Audit Uncertainty, Bayesian Updating, and Tax Evasion Arthur Snow University of Georgia Ronald S. Warren, Jr.* University of Georgia
Send page proofs and reprints to: Arthur Snow Department of Economics University of Georgia Athens, GA 30602 Email:
[email protected] Tel.: 706-542-3752
We extend the standard, one-period model of tax evasion to an intertemporal framework in which an expected-utility-maximizing taxpayer updates expectations about the probability of a future audit based on past audit experience. This framework provides a theoretical grounding for the empirical evidence indicating that tax evasion is affected by taxpayers’ perceptions of audit probabilities and is influenced by taxpayers’ prior audit experience. We show that, for a variety of risk preferences, Bayesian updating increases present and expected future tax evasion, and reduces tax payments, inclusive of expected fines. These findings call into question the usefulness of IRS secrecy about audit probabilities for raising taxpayer compliance and expected tax revenue.
Keywords: Subjective expected utility, audit probability, tax compliance, tax revenue JEL classification: H26; D81; D83
1. INTRODUCTION The low probability of being audited, coupled with small penalties for detected evasion, provide US taxpayers with ample opportunity to evade their tax liabilities.1 Given the existing incentives for evasion, it is not surprising that the Internal Revenue Service (IRS) estimates that $345 billion of federal income tax was evaded in 2001, amounting to more than sixteen percent of total revenue actually collected.2 To encourage compliance with the tax code, the IRS relies on two related strategies. First, a small fraction of income-tax returns is periodically audited on a purely random basis to establish benchmarks for reported income, deductions, and credits. Second, a closely guarded discriminant-analysis formula is used to trigger the auditing of specific returns that exceed certain thresholds. In successfully defending challenges to these strategies brought under the Freedom of Information Act, the IRS testified to the importance of both randomness and secrecy in its audit policy for increasing compliance when auditing all returns would not be cost-effective.3 Various theoretical studies of the tax-evasion decision have examined the usefulness of uncertain audit policies for increasing taxpayer compliance, reaching opposing conclusions. Some analyses of compliance behavior support the IRS view that tax evasion is reduced by greater uncertainty regarding either the proportion of undeclared income that will be detected by an audit (Alm 1988) or the amount of income that is taxable (Scotchmer and Slemrod 1989). However, other studies conclude that uncertainty may increase evasion. Reinganum and Wilde (1988) assume that taxpayers are uncertain about the cost incurred by the tax authority when an audit is undertaken, and hence are uncertain about the level of suspected evasion that will trigger an audit.
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They show that, under current penalty rates, as this uncertainty increases compliance at first increases, but ultimately declines. Snow and Warren (2005a) assume that taxpayers are uncertain about the amount of evasion that will be detected once an audit is undertaken, and conclude that, as this uncertainty increases, compliance decreases, given the relatively low current penalty rates. Cronshaw and Alm (1995) examine a gametheoretic model in which taxpayers are uncertain about both the tax authority’s audit cost and the probability that evasion will be detected when an audit is undertaken, and conclude that the introduction and maintenance of audit uncertainty may be counterproductive. Low penalty and audit rates, coupled with conflicting predictions concerning the effect of audit uncertainty on the evasion decision, have led some researchers to explore the role of uncertainty about the probability of being audited (i.e., ambiguity) in explaining the observed extent of tax compliance. For example, Spicer and Thomas (1982) present experimental evidence showing that the strength of the (negative) correlation between the fraction of taxes evaded and the probability of an audit falls as taxpayer information about the risk of being audited becomes less precise. Alm, Jackson, and McKee (1992) report results from several experiments suggesting that uncertainty about the probability of being audited affects compliance, although the direction of the effect turns on whether the tax revenue is to be spent on projects that would benefit those being taxed. Employing a version of cumulative prospect theory, Snow and Warren (2005b) show that greater uncertainty about audit probability increases the compliance of ambiguity averse taxpayers.
