The profit tax evasion and monopoly output decisions are examined in the uncer- Abstract tainty model with endogenous probability of detection. When a rational but amoral, profit-understated firm in advance considers the probability of detection and punishment, the optimal output rate and the optimal cost-overstating factor will be deliberately determined The analysis shows that the effect of profit tax may be either production in excess, or less than, the conventional monopoly output level. The result suggests that the profit tax cannot be relied on for reducing the monopoly distortion. TAX EVASION AND MONOPOLY OUTPUT DECISIONS WITH ENDOGENOUS PROBABILITY OF DETECTION LEONARD F. S. WANG Indiana State University There individual has been a recent proliferation of studies analyzing tax evasion and tax avoidance behavior. The seminal paper is that of Allingham and Sandmo (1972) who followed Becker (1968) by placing the taxpayer s decision concerning the level of declared income in a standard choice under the 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. Subsequent theoretical work has extended the basic analysis to consider alternative penalty and tax functions, and endogenous income and labor choices. I Kreutzer and Lee (1986) recently explored the possibility of using a profit tax to reduce monopoly distortion. They used a simple model in which a profit-maximizing monopolist can reduce tax liability by underreporting profits through an overstatement (by a fixed proportional factor) of production costs. They assert that this tax-evading AUTHOR S NOTE: The author thanks Professor J. Ronnie Davis and the anonymous referee for helpful comments and suggestions. Remaining errors are exclusively his own responsibility. PUBLIC FINANCE QUARTERLY, Vol. 18 No. 4, October 1990 480-487 0 1990 Sage Publications, Inc. 480
481 activity, in the absence of a penalty, will induce the firm to expand output beyond the no-tax (and pure-profit tax, no evasion) level. 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. Wang and Conant (1988) added realism by formulating an uncertainty model in the tradition of Allingham and Sandmo (1972) in which there is a positive probability of detection and a penalty proportional to the tax evaded. With the quantity of production and the proportion of cost overstatement as choice variables for an expected-utility-maximizing monopolist, they showed that firms optimal production will be determined independent of the cost-overstatement decision. That is, the conventional view of a neutral profit tax still holds under an uncertainty framework with cost overstatement. This is different from what is derived in Kreutzer and Lee (1986, 1988).2 Wang and Conant have assumed the probability of detection to be exogenously given; consequently, it is independent of the amount of cost overstatement (tax evasion). However, this is not entirely satisfactory. In the real world, tax authorities formulate rules according to which the probability of their investigating and the penalty imposed vary with the amount of cost overstatement. The purpose of this article of detection is to generalize the model by endogenizing the probability and allowing a variable penalty. With such modifications, it shows that whether the profit tax is neutral or not crucially depends on whether the probability of detection and penalty imposed vary with the amount of cost overstatement. THE BASIC MODEL AND ANALYSIS This article adopts Becker s approach of analyzing decision making under uncertainty. The monopolistic firm can evade profit tax liability by cost overstatements (5), which are either detected (with probability p), or remain undiscovered (with probability 1 - p). The probability of detection is an increasing function of the amount of cost overstatement, p (8C) > 0. The firm s after-tax profit when the tax authority does not detect the cost overstatement is TII, and the after-tax and
482 after-penalty profit when the authority successfully detects the evasion is Hz. Detection of underreported profits involves a penalty (s). It is assumed that this penalty is positively associated with the amount of cost overstatement, so that the penalty tax rate is s(ôc)t (with s > 1 and s > 0 reflecting the higher evading cost for the firm overreporting a greater percentage of production cost). The penalty 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 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 It is assumed that the preference function of the monopolist is given by a Von Neumann-Morgenstern utility function (i.e., U(II) with u (n) = du(n)/dn > 0, U&dquo;(II) = dzu(ii)/dii2 < 0. Here, u&dquo;(n) < 0 implies that the monopolist is risk averse in the sense of Tobin. The firm is assumed to choose its optimal output level (Q) and cost overstatement factor (ô) so as to maximize its expected utility function, EU, ex ante; which can be written as The first-order conditions for an interior maximum of (3) be written as can then
483 and where R is marginal revenue, C is marginal cost, and U (II1), U (II2) are the first derivatives of the utility function with respect to II, and n,, respectively. The optimal values of Q and Ô are determined from [4] and [5]. Assume that the second-order conditions are satisfied everywhere. Equation [4] can be written as 0, and the second term on Because (1 - p),p,p, U (II,) and U (TI2) > the left-hand side is negative via the concavity of the utility function, equation [4 ] will be verified only if [(R - C )(1 - t) + t8c ] > 0 and [(R - C )(1 - t) + (1 - s - s 8C)t8C ] < 0. That is, at the optimum, the firm will find its equilibrium point somewhere between the quantities for which and Condition [6a] is the same as Kreutzer and Lee s condition (2) which demonstrated that the after-tax profit maximizing output will be larger than the profit-maximizing output in the absence of a tax. Their model, however, fails to recognize the probability that the cost overstatement would be detected by the tax authorities who would then impose a penalty on the amount of unreported profit. With p = 0, as long as both t and 8 are positive, the marginal pretax profit, R - C will be negative. And as a result, the after-tax profit maximizing
484 output, Õ, is greater than the output, Q which would maximize profit in the absence of a tax.3 From (4) we have where A =- [U(n,) - U(IL;)]/U (n,) > function. And from [5] we have 0 via the concavity of the utility Note that the second term on the right hand side is negative, hence equation [8] implies the following inequality 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 = 8* and then equation [7] can be written as Substituting [7 ] for the left-hand side of [8 ] leads to which is equivalent to The first term of [10] indicates the expected marginal net profit and the second term represents the expected marginal punishment from
485 overreporting the production cost (tax evasion). Viewing the tax evasion as premeditated crime, a rational but amoral firm will decide in advance whether to commit a crime and will weigh the expected benefit against the expected punishment. As one would expect in a general model, the dishonest monopolist will adjust both his output and cost-reporting strategies to find the optimal solution. In view of equation [10], a variable probability of detection makes choices of production decision and tax evasion no longer separable. The second term in [10] is the product of detection probability, the imposed penalty rate and the amount of tax evasion. Because the second term is unambiguously negative, the first term can either be positive or negative for equation [10] to exhibit a negative sign. The sign of the first term is crucial for our results and arises because changes in output affect the cost of production--the reported profits that the tax authority monitors, which in turn affect the probability of detection. The effect of profit tax on the output decision is ambiguous and depends on the relative magnitudes of the expected marginal net profit and expected marginal punishment. Tax evasion has commonly been thought to reduce the distortion created by taxation, thus increasing efficiency. Consider the case that (1 - p)(s + s 8*C)(1 - t)(r - C ) < 0 in [10], and the marginal pretax profit, R - C, will be negative. Making standard assumptions on the slopes of R and C, this implies that an uncertain monopolist will expand output beyond the no tax level. The intuition behind this result is provided as follows. For a dishonest monopolist, the marginal gain from reduced tax liability on overstated cost equals (1 - p)8*c(l - t) (R - C ) if he is not detected. For any optimally chosen evasion level, the marginal cost of production must be expanded to the point at which it equals marginal net of tax revenue. With endogenous probability of detection, evasion raises the effective marginal tax rate for a detected firm and may result in production inefficiency. It can be shown from [10] that when the first term is positive but predominated by the negative second term, the marginal pretax profit is positive. That implies that a rational monopolist restricts output, which is devastating because it deepens the monopoly distortion. This result is closely linked to the interaction between penalties and output. Evasion raises the penalty, which increases the effective marginal tax rate and creates
486 the incentive to reduce detection probability. Output will then decline and production inefficiency will deteriorate. It is easily shown that the neutrality result obtained by Wang and Conant emerges from the generalized uncertainty model here with the assumptions of fixed probability of detection and fixed level of penalty. Equations [7 ] and [8] will then reduce to equations (7) and (8) in Wang and Conant (1988); the production decision and the evasion decision are, in this case, separable. The result is similar to the separation theorem of finance literature. CONCLUSION This article has examined the profit tax evasion and monopoly output decisions within the context of the uncertainty model with endogenous probability of detection. When a rational firm considers the probability of detection and punishment in its expected utility function of profit, then the uncertain monopolist s optimal rate of output is affected by the profit tax and the penalty rate. It is shown that the profit tax cannot be relied on for reducing the monopoly distortion when a monopolist understates its profits. This result is different from what is suggested by Kreutzer and Lee (1986, 1988). NOTES 1. See Cowell (1987), in his excellent survey, for detailed discussion of these issues. 2. Marrelli (1984) analyzed a monopolist firm s decision about whether, and to what extent, to avoid an ad valorem (sales) tax by underreporting revenues. 3. If we ignore the strategy of overreporting costs = (δ 0), Π 1 and II 2 are identical, and thus U(II 1) U(Π 2) = and equation [4 ] reduced to 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.
487 REFERENCES Allingham, M. G., and A. Sandmo. 1972. Income tax evasion: A theoretical analysis. Journal of Public Economics 1.323-38. Becker, G. S. 1968. Crime and punishment: An economic approach. Journal of Political Economy 76:169-217. Cowell, F. A. 1987. The economic analysis of tax evasion. In Surveys in the economics of uncertainty, edited by J. D. Hey and P. J. Lambert. New York: Basil Blackwell. Kreutzer, D., and D. R. Lee. 1986. On taxation & understated monopoly profits. National Tax Journal 39:241-43.. 1988. Tax evasion and monopoly output decisions: A reply. National Tax Journal 41:583-84. Marrelli, M. 1984. On indirect tax evasion. Journal of Public Economics 25:181-196. Wang, L.F.S., and J. L. Conant. 1988. Corporate tax evasion and output decisions of the uncertain monopolist. National Tax Journal 41:579-82. Leonard F. S. Wang is Assoccate Professor ofeconomics at Indiana State University. His current research examines tax evasion, tax cnccdence, countertrade transactions, and economic liberalczation cn developing countries.