that the tax structure must be such that the (gross income, net income) pair of the low-ability household is not attractive for the high-ability household (in the ‘normal’ case in which the redistribution goes from the high- to the low-ability household – more on this later). This will prevent the type-2 households from reducing their labour supply to mimic the behaviour of type 1 households. Here, however, the government can only observe reported income, and thus mimicking takes the form of reporting the same income of the low-ability person, not of earning the same gross income. As a consequence, mimicking may be accomplished either working less or by concealing more income. Then, in the present setting, the self-selection constraint prevents the high-ability households from reporting the same gross income as the low-ability ones. To misrepresent its type, a high-ability household should obey: y 1 s 1 w 1 l 1 ¼ ^s 2 w 2 l^2 y^ 2 ;

(3)

where the ‘hat’ denotes the variables pertaining to the mimicker. (3) tells us that, in order to have the same reported income as the low-ability household, the mimicker chooses either its labour supply or the fraction of income to report, as the two are not independent. Its utility is u^ 2 ¼ u(^ x 2 ; l^2 ; G), and its budget constraint is: s 2 )]w 2 l^2 T 1 ; x^ 2 ¼ [1 t 1 s^2 c(^ ^2

1

(4)

^2

2 ^2

where s ¼ y =(w l ) depends on l because of (3). The mimicker will choose x^ 2 and l^2 to maximize its utility subject to (4). This yields the first order condition:

u^ 2l u^ 2x

¼ w 2 (1 þ c^2s s^2 c^2 );

(5)

s 2 =l^2 . (5) reflects the where we have substituted for @^s 2 =@ l^2 ¼ ^ interdependency between the choice of l and s for the mimicker: it represents the trade-off between the disutility of labour supply and the advantages (net of costs) of income concealment. The mimicker’s indirect utility will be v^2 (t 1 ; T 1 ; G). We can now characterize the behaviour of the mimicker relative to that of the low-ability household, establishing a result which will be useful in what follows.5 5

Our Lemma 2 is an extension of Lemma 1 in Boadway et al. (1994).



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Lemma 2: If x is a normal good, then s^2 < s 1 , x^ 2 > x 1 and w 2 l^2 > w 1 l 1 . Proof: Consider what happens when ^s 2 ¼ s 1 . We have that l^2 ¼ l^20 < l 1 such that w 2 l^20 ¼ w 1 l 1 . Hence, c^2 ¼ c 1 and x^ 2 ¼ x 1 . If x is a normal good, then x^ 2 ¼ x 1 and l^20 < l 1 imply that the indifference curve of the mimicker is flatter than that of type 1 in the (x; l)-plane (this follows because normality of x requires @MRSxl =@l > 0). Also, the slope of the mimicker’s budget constraint is (at least locally) steeper than the budget constraint of type 1, since w 2 > w 1 ; s^2 ¼ s 1 ; w 2 l^20 ¼ w 1 l 1 and c^2 ¼ c 1 . Formally: ul (^ x 2 ; l^20 ; G) ul (x 1 ; l 1 ; G) l^20 and s^2 < s 1 , and thus also x^ 2 > x 1 and w 2 l^2 > w 1 l 1 . & Notice that nothing can be said as to whether l^2 > l 1 or vice-versa; this fact bears important implications for the results that follow.

III.

OPTIMAL TAXES

We study Pareto-optimal arrangements, whereby the policy variables are chosen so as to maximize the utility of one household type, subject to the other type’s utility being constant, plus the self-selection and revenue constraints. We focus on the ‘normal’ case, in which the social planner wishes to redistribute from the high-ability to the low-ability households. Furthermore, we impose the standard single-crossing condition, that the high-ability consumer indifference curves in the (pre-tax reported income, post-tax reported income)-space are flatter.6 The combined effect of these assumptions is that the only binding incentive constraint is that stopping the high-ability type from mimicking the low-ability one by reporting the same income. The optimal policy problem is (the Lagrange multipliers for the three constraints are in brackets): max

{t h ; T h }; G

s:t:

v 1 (t 1 ; T 1 ; G) v 2 (t 2 ; T 2 ; G) Zv 2 ; 2

2

2

2

[] 1

1

v (t ; T ; G) v^ (t ; T ; G); 2 X

n h (t h s h w h l h þ T h ) ¼ G:

[]

(6)

[]

h¼1 6 Adapting the proof provided by Boadway et al. (1994), we can show that the singlecrossing condition holds when the private good and leisure are non-inferior.