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There is considerable evidence consistent with taxpayers updating their estimates of the probability of audit based on prior experience. Spicer and Hero (1985) report on an experiment in which the participants file tax returns in each of ten rounds of play based on their given income, a random audit probability, and a preset penalty rate. The results reveal a statistically significant, negative relationship between tax evasion in the final round and the number of prior audits. Benjamini and Maital (1985), reporting on a similar experiment, also find that being audited in previous rounds reduces subsequent evasion. Long and Schwarz (1987), examining panel data on the outcomes of audits conducted by the Internal Revenue Service (IRS) in 1969 and 1971 on the same taxpayers, conclude that the 1969 audit was marginally effective in reducing the frequency of evasion in 1971. Erard (1992) finds a strong tendency for increased compliance among taxpayers with substantial prior-year audit assessments. While these studies provide empirical evidence that past adverse audit outcomes deter future tax evasion, Bayesian updating implies, by the same token, that successful previous evasion encourages greater future evasion. We show that uncertainty about the probability of being audited can increase the present value of expected tax evasion and reduce expected tax revenue when taxpayers treat their audit experiences as informative about the probability of being audited in the future. A simple example shows how Bayesian updating of subjective beliefs about the probability of an audit can result in more tax evasion than would occur in the absence of such updating. Suppose the expected subjective probability of an audit is at the lower bound of the prohibitive range, so that the expected return to a dollar of evaded tax equals zero. In the absence of updating, there is no tax evasion in either the present or the future. However, when the
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taxpayer updates expectations based on experience, the expected subjective probability of being audited in the future falls below the prohibitive range if there is no audit in the present. In that event, some tax evasion occurs in the future. Hence, expected future tax evasion in this example is positive when Bayesian updating occurs, while there would be no tax evasion if expectations were not revised on the basis of experience. Analyzing a two-period model of the evasion decisions of a risk averse taxpayer who maximizes subjective expected utility, we show that the essential features of this example carry over to the case in which there is tax evasion in the present. Our model provides a theoretical basis for the empirical evidence that taxpayers update their perceived probability of an audit based on prior audit experience. Moreover, our results indicate that policies intended to increase taxpayer compliance by creating uncertainty about the probability of being audited could be counterproductive. In the next section we set out a model of tax evasion, and in section 3 we analyze the effect of Bayesian updating on tax evasion decisions. In section 4, we examine the implications of Bayesian updating for expected tax revenue. Our conclusions are presented in section 5. 2. A Model of Tax Evasion with Bayesian Updating We abstract from the labor-supply decision and assume that the taxpayer earns an exogenous taxable income in the first period and is retired in the second period. In the first period, saving and tax evasion decisions are based on the taxpayer’s prior expectation about the probability of being audited. The true probability, which we denote by p, is fixed and invariant with respect to either the level of evasion or the outcome of prior audits. However, the taxpayer is uncertain about the value of p and so faces
4
ambiguity. The taxpayer’s subjective beliefs about p are represented by a cumulative distribution function F (q ) defined on the subjective probability of being audited, denoted
by q. As a distribution function defined on probabilities, F (q ) is known in the decision theory literature as a second-order probability (SOP) distribution. (See Camerer and Weber 1992 for a survey of several SOP models.) We assume that the taxpayer maximizes subjective expected utility, and therefore the taxpayer’s decisions are influenced only by the mean of the SOP distribution, which we denote by4 1 0
π ≡ ∫ q dF (q) .
(1)
After the audit experience of the first period, the taxpayer’s posterior beliefs are revised in accordance with Bayes’ rule to the conditional SOP distribution F (q | i) , where i = a in the event an audit occurred in the first period and i = n if no audit occurred.5 Since we are concerned only with the qualitative effect of Bayesian updating on tax evasion, we do not specify an explicit model of the informational content of the audit experience. Indeed, a tax audit need not be truly informative about the likelihood of future audits. We only assume, in line with the experimental evidence cited earlier, that the taxpayer perceives an audit to be informative and gleans some information from having been audited or not in the first period concerning the likelihood that the tax return filed in the second period will fit a profile that triggers an audit. Therefore, the expected 1 0
conditional probability of being audited in the second period, π i ≡ ∫ q dF (q | i ) , is assumed to be greater than π if an audit occurred in the first period, but less than π if no audit occurred, so that π a exceeds π n . Moreover, we require that (1 − π )π n + ππ a = π 5
(2)
