a fairness approach to income tax evasion

Journal of Public Economics 52 (1993) 345-362. North-Holland

A fairness approach to income tax evasion

Massimo Bordignon”

Catholic University of Milan, Largo Gemelli I, 20123 Milano, Italy

Received June 1990, final version received April 1992

The tax that a taxpayer wishes to evade is determined on the basis of his perception of the fairness of his liscal treatment, with respect to both governmental supply of public goods and the perceived behaviour of other taxpayers. The coercive powers of the State and the taxpayer’s attitude toward risk determine the extent to which this desired level is undertaken in practice. This approach is able to produce implications for the relationship between public expenditure, tax rates and tax evasion which are more consistent with empirical evidence than the results of the conventional portfolio choice approach.

1. Introduction

A common feature of the economic literature on income tax evasion is that it views the relationship between taxpayers and the state simply as one of coercion [Cowell (1990)]. Taxpayers wish to evade their income tax entirely and the only reason they might not do so is that there is some non-zero probability of being caught and punished by the government. The main problem with this approach is that it does not easily come to terms with observed tax behaviour. First, it is hard to reconcile this view with the high rate of tax compliance experienced in most countries [Graetz and Wilde (1985), Hansson (1985), Skinner and Slemrod (1985)] and with the experimen tal literature which suggests that some people never evade, even when the tax evasion gamble is clearly better than fair [Baldry (1986)]. Second, the portfolio choice model of tax evasion predicts, under reasonable assumptions on preferences and penalty system, that the level of evaded tax is negatively sloped in the tax rate [Yitzhaki (1974), Christiansen (1980)]. This prediction is counter-intuitive and contradicted by most of the empirical [Clotfelter (1983), Slemrod (1985), Crane and Nourzad (1986)] and experimental

Correspondence to: M. Bordignon, Department of Economics, Catholic University of Milan, Large Gemelli 1. 20123 Milano. Italv.

*This paper is a shortened version of chapters V to VII of my Ph.D. thesis. I wish to thank P. Weller, C. Beretta, A. Petretto, M. Marelli, F. Cowell, C. Frazer, G. Myles, P. Natale, P. Giarda and two anonymous referees for helpful and constructive comments. Errors remain my own.

0047-2727/93/$06.00 0 1993-Elsevier Science Publishers B.V. All rights reserved

346 M. Bord;gnon, Income tax evasion

[Friedland et al. (1978), Baldry (1987)] evidence. Third, experiments and attitude surveys’ suggest that tax evasion is positively affected by the perceived inequity in the trade-off between tax payments and public expenditure benefits [Spicer and Lundstedt (1976), Wallschutzky (1984), Becker et al. (1987)], and by the perceived evasion by other taxpayers [Spicer and Hero (1985), Porcano (1988)]. These features are not rationalized by standard individualistic models of tax evasion.

These shortcomings suggest that a more satisfactory explanation of tax evasion may be attempted by taking into account other motivations beyond selfishness. With few exceptions, economists have so far seemed reluctant to follow this route. Cowell and Gordon (1988) and Falkinger (1988), for example, attempt to explain the links between public expenditure and tax compliance by introducing a government financed public good in the standard portfolio choice model of tax evasion. However, their results imply a relationship between public expenditure and tax evasion that goes in the opposite direction to that suggested by the empirical literature. Gordon (1989), building upon Benjamini and Maital (1985), introduces ethical and social norms supporting tax compliance in terms of fixed ‘stigma’ costs on evasion. However, stigma costs are exogenous to the analysis so that they can at best rationalize, but not explain, differences in tax behaviour across consumers or social groups.

In this paper we propose a different approach, which builds directly upon the weaknesses of these previous attempts. We introduce ‘fairness’ considerations as an additional motivation to the evasion decision, but we rationalize the ethical norms supporting tax compliance by making them dependent on tax structure, public expenditure and perceived evasion by other taxpayers, This is done by extending and formalizing the fair-exchange approach of Spicer and Lundstedt (1976). According to this approach, the taxpayer perceives his relationship with the state not only as a relationship of coercion but also as one of exchange, where he gives up purchasing power in return for publicly supplied goods. In this paper it is assumed that the taxpayer can compute the ‘fair’ terms of trade between his private consumption and government provision of public goods. If the terms of trade offered by government through the tax system differ from these ‘fair’ terms of trade, the taxpayer wishes to evade in order to re-establish fairness in his relationship with the other actors of the fiscal system. However, since evading is risky, we allow the taxpayer to evade less than his desired level if he perceives that by doing otherwise he could put in danger this desired goal.