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Since the subsequent analysis of public-good provision presupposes that an optimal tax system is implemented, we briefly note a few results concerning the tax rates. Following the procedure outlined by Boadway et al. (1994), it is possible to show that the marginal tax rates are as follows: t1 ¼

n 1

^ 2 (c 1s c^2s ) > 0;

t 2 ¼ 0:

(7)

(7) states that the top-earners should be undistorted and therefore that ^ 2 ; , and they do not misreport their income – see (2); and that, since , 1 n are all greater than zero, the marginal tax rate on the low-earners has the same sign as the term (c 1s c^2s ). This term represents the difference between the marginal cost of (mis)reporting for the low-ability households and for the mimicker, and is strictly positive, since s^2 < s 1 (see Lemma 2) and css > 0 imply that c^2s < c 1s < 0; thus the low-ability households will be taxed at the margin (t 1 > 0).7 An intuitive understanding of this latter result is easily gained by noting that it is a simple consequence of the fact that marginal avoidance costs are higher for the mimicker. Indeed, it is desirable to set t 1 > 0 and to induce tax avoidance – see (2) – just because this will hurt the mimicker more than the true low-earner, and therefore the self-selection constraint will be relaxed. By contrast, setting t 2 > 0 would be useless, as there is nobody interested in mimicking the high-ability type.

IV.

A MODIFIED SAMUELSON RULE FOR PUBLICGOOD PROVISION

The rule for optimal public-good provision can be obtained from problem (6) by manipulating the first-order conditions in a way similar to Boadway and Keen (1993)8 – see also Hindricks (1999): n 1 MRS 1Gx þ n 2 MRS 2Gx ¼ 1 þ

^2

2

1 d (MRS Gx MRS Gx ):

(8)

7 Actually, without the assumption that the cost-of-avoidance function is linearly homogeneous, we could not prove that t 1 is necessarily positive. However, the interpretation of the income tax rules would continue to be valid, and our subsequent results on public-good provision would go through unchanged (indeed, they only depend on the presence of an optimal income tax, not on the specific form taken by the income tax schedule). 8 Actually, Boadway and Keen (1993) derive a Samuelson rule also for the case in which labour, as opposed to consumption, is used as the numeraire, and show that the choice of the numeraire is not without consequences for the analysis. An explicit consideration of these issues would, however, take us too far afield; a comprehensive treatment can be found in Gaube (1999).



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(8) differs from the Boadway-Keen modification of the Samuelson rule in that the MRS for the mimicker and the low-ability household are not evaluated at the same after-tax income (i.e. consumption), as was the d 2 j 1 denote the case in the Boadway and Keen (1993) model. Let MRS Gx x mimicker’s marginal valuation of G evaluated at the low-ability type consumption level. Since x^ 2 6¼ x 1 and l^2 6¼ l 1 by Lemma 2, the second term on the r.h.s. can be decomposed by adding and subtracting d 2 j 1 so as to obtain: MRS Gx x n 1 MRS 1Gx þ n 2 MRS 2Gx ¼ 1 þ

^2

d 2 j 1 MRS 1 ) þ (MRS Gx Gx x

^2

d 2 MRS d 2 j 1 ): (9) (MRS Gx Gx x

The second term on the r.h.s. of (9) (call it the ‘BK-term’) is similar to that in Boadway and Keen (1993), as labour supply is different while consumption (net income) is the same for type-1 and the mimicker. The third term (‘TA-term’, from ‘tax avoidance’) is new, as labour supply is the same but consumption is different. Note that, by Lemma 2, x^ 2 > x 1 ; instead, l^2 can be lower or higher than l 1 . Before discussing and interpreting this condition, we need two pieces of terminology: *

*

we say that G is overprovided (underprovided) with respect to the first best-rule if the sum of the marginal rates of substitution – the l.h.s. of (9) – falls short of (exceeds) the marginal rate of transformation – unity in our case, the first term on the r.h.s of (9);9 we say that G is an L-substitute (L-complement) if MRSGx decreases (increases) as leisure, denoted by L, increases.