so that the taxpayer’s beliefs are internally consistent; that is, the ex ante expected value of the taxpayer’s posterior beliefs about the probability of an audit is equal to the unconditional, prior value. We assume that the first-period saving and tax evasion decisions are made before the tax filing is subject to being audited. This assumption is intended to capture in a tractable formulation a prominent feature of IRS tax audits, namely that they occur after current evasion and saving decisions have been made.6 Therefore, penalties arising from an audit reduce consumption rather than the income that is allocated to consumption and saving.7 The income from saving that accrues in the second period is also taxable, and the taxpayer decides how much of this tax liability to evade based on the posterior expectation of being audited. Consumption in the second period then depends on whether the second-period tax filing is audited. We show that, under widely accepted conditions on risk preferences and empirically relevant assumptions about the probability of an audit, tax evasion is greater in this environment than it would be if the taxpayer did not update beliefs about the probability of being audited based on previous audit experience. In principle, the taxpayer’s beliefs about the probability of an audit, captured in the prior and posterior SOP distributions, could be biased in either direction. However, there is considerable empirical evidence indicating that taxpayers overestimate the true probability of being audited.8 To the extent that the IRS policy of secrecy fosters this upward bias, this policy increases taxpayer compliance. However, insofar as taxpayers treat their audit experiences as informative about the probability of being audited, IRS secrecy can also engender a countervailing reduction in compliance. Consequently, the
6
usefulness of this policy of secrecy for raising compliance and expected tax revenue is called into question. Using R to denote the after-tax gross rate of return to saving, we write the taxpayer’s indirect utility function for the second period as9 V ( s, π i ) ≡ max (1 − π i )u ( Rs + x i ) + π i u ( Rs − θ x i ) , xi
(3)
where u (⋅) is the taxpayer’s von Neumann-Morgenstern utility function, s is the amount saved in the first period, x i is the amount of tax evaded in the second period, and θ is the gross penalty rate assessed if the second-period tax filing is audited. We assume that the penalty rate and the expected probability of an audit are always low enough to make some tax evasion profitable. Finally, the taxpayer’s first-period decision problem is max (1 − π )[u ( y + x − s) + δ V ( s, π n )] + π [u ( y − θ x − s) + δ V ( s, π a )] ,
(4)
x, s
where y is the exogenous, first-period, after-tax income with no evasion, x is the amount of tax evaded in the first period, and δ is the taxpayer’s discount factor. 3. The Extent and Timing of Tax Evasion
To examine the implications of Bayesian updating of audit probabilities for tax evasion decisions, we derive the comparative statics effects of an increase in π a on the optimal choices for x , x n , and x a , while requiring that π n change to maintain the equality in (2). This approach is equivalent to studying the effect on tax evasion of changes in expected conditional audit probabilities as the first-period audit experience becomes more informative about the probability of being audited in the second period.10
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3.1. First-Period Tax Evasion
Beginning with the first-period evasion decision, let L denote the objective function in problem (4). The first-order condition for the evasion decision is
0 = L x = (1 − π )u ′(c n ) − π u ′(c a )θ
(5)
and for the saving decision is 0 = Ls = (1 − π )[−u ′(c n ) + δ Vs ( s, π n )] + π [−u ′(c a ) + δ Vs ( s, π a )] ,
(6)
where c i denotes first-period consumption in audit event i ∈ {n, a} , and the subscripts on L and V, and the prime on u, indicate partial derivatives. Observing that L xπ a = 0 , we
find that the second-order sufficient conditions for a maximum imply that first-period tax evasion increases (decreases) as π a increases, while maintaining condition (2), if 11 L xs Lsπ a > ( c an
8
and c na < c aa . It follows that Lsπ a is negative since u ′′ is negative. Hence, firstperiod evasion increases (decreases) if L xs is negative (positive). First-order condition (5) implies
L xs = −(1 − π )u ′′(c n ) + π u ′′(c a )θ
(10)
as well as (1 − π )u ′(c n ) = π u ′(c a )θ > 0 . As a consequence, we have L xs /[(1 − π )u ′(c n )] = An − Aa ,
(11)
where Ai ≡ −u ′′(c i ) / u ′(c i ) denotes the state-contingent Arrow-Pratt index of absolute risk aversion. The expression in (11) vanishes when the decision maker’s preferences exhibit constant absolute risk aversion (CARA), and is negative under decreasing absolute risk aversion (DARA). Thus, the anticipation of Bayesian updating based on audit experience increases first-period tax evasion when preferences exhibit DARA, but has no effect on first-period tax evasion when preferences exhibit CARA. The explanation for this result is straightforward. As the first-order condition (5) for tax evasion reveals, the taxpayer’s second-period expectation of an audit has only an indirect income effect (channeled through a change in the saving decision) on the willingness to evade taxes in the first period and bear the risk of an audit. From the firstorder conditions (5) and (6), we find that the effect of an increase in π a on the saving decision is dictated by the sign of − L xx Lsπ a , which is negative. Thus, first-period income net of saving increases as the audit experience becomes more informative about the probability of being audited in the second period. However, with CARA this increase in income has no effect on the willingness to bear risk, so there is no change in the tax
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evasion decision. In contrast, a taxpayer who exhibits DARA is more willing to bear risk and therefore increases tax evasion in the first period. The explanation for the decline in saving is found in the effect of an increase in
π a on expected future utility. Partially differentiating L with respect to π a , while maintaining condition (2), yields Lπ a = δπ [− Vπ n ( s, π n ) + Vπ a ( s, π a )] = δπ [u (c nn ) − u (c na ) − u (c an ) + u (c aa )],
(12)
which is positive since c nn > c an and c aa > c na . Thus, second-period expected utility increases as the audit experience becomes more informative since the taxpayer adapts the second-period evasion decision in accord with updated expectations about the probability of an audit. As a consequence, saving declines to reestablish equality between the marginal utility of income in the first and second periods. 3.2. Second-Period Tax Evasion
Turning to the second-period decision, the first-order condition for optimal evasion after the first-period audit event i ∈ {n, a} is (1 − π i )u ′(c in ) − π i u ′(c ia )θ = 0 ,
(13)
which can be rewritten as MRS i ≡
1 − π i u ′(c in )
π i u ′(c ia )
=θ ,
(14)
implying the comparative statics equation dx i dπ a
=−
dMRS i / dπ a dMRS i / dx i
10
(15)
for the effect of Bayesian updating on second-period evasion. After taking account of equality (14), we obtain dMRS i dπ a
= −θ
∂π i / ∂π a
ds + θ ( Aia − Ain ) R π i (1 − π i ) dπ a
(16)
and dMRS i dx i
= − θ ( Ain + θAia ) .