With this approach we are able to show that: (1) a share of the population do not evade taxes even if it would be in their self-interest to do so; (2)

‘No attempt is made here to provide a discussion of the results and methodology of the sociological and psychological literature on tax evasion. For recent critical surveys see Jackson and Milliron (1986), Weigel et al. (1987), and the relevant chapter in Cowell (1990).

M. Bordignon, Income tax evasion 347

evaded tax is an increasing function of the tax rate, at least over some range; and (3) public expenditure effects on tax compliance are in line with the empirical literature.

The paper is organized as follows. Section 2 presents the basic structure of the model, which draws heavily on Cowell and Gordon (1988), and discusses the links with the pre-existing literature. Section 3 discusses equilibrium and performs a number of comparative statics experiments in the simpler case of exogenous public expenditure. Section 4 introduces endogenous public expenditure by means of a simple example. Section 5 discusses more fully equilibrium and comparative statics results in the case of endogenous public expenditure and identical individuals. Section 6 closes the paper by indicat- ing open questions and possible extensions. All proofs are in the appendix.

2. The model

2.1. Preliminaries

We use the Cowell and Gordon (1988) model of income tax evasion with public expenditure extended to two types of identical individuals, indexed by i = 1,2. This extension is introduced because we wish to investigate the effects of income differences and distributional characteristics of public expenditure on tax behaviour. For simplicity, it is assumed that there are N/2 individuals of each type. When needed, an individual is distinguished from his type by adding a suffix, h. For example, Cih is the private consumption for individual h of type i, where h = 1 . . N/2 and i= 1,2. Each type’s preferences can be represented by a strictly concave, twice differentiable utility function which obeys the axioms of expected utility theory. Such a utility function, denoted U’(‘), i= 1,2, is defined on two types of goods, private consumption, Cih, and a public good, G. The utility function of a representative consumer h of type i is

Ui=Ui(Ci,,G), i=1,2,h=l...N/2.

Each type is endowed with an exogenously given income, Ii, i= 1,2. The supply of G is proportional to the amount of the private good used as input in its production:

where qih is h’s private consumption given up for the production of G and $(*) is the marginal rate of transformation (henceforth MRT). Note that in

348 M. Bordignon, Income tax evasion

(2) the MRT is a function of the total number of individuals, with dl(/(N)/dN>O. This can be thought of as a crowding assumption: G is not pure and some amount of rivalry is present in its consumption. Following Cowell and Gordon (1988) two limiting assumptions are imposed on the production function for the public good:

lim l/$(N) = 0; lim N/$(N) = l/Y >O. (3) N-02 N-m

In large economies, where N-+co, (3) implies that aG/aqih= l/II/(N) =O; that is, for each h, G is fixed and cannot be modified by unilaterally changing h’s contribution. In what follows, we will always consider an economy where N is ‘large’.

2.2. The tax evasion choice

Tax evasion is motivated by the attempt of the taxpayer to restore fair terms of trade with respect to the government and the other taxpayers. The coercive powers of the state play a role only in determining the extent to which desired tax evasion is undertaken in practice. Taxpayer behaviour is thus modelled as the result of a constrained maximization process, in which the taxpayer maximizes an expected utility function - which takes into account the riskiness of evading - subject to the constraint represented by the amount of (feasible) desired tax evasion - which takes into account the fairness of evading. Calling pi the probability of detection for an individual of type i, IL (n>O) the penalty imposed on tax evaded, and t the proportional tax rate, taxpayer h’s problem is

max( 1 -pJU’((l - t)li+xih; G) +piUi(( 1 - t)Zi--xi,,; G) Xih

(44

S.t. O 5 Xih 5 Xih( t, Gy XT, Xj), (W

where xi,,, the choice variable, is tax evaded by individual h. Clearly, the solution to the expected utility maximization in (4a), without taking into account the constraint in (4b), is tax evasion as studied in the conventional portfolio-choice model. We will indicate by x:~ this standard solution. The novelty of our approach derives from the fairness constraint, Xih, which is determined endogenously in the model as a function of the fiscal parameters t and G and tax evasion by the other taxpayers (x:,xj). Leaving to the next section the formal setting for Xi,,, eqs. (4a) and (4b) can be used to present intuitively our approach and to discuss the links with the previous literature.

