National Tax Journal, Vol. 41, no. 4,
(December, 1988), pp. 579-81
CORPORATE
TAX EVASION AND OUTPUT DECISIONS
UNCERTAIN
MONOPOLIST**
OF THE
LEONARD F. S. WANG* AND JOHN L. CONANT*
1. Introduction
N their recent article, Kreutzer and Lee
I (1986) explore the possibility of using a
profit tax to reduce monopoly distortion.
They conclude that monopolists can reduce their tax liability by overstating the
costs of production, and that this cost
overstatement (tax evasion) induces them
to increase their output. Kreutzer and
Lee's conclusion casts some doubt on the
conventional view that profit taxes are
neutral with respect to a monopolist's
profit maximizing rate of output. Marrelli (1984) on the other hand, analyzed a
monopolistic firm's decision on whether,
and to what extent to avoid an ad valorem (sales) tax by under-reporting
revenues.
A considerable literature has followed
the pathbreaking paper by Allingham and
Sandmo (1972) focusing on the phenomenon of individual income tax evasion.
Allingham and Sandmo pioneered the
analysis by placing the taxpayer's decision concerning the level of declared income in a standard choice under uncertainty model. In these models, individual
taxpayers analyze at the margin their expected benefits from undetected tax evasion and their expected costs from the
penalty that accompanies a detected evasion.
The purpose of this paper is to formulate an uncertainty model for monopolists
along the lines of Allingham and Sandmo
which incorporates the incentives to reduce tax liability by under-reporting
prof_
its through an overstatement
of production costs. This model will invalidate the
conclusion reached by Kreutzer and Lee,
and reinforce the generally held view that
profit taxes are neutral with respect to the
profit maximizing rate of output. The
problem
we consider
here is important
since a monopolist has the incentive to
conceal the true cost of production when
*Indiana
State University,
Terre Haute,
IN 47809.
579
the average cost is declining
evant range of output.
over the rel-
II. The Basic Model and Analysis
We adopt the standard approach of
analyzing decision making under uncertainty. The monopolistic firm can evade
profit tax liability by cost overstatements
(8), which are either detected (with probability p), or remain undiscovered (with
probability 1
p). The firm's after-tax
profit when the tax authority does not detect the cost overstatement
is HI, and the
after-tax and after-penalty profit when the
authority successfully detects the evasion
is F[2. Being detected
in under-reporting
of profits involves a penalty (s). It is assumed that this penalty increases the tax
rate (t), so that the penalty tax rate is st
(s > 1) which is applied by the tax authority to the unreported portion of actual profit. The firm's net profit when it
overstates its costs and is not detected by
the authorities is:
Fli
[R(Q) - C(Q)j
t[R(Q) (I + 6)C(Q)l
(1
t)[R(Q)
C(Q)l + tbC(Q)
(1)
where R(Q) is total revenue and C(Q) is
total cost. On the other hand, if the firm's
cost overstatement
is discovered by the
taxing authorities, its net profit is:
F12
III
St[8c(Q)l
(2)
It is assumed that the preference function of the monopolist is given by a von
Neuman-Morgenstem
utility function, i.e.,
dU(Fl)/dFl > 0, U"(FI)
U(fl)2U(II)
with u,(n)
/dF12 < 0. Here, U"(rI) < 0 imd
plies that the monopolist is risk averse in
the sense of Tobin. In our model there is
a fixed probability (p) of tax evasion being
discovered, and the firm is assumed to determine its optimal output level (Q) and
National Tax Journal, Vol. 41, no. 4,
(December, 1988), pp. 579-81
NATIONAL
580
TAX JOURNAL
[Vol. XLI
their result that the after-tax profit maximizing output will be larger than the
output which would maximize profits in
the absence of a tax. Their model, how(3) ever, did not recognize the probability that
MAX EU - (I - P)U(rll) + PU(112)
Q b
the cost overstatement
would be detected
by the tax authorities
who would then
The first-order conditions for an interior
impose a penalty on the amount of unmaximum of (3) can then be written as:
reported profit (i.e., p = 0).
With p = 0, as long as both t and 8 are
positive, the marginal pre-tax profit, R' aEU
(I p)[(R' C')(1
t)
C' will be negative. And as a result,. the
-aQ
after-tax profit maximizing output, Q, is
+ t8c,]U'(Fil) + p[(RI - Cl)(i - t)
greater than the output, Q which would
(4) maximize profit in the absence of a tax.'
+ (I - S)tbC']U'(112)
- 0
From (6a) and (6b) it is clear that the
optimal output of the uncertain
monopoand
list, Q*, lies between Q and
From (4) we have:
aEu - (i - p)[tC(Q)]U'(fll)
a8
U'(F12)
(5)
+ P[(l - S)tC(Q)]U'(rl2) - 0
cost overstatement factor (b) so as to maxiniize its expected utility function EU ex
ante; which can be written as:
where R' is marginal revenue, C' is marginal cost, and U'(fli), U'(112) are the first
derivatives of the utility function with respect to fl, and fl2, respectively.
From (4) arid (5) we will be able to find
the optimal values of Q and B. We assume
that the second-order conditions are satisfied everywhere.