(17)
Substituting from equations (16) and (17) into (15), we arrive at
dx i dπ a
− =
∂π i / ∂π a
ds + ( Aia − Ain ) R π i (1 − π i ) dπ a Ain + θAia
(18)
as the effect of an increase in the informational content of the first-period audit experience on the second-period tax evasion decision after experiencing audit event i in the first period. Equation (2) implies ∂π a / ∂π a = 1 and ∂π n / ∂π a = −π /(1 − π ) . We wish to determine the sign of E[dx / dπ a ] ≡ (1 − p )(dx n / dπ a ) + p (dx a / dπ a ) ,
(19)
where p is the true probability that the taxpayer’s filing will trigger an audit. In keeping with the evidence cited earlier that taxpayers overestimate this probability, we assume that π is greater than or equal to p . We show in the Appendix that when π exceeds p , the expected change in second-period tax evasion has a smaller value when π is used in place of p . Therefore,
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1 1 ⎧ ⎫ ⎪ n ⎪ n a a ⎪ π (1 − π ) − π (1 − π ) ⎪ n a dx dx ⎪ ⎪ (1 − π ) +π = π ⎨ Ann + θAna Aan + θAaa ⎬ (20) dπ a dπ a ⎪ ⎪ ⎡1 − π Ana − Ann A − Aan ⎤ ds ⎪ ⎪ R +⎢ + aa ⎥ ⎪⎩ ⎣ π Ann + θAna Aan + θAaa ⎦ dπ a ⎪⎭ is less than or equal to E[dx / dπ a ] . The change in saving introduces an income effect that vanishes in the case of CARA. In that case, the right-hand side of equation (20) reduces to ⎤ ⎡ 1 1 − ⎥, ⎢ A(1 + θ ) ⎢⎣π n (1 − π n ) π a (1 − π a ) ⎥⎦
π
(21)
which is positive, assuming realistically that the expected probability of an audit is always less than one half.12,13 Thus, with CARA first-period tax evasion is unaffected by updating, but expected second-period evasion increases, implying that expected tax evasion is higher overall as a consequence of the taxpayer’s updating expectations of an audit based on previous experience. With DARA, first-period tax evasion is higher when the taxpayer updates expectations of an audit, but the expected value of second-period tax evasion may be higher or lower. The principal reason for this result is that saving declines and, with DARA, the lower value of second-period income dampens the taxpayer’s willingness to bear risk. For the case of a taxpayer with risk preferences exhibiting a constant ArrowPratt index of relative risk aversion (CRRA) equal to one, the optimal amount tax evasion equals [1 − π (1 + θ )] θ multiplied by the amount of after-tax income (net of any saving).14,15 Thus, expected tax evasion, using the taxpayer’s prior beliefs, is
12
( y − s)
1 − π (1 + θ )
θ
⎡ 1 − π n (1 + θ ) 1 − π a (1 + θ ) ⎤ + ρ ⎢(1 − π ) +π ⎥ Rs , θ θ ⎥⎦ ⎢⎣
(22)
where ρ is the social discount factor. By exploiting equation (2), the term within brackets in expression (22) reduces identically to [1 − π (1 + θ )] θ . As a result, increasing π a while maintaining equation (2) yields −
ds dπ a
(1 − ρR )
1 − π (1 + θ )
θ
⎡ ∂π n ∂π a ⎤ 1 + θ + ⎢− (1 − π ) −π Rs ⎥ ⎢⎣ ∂π a ∂π a ⎥⎦ θ
(23)
for the change in the taxpayer’s expected evasion as the first-period audit experience becomes more informative. Recalling that ∂π n / ∂π a = −π /(1 − π ) and ∂π a / ∂π a = 1 , we find that the bracketed term in expression (23) vanishes. Thus, for the case of CRRA equal to one, Bayesian updating induces only income effects on the taxpayer’s expected evasion. Since CRRA implies DARA, ds / dπ a is negative, so that tax evasion is higher in the first period, but expected evasion is lower in the second period. Nonetheless, expected tax evasion is higher overall as a result of Bayesian updating if the social discount rate equals the before-tax rate of return, so that ρR is less than one. If the social discount rate equals the after-tax rate of return, then ρR equals one and expression (23) equals zero, but would be positive if the true probability of audit p were used in the calculation in place of the taxpayer’s expectation π , assuming π exceeds p . We conclude that expected tax evasion is higher with Bayesian updating for taxpayers whose risk preferences exhibit CRRA equal to one.
