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'APER NO. 1467 

How Tax Agency Audit Policies, Tax Rates, 
and Uncertainty Affect Taxpayer Compliance 

Paid J. Beck 
Jon S. Davis 
Woon O. Jung 

^u G 








College of Commerce and Business Administration 
Bureau of Economic and Business Research 
University of Illinois. Urbana-Champaign 



College of Commerce and Business Administration 

University of Illinois at Urbana- Champaign 

June 1988 

How Tax Agency Audit Policies, Tax Rates, 
and Uncertainty Affect Taxpayer Compliance 

Paul J. Beck, Associate Professor 
Department of Accountancy 

Jon S. Davis, Assistant Professor 
Department of Accountancy 

Woon 0. Jung, Assistant Professor 
Department of Accountancy 

Digitized by the Internet Archive 

in 2011 with funding from 

University of Illinois Urbana-Champaign 

How Tax Agency Audit Policies, Tax Rates, 
and Uncertainty Affect Taxpayer Compliance 


The present study investigates the effects of tax audit policies 
on taxpayers' reporting strategies. Previous research is extended by 
incorporating taxpayers' uncertainty about their tax liabilities and 
by developing two alternative models of tax agency audit policies. An 
evaluation of the comparative statics properties of each model indi- 
cates that taxpayers' expectations about audit policies can have an 
important role in mediating the effects of the tax rate structure and 
uncertainty on taxpayer compliance. A noteworthy result is that 
increasing the tax rate will strengthen compliance incentives when 
taxpayers expect the tax enforcement agency to modify its audit poli- 
cies in response to individual taxpayers' reporting, but have no 
effect when audit policies are expected to be fixed. We also show 
that uncertainty about taxable income and penalties affects compliance 
even when taxpayers are assumed to be risk-neutral. 

1. Introduction 

Numerous studies [e.g., Allingham and Sandmo (1972), Srinivasan 
(1973), Singh (1973), Yitzhaki (1974), Kakwani (197a), Landsberger and 
Meilijson (1982), Koskela (1983), Greenberg (1984), Reingauum and 
Wilde (1985, 1986), and Graetz, Reinganum, and Wilde (1986) and Aim 
(1988)] present models of tax evasion and taxpayer reporting behavior. 
To date this literature has focused primarily on the effects of tax 
and penalty rate structure, while largely ignoring the impact of tax- 
payer perceptions of tax agency audit policies on reporting behavior. 

Tax agency audit policies have been represented in two ways in the 
literature. Traditionally, most modelling studies have adopted a 
partial equilibrium framework, in which taxpayers viewed the probabil- 
ity of audit as fixed and unresponsive to reported income (i.e., the 
selection of returns for audit by random sampling). The assumption of 
a fixed audit probability has recently been criticized [see Greenberg 
(1984), Graetz and Wilde (1985)] for ignoring strategic interplay be- 
tween tax auditing and taxpayer reporting decisions. More recently, 
several studies [e.g., Greenberg (1984), Graetz, Reinganum and Wilde 
(1986), and Reinganum and Wilde (1985, 1986)] have incorporated such 
interplay by adopting a game-theory framework in which taxpayers and 
the tax agency concurrently make their respective reporting and 
auditing decisions. 

While empirical evidence is somewhat limited with respect to tax- 
payers' beliefs regarding the tax agency's audit decision, one recent 
survey [Aitken and Bonneville (1980)] commissioned by the Internal 
Revenue Service (IRS) suggests that taxpayer perceptions are mixed. 


Aitken and Bonneville report that 29.3% of respondents indicated that 
they believe the IRS randomly selects tax returns for audit. An addi- 
tional 30.4% of the respondents reported that they believe that attri- 
butes of their tax return (e.g., reported income level, the level of 
deductions taken, irregularities, etc.) provide the basis for audit 
selection. The remainder of respondents either stated that they did 
not know how the IRS selected returns for audit or responded with some 
other miscellaneous audit selection rule. These results provide 
limited empirical support both for the traditional modelling approach 
which assumes taxpayers view the audit selection decision as random 
and for the more recent game-theoretic models which represent the tax 
agency as responsive to taxpayer reporting decisions. 