44. Bordignon, Income tax evasion 349

To this aim note first that the solution to problem (4) can be written simply as iih=min(xz; Xi,,)_ This equation illustrates the intuition behind our approach. For, suppose that & > Xih; in this case selfishness dictates a higher level of tax evasion than h would consider fair. But, according to the fairness approach, no individual would ever evade more than Xi,,, simply because he would consider it ‘unfair’ to do so. Therefore, &,,=Xih. Suppose next that & < xi,. Then h would be prepared to evade up to Xi,,; but evading any amount above x& could be self-defeating in terms of h’s own perceptions of the risk involved, thus undermining the re-establishment of fair terms of trade that he is trying to achieve by evading. Then, &,=x$,. Second, our approach encompasses the standard portfolio choice model; as (4) makes clear, the latter is simply a special case of the former. Third, in line with what has been suggested by several authors [e.g. Sen (1977)], we model ethical behaviour in terms of a system of constraints imposed on the pursuit of selfish goals. This is intuitively convincing, but it does not allow, differently from the stigma costs model, trade-off between morality and selfish benefits. This is partly a modelling choice: but it should be noted that there is some experimental evidence which supports this choice [Friedland et al. (1978)].

2.3. The fairness constraint

Spicer and Lundstedt (1977) argue that there are two dimensions according to which a taxpayer h could judge the fairness of the terms of trade offered to him by the state. First, h may perceive his terms of trade with government as ‘unfair’ simply because the quantity/quality of goods received from government are considered inadequate with respect to his tax payment. Second, h may perceive the distribution of the tax burden across taxpayers as unfair, either because the tax structure itself is considered unfair or because the other taxpayers evade taxes, or both. This suggests that the tax h considers fair to pay should be determined as a function of three elements: public good supply, G; tax rate, t; and perceived tax evasion by the other taxpayers. Let us call 4: this fair tax; the desired level of tax evasion is then simply the difference between the sum h is asked to pay and the sum he would wish to pay, i.e. zih= tli-9%.

In order to model qz we turn to the literature on private supply of public goods. We do so because in our approach the fair tax can be thought of as an example of voluntary financing of a public good. An ethical rule that has been widely studied in this literature is the so-called Kantian rule, according to which an individual considers it fair to pay as much as he would wish other individuals to pay [Laffont (1975)]. The Kantian rule has a number of desirable characteristics for our purposes; it is a simple and well-known rule of


morality, so that one can assume that individuals know and are influenced by it, and it is easy to treat in a formal model. However, it is also an extreme rule of morality. Following Sugden (1984), we will then weaken it by introducing reciprocity in tax behaviour. We do so by assuming that a taxpayer considers it fair to pay his Kantian tax if and only if he perceives that everybody else does the same and that he revises his desired payment otherwise.

The Kantian tax is modelled by using Bordignon’s (1990) approach, who shows that an individual behaving according to the Kantian rule selects fair contributions to a public good by following a two-step process. First, he chooses an optimal amount of public good supply and a unique level of private good consumption for all the members of the community as if he were endowed with average income and faced a price of the public good equal to the MRT divided by N. Second, he distributes the burden of paying for the selected amount of public good among individuals so as to equalize private consumption. The latter results, which may seem unrealistic even for an ethical rule, are a consequence of the assumption of fixed endowments and will be carried on into the present context.2

Extending this approach to our model, it has to be taken into account that in this case what has to be chosen is the fair price to be paid for the given G supplied by the state. Thus, we model the process of reaching a Kantian contribution by assuming that first the taxpayer selects a fair price for G as if he were endowed with average income and had to pay a ‘price’ for G equal to the MRT divided by N. By multiplying this fair price for G we get the contribution that, on average, the taxpayer would wish that all individuals paid to the state. Finally, we redistribute this average contribution across individuals so as to equalize private consumption. As for the fair price itself we assume that it is equal to the taxpayer’s marginal willingness to pay:

w’( G, i) = U&/U;( G; i- 'PC). (5)

That is, the price a taxpayer of type i perceives as fair, wi, is equal to his marginal rate of substitution evaluated at G and at the level of private consumption which would result if he were endowed with average income, i, and had to pay for G a ‘price’ equal to the MRT divided by N. By multiplying wi by G and by redistributing the resulting amount across individuals so as to equalize private consumption, as indicated above, we get

‘Formulating rules of fairness for economies with variable labour supply raises some difficult theoretical issues, because one should take into account the fact that individuals may feel that they are entitled to a larger share of private consumption if they have worked harder or risked more than other individuals. For some suggestions on how to tackle these problems in a formal model, see Bordignon (1990).