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In the empirically relevant case of risk preferences exhibiting DARA, the taxpayer’s expected future evasion could be lower with Bayesian updating, as it is with CRRA equal to one, but expected evasion could nonetheless be higher overall. When expected future evasion is lower, the expression within braces in equation (20) is negative. However, that expression is multiplied by the taxpayer’s prior probability of an audit, and must also be discounted to obtain its present value. Moreover, when the taxpayer overestimates the probability of an audit, equation (20) overstates the decline in expected future tax evasion. The implication is that the effect of Bayesian updating on the present value of expected tax evasion is almost surely dominated by the increase in first-period evasion. When the saving and tax evasion decisions are made contemporaneously, as we assume, tax evasion in the first period is a gamble on first-period consumption. In this context, Bayesian updating enhances second-period utility leading to a reduction in saving that, through income effects, influences tax evasion and the willingness to gamble on first-period consumption. By contrast, when the saving decision can be postponed until after the assessment of any audit penalties, evasion is a gamble on first-period wealth. In that case, the marginal benefit of successful evasion is lower since saving is reduced because second-period tax evasion is more likely to be successful. By the same token, the marginal cost of unsuccessful evasion is higher since saving is increased because second-period tax evasion is less likely to be successful. As a consequence, firstperiod tax evasion is lower with Bayesian updating when saving occurs after audit penalties are imposed. We explore this scenario formally in an appendix available from the authors, where we show that the tendency for second-period evasion to increase with
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Bayesian updating is strengthened by the opportunity to pay audit penalties before making the saving decision in the first period. We also demonstrate that the present value of tax evasion unambiguously increases with Bayesian updating for taxpayers whose risk preferences are characterized by CRRA equal to one. 4. Expected Tax Revenue
The expected loss in tax revenue is positive when taxes are evaded, even after expected fines are included.16 With CARA, first-period tax evasion is unaffected by Bayesian updating but saving is lower, resulting in a lower second-period tax base, while second-period expected tax evasion is greater. Hence, expected tax revenue is lower. With CRRA equal to one, expected tax revenue is also lower with Bayesian updating, since saving is lower and expected tax evasion is greater. For the more general case of DARA preferences, expected tax revenue in the first period is lower because tax evasion is greater. In the second period, expected tax evasion could be lower, but expected tax revenue could also be lower with Bayesian updating since the tax base is lower as a result of the decline in saving. We conclude that fostering taxpayer uncertainty about audit probabilities may not be a useful instrument for raising either taxpayer compliance or expected tax revenue. 5. Conclusions
We analyze the implications for expected tax evasion and tax revenue of taxpayers’ facing ambiguity about the probability of being audited and updating their beliefs on prior audit experience. With Bayesian updating, tax evasion decisions in the future will be adapted to updated expectations. As a result, a taxpayer’s expected future utility is higher and the taxpayer saves less in the present. With constant absolute risk
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aversion, this change in saving has no effect on the willingness to bear risk, so tax evasion in the present is unaffected. However, the present period’s audit experience has an asymmetric influence on future tax evasion such that expected evasion is higher with updating. With an Arrow-Pratt index of constant relative risk aversion equal to one, optimal tax evasion is equal to a proportion of income that depends directly on the expected return to a dollar of tax evaded. Hence, updating influences tax evasion only through income effects. The reduction in saving increases present tax evasion, and this response dominates the present value of the total change in expected future tax evasion. In the more general case of decreasing absolute risk aversion, tax evasion in the present period is higher with Bayesian updating because of the income effect induced by the decline in saving. The income effect works in the opposite direction in the future period, so that expected future tax evasion could be lower with Bayesian updating. Nonetheless, expected future tax revenue could also be lower, since the decline in saving reduces the tax base. Our analysis provides a theoretical underpinning for the empirical evidence indicating that tax evasion responds to the updating of beliefs about the probability of an audit based on prior audit experience. IRS secrecy about audit policies could be responsible for taxpayers’ overestimating the likelihood of being audited, leading to greater compliance with the tax code. However, we show that taxpayer uncertainty about the probability of being audited can also increase expected tax evasion and reduce expected tax collections. Thus, our findings call into question the usefulness of policies that foster taxpayer uncertainty about audit probabilities.