In the present study, we investigate how expectations about the 
tax agency's audit policies affect taxpayer compliance. Consistent 
with the past analytical work and the Aitken and Bonneville (1980) 
survey evidence discussed above, two economic models of taxpayer 
reporting are developed. The first is representative of an environ- 
ment in which taxpayers expect the tax enforcement agency's audit 
policies to be based upon random selection within audit classes (e.g., 
groups of taxpayers sharing observable characteristics such as occupa- 
tion and income sources) rather than strategically directed at indivi- 
dual taxpayers. In the second model, the tax agency is assumed to 
interact strategically with individual taxpayers. Hence, each tax- 
payer must anticipate the effect which his (her) reporting decision 
has on the probability of receiving an audit. The framework 
underlying both models differs from most previous studies in that we 


assume that taxpayers have uncertainty about their tax liabilities and 
the imposition of penalties. Such uncertainty could arise due to the 
complexity of the tax laws applicable to taxpayers' transactions 
and/or frequent changes in the tax code. Our analysis shows that the 
effects of the tax rate structure and uncertainty about taxable income 
on taxpayers' reporting incentives depend crucially upon their expec- 
tations regarding the tax agency's audit policies. Hence, policy 
makers' predictions regarding the compliance effects of changes in the 
tax rate structure and tax simplification should take into account 
concurrently the role of the tax enforcement agency. 

Our study is organized in four sections. In Section 2 we present 
a benchmark model of taxpayer reporting under the assumptions that 
audits are random and the audit probability is independent of the 
amount of taxable income reported. Consideration of the simple model 
facilitates comparisons with traditional taxpayer reporting models 
and, thus, highlights the effects of taxpayer uncertainty as well as 
providing a foundation for subsequent extension. Section 3 incor- 
porates the tax agency as a strategic player in an extensive form game 
with taxpayers. In particular, the tax agency is assumed to determine 
its tax audit policies based upon a cost-benefit analysis. Hence, as 
the audit benefit to the tax agency is directly related to the amount 
of reported income, tax audit policies are maximally responsive, given 
the attendant costs of performing audits. Another desirable feature 
of the sequential equilibrium framework is that the natural time 
ordering of the reporting/audit process is modelled explicitly. 


Furtherraore, the tax agency's actual audit policies are fully con- 
sistent with the audit policies that taxpayers anticipated when 
making their reporting decisions. By considering both the fixed and 
strategic tax agency models, we can determine the extent to which the 
effects of other environment factors are mediated by expected tax 
auditing policies. Section 4 presents the conclusions and policy 
implications of our study. 

2. Taxpayer Model 

Consistent with previous research, the tax reporting decision is 
modelled for a representative taxpayer. Furthermore, we temporarily 
assume that the audit probability (p) depends only upon the taxpayer's 
audit class membership and, thus, does not vary with the amount of 
reported taxable income. A fixed audit probability assumption is con- 
sistent with an environment in which the tax enforcement agency ran- 
domly selects taxpayers belonging to a given audit class that shares 

observable characteristics. As many previous studies have also 

relied upon this assumption, a direct comparison of results is facili- 

Another modelling assumption throughout the study is that tax- 
payers are risk-neutral. This assumption is based upon several con- 
siderations. First, previous studies that investigated the effects of 
taxpayer uncertainty [e.g., Aim (1988) and Scotchmer and Slemrod 
(1988)] assumed taxpayer risk aversion. Hence, analysis of risk- 
neutral taxpayers serves to fill an existing void in the literature. 
A second, but related reason is that while casual intuition might 


suggest chat changes in the uncertainty level would either not affect 
the reporting decisions of risk-neutral taxpayers or would be the same 
as for risk-averse taxpayers, our results indicate the opposite. A 
third reason for assuming risk-neutrality is to provide comparability 
with the assumptions required by the game-theory analysis and to per- 
mit the taxpayer's decision problem to be formulated as the minimiza- 
tion of the expected value of the tax liability and penalties. 

A distinguishing feature of our characterization is that the tax- 
payer is assumed to be uncertain about his (her) taxable income and 
the associated tax liability. Such uncertainty could arise due to the 
complexity of the tax laws applicable to the taxpayer's particular 
circumstances and/or changes in the tax statutes. Uncertainty about 
the post-audit taxable income (x) is modelled by means of a probabi- 
lity density function, f(x). The density is assumed to have a finite 
interval support, [L,H] and a mean denoted by y. 

Assuming that the taxpayer files a return on which a taxable in- 
come of R is reported, there are four possible events which result in 
distinct liabilities. The first is that the taxpayer's return is not 
audited, in which case, the taxpayer's liability remains T(R), where 
T(«) denotes the tax rate structure. Alternatively, when x < R, the 
taxpayer's post-audit tax liability will be revised downward from T(R) 
to T(x), thereby resulting in a refund. However, if x > R, the tax- 
payer will have to pay additional taxes. Furthermore, a monetary 
penalty could be imposed for underpayment of taxes. 