M. Bordignon, Income tax et&on 351

qj=(Zj--)+w’(G,r)G, i=1,2;j=l,2, (6)

where qj is the Kantian contribution that an individual of type i would wish that an individual of type j paid to the state.3 Using (6) we then define the ‘Kantian taxes’ as the average rates resulting by dividing qj by j’s income:

t; = q& i= 1,2; j= 1,2, (7)

where t: is the Kantian rate of tax. Assume now that each individual knows the average level of tax evasion inside each type, which in terms of large aggregate may not be too unrealistic. Reciprocity considerations can then be introduced by assuming that the fair tax an individual h of type i wishes to pay depends, in addition to his Kantian tax, on the difference between the Kantian tax that h would like the other taxpayers to pay and what they actually pay on average. Selecting a linear specification for simplicity, we write the fair tax as

(8)

where x: is the average level of tax evaded by all the individuals of type i except h, and xj is the average level of tax evaded by all individuals of type j.

Bf, t = 1,2, 05$5 1, PI +fl’, 5 1, are the reciprocity weights on other contributions.4 Note that (8) implies that h would wish to pay his Kantian tax only if everybody else did the same. From (8) we obtain the desired level of tax evasion as a function of G, the tax rate t, and the level of tax evasion by the other taxpayers:

The linear specification of zih in (9) offers a clear advantage: depending on the values attributed to the reciprocity weights, it allows us to obtain, as special cases, alternative models of tax evasion. For example, for /?\ =pi =O, we have a case of pure Kantian behaviour and for /?y = 1 and /I\ = 0 we get a

3Eq. (6) requires individual i to have accurate estimates of the production function for the public good. This certainly requires more information than a taxpayer typically possesses. However, many researches show [see Spicer and Lundstedt (1976)] that people make judge- ments regarding the fairness of the relationship between what they pay to the state and what they receive back. Furthermore, most of the so-called state-provided public goods are really private goods with public good characteristics. For this type of goods our assumption is not too unrealistic, because typically such goods are also privately supplied. Then, simply by looking at the existing terms of trade in the private market, an individual is likely to obtain a rough knowledge of the production function for these goods.

41n (7) we allow the reciprocity weights to differ according to the type because, as suggested by the empirical literature, a taxpayer may be differently influenced by the tax behaviour of individuals belonging to his and other social groups.


pure reciprocity model inside a group. For future reference, note that in the limiting case fii + /?\ = 1 (or 9’~ pi/( 1 --pi) = 1) the Kantian taxes disappear and we get a model of reciprocity among social types corrected by differences in income distribution [this can be easily verified by substituting (6) and (7) in (9)].

Finally, observe that there is no reason to expect Oszihs tli. Since a taxpayer would not in general evade a negative amount of income or be able to force the government to pay him a subsidy, we choose to model these two cases as corner solutions. We then define the fairness constraint Xi, as

x, = Zih, if 0 5 zih 5 tfi;

xi, = 0, if zih < 0; (10)

xi, = tli, if zih > tli.

Substitution of (10) into (4b) closes the model. A unique solution to (4) and the second-order condition are guaranteed by strict concavity of the utility function and by the convexity of the constraint. Furthermore, holding G constant, O<xyh<tZi if (l-_~,)-rrp~~~~>O and (l-pi)@(l,;G)+ piUi(( 1 -zt)l,; G) <O, where the suffix in the utility function indicates the partial derivative with respect to private consumption [Allingham and Sandmo (1972)]. In what follows these two conditions will always be assumed to hold.

3. Equilibrium and comparative statics analysis with exogenous public expenditure

In order to investigate the results of the model we have to solve it for the equilibrium. We consider first the special case where public expenditure is taken as exogenous, in the sense that government chooses t and G independently. The more general case with endogenous expenditure will be investigated later. Let us then indicate by A = {p1,p2, t,n, G} the parameters selected by government. Given A, an equilibrium in the model is a vector of evaded taxes, xz, such that xz=&,, Vi, Vh. That is, an equilibrium is a vector of evaded taxes such that each individual, taking the behaviour of any other agent in the economy and the parameters set up by government as given, maximizes his (expected) selfish utility function subject to his fairness constraint. Thus, an equilibrium is a Nash equilibrium in evaded taxes of a non-cooperative game among taxpayers. By using the standard techniques of game theory [Friedman (1986)] it is then easy to establish existence. Furthermore, if 9’~ 1, i= 1,2, the equilibrium is unique and symmetric (i.e.