We can now decompose our provision rule in three parts. If we consider only the first term on the r.h.s, we just have the standard Samuelson condition. Adding the second term, we have the BoadwayKeen modification, which states that deviations from the first best rule are justified inasmuch as the mimicker’s marginal valuation of the public good differs from that of the low-ability household, due to the difference in their labour supply levels (l^2 6¼ l 1 ); if the BK-term is negative (positive), G should be overprovided (underprovided). The third term is specific to our model and represents the effect of tax avoidance, namely 9 Given that the standard concavity properties are not generally satisfied in this sort of second best policy problems, under- or overprovision in this sense does not imply that the provision level is necessarily lower or higher than in first best. Recently, Gaube (1999, 2000) has, however, identified analytically the conditions which determine under- and overprovision in level terms.




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the fact that the mimicker’s marginal valuation of the public good is not evaluated at the same consumption level of the low-ability household (^ x 2 6¼ x 1 ). Overprovision (underprovision), then, requires that the algebraic sum of the two last terms is negative (positive). What can we say about the sign of the BK- and the TA-term? Taking the former first, note that: i) G can be an L-substitute or an Lcomplement, and ii) the labour supply of the mimicker can be lower or higher than type-1’s labour supply. If the mimicker’s labour supply is lower than that of the low-ability household, then, for equal consumption, the BK-term is positive if G is an L-complement (thus tending to favour underprovision of G); on the other hand, this term is negative if G is an L-substitute (thus tending to favour overprovision of G). This corresponds exactly to what happens in the standard case, that is optimal income taxation without tax avoidance. However, the opposite signs for the BK-term, and thus the opposite implications for the provision of G, are obtained when the mimicker’s labour supply is higher than type-1’s labour supply. Moreover, if taxes can be avoided, net incomes (consumption levels) differ for the mimicker and the low-ability household. This is reflected in the TA-term, whose sign depends on whether the mimicker values G more at her net income level or at the low-ability household’s net income level. Using Lemma 2, we can prove the following: Lemma 3: If G is a normal good, then the TA-term is positive. Proof: If G is a normal good, then: @MRSGx @x

¼

ux uGx uG uxx (ux ) 2

> 0:

Therefore, since x 1 < x^ 2 by Lemma 2, we have that: MRS 2Gx

uG (^ x 2 ; l^2 ; G) uG (x 1 ; l^2 ; G) > MRS 2Gx jx 1 : 2 2 1 2 ^ ^ x ; l ; G) ux (x ; l ; G) ux (^ &

In words, the mimicker’s marginal willingness to pay for the public good (in terms of the consumption good) is higher at its own income level. This means that tax avoidance always tends to favour the underprovision of G. To interpret this discussion of (9), we can adapt the intuition provided by Boadway and Keen (1993). Suppose that we start from a situation in which the public good is provided according to the Samuelson condition. We now reduce G infinitesimally, and simultaneously reduce the income # Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research 2003.


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tax liabilities of both types by their MRS. This way, neither the welfare levels nor the government budget will be affected. Suppose, however, that had the mimicker the same net income as the low-ability household, she would value G more; then, the BK-term is positive, that is the reduction in the mimicker’s tax liability (MRS 1 ) does not compensate its loss from the reduced availability of G. Moreover, we know that the mimicker’s actual MRS, that is the one evaluated at its income level, is larger than the one evaluated at the low-ability household income level, so that the mimicker’s total loss is underestimated by the BK-term: adding the TA-term gives us the correct measure. Therefore, the overall effect of this policy change is that of making the mimicker worse-off: the reduction in public good provision relaxes the self-selection constraint, enabling the realization of a Pareto-improving change in the income tax schedule. A similar intuition applies to the case in which the policy change consists of an increase in the provision of G. In the Boadway and Keen (1993) setup, when preferences are identical, differences in the valuation of the public good arise only because the mimicker enjoys more leisure than the low-ability household (they have the same income, but the mimicker has a higher wage). Therefore, the mimicker values G less than the low-ability household if the public good is an L-substitute, in which case G will be overprovided (by a symmetric argument, it will be underprovided if it is an L-complement).10 In our case, the relations between the various forces at work is less straightforward, for two reasons. First, we do not know whether the mimicker has a larger or a smaller labour supply than the low-ability household; second, they have different consumption (net income) levels. The first effect implies that the sign of the BK-term may be reversed compared to the standard case; the second effect, instead, offers a rationale for distorting the provision of the public good (in the direction of underprovision) which, remarkably, does not depend on L-complementarity or L-substitutability. Indeed, with tax avoidance, underprovision occurs even if G is L-independent (i.e. MRSGx does not depend on L).11 It is easy to see that, in that case, the BK-term is zero, while the TA-term is, as always, positive. We provide an overview in Table 1. The BK-term and the TA-term may have opposite signs (when l 1 > l^2 and G is an L-substitute, or when 10 Boadway and Keen (1993) develop their model without imposing identical preferences, and then derive the correspondence between L-substitutability (L-complementarity) and overprovision (underprovision) for the case in which households have the same utility function in their Corollary 2. 11 The condition of L-independence corresponds to the case in which the utility function is weakly separable between leisure and all the other goods; in a model without tax avoidance that type of utility function ensures that the Samuelson condition holds at the second best optimum (cf. Corollary 1 in Boadway and Keen, 1993; the result is originally due to Christiansen, 1981).