Most tax reporting models have employed two different penalty 
structures. Under the first, the monetary penalty rate (q) is applied 


directly to the reported income deficiency (x-R) [see Allingham and 
Sandmo (1972), McCaleb (1976), Srinivasan (1973), Koskela (1983), 
Greenberg (1984), and Aim (1988)]. Other studies [e.g., Yitzhaki 
(1974), Scotchmer (1987a), and Reinganum and Wilde (1987)] have 
assumed that the penalty is proportional to the tax deficiency, 
T(x) - T(R). As the penalties for underpayment of taxes in the United 

States are based on the tax deficiency, we employ the second penalty 



In the United States, the Internal Revenue Code provides a mone- 
tary penalty for the substantial underpayment of taxes. However, 

such penalties can be avoided if the taxpayer has substantial authority 

for the tax treatment taken. Since the taxpayer (and the tax agency) 

could have uncertainty with respect to whether substantial authority 

exists [see Ayres, Jackson, and Hite (1987), and Chow, Shields, and 

Whittenburg (1987)], we assume that the imposition of penalties is 

uncertain and denote the associated probability by <£. The liabilities 

together with their probability of occurrence are summarized below: 

Event Liability Probability 

1. No Audit T(R) (1-p) 


2. Audit/No Deficiency T(x) p / f(x)dx 



3. Audit/Penalty Waived T(x) p( !-<!>)/ f(x)dx 


4. Audit/Penalty Imposed T(x) + q[T(x)-T(R)] p<J> / f(x)dx 



The expected tax and penalty liability can be written as: 

ET = (l-p)T(R) + p{/ T(x)f(x)dx 


+ $/ [T(x)+q(T(x)-T(R))]f(x)dx 

+ (l-<f>)/ T(x)f(x)dx}. (1) 


Differentiating (1) with respect to R, we obtain the following 

optimality condition: 

££ = (l-p)T'(R*) - p4,qT'(R*)/ f(x)dx = 0, (2) 

a * R* 

where R* denotes the solution to (2). 

The first group of terms on the RHS of (2) represents the expected 
marginal benefit to the taxpayer from reducing reported taxable 
income, while second group represents the expected marginal cost 
(penalty). Hence, the optimality condition indicates that the tax- 
payer will have an incentive to increase reported taxable income up to 
the point where the expected marginal benefits (tax savings) are 
equated with expected marginal costs. Simplifying (2), the following 
analogous optimality condition is obtained: 

F(R*) - 1 - (l-p)/(p«j>q). (3) 

Since F(0 represents the cumulative probability distribution for 
post-audit taxable income, the first-order condition has a direct 
interpretation. In particular, (3) indicates that the taxpayer's 
optimal strategy is to report at the 1 - (l-p)/(p<frq) fractile of the 


post-audit income distribution. As F(») is a strictly increasing 
function, it is apparent that the taxpayer's reported taxable income 
is an increasing function of the audit probability and the expected 
monetary penalty rate (<f>q) for underpayment of taxes. Proposition 1 
presents an analysis of the effects of the tax rate structure. 

Proposition 1 : 

Assuming that the audit probability is independent of the amount 
of taxable income reported, tax compliance will be the same under pro- 
portional and progressive tax regimes. Furthermore, changes in the 
tax rate have no effect upon compliance. 

Proof : 

Since assumptions were not made about the tax rate structure 
(other than differentiability), any strictly convex tax rate structure 
T.(») can be substituted for T(») in (1) to obtain the same first- 
order condition as in (3). Similarly, the first-order condition 
corresponding to the proportional rate structure: T 2 (x) ■ tx also 
will be the same as (3). Hence, as both first-order conditions are 
identical under T ,(•) and !_(•)> taxpayers' optimal reporting frac- 
tiles and reported taxable income levels will be the same. Also note 
that, as T(») does not appear in the first order condition, changes in 
the tax rate itself have no effect upon taxpayer compliance. 


The result in Proposition 1 concerning the proportional tax rate 
contrasts with Yitzhaki's [1974] findings that risk-averse taxpayers' 
compliance increased with the tax rate. An explanation for this dif- 
ference is that, under decreasing absolute risk aversion, a tax rate 


increase has a compliance-enhancing income effect. However, under 
risk-neutrality, the income effect is absent. Further, when penalties 
are proportional to the amount of the tax deficiency (as assumed in 
Proposition 1), the substitution effect is neutralized since penalties 
increase concurrently with the tax rate [see Yitzhaki (1974)]. There- 
fore, with both the income and substitution effects absent, taxpayers' 
compliance incentives with respect to changes in tax regimes or the 
rate structure are unaffected. 

Given the absence of tax regime and rate effects, one might be 
tempted to conclude that changes in the uncertainty level also would 
have no effect upon a risk-neutral taxpayer's compliance. However, we 
show that changes in the uncertainty level will generally affect com- 
pliance. Furthermore, in contrast with Aim (1988) who found that 
increased uncertainty will either have ambiguous effects or enhance a 
risk-averse taxpayer's compliance, we are able to identify conditions 
under which compliance will decline. 