~ih=~.i, Vh, i= 1,2).’ The economic intuition behind this result is clear. As noted above, if 9’= 1 the Kantian taxes disappear from the computation of desired tax evasion and we are left with a simple model of reciprocity across social types. It is then obvious that in such a model we may have multiple equilibria (but see Proposition 2). In what follows we stick to the case 9’~ 1, with both jIf>O, i, t= 1,2.

If each individual of the same type evades the same amount of tax in equilibrium, this can only lie in one of the four regions derived by combining constrained and unconstrained behaviour. The comparative static results of the model depend on the region in which the equilibrium lies. Changes in p1,p2 and rr will have the usual negative sign in the region where both types are unconstrained, and no effect in the region where both types are constrained. Note, however, that in the regions where only one type is unconstrained, say type j, a change in pj or in n will also affect the tax evaded by type i, through the reciprocity weights. In order to sign the effects of changes in t and G on tax evasion, extra assumptions on preferences have to be imposed. In what follows we make the following assumptions:

Assumption I. Taxpayers’ preferences are characterized by decreasing absolute risk aversion.

Assumption 2. There are no income effects on the demand for the public good.

The two conditions are crucial to sign, respectively, the effects of changes in t and G on x0 [Yitzhaki (1974) Cowell and Gordon (1988)]; they are, however, standard assumptions in the literature on public goods and consumer behaviour under risk.

Thus, by imposing Assumptions 1 and 2, solving for the symmetric equilibrium, and totally differentiating with respect to t and G, respectively, we obtain the results summarized in tables 1 and 2. Inspection of table 1 reveals that the effect of an increase in t on tax evasion is unambiguously positive in the fully constrained region. At unchanged G the taxpayers, who were already constrained in their behaviour by considerations of fairness, perceive the increase in t as ‘unfair’: thus, since an increased amount of tax evasion pays off in terms of expected costs and benefits, they react by increasing tax evasion. This result offers an explanation for Baldry’s (1987) experimental evidence.

The results in table 2 indicate that the effect of a change in G on tax evasion in the constrained region can be positive or negative according to

‘This follows from the fact that with exogenous public expenditure fl, +b\ < 1 ensures that each individual’s best reply functions is a contraction; see Friedman (1986, p. 46).


Table 1

Effects of changes in t.

Type 1 Type 2

Constrained

Unconstrained

Unconstrained

Type I

Table 2

Effects of changes in G.

Type 2

Constrained 1-- Unconstrained

Constrained I II

dx,/L?GsO symmetric iiXJ3G 2 0’“” 2’ to III

Unconstrained

._

the quantity elasticity of the fair price si=(wbG/wi), i= 1,2. In fact, if I.?< I a 1 percent increase in G would reduce the fair price by less than 1 percent: then the total income that individual i considers fair to pay increases as G increases and consequently desired tax evasion falls; vice versa for the case l&ii > 1.

The main limit of the results in this section is that for G exogenous we cannot predict where the equilibrium lies with respect to different values of the parameters selected by government. In the next two sections we address this question. Here we limit ourselves to stating the following result (see the appendix):

PropositionI. Suppose (1) U’=U’=U and S1=S2=9<1; (2) s1=s2=s, and that Assumption I holds. Then, a regime where rich people are unconstrained and poor people are constrained by fairness considerations is impossible. That is, either region II or region III disappears.

Hence, if differences in preferences and in opportunities for evading across social types are not relevant, poor people can never be constrained if rich people are not constrained by fairness considerations. Proposition 1 then suggests that, if differences in preferences and in opportunities for evading


are small, equilibrium will shift from one region to another, following a change in the rate of tax and/or in public good supply, without ever entering region II (for I, > II). The example in section 4 confirms this conjecture.

4. An example: Cobb-Douglas utility functions

In this section we present a simple example of the model discussed in the previous sections, but with public expenditure linked to tax revenue. In what follows we assume that s1 =s2=s and S1 = 9* = 9. Government selects the rate of tax according to the rule t = 6 YG/i, where 15 6 5 l/( 1 -s). The idea is that the government decides, ex-ante, the supply of G and selects t so as to guarantee a balanced budget on the basis of its beliefs about expected tax evasion. 6 is the parameter that captures these beliefs. If government expects no tax evasion, 6 is set equal to 1; if government expects some tax evasion, 6 is raised proportionally, and if government expects everybody to evade the taxes completely, 6 = l/( 1 -s). Both types of individuals are characterized by the same log-linear utility function:

U’=lOgC,+crilOgG, h=l,...,N/2; i=1,2. (11)

The parameters ai, 0 <ai< 1, capture the differences in preferences for G across social groups. Note that (11) implies separability of preferences in public and private goods. This entails that x: is independent of G. Without loss of generality, let I, &I1 and let us simplify the notation by writing I, =I and I, = kl, with k 2 1. Then from (1 l), and (3)-(10) we can write the choices of the two types as

f,=Zmin[s(l-t)/n;max{0;(1-_)(9k-1)

+(l-9)(1-cr,(l-t/@)(l+k)/2+9x,)],

(12) a,=Zmin[s(l-t)k/n;max{O;(l-t)(&k)

+(l-9)(1-a,(l-t/6))(1+k)/2+9x,)].

In this simplified version of the model the decision to evade by the two types depends on four parameters in addition to s: the tax rate, t; the distribution of preferences for G, al, az, . the income distribution, k; and the reciprocity weights, 9. Using (12) we can investigate the characteristics of equilibrium corresponding to different values of the four parameters. Let us first assume 9< 1 so that G matters in the fairness constraint. In this case the model offers a number of interesting results. For example, it is easy to check that k= 1 and aj< ai entail gjzgi everywhere: that is, as suggested by


experimental evidence [Becker et al. (1987)], with identical incomes, individuals with a higher preference for G would never evade more than people with a lower preference. Similarly, in line with Proposition 1, if k> (9A,+A,)/(9A,+A,), [A,=(l-ai(l-t/6)), i=1,2] the case of ?r=Xr and iz =x: is not possible. This also implies that poor people will tend to evade more than rich people, in absolute terms, at low levels of t, while the opposite will hold at higher levels of t. Finally, note that xy is decreasing and Xi increasing in t, and if t56(D- 1 +Cr)/(D6+cC) [Cr=(ccr +a,)/2 and D=(s+ n)/rr] at least one type is constrained in his tax behaviour. That is, the higher is the average preference for G, the higher t must be for individuals to enter into the unconstrained region. This suggests that individuals should be constrained in their tax behaviour at low levels of t and unconstrained at high levels, which also implies that tax evasion may become negatively sloped in t only for relatively high levels of the tax rate. As we will see in the next section, this relation between tax rates and tax evasion also emerges in the case of endogenous public expenditure and non- separable preferences.

Let us then finally consider the extreme case where the Kantian tax, and therefore public expenditure, do not play any role in the fairness constraint, 9= 1. Recall that in the previous section we used the condition 3 < 1 to establish uniqueness of equilibrium. But that condition was only a sufficient condition as the next result shows (see the appendix):

Proposition 2. Suppose Y= 1. Then, if k > D, the only symmetric equilibrium for any t is one where ?I = xy and 1, = 2, = 0.

k>D is a mild condition. In fact, since s < 1 and rr is usually larger than 1, D=(s+x)/rc is smaller than 2. Then if 9= 1, a small amount of income inequality is enough to enforce in this example an equilibrium where rich people never evade and poor people are never constrained in their tax behaviour. For 1 <k < D there are multiple symmetric equilibria; an equilibrium where ~?r =xy and i2 = Xz =0 is still possible but there are also equilibria for any value of 0 5 xi < xp, i = 1,2.

5. The general case: endogenous public expenditure

Let us now turn to the analysis of the model with endogenous public expenditure. Space constraints force us to discuss only the simpler case with identical taxpayers.6 Let us first derive an expression for G in terms of t and individuals’ tax behaviour. Following Cowell and Gordon (1988) let R be

6For a detailed analysis of the general case with two types of taxpayers, see Bordignon (1989).


total expected tax revenue and R-, total expected revenue if h fully paid his tax. With identical taxpayers, R_, is

R -,, = NCtI - c(p)1 - c E(x,J, k#h

(13)

where E(*) is the expected value operator and c(.) is the average cost to enforce probability of detection, p. Of course, R = R _h - E(xh). Using (2):

G = N[tl- c(P)l/$(N) - 1 EC ) [k+h xk]/

‘h(N) - [E(Xdl/‘!‘(N). (14)

Taking the limit of (14) for N*cc, using (3) and the fact that for a large N the probability of being detected tends to coincide with the actual share of taxpayers caught evading, we get