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TABLE 1 Public-good provision BK term

TA term *

Provision of G

l 1 > l^2

L-compl. L-subst. L-indep.

þ 0

þ þ þ

Underprovison Ambiguous Underprovision

l 1 > l^2

L-compl. L-subst. L-indep.

þ 0

þ þ þ

Ambiguous Underprovision Underprovision

0

þ

Underprovision

l 1 ¼ l^2 *Assuming normality of G.

l 1 < l^2 and G is an L-complement), so that the net effect on the provision of G is ambiguous. In these cases tax avoidance tends to weaken the distortion required by self-selection, reducing the extent to which the public good is overprovided (and possibly reverting the direction of the distortion if the TA-term dominates the BK-term). In the other cases, the BK-term and the TA-term work in the same direction, leading to underprovision of the public good.

V.

TAX AVOIDANCE VS. TAX EVASION

So far, we have modelled tax dodging in terms of tax avoidance, rather than tax evasion. As mentioned in the Introduction, the distinction is customary, and can be based on two alternative criteria. The first, and the one popularly used, is based on a legal distinction: both activities have the aim of reducing the tax liability by hiding part of the tax base from the government, but tax evasion is an illegal activity whereas tax avoidance is a legitimate activity. With the former, the taxpayer breaks the law; with the latter, he or she takes advantage of special provisions or loopholes in the tax code. The second criteria is based on the economic concept of risk (Cowell, 1990b, pp. 10 – 14), and it is the one adopted in most of the economic literature. Tax avoidance is costly12 and riskless, whereas tax evasion is risky because taxpayers face the possibility of an audit and, in case of discovered tax evasion, are subject to the payment of a fine. According to this criterion, therefore, tax avoidance can be 12 To conceal income from tax authorities is costly since the taxpayer needs to ‘study’ the tax code, seek advice from tax practitioners, or keep a system of double accounting books, one for regular and one for irregular transactions.



48


either legal or illegal: what only matters is that the taxpayer can get away with it without taking any risk. Having clarified the differences between the two forms of tax dodging, we now discuss to which extent our model and results concerning the impact of tax avoidance on public-good provision can be extended to the case of tax evasion. To this end, we employ the concept of the cost of evasion introduced by Cowell (1990a). We can conveniently solve the household’s problem in two stages. In the first, the optimal labour supply, l~h , is chosen as a function of (1 t h )w h and T h only – in Cowell (1990a) gross income, wl, is exogenous, i.e. l is fixed. In the second, decisions concerning tax evasion are taken.13 This second stage is modelled following the standard Allingham and Sandmo (1972) view of tax evasion as a choice under uncertainty. Each taxpayer faces a probability p, 0 < p < 1, of being audited, in which case tax inspectors costlessly observe her true income and, in case of tax evasion, a fine f per unit of evaded tax is levied, f > 0. Formally, and using the same notation as in Section II, the taxpayer’s net income (consumption) in state d (tax evasion is discovered and punished) and in state nd (tax evasion is not discovered) is given by:14 x hd ¼ (1 t h )w h l~h ft h H h T h ; x hnd ¼ (1 t h )w h l~h þ t h H h T h :