A taxpayer's uncertainty about taxable income is likely to depend 
upon the taxpayer's particular circumstances (e.g., sources of income) 
and the complexity of applicable tax laws and their susceptibility to 
change. For modelling purposes, we now assume that the taxpayer has a 
uniform distribution for taxable income. Given a uniform distribu- 
tion, uncertainty can be manipulated in our model by introducing a 
second uniform distribution, J(x) having the same mean (y) as F(x), 
but a larger range, [L-A.H+A], where < A < L represents a perturba- 
tion parameter. Before comparing the reporting decisions under 
distribution J(x) with those under F(x) , several additional features 


of the distributions should be noted. First, as both uniform distri- 
butions are symmetric and have the same mean, it is apparent that 
J(s) ■ F(s) = .5 for s ■ ]i, A second feature is that J(s) > (<) F(s) 
for all s < (>) u. Proposition 2 indicates how the amounts of taxable 
income reported are affected by changes in the taxpayer's uncertainty 
about taxable income. 

Proposition 2 : 

Assuming a uniform income distribution, a risk-neutral taxpayer's 
compliance will increase (decrease) in response to changes in the 
uncertainty level depending upon whether p/(l-p) > (<) 2/<j>q. Only in 
the special case in which p/(l-p) = 2/^q will compliance be unaf- 

Proof : (See the Appendix.) 

The results in Proposition 2 are in contrast with Aim (1988) and 
Scotchmer and Slemrod (1988) who found that (decreasingly) risk-averse 
taxpayers will report higher levels of income. The differences in 
results appear to be based upon differences in technical modelling 
assumptions and by the presence of a compliance-enhancing income 
effect. Given our assumption of risk-neutrality, uncertainty level 
changes influence reporting incentives through the expected penalty. 
In particular, increasing the range of the income distribution can be 
shown to increase the taxpayer's expected marginal penalty when the 


initial level of reported income is above the mean. Such a penalty 
increase creates incentives for the taxpayer to report a higher income 
level to maintain the previous equilibrium relationship between ex- 
pected benefits (tax savings) and marginal costs (penalties). A 


parallel argument will show that, when the initial level of reported 
income is below the mean, increased uncertainty will reduce the 
expected marginal penalty, thereby resulting in a lower reported 
income. Hence, elevating the taxpayer's uncertainty level will have a 
mean-diverging effect on reported income, except when the taxpayer's 
optimal decision is to report at the mean of the income distribution. 
In the following section, we determine whether these results can be 
extended to an environment in which the tax agency operates strate- 

3. Game Theory Model 

Strategic tax auditing policies are now modelled by incorporating 
the tax enforcement agency as an active player in an extensive form 
game. The taxpayer is assumed to make the reporting decision first 
and then the tax enforcement agency decides whether or not to audit 
upon receipt of the taxpayer's return. Consistent with the concept of 
a sequential equilibrium, each player in the game is assumed to employ 
optimal strategies given the previous moves of the other players. An 
important advantage of this approach is that the tax agency will have 
no incentive to deviate ex post from the audit strategy anticipated by 
taxpayers when making their reporting decisions. Reinganum and Wilde 
(1987) also employ the sequential equilibrium framework in modelling 
the interplay between taxpayer reporting and tax agency audit poli- 
cies. The present model differs from theirs in that, prior to the 
audit, the taxpayer and the tax agency are assumed herein to be 
uncertain about the taxpayer's actual taxable income and associated 
tax and penalty liabilities. 


The tax agency's audit decision is assumed to be based upon a 
cost-benefit analysis in which the expected values of incremental 
revenues from tax collections and penalties are balanced against audit 
costs. Given our results in Proposition 1, we simplify by assuming a 
proportional tax rate structure. Such an assumption is arguably a 
reasonable approximation of the present income tax rate structure in 
the U.S. following the Tax Reform Act of 1986. Assuming a propor- 
tional tax rate, the incremental expected total revenue that would be 

collected from a taxpayer known to have an income distribution f(x), 


but reporting a taxable income of R, is given by: 

H H 

B - / txf(x)dx + <frqt/ (x-R)f(x)dx - Rt, (4) 

L R 

where t denotes the proportional tax rate. 

The first term in (4) is the expected value of tax collections, 
while the second represents the expected value of penalties. As 
audits are costly, however, a risk-neutral tax agency will have incen- 
tives to perform an audit only when the expected benefit exceeds the 
cost (i.e., c < B). Note that the expected audit benefit (B) is mono- 
tone decreasing in R, having a maximum value when R equals the lowest 
possible income level L. 

Since t and q are assumed to be known by taxpayers, the expected 
benefit (B) from an audit can be computed. Thus, if the tax agency's 
cost (cutoff point) were known, there would be no uncertainty about 
whether or not an audit would be performed. Taxpayers in our model, 
however, are uncertain about audit costs. Hence, the tax agency's 


audit decision is uncertain from the perspective of taxpayers. Tax- 
payers, however, are assumed to assess a uniform probability density 
function for audit costs, g(c), defined over the interval, [C.,C U ]. 
Throughout the study, the maximum expected audit benefit (i.e., B when 
R - L) is assumed to be greater than the lowest possible audit cost 
C . Without this assumption a trivial equilibrium would exist in 
which taxpayers report the lowest possible income level L and the tax 
agency never performs an audit. 