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Appendix
To show that the change in expected second-period tax evasion is smaller when
π > p replaces p in equation (19), we establish the following inequality ∂E[dx / dπ a ] ∂p = (dx a / dπ a ) − (dx n / dπ a ) < 0 ,
(A.1)
which implies the desired result. Substituting for dx i / dπ a from equation (18), and for
π n from equation (2), and then factoring out B ≡ ( A an + θA aa ) − 1 ( A an + θA na ) − 1 , we arrive at Aan + θAaa ⎧ Ann + θAna ⎫ + ⎪ ⎪ a a n a ∂E[dx / dπ a ] ∂p = − B ⎨ π (1 − π ) (1 − π )(1 − π ) ⎬, ⎪ a ⎪ ⎩ + ( Ana Aan − Ann Aaa )(1 + θ ) R(ds / dπ )⎭
(A.2)
which is negative if the term within parentheses is positive. This term is positive with CARA since ds / dπ a is then equal to zero, but with DARA ds / dπ a is negative while the terms multiplying ds / dπ a are positive. To eliminate ds / dπ a from equation (A.2), we use the fact that expected utility in the second period, EV ≡ (1 − π )V ( s, π n ) + πV ( s, π a ) , increases as the audit experience is viewed as more informative. Thus, we have 0 < E[dV / dπ a ] = (1 − π )[u (c na ) − u (c nn )](∂π n / ∂π a ) + π [u (c aa ) − u (c an )](∂π a / ∂π a ) + E[Vs ](ds / dπ a ).
(A.3)
Using the first-order conditions for evasion in the second period, (1 − π i )u ′(c in ) − θπ i u ′(c ia ) = 0 , we find
17
(A.4)
E[Vs ] = [(1 − π )π n u ′(c na ) + ππ a u ′(c aa )](1 + θ ) R .
(A.5)
Substituting for ∂π n / ∂π a = −π /(1 − π ) and ∂π a / ∂π a = 1 in equation (A.3), and for (1 − π )π n in equation (A.5) using equation (2), and then substituting the result into equation (A.3), we obtain u (c na ) − u (c aa ) + u (c an ) − u (c nn ) a (1 + θ ) R(ds / dπ ) > . (1 − π a )u ′(c na ) + π a u ′(c aa )
(A.6)
With c na < c aa < c an < c nn , there exist scalars u1′ ∈ (u ′(c aa ), u ′(c na )) and u 2′ ∈ (u ′(c nn ), u ′(c an )) such that u (c aa ) − u (c na ) = u1′ (c aa − c na ) = u1′θ ( x n − x a ) and u (c nn ) − u (c an ) = u 2′ (c nn − c an ) = u 2′ ( x n − x a ) . Substituting these expressions into the numerator of (A.6), and substituting for
π a u ′(c aa ) from the first-order condition (A.4), we obtain (1 + θ ) R(ds / dπ a ) > −θ
θu1′ + u 2′
xn − xa
1 − π a θu ′(c na ) + u ′(c an )
> −θ
xn − xa 1−π a
,
(A.7)
where the second inequality follows from the assumptions u1′ < u ′(c na ) and u 2′ < u ′(c an ) . It follows from (A.7) that the term within parentheses in (A.2) must have a value greater than ⎧ Ann + θAna Aan + θAaa ⎫ − θ ( x n − x a )( Ana Aan − Ann Aaa )⎬ , + ⎨ 1−π a ⎩ 1−π n πa ⎭ 1
and the term within braces in this expression must be greater than
18
Ann + θAna + Aan + θAaa − θ ( x n − x a )( Ana Aan − Ann Aaa ) ,
(A.8)
which we wish to show is positive. To this end, we first introduce a scalar λ and rewrite (A.8) as Ann + θAna [1 − (1 − λ ) Aan ( x n − x a )] + Aan [1 − λAnaθ ( x n − x a )] + θAaa [1 + Ann ( x n − x a )] ,
which is positive if the first two terms in brackets are nonnegative. This requires 1 Anaθ ( x n − x a )
>λ >
1 − Aan ( x n − x a ) Aan ( x n − x a )
,
(A.9)
for which it is sufficient to have Aan ( x n − x a ) ≥ 1 .
(A.10)
We next introduce the scalar γ and rewrite (A.8) as Ann + θAna [1 − γAan ( x n − x a )] + Aan − (1 − γ )θλAna Aan ( x n − x a )] + θAaa [1 + Ann ( x n − x a )] ,
and set γ equal to 1 / Aan ( x n − x a ) . The resulting expression is positive if we have
γ ≥ 1 , that is, if Aan ( x n − x a ) ≤ 1 .
(A.11)
Since either (A.10) or (A.11) must hold, we conclude that the term within parentheses in (A.2) is positive, implying that ∂E[dx / dπ a ] ∂p is negative as desired.
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Notes *
We thank two anonymous referees and the Editor for helpful comments on an earlier version.
1
The IRS approximates the audit (examination) rate for a given tax year by calculating
the number of examinations closed in a given fiscal year divided by the number of returns filed in the previous calendar year. (See Department of the Treasury, Internal Revenue Service, Report to Congress: IRS Tax Compliance Activities July 15, 2003.) For examinations closed in fiscal year 2002, the IRS reports an examination rate of less than 0.6 percent (irs.gov/pub/irs-soi/rtctab6a.xls). Andreoni, Erard, and Feinstein (1998, 820) report that U.S. taxpayers face a gross penalty rate ranging from 1.2 for negligent understatement of liabilities to 1.75 for intentional fraud. The penalty for fraud is so low that the expected return to a dollar of fraudulent tax evasion is positive as long as the probability of being detected is less than thirty-six percent. Moreover, Andreoni, Erard, and Feinstein (1998, 821) report that in 1995 only 4.1 percent of taxpayers whose returns were reassessed following an audit received any penalty at all, and in a number of these cases penalties were eventually reduced.
2
See IRS News Release IR-2006-28 February 14, 2006.
3
See Roberts v. Internal Revenue Service, 548 Federal Supplement 1241 (Eastern
District, Michigan), 1984.
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4
Although facing ambiguity about the probability of an audit, the taxpayer is assumed to
be ambiguity neutral, and therefore maximizes subjective expected utility. This assumption allows us to focus on the implications of Bayesian updating while abstracting from the effects of a non-neutral attitude toward ambiguity, which would be introduced if the integral in (1) were replaced by
1
∫ 0 ϕ (q) dF (q)
with a nonlinear weighting function
ϕ , as in Snow and Warren (2005b).
5
Viscusi and O’Connor (1984) provide evidence from the workplace showing that
individuals engage in a Bayesian updating process in which they learn about the health risks of their jobs. Anderson and Holt (1997) report results from an experimental setting in which individuals’ decisions in response to others’ decisions were generally consistent with rational Bayesian updating.
6
In 2002, the average time elapsed between the filing of a personal income tax return and
the initiation of an audit was eleven months, according to data provided by the IRS (Research, Analysis, and Statistics, Office of Research, Compliance Data Warehouse, Enforcement Revenue Information System September 2005.) Moreover, the average time for completion of an audit in 2002 was approximately ten months, excluding appeals and collection proceedings (Joint Committee on Taxation of the US Congress, JCX-53-03 May 19, 2003.) Thus, the time elapsed between filing a tax return and the assessment of any audit penalties was almost two years on average.
21
7
An alternative modeling strategy would have the saving decision made after the
assessment of any audit penalties, leading to a more complex analysis without substantially modifying our qualitative conclusions. We discuss the implications of this modeling strategy at the end of the next section.
8
For example, Alm, McClelland, and Schultz (1992) report results from several
experiments suggesting that many subjects overestimate the low probability of being audited, leading to less evasion than predicted by the standard model. Scholz and Pinney (1995) also provide evidence that taxpayers have upwardly biased subjective estimates
π of the true audit probability p , with the magnitude of the bias negatively correlated with their expected gain from evasion behavior.
9
In stating the taxpayer’s decision problem, we suppress the tax rates notationally since
they are not of interest in our analysis.
10
Greenberg (1984) derives a feasible, revenue-maximizing audit strategy for the tax
authority in which, for a given tax function, penalty rate, and audit probability, the proportion of taxpayers who evade is arbitrarily small. In Greenberg’s model, evading taxpayers revise upward their expected probability of audit to unity in the event they are audited and so thereafter truthfully reveal their incomes.
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11
The first-order conditions (5) and (6) yield the differential equations
L xx dx + L xs ds = − L
xπ a
dπ a and Lsx dx + Lss ds = − L a dπ a , where it is understood sπ
that equation (2) is maintained. After taking account of the fact that L system of equations yields dx / dπ a = L xs L
sπ a
xπ a
= 0 , this
/( L xx Lss − L2xs ) . The second-order
sufficient conditions for a maximum require that the denominator is positive, so that the comparative statics effect has the same sign as the numerator.
12
If taxpayers’ estimates are realistic, then their expected audit probabilities are below
one percent.
13
Observe that π (1 − π ) increases as π increases if π is less than one half. Hence, with
π a greater than π n , π a (1 − π a ) exceeds π n (1 − π n ) if π a is less than one-half, and the term within brackets at (21) is then positive.
14
With risk preferences exhibiting CRRA equal to one, we have u (c) = ln(c) . The first-
order condition for maximizing (1 − π ) ln(Y + x) + π ln(Y − θ x) with respect to x yields x = Y [1 − π (1 + θ )] θ as the optimal amount tax evasion.