G = [tZ -c(p) - sx*]/!Z’, (15)

where x* =I k+h xk/(N - 1). By substituting (15) in (4a) and (4b) we eliminate G and express h’s maximization problem as a function of the parameters set by the government, A’ = [p, TC, t], and of perceived tax evasion by the other individuals. It is easy to check that most of our previous results will go through even in this more general case.’ The main difference is that with endogenous G, taxpayers’ interdependence also derives from the effects that taxpayers’ choices have on tax revenue and therefore on G. To see the implications of this for the evasion choice, let us determine the sign of the optimal response of taxpayers to a change in t (and therefore in G) in a symmetric equilibrium. Let us assume first that the equilibrium lies in the region where all taxpayers are unconstrained by fairness considerations. In this region, with identical taxpayers, our model replicates that of Cowell and Gordon (1988); under Assumptions 1 and 2 above their main comparative statics result can be stated as follows [see Cowell and Gordon (1988, p. 312)]:

sign (ax’/&) = sign (w(G) - Y). (16)

To interpret this condition, note that w(G*) = Y identities the Pareto- optimal level of G. In fact, by (3) Y = $(N)/N; therefore w = Y can be written

‘In particular, note that, by invoking (3), the usual Allingham and Sandmo (1972) conditions would still imply O<xi< tl. Furthermore, it is easy to prove existence of the equilibrium, and one can impose conditions which would guarantee uniqueness. See Cowell and Gordon (1988) and Bordignon (1989).


as Nw=$(N), which is the Samuelson condition for efficiency in public good provision. Eq. (16) then implies that evaded tax in the unconstrained region is increasing (decreasing) in t if G < G* (G > G*) so that x0 obtains a (global) maximum at G= G*. This result is clearly in sharp contrast with the fair- exchange approach and the empirical literature supporting it.*

Let us now consider the opposite case of an equilibrium lying in the fully constrained region. From (+OO) the fairness constraints in a symmetric equilibrium with identical individuals is simply

zZ=max[O;tl-w(G)G]. (17)

Assuming first X>O, substituting (15) into (17) and differentiating with respect to t, we get

ax/at=z- T[I(I -s)~y(i +T), (18)

where T = W(E+ 1)/Y and, as above, s=(wcG/w). To sign (18) note that if we assign some weight to the Kantian taxes (i.e. fl< l), local stability of the Nash equilibrium requires 1 > sT;~ therefore the denominator of the second term in (18) is positive. Hence if IsI > 1, T ~0 and &i/at >O. If 1.s < 1, we get an ambiguous sign. The intuition behind these results is straightforward. From (9) an increase in t has two effects on z,,: a direct one and an indirect one through the effect of the change in t on G.” The first element in (18) captures the direct effect which is always positive in the fully constrained region, and the second element the indirect one, which hinges on the elasticity of w (see section 3). Note, however, that even if IE~< 1, &/LJt<O

would require (~1 <(w- Y)/w; this can only be possible of G is strongly underprovided. Thus for G 2 G*, c?Z/& > 0 certainly, while for G < G* we may get either sign. However, this result was obtained by assuming X>O. It is now time to inquire for which level of G this is possible. To see this, let us fix G at G and use (15) to rewrite (17) as follows:

.C=max{O;[(Y-w(G))G+c(p)]/(l-s)}. (19)

Suppose, for simplicity, that enforcement costs are negligible with respect

*The same perverse results emerge in the work of Falkinger (1988) and Cowell (1992) who both attempt to capture issues of equity in a tax evasion context by simply introducing public expenditure or inequality index in a selfish utility function. This implicitly supports our attempt to capture issues of fairness in terms of a constraint on selfish utilty maximization.

‘Local stability of a symmetric Nash equilibrium with identical individuals requires - 1 i d&,/ax,< l/(N - 1) at the equilibrium [see Cornes and Sandier (1986, p. 94)]. In the constrained region, as a&/ax, = [( 1 - P)sT + b]/(N - 1), this condition for b < 1 reduces to ST < 1.

“‘As can be easily verilied, since s < 1, aG/dt >O in each region.


to tax revenue so that we can put c(p) =0 in (19). Since w(G) 2 Y for G 5 G* it immediately follows:

Proposition 3. In an economy with identical individuals, taxpayers will never evade for G 5 G*.