Letting G(«) denote the cumulative probability distribution for 
audit costs (as assessed by a representative taxpayer), the probabil- 
ity of an audit is given by G(B), where B is a function of R as defined 
in (4). Accordingly, the taxpayer's expected liability from reporting 
taxable income of R is given by: 

H H 

ET = G(B){/ txf(x)dx + <frqt/ (x-R)f(x)dx) 

L R 

+ (l-G(B))tR. (5) 

Rearranging terms and making use of the definitions: G(B) = 

(B-C L )/(C -C.) and u = / xf(x)dx, (5) can be rewritten as: 


(B-C ) H 

ET - -rz — rr-r [ut + <frqtf (x-R)f(x)dx - Rt ] + Rt. (6) 

^H~V R 

Upon simplification, the first-order condition for the taxpayer's 

optimal reporting decision is given by: 

(2B-C ) 

(c _ c } [1 + ♦qd-FCR*))] = 1, (7) 

H L 

where B is defined in (4). 


Two important comparative statics properties of the model concern 
the effects of changes in the penalty and tax rate. Proposition 3 
verifies that, consistent with the results obtained in the fixed audit 
probability model, taxpayer compliance increases concurrently with the 
penalty rate for underpayment of taxes. 

Proposition 3 ; 

Taxpayer compliance is an increasing function of the expected 
penalty ($q) on the tax deficiency. 

Proof : (See the Appendix.) 

A somewhat more interesting result concerns the effects of changes 
in the tax rate. While we previously found that tax rate changes had 
no effect upon compliance when taxpayers expected a fixed audit proba- 
bility, such is not the case when the tax agency is expected to behave 

Proposition 4 : 

Taxpayer compliance is an increasing function of the tax rate. 

Proof : (See the Appendix. ) 

Propositions 3 and 4 arise from the impact that changes in the tax 
and penalty rates have on the audit benefit. In particular, increas- 
ing the tax and/or penalty rate results in a larger audit benefit and 
higher equilibrium audit probability. Such an increase in the equili- 
brium audit probability enhances taxpayers' compliance incentives vis- 
a-vis the previous environment in which the audit probability was 


fixed. Hence, the amount of taxable income reported by the taxpayer 
increases even when both the substitution and income effects are 

A further issue concerns the role of uncertainty about taxable 
income. We show in Proposition 5 that uncertainty effects are 
identical to those in Proposition 2 for cases in which the initial 
reporting level is above the mean. When the initial report is below 
the mean, however, the effects are not necessarily the same as in 
Proposition 2. 

Proposition 5 : 

Assuming a uniform income distribution, when taxpayer's initial 
reported income level is above the mean, increasing taxpayer uncer- 
tainty (range of possible taxable income levels) will enhance com- 
pliance. Otherwise, the effects are potentially ambiguous. 

Proof : (See the Appendix.) 

Once again, the intuition underlying this result is that mean- 
preserving changes in the dispersion of audit outcomes can affect the 
audit benefit and equilibrium audit probability. Unlike penalty and 
tax rate changes, however, the specific effects depend upon the ini- 
tial reporting level. Note that when the taxpayer's initial reporting 
level is above the mean, for a given R value, the expected value of 
the audit penalty (partial expectation over the interval, [R,H+A]) 
will increase with the range of taxable income levels since the lower 
endpoint remains fixed, while the upper endpoint is raised. Since 
this will lead to an increase in the audit probability, the enhanced 
compliance incentives that were already present with a fixed audit 
probability will be further reinforced. When the initial reported 


taxable income level is below the mean, however, the expected value of 
penalties will increase, but to a lesser extent as low income realiza- 
tions also become more likely. Consequently, when the initial 
reporting level is low, the marginal effect upon the audit benefit and 
equilibrium audit probability will be smaller than in the previous 
case. Since the incentives which would otherwise exist for taxpayers 
to reduce their compliance levels (see Proposition 2) under a fixed 
audit probability may not be offset, the effects of increased income 
uncertainty become ambiguous. 

4. Conclusions 

The present study has provided an analysis of taxpayer reporting 
decisions under alternative tax auditing policies. With the exception 
of penalty rate increases that were found consistently to enhance 
taxpayer compliance, the other comparative static properties were 
found to be sensitive to taxpayer expectations regarding the tax 
agency's audit policies. Under the assumption of a fixed audit proba- 
bility, tax rate increases had no effect upon risk-neutral taxpayer 
compliance. Alternatively, when taxpayers expected the tax agency to 
behave strategically in response to reported income, tax rate 
increases induced greater compliance. 