15
The assumption that risk preferences exhibit CRRA is widely adopted in the literature.
Arrow (1965) points out that any utility function satisfying the von Neumann-
23
Morgenstern axioms must be bounded above and below, and in that case CRRA implies a value of one for the index of relative risk aversion.
16
The loss in expected tax revenue per dollar of tax evaded is (1 − p) − pθ
= 1 − p(1 + θ ) , where p is the true probability of an audit. With π ≥ p , the expected loss is greater than or equal to 1 − π (1 + θ ) , which must be positive as a necessary condition for positive tax evasion.
24
References
Alm, James. 1988. Uncertain Tax Policies, Individual Behavior, and Welfare. The American Economic Review 78 (1): 237-245.
Alm, James, Betty Jackson, and Michael McKee. 1992. Deterrence and Beyond: Toward a Kinder, Gentler IRS. In Why People Pay Taxes: Tax Compliance and Enforcement, Joel Slemrod, ed., 311-329. Ann Arbor: University of Michigan Press.
Alm, James, Gary H. McClelland, and William D. Schultz. 1992. Why Do People Pay Taxes? Journal of Public Economics 48 (1): 21-38.
Anderson, Lisa R., and Charles A. Holt. 1997. Information Cascades in the Laboratory. The American Economic Review 87 (5): 847-862.
Andreoni, James, Brian Erard, and Jonathan Feinstein. 1998. Tax Compliance. Journal of Economic Literature 36 (2): 818-860.
Arrow, Kenneth J. 1965. Aspects of the Theory of Risk Bearing. Yrjo Jahnsson Lectures, Helsinki.
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Benjamini, Yael, and Shlomo Maital. 1985. Optimal Tax Evasion and Optimal Tax Evasion Policy: Behavioral Aspects. In The Economics of the Shadow Economy, Wulf Gaertner and Alois Wenig, eds., 245-264. Berlin: Springer-Verlag.
Camerer, Colin F., and Martin Weber. 1992. Recent Developments in Modeling Preferences: Uncertainty and Ambiguity. Journal of Risk and Uncertainty 5 (4): 325-370.
Cronshaw, Mark B., and James Alm. 1995. Tax Compliance with Two-Sided Uncertainty. Public Finance Quarterly 23 (2): 139-166.
Erard, Brian. 1992. The Influence of Tax Audits on Reporting Behavior. In Why People Pay Taxes: Tax Compliance and Enforcement, Joel Slemrod, ed., 95-114. Ann Arbor: University of Michigan Press.
Greenberg, Joseph. 1984. Avoiding Tax Avoidance: A (Repeated) Game-Theoretic Approach. Journal of Economic Theory 32 (1): 1-13.
Long, Susan B., and Richard D. Schwartz. 1987. The Impact of IRS Audits on Taxpayer Compliance: A Field Experiment in Specific Deterrence. Paper presented at the annual meetings of the Law Society Association, Washington, DC.
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Reinganum, Jennifer F., and Louis L. Wilde. 1988. A Note on Enforcement Uncertainty and Taxpayer Compliance. Quarterly Journal of Economics 103 (4): 793-798.
Scholz, John T., and Neil Pinney. 1995. Duty, Fear, and Tax Compliance: The Heuristic Basis of Citizenship Behavior. American Journal of Political Science 39 (2): 490-512.
Scotchmer, Suzanne, and Joel Slemrod. 1989. Randomness in Tax Enforcement. Journal of Public Economics 38 (1): 17-32.
Snow, Arthur, and Ronald S. Warren, Jr. 2005a. Tax Evasion under Random Audits with Uncertain Detection. Economics Letters 88 (1): 97-100.
Snow, Arthur, and Ronald S. Warren, Jr. 2005b. Ambiguity about Audit Probability, Tax Compliance, and Taxpayer Welfare. Economic Inquiry 43 (4): 865-871.
Spicer, Michael W., and Rodney E. Hero. 1985. Tax Evasion and Heuristics: A Research Note. Journal of Public Economics 26 (2): 263-267.
Spicer, Michael W., and J. Everett Thomas. 1982. Audit Probabilities and the Tax Evasion Decision. Journal of Economic Psychology 2 (3): 241-245.
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Viscusi, W. Kip, and Charles J. O’Connor. 1984. Adaptive Responses to Chemical Labeling: Are Workers Bayesian Decision Makers? The American Economic Review 74 (5): 942-956.
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Ronald S. Warren, Jr. is associate professor of economics at the University of Georgia. His research interests are primarily in empirical labor economics and theoretical public finance.
Arthur Snow is professor of economics at the University of Georgia. He has published numerous articles in the areas of public finance, information economics, and decision making under uncertainty.
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