Equation (4) can be written as:
(i - p)[(R' - C')(1
t) + tbc,l U'(fll)
P[(R' - C')(i - t)
+ (I
- S)tbC']U'(fl2)
(4')
> 0,
p), p, U'(fil) and U'(rl2)
Since (1
equation (4') will be verified only if [(R'
- C,)(1 t) + tbcll > 0 and [(R' - C')(1
- t) +
s)tBCII < 0. That is, at the
optimum, the firm will find its equilibriun, point somewhere between the quantities for which:
(i - t)R' - [1
(1 + 8)t]C'
p)[(R'
P[(R'
C')(1
+ 6)t + sbt]C'
t) + (I
s)tBC'l
And from (5) we have:
(I - P)
U'(F12)
U'(Fll)
p(l
(8)
S)
From (7) and (8), we see that the choice
of 8 considerably
modifies the optimum
condition for the firm's production
decision. From (8) we find 8 - 6* and then
equation (7) can be written as:
U'(fl2)
U,(rll)
(1
p)[(R' - C')(1 - t) + tb*C'l
p[(R' - C')(1 - t) + (1
s)ta*C'l
(7')
By equating the right
and (8) we get:
[1
C')(I - t) + tBC'l
(7)
(6a)
and
(1 - t)RI
(I
hand sides of (7')
(6b)
Condition (6a) is the same as Kreutzer
and Lee's condition (2) which provided
s(R' - C')(1 - t) - 0
(9)
Since 0 - t :s 1 and s > 1, there exists
National Tax Journal, Vol. 41, no. 4,
(December, 1988), pp. 579-81
No. 41
THE UNCERTAIN
a Q - Q such that R' - C'; that is, an
uncertain monopolist who is able to find
the optimal cost overstatement
factor, and
set its production level where marginal
revenue equals marginal cost. This result
is particularly
interesting
because it implies that neither the profit tax nor the
penalty rate affect the firm's output decision whatsoever.
In general, it can be said that if the
profit-tax-paying
firm is able to equate the
marginal rate of substitution
between rll
(escape) and n2 (detected) to the real price
of evasion, -(I - p)/p(l - s), then tax
evasion (overstating
costs) has no influence on the output decision of the uncertain monopolist. That is, the production
decision and the evasion decision are, in
this case, separable.
This result significantly differs from the
certainty model of Kreutzer and Lee which
does not consider the probability
of detection and punishment
where the monopolist increases production beyond the
point where marginal
cost equals marginal revenue. In their model the firm increases output until the marginal
loss of
production is equal to the marginal
gain
from reduced tax liability on overstated
costs. An uncertain
monopolist
in our
model sets the level of production at the
traditional optimum (MR - MC), and then
considers the tax rate, penalty rate, and
the probability
of detection in determining the optimal level of cheating which
maximizes its expected utility of after-tax
profits. The well-known result that profit
taxes are neutral under certainty
also
holds under an uncertainty
framework,
and thus profit taxes cannot be used by
public policy analysts to reduce monopoly
distortion.
A comparative
statics analysis of the
model reveals that increases in the profit
tax rate, the penalty rate and/or the
probability of detection reduce the optimal level of profit tax evasion.2
III. Conclusion
We have examined the profit tax evasion and output decisions of the uncertain
581
MONOPOLIST
monopolist. We have shown that contrary
to the results of Kreutzer and Lee, that
when the firm considers the probability
of detection and punishment
in its expected utility function of profit, that the
uncertain
monopolist's
optimal rate of
output is not affected by either the profit
tax or the penalty rate. The firm's decisions concerning the level of output and
the extent of tax evasion are separable.
Although not derived here, a comparative static analysis of our model shows
that variations
in the tax rate, the penalty rate, and the probability of detection
affect the level of tax evasion. As in the
literature on individual income taxes, increases in these parameters
reduce the
optimal amount of tax evasion.
ENDNOTES
**We wish to thank two anonymous referees for their
helpful suggestions.
'If we ignore the strategy of over-reporting
costs (b
0), FI, and fl2 are identical, and thus U(Hl) = U(F12)
and equation (4') reduces to:
U'(H)[(R'
C')(1
t)]
0
Thus, in the absence of tax evasion the equilibrium
output level will be found where MR
MC, yielding
the familiar result that a profit tax is neutral.
'The comparative
statics analysis is available from
the authors on request.
REFERENCES
Allingham,
M. G. and A. Sandmo, "Income Tax Eva
sion: A Theoretical
Analysis," Journal
of Public
Economics
1 (1972), 323-338.
Becker, G. S, "Crime and Punishment:
An Economic
Approach," Journal of Political Economy 76 (1968),
169-217
Cowell, F. A., 'The Economic Analysis of Tax Evasion" in Surveys in the Economws of Uncertainty,
eds. John D. Hey and P. J Lambert, New York:
Basil Blackwell Inc., 1987, 173-203.
Hey, J. D., Uncertainty
in Microeconomics,
Oxford:
Martin Robertson, 1979.
Kreutzer, D and D. R. Lee, "On Taxation and Un
derstated Monopoly Profits," National Tax Journal
39 (1986), 241-243
Marrelli, M., "On Indirect Tax Evasion," Journal of
Public Economics 25 (1984), 181-196.
Paulsen, J W. and R. D. Adams, "Optimal Taxation
of a Monopoly," National
Tax Journal 40 (1987),
121-125.
Sandmo, A, "On the Theory of the Competitive
Finn
under Price Uncertainty,"
American Economic Re
view 61 (1971), 65 73.