The effects of changes in taxpayers' uncertainty about taxable 
income also were found to be sensitive to taxpayer expectations about 
the tax agency's audit policies and to be dependent upon the initial 
reporting level. In particular, when the audit probability and the 
penalty rate were sufficiently high to induce taxpayers to report ini- 
tially above the mean of their taxable income distribution, increased 


uncertainty further enhanced compliance under both fixed and strategic 
tax audit policies. However, under the complimentary circumstances in 
which taxpayers initially reported below the mean, taxpayer compliance 
was shown to decline in response to increased uncertainty under fixed 
audit policies, but have ambiguous effects under strategic audit poli- 

The above results have potential implications for policy-makers. 
First, frequent changes in tax statutes that create or elevate tax- 
payer uncertainty can have direct effects upon taxpayer compliance. 
Second, predictions regarding the specific effects on compliance must 
take into account taxpayer expectations regarding tax audit policies. 
Thus, tax policy and tax enforcement issues should be analyzed con- 

Several simplifying assumptions were introduced in modelling tax- 
payer compliance. One important assumption is that taxpayers are 
risk-neutral. As noted previously, incorporating taxpayer risk aver- 
sion would create an income effect. In an unpublished study, Beck and 
Jung (1988a) demonstrated that tax rate increases will predictably 
enhance risk averse taxpayers' compliance. Another key assumption is 
that taxpayers and the tax agency have symmetric uncertainty about 
taxable income. In some cases, however, the information sets of tax- 
payers and the tax agency could differ. To the extent that taxpayers 
have additional information about their actual tax liabilities, 
reported income will become a potential signal as in the Reinganum and 
Wilde (1986) and Beck and Jung (1988b) models. However, based upon 


their findings, we would not expect the presence of information asym- 
metry to have a qualitative effect upon the results of the present 

An interesting extension of the present study would be to incor- 
porate a tax advisor as a means of reducing taxpayer uncertainty. 
Reinganum and Wilde (1988) have already obtained some preliminary 
results in a tax advisor model. Another extension would be to test 
experimentally the model-based implications in an experimental eco- 
nomics context. Such testing would be particularly useful in deter- 
mining the robustness of results to various modelling simplifications 
and permit additional evidence to be gathered in cases where model- 
based predictions are ambiguous. 



^lm (1988) and unpublished studies by Scotchmer (1987b), 
Scotchmer and Slemrod (1988), and Beck and Jung (1988a) also have 
modelled taxpayer uncertainty about taxable income. Our models differ 
from theirs in that we incorporate the interplay between the tax 
agency's audit policies by means of a sequential equilibrium model. 
The present study also differs from Beck and Jung (1988b) in several 
respects. First, they assume asymmetric information between taxpayers 
and the tax agency, but utilize a discrete probability distribution, 
while we assume symmetric information and model income as being con- 
tinuous [see Slemrod (1988) for a discussion of the contrast between 
the discrete and continuous income models]. A further difference is 
that they define compliance in terms of the taxpayer's type (cut off 
probability for reporting the high income level), whereas we define 
compliance in terms of the amount of income reported by the taxpayer. 

examples of such characteristics could include the taxpayer's 
occupation, sources of income, residential location (zip code), and 
the particular schedules filed together with the tax returns. 

Section 6661(a) of the United States Internal Revenue Code (IRC) 

provides a penalty equal to 20 percent of additional taxes due when 
the tax liability is substantially understated. This occurs when the 
reported tax liability is understated by $5,000 or 10 percent of the 
post-audit tax liability, whichever is larger. Several other 
penalties that are proportional to additional taxes due may also be 
assessed. IRC Section 6653(a)(1) imposes a penalty equal to 5 percent 
of any underpayment of tax together with a non-deductible interest 
charge (essentially a penalty) equal to 50 percent of the interest 
relating to any underpayment attributable to taxpayer negligence. In 
addition, IRC Section 6651(a)(2) provides for a maximum penalty of 25 
percent of additional taxes due for failure to pay. If the balance 
due after filing is more than 10 percent of the tax shown on the 
taxpayer's return, such a penalty will be imposed unless reasonable 
cause can be shown. When the taxpayer can be shown to have de- 
liberately understated taxable income by failing to report income or 
by knowingly taking inappropriate deductions, criminal penalties also 
can be imposed for fraud. Since taxpayers are assumed to be uncertain 
about their taxable income, criminal issues are not addressed in our 

In the United States, Section 6661(b)(2)(B) of the Internal 

Revenue Code (1987) states that the amount of understatement [of 

taxes] under subparagraph (A) shall be reduced by that portion of the 

understatement which is attributable to the tax treatment of any item 

by the taxpayer if there is or was substantial authority for such 

treatment. When the various sources of authority (e.g., judicial, 

statutory, and administrative systems) conflict, however, taxpayers 

could have uncertainty about whether there is substantial authority 


for their position. A recent study by Chow, Shields, and Whittenburg 
(1987) examined the judgments of experienced tax practitioners and 
found a high level of consistency, but only a moderate level of 
consensus regarding the presence of substantial authority in the 
specific cases analyzed. 