The intuition is straightforward: if G 5 G*, w(G) is larger than the cost of providing G; therefore if government sets tl= YG, individuals would never evade. This result may look very strong, but it should be recalled that here we have assumed away any problem concerning the distribution of the tax burden across social groups. Therefore taxpayers actually act as Kantian. Moreover, if c(p) >O is introduced back into (19), tax evasion is positive, at G= G*. Returning to (18) above, the ambiguity about the sign of dZ/lat disappears: with identical individuals and negligible enforcement costs, as X > 0, &Z/& > 0 certainly. l1 Putting together the two pieces of analysis for x0 and X, it then follows that, in an economy with identical individuals, tax evasion is zero for all G 5 G*, it is positive and increasing in t for a range of values of t such that G > G*, and might become positive and decreasing in t for high values of t, if such high values were ever to be reached. Even though we cannot rule out the possibility that at high tax rates evaded tax may become a decreasing function of t, it is clear that the relationship between tax evasion and the tax rate implied by our mode1 is more in agreement with intuition and observed behaviour than the implications of the traditional model. It is perhaps worthwhile to stress again the intuition behind this result: for G 2 G* an increase in t would increase the welfare of taxpayers; therefore, the latter do not resist this increase and keep evaded tax at zero. If G>G*, an increase in t would instead reduce individual welfare and therefore individuals attempt to resist by increasing tax evasion.

6. Concluding remarks

The mode1 discussed in this paper is primitive and it must be considered only as a first attempt to address formally the complex issue of taxpayers’ perceptions of the fairness of the fiscal system and its effects on tax compliance. One would certainly want to provide a fuller discussion of the informational constraints on taxpayers and to consider alternative formaliza- tions to the Kantian rule of fair behaviour. In spite of its limitations, the approach presented in this paper is able to offer a sounder explanation of

“As can easily be checked even if c(p)>O, the mild condition lel>c(p)/rl ensures that whenever X > 0, ax/& > 0.


income tax evasion than traditional theory and produce results that are consistent with a number of stylized facts about tax evasion. As was shown in the previous sections, our model is consistent with the fact that some people do not evade even if it would be in their self-interest to do so; and at least in some regions the effects of the tax rate and public expenditure on tax evasion is in line with what is suggested by the empirical and experimental literature. Furthermore, a number of extensions to the present model could be considered [see Bordignon (1989) for some examples]. In particular, our approach may be suited to offering a useful starting point for that ‘appropriate treatment of notions of justice and normality’ which Sandmo (1981) considers necessary to address problems of optimal taxation in economies where agents can evade.

Appendix

Proof of Proposition 1. Without loss of generality let I, > I,. Note from (5) that U’ = U2 implies t;= tj, i= 1,2, j= 1,2, i#j. Suppose that gi-, =x; and k1 =X1 is an equilibrium. Then, from Assumption 1 and (1) and (2) above, in equilibrium X2 2 xi > xy 2 X1, where xy > 0 by (2). Substituting from (9) and (10) in the inequalities above and solving for x: in equilibrium we obtain

(A.l)

(A.2)

Suppose first that z,>O. Then, by substituting for z1 in eq. (A.l) and by solving both eqs. (A.l) and (A.2) for x4 we obtain

z2(t-t;)>z1(t-t:). (A.3)

By substituting from (5) and (6) into tj, i= 1,2, the inequality in (A.3) requires I, < I,, which is impossible by assumption. Next, suppose z1 50: then substituting into eqs. (A.l) and (A.2) and repeating the steps above we obtain

12(t-t:)>Il(t-t:)[1+g-92]>I~(t--t:), (A.4)

since 9~ 1 by assumption. This is again impossible; we then conclude that f, =xi and i1 =X1 cannot be an equilibrium. QED.

Proof of Proposition 2. Substituting for 9 = 1 in (12) we obtain

z,=(l-t)(k-1)1+x,, (A.5)


zz= -(l-t)(k-1)1+x,. (A.6)

By definition of pi and from (A.6) it follows that

ZZ(X1=~21)~ZZ(X1=X~)=z[S(1-t)/7c-(1-r)(k-l)]. (A.7)

Differentiating z,(xy) with respect to t:

az,(xy)/at = Z(k - D). (A.81

Then, if k> D, x,(x:) is everywhere an increasing function of t. Hence it reaches a constrained maximum at the highest admissible value for t, t= 1. At t=l, z,(xy)=O. Hence, zz(~2~)SO for OSt51. Then, for all Osrtl, x2,=X, =O. Substituting back into (AS), we then get that xi-, =x7 if z,(x,=O)~x~; that is x1=x: if

z,(x,=O)-x:=(1--)Z(k-D)zO, (A.9)

which certainly holds for k > D. Q.E.D.

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a fairness approach to income tax evasion

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