(10)

Letting EU denote the expected utility, the household’s tax evasion problem is then the following: max EU h ¼ pu(x hd ; l~h ; G) þ (1 p)u(x hnd ; l~h ; G); Hh

(11)

subject to (10). Note that here we frame the taxpayer choice in terms of H instead of s ¼ (wl H)=wl as we did in Section II; this choice is only made for expositional convenience, as the two approaches are equivalent. Cowell’s reinterpretation of the Allingham-Sandmo approach makes use of the concept of the cost of evasion, D, which is defined as ‘the monetary amount that the person would just be prepared to pay in order to be guaranteed that he will get away with tax evasion’ (Cowell, 1990a, p. 232). Formally, D is defined by solving the following equation: u(x hnd D; l~h ; G) ¼ pu(x hd ; l~h ; G) þ (1 p)u(x hnd ; l~h ; G):

(12)

13

Cowell (1985) illustrates in detail the restrictions on preferences that allow the separation of total labour supply and tax evasion decisions. 14 Contrary to our general rule, the subscripts d and nd will thus not denote partial derivatives. # Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research 2003.



49

In principle, this can be solved to give the cost of evasion function D(H h ; w h l~h ; G; F), where F ¼ {t h ; T h ; p; f } is the set of fiscal instruments. The tax evasion problem (11) can then be alternatively written as: max u(x hnd D; l~h ; G): Hh

(13)

Cowell (1990a, p. 232) shows that DH > 0, DHH > 0 for t > 0; hence, D(:) is increasing and convex in H as in the tax avoidance case.15 Since DH ¼ p(1 þ f )t at H ¼ 0, the optimal H that solves (13) is strictly positive provided that p(1 þ f ) < 1, and is uniquely determined by the following first order condition: DH ¼ t:

(14)

In (14), the marginal cost of evasion equates the marginal benefit of evasion, in the same manner as the marginal cost of avoidance equates the marginal benefit of avoidance in equation (2). In this respect, the tax evasion and the tax avoidance approaches look similar. There is however a crucial difference: the cost-of-evasion function depends on all policy parameters,16 whereas we assumed that the cost-of avoidance function was independent of them.17 This has two major implications. The first concerns the signs of the comparative statics effects. In particular, tax avoidance is increasing in t, since the marginal cost of avoidance is independent of t, while marginal benefits are increasing in t. As for tax evasion, Cowell proves that both marginal costs and marginal benefits are non-decreasing in t, with the result that tax evasion is, under decreasing absolute risk aversion, decreasing in t.18 More importantly in the present context, the second implication is that the optimal tax rules and the optimal public-good provision rule differ once we move from the cost-of-avoidance to the cost-of evasion 15 Cowell also shows that DHp > 0 and DHf > 0, from which it follows that the optimal H is decreasing in the probability of audit and in the severity of sanctions. In our tax avoidance model in Section II we abstracted from tax enforcement. But, had we considered it, we could have written C(H; wl; A) with A representing a measure of the administrative costs devoted to tax enforcement (e.g. efforts to reduce loopholes in the tax code). Then, it would have been most natural to impose CA > 0 and CHA > 0, just as in the tax evasion case. 16 Actually, it is easy to show that D does not depend on G if the latter is weakly separable from x and l in the utility function. 17 We also assumed that the cost-of-avoidance function is homogeneous of degree one; clearly, it is not easy to find plausible restrictions on the utility function that lead to a linearly homogeneous cost-of-evasion function. However, we already remarked in the Introduction that this assumption is for expositional convenience only and its sole effect is to help us to sign the marginal tax rate for the low-ability households (see fn. 7). 18 This result, which is clearly at odds with empirical observations, represents the wellknown Allingham-Sandmo paradox. The intuition is simple. With a sanction proportional to the evaded tax, an increase in t causes a negative income effect: hence, under decreasing absolute risk aversion, the taxpayer is induced to reduce risk by evading less.



