How tax agency audit policies, tax rates, and uncertainty affect taxpayer compliance

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FACULTY WORKING
'APER NO. 1467

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

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

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

BEBR

FACULTY WORKING PAPER NO. 1467

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

http://www.archive.org/details/howtaxagencyaudi1467beck

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

Abstract

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.

-2-

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

-3-

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.

-4-

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

2
observable characteristics. As many previous studies have also

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

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

-5-

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

-6-

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

3

structure.

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

4
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)

R

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

L

H

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

R
H

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

R

-7-

The expected tax and penalty liability can be written as:

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

L

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

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

R

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

-8-

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: T2(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.

Q.E.D

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

-9-

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 0 < A < L represents a perturba-
tion parameter. Before comparing the reporting decisions under
distribution J(x) with those under F(x) , several additional features

-10-

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-
fected.

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

Q

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

-11-

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-
gically.

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.

-12-

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),

9

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

-13-

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.,CU].
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) =

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

u

(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).

-14-

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
strategically.

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

-15-

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

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

-16-

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

-17-

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-
cies.

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-
currently.

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

-18-

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

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.

-19-

Footnotes

^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.

3
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
study.

4
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

-20-

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.

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

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

R**

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

-21-

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

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

R**

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

R*

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)
that

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.

9
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.

-22-

Ref erences

Allinghatn, M. and A. Sandmo. 1972. Income Tax Evasion: A Theoretical
Analysis. Journal of Public Economics, 323-338.

Aitken, S. and L. Bonneville. 1980. A General Taxpayer Opinion Survey.
Prepared for the Office of Planning and Research, Internal Revenue
Service, March.

Aim, J. 1988. Uncertain Tax Policies, Individual Behavior, and Wel-
fare. American Economic Review, 237-245.

Ayres, F., B. Jackson, and P. Hite. 1987. Factors Related to the
Degree of Aggression Recommended by Professional Tax Preparers:
An Empirical Analysis. Unpublished Manuscript, University of
Oklahoma, April.

Beck, P. and W. Jung. 1988a. An Economic Model of Taxpayer Compliance
under Complexity and Uncertainty. Unpublished Manuscript,
University of Illinois, March.

Beck, P. and W. Jung. 1988b. Taxpayer Compliance and Auditing under
Asymmetric Information: A Game-Theoretic Approach. Unpublished
Manuscript, University of Illinois, April.

Chow, C, Shields, M. , and G. Whittenburg. 1987. An Examination of

Tax Practitioners' Judgment Quality Regarding Substantial Authority.
San Diego State Working Paper, June.

Graetz, M. , J. Reinganum, and L. Wilde. 1986. The Tax Compliance
Game: Toward an Interactive Theory of Law Enforcement. Journal
of Law Economics and Organization, 1-32.

Graetz, M. , and L. Wilde. 1985. The Economics of Tax Compliance:
Fact and Fantasy. National Tax Journal, 355-363.

Greenberg, J. 1984. Avoiding Tax Avoidance: A (Repeated) Game-
Theoretic Approach. Journal of Economic Theory, 1-13.

Kakwani, N. 1978. Income Tax Evasion and Income Distribution. New
York: Frank Cass and Company, 161-173.

Koskela, E. 1983. A Note on Progression, Penalty Schemes and Tax
Evasion. Journal of Public Economics, 127-133.

Landsberger, M. and I. Meilijson. 1982. Incentive Generating State
Dependent Penalty System. Journal of Public Economics, 333-352.

McCaleb, T. 1976. Tax Evasion and the Differential Taxation of Labor
and Capital Income. Public Finance, 287-294.

-23-

Reinganum, J. and L. Wilde. 1985. Income Tax Compliance in a

Principal-Agent Framework. Journal of Public Economics, 1-18.

Reinganum, J. and L. Wilde. 1986. Equilibrium Vertif ication and
Reporting Policies in a Model of Tax Compliance. International
Economic Review, 739-760.

Reinganum, J. and L. Wilde. 1987. A Note on Enforcement Uncertainty
and Taxpayer Compliance. Unpublished Manuscript: California
Institute of Technology, August.

Reinganum, J. and L. Wilde. 1988. Tax Practitioners and Tax Com-
pliance. Unpublished Manuscript, University of Iowa, April.

Scotchmer, S. l987a. Audit Classes and Tax Enforcement Policy.
American Economic Review, 229-233.

Scotchmer, S. 1987b. Who Profits from Taxpayer Confusion? Unpub-
lished Working Paper, University of California, Berkeley,
November.

Scotchmer, S. and J. Slemrod. 1988. Randomness in Tax Enforcement.
Unpublished Manuscript, University of Michigan, January 1988.

Singh, B. 1973. Making Honesty the Best Policy. Journal of Public
Economics, 257-263.

Slemrod, J. 1988. Complexity, Compliance Costs, and Tax Evasion.
National Academy of Sciences Conference on Taxpayer Compliance,
January 1986 (volume to appear in 1988).

Srinivasan, T. 1973. Tax Evasion: A Model. Journal of Public
Economics, 339-346.

U.S. Congress. 1987. Internal Revenue Code of 1986 (Paramus, New
Jersey: Prentice Hall Information Services).

Yitzhaki, S. 1974. A Note on Income Tax Evasion: A Theoretical
Analysis. Journal of Public Economics, 201-202.

D/499

-24-

Appendix

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:

-25-

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

3 t $q ) " 3q

where:

H

r

R

e ■ ^r - ^(1-FCR*»w + «/<***>««>*.

3R*
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-CT)*qf(R*)}
3 k$q ) "

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

As 0 < 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

3R*

first set of braces are negative, it must be the case that —, r- > 0

8($q)

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-CL)[-*qf(R*>|^], (6A)

where:

u

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

-26-

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

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

R*

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

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

3R*
the second set of braces are positive, (8A) requires that - — > 0.

o t

Q.E.D.

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).-^

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

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

(9A)

where:

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:

-27-
3R**

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

9A l <H

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

3R**
Note that, as the coefficient of ■ is negative, the sufficient

da

3R**
condition for — — — > 0 is that the remaining terms in (12A) be positive.
da

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.

3R**
Thus, a sufficient condition for _ . > 0 is that R** > y. Q.E.D.

3q —

IECKMAN

,|NDERV INC.

JUN95