Differentiating (1) a second time with respect to R, one can 
verify that the second-order (sufficient) condition for an interior 
solution will be satisfied provided that: 

[p,frq(l-F(R))-(l-p)]T"(R)/T'(R) < p<frqf(R). 

Since the terms inside the brackets on the LHS of the inequality are 
zero due to the first-order condition, while the terms on the RHS are 
positive, the latter inequality is clearly satisfied. 

The assumption of a uniform distribution is not essential to our 
analysis. Our proof requires that the cumulative distributions cross 
at only one point (x.q) (i.e., G(x) > F(x) for x < x^ , with the in- 
equality reversed for x > xq) . The uniform distribution family is 
particularly convenient to employ as the crossing point occurs at 
X(] = y = (H+L)/2. We have also analyzed other income distributions 
having the single crossing point property and have verified that tax- 
payer compliance will increase (decrease) depending on whether the 
initial reporting fractile is above (below) the crossing point. 

Aim found that, in general, mean-preserving changes in the dis- 
persion of the distribution of evaded income will have ambiguous 
effects upon taxpayer reporting. However, under the assumptions of 
decreasing absolute risk aversion and non-increasing relative risk 
aversion of less than one, Aim showed that increased uncertainty would 
increase declared income. While we focus on mean preserving changes 
in the dispersion of the total income distribution, Aim (1988) con- 
siders mean preserving changes in the distribution of evaded income. 
In our model this would correspond to the truncated income (penalty) 
distribution defined over the interval [R,H] whose expectation can 
(and in fact usually will) change with increased uncertainty. 
Scotchmer and Slemrod (1988) also focus on mean preserving changes 
in the total income distribution, but assume a discrete income dis- 
tribution. Thus, changes in uncertainty in their model will change 
the expected value of penalties in their model, but not affect the 
probability of penalty occurrence as in our model. Slemrod (1988) 
provides a further discussion. 

The taxpayer's optimality condition function under distribution 
J(x) is given by: 

(l-p)T'(R**) - p4,qT'(R**)J j(x)dx » 0, (A) 


where j(x) - J'(x) and R** denotes the optimal solution to (A). 


As the right hand sides of (A) and (2) are equal, it follows that: 

(l-p)T'(R**) " p$qT'(R**)/ j(x)dx 


= (l-p)T'(R*) - p$qT(R*)J f(x)dx. (B) 


We will now show that if R* > y, then R** > R*. Suppose to the 
contrary that R** = R*. Given this supposition, it follows from (B) 

H+A H+A 

/ j(x)dx = / f(x)dx. (C) 

R* R* 

Making use of the facts that f(x) = 1/(H-L) and j(x) = 1/(H-L+2A), one 
can verify that (C) is equivalent to: 

[(R*-L)(H-L+A) - (H-L)(R*+A-L)]/[(H-L+2A)(H-L)] = 0. (D) 

Upon simplification, (D) can be rewritten as: 

| (R*-y) - 0. (E) 

Note that when R* > u, (E), (D), and (C) imply by transitivity 
that (B) will be violated. Hence, we have obtained a contradiction. 
In particular, the RHS of (B) will be smaller than the LHS since the 
expected marginal penalty is larger under distribution j(x). In order 
to restore the optimality condition in (B) the taxpayer must, there- 
fore, reduce the expected marginal penalty by increasing reported 
income (i.e., R** > R*). A parallel argument will show that if R* < 
U, the expected marginal penalty term under j(x) will be smaller than 
the penalty under f(x), so the taxpayer will have an incentive to 
reduce the reported income level. 

An implicit assumption is that taxpayers and the tax agency have 

symmetric information about taxable income. In practice, one might 
argue that taxpayers and the tax agency could have different sources 
of uncertainty. The taxpayer could be uncertain about the appropriate 
tax treatment, while the tax enforcement agency could be uncertain 
about factual circumstances. Under such conditions, the taxpayer's 
report and the tax agency's audit decision both could provide signals 
regarding their respective private information. Reinganum and Wilde 
(1986) and Beck and Jung (1988b) have developed models in which tax- 
payers have private information that is communicated by their reported 
income. In the present study, we suppress the signalling value of the 
report to simplify the model and to facilitate a more direct comparison 
with the fixed audit probability model in the previous section. 


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Proof of Proposition 2 ; 

In proving Proposition 2, it is useful to make a preliminary 
observation. From the optimality condition in (3), one can verify 
that a risk-neutral taxpayer will report at the .5 fractile (mean) of 
his (her) income distribution when p/(l-p) " 2/(<J>q) and above (below) 
the mean when p/(l-p) > (<) 2/($q). 