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approach. However, it turns out that, despite these differences, the mechanism leading to the underprovision of public goods is at work also under tax evasion. To see this, suppose that we rewrite the cost-of-avoidance function adding the policy instruments to the list of the arguments, C(H; wl; t; T; G); this way, we may think of it as a reduced form of the cost-of-evasion function D(). Reworking the whole analysis of optimal taxes, it is possible to show that the expressions of the marginal income tax rates now include an additional term reflecting the impact of changes in t and T on avoidance=evasion costs. For example, although there is still no reason to distort the labour supply of the high-ability households for selfselection purposes, it will in general be desirable to do so in order to discourage imperfect tax compliance (e.g. by setting t 2 > 0 if this increases the cost of hiding the tax base from the fiscal authorities). Similarly, the sign of t 1 will depend on a mix of self-selection considerations and the need to increase tax compliance. Once the expressions for the tax rates have been determined, it is possible to use them, exactly in the same way as before, to arrive at a public-good provision rule. We can then show that the original mechanism favouring the underprovision of the public good is still operating: intuitively, Lemmas 2 and 3 continue to be valid, and these are enough to determine the working of the said mechanism. However, there are two additional factors affecting the direction in which publicgood provision is optimally distorted. First, there will again be an extraterm, reflecting the impact of changes in G on avoidance=evasion costs; intuitively, if an increase in G makes imperfect tax compliance less (more) costly, then this yields an independent rationale for underprovision (overprovision). Second, the valuation of the public good is now given by weighted marginal rates of substitution, where the weights account for the fact that the (social) cost of revenue collection also now includes the induced changes in non-compliance costs. So, for example, if the cost of revenue collection is exacerbated by the fact that it helps to discourage tax compliance (that is, if higher taxes reduce non-compliance costs), the social value of public expenditure is reduced; whether this will result in a tendency to underprovision depends however on the effect of the weights on the costs side, and in particular on their relative impact on the valuation of the public good by the mimicker and by the true low-ability households. Since we argued that our results on public-good provision under tax avoidance carry over, to some extent, to the case of tax evasion, it may be of same interest to conclude by comparing them with those obtained by Falkinger (1991) in an early contribution to the subject. In a model with linear income taxation, identical consumers and fixed labour supply, Falkinger shows that if G is normal, then ‘tax evasion may lead to less public expenditure but may also imply a higher optimal level of public good provision’ (Falkinger, 1991, p. 130). The conditions he # Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research 2003.



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derives for under- or over-provision involve the third derivative of the utility function. In our model, instead, normality of G implies that imperfect tax compliance unambiguously favours under-provision of the public good. Falkinger also shows that if G is neither normal nor inferior (a public good with zero income effects) tax evasion does not affect the Samuelson rule. In our model, the TA-term is zero if G has no income effects, but we obtain Falkinger’s result of no public-good distortion only if there is L-independence (which produces the same effects as Falkinger’s assumption of fixed labour supply).

VI.

CONCLUDING REMARKS

In the real world, people often do their best to evade or avoid taxes, and most governments fight a constant battle against these activities. While several early and current contributions on optimal taxation have focused on the case in which the tax agencies ascertain the tax liability at no cost, there is now a well-established body of knowledge on the economics of tax evasion and avoidance. We have built on this stream of work in order to investigate the question of whether imperfect tax compliance affects the optimal provision of a public good. At a time when the legitimacy of government intervention is questioned, and a thorough reform of the Welfare state is called for, it is interesting to know whether the explicit consideration of imperfect tax compliance gives us some theoretical reasons for advocating the expansion or the reduction of the scope of public action. We have approached this issue using a rather general framework in which the main instrument for redistribution is the non-linear income tax. Our analysis suggests that it is optimal to reduce public-good provision below its first-best level and also below the level which would arise in a second-best world with full tax compliance. The main reason is that, under a simple assumption of normality of private and public consumption, underprovision of the public good relaxes the self-selection constraint, thereby enabling a Pareto-improving change in the income tax schedule. This follows because the mimicker, that is the high-ability household which modifies its level of reported gross income in order to misrepresent its type, ends up having a larger disposable income than the true low-ability type, and therefore has a stronger preference for the public good. This way, underprovision will hurt the mimicker more than the true low-ability household.

REFERENCES

Agell, J. and Persson, M. (2000). ‘Tax arbitrage and labour supply’, Journal of Public Economics, vol. 78, pp. 3– 24. # Blackwell Publishing Ltd and the Board of Trustees of the Bulletin of Economic Research 2003.


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