Tax evasion with psychic costs and penalty renegotiation.

I. Introduction

The