Now suppose that the taxpayer's uncertainty is elevated so that 
J(x) represents the taxpayer's income distribution. The optimality 
condition corresponding to J(x) is given by: 

J(R**) - 1 - (l-p)/(p<frq), (1A) 

where R** denotes the solution to (1A). 

Since the RHS of (1A) and (3) are equal, it follows that: 

J(R**) - F(R*). (2A) 

Observe that when p/(l-p) > 2/($q), R* > u. Therefore, as 
F(R*) > J(R*) for R* > u, by transitivity, (2A) implies that J(R**) > 
J(R*). As J(») is a strictly increasing function, R** > R* as claimed. 
A parallel argument will establish that when p/(l-p) < 2/($q), R* < u 
and that: J(R*) > F(R*) - J(R**). Only in the special case in which 
p/(l-p) - 2/($q) will R* - R**. Q.E.D. 

Proof of Proposition 3 : 

Differentiating the first-order condition in (7) with respect to 
$q , we obtain: 


2(^||-T)U+*q(l-F(R*))] + (2B-C. ) [ ( l-F(R*))-*qf (R*>|~] = 0, (3A) 

3 t $q ) " 3q 





e ■ ^r - ^ (1 - FCR *»w + «/<***>««>*. 

Collecting the terms involving -r-r- — r, substituting (4A) into (3A), 

3 v<J>q J 

and rearranging terms, (3A) is equivalent to: 

T77^ T {2[-t-*qt(l-F(R*))](l+*q(l-F(R*)))-(2B-C T )*qf(R*)} 
3 k$q ) " 

+ {2(t/ (x-R*)f(x)dx)(l+*q(l-F(R*))+(2B-C T )(l-F(R*))> = 0. (5A) 

As < F(R*) _< 1 and B > C. , it follows that the second term en- 
closed by braces in (5A) is positive. Since the terms inside the 


first set of braces are negative, it must be the case that —, r- > 


in order to satisfy (5A). Q.E.D. 

Proof of Proposition A : 

Differentiating (7) with respect to t, 

(2 ||)(l+*q(l-F(R*))+(2B-C L )[-*qf(R*>|^], (6A) 



I7 - U - R* - *qt |~ • (1-F(R*)) + *qj (x-R*)f Cx)dx. (7A) 
3t at R 

Substituting (7A) into (6A) and simplifying, one can verify that 


|^{-2<J,qt(l-F(R*))(l+^l-F(R*)))-<t,qf(R*)(2B-C T )} 
3 1 Li 

+ {2[ u -R*+<|>q/ (x-R*)f(x)dx](l+$q(l-F(R*))} = 0. (8A) 


Note that: u - R* + 4>q/ (x-R*)f (x)dx = B/t > 0. Hence, as the 

terms inside the first set of braces are negative, while those inside 

the second set of braces are positive, (8A) requires that - — > 0. 

o t 


Proof of Proposition 5 : 

Proposition 5 is established by perturbing both the lower and 
upper support of f(x) by A to obtain the distribution, J(x) ■ 
(x-L+A)/(H+A-(L-A)). Substituting into (7), the revised optimality 
condition becomes: 

(2B "°L ) .. A (H-R**+A) , . , n 
TC^T [1 + W (H-L)+2A ] " l '°> 

where B - t(y-R**) + <f>qt . (H-R**+A) 2 / [2(H-L+2A) ] . 

Differentiating (9A) with respect to A, we obtain: 

SB (H-R**-hO (2R**-H-L)-(H-L+2A).-^ 

2 H [1 ^ (H-LH2A ^(2B-C )4> q[ 5 ^-] - 0, (10A) 

3A (H L)+2A L ((H-D+2A) 2 



M *aL_^ H - R ** +A)[(R **- L+A) - (H -L+2A).-^] - t- 3 -^ (11A) 

3A (H-L+2A) 2 3A 3A 

Upon rearranging terms, (10A) can be rewritten as: 


{-2t(H-L+2 A )[l+ (H "q +2A) ]U^q ("l^I)* 1 " (H-L+2 A )(2B-C L )> 

9A l <H 

+ 2t(H-R**+A)(R**-L+A)[l+ ^("1^1) ^ + (2B " CL) * (2R**-H-L) = 0. (12A) 

Note that, as the coefficient of ■ is negative, the sufficient 


condition for — — — > is that the remaining terms in (12A) be positive. 

While these terms could be either positive or negative, observe that 
they will be positive when 

2R**-H-L > 0. (13A) 

Simplifying, (13A) is equivalent to 

R** ^ (H+L)/2 - y. 

Thus, a sufficient condition for _ . > is that R** > y. Q.E.D. 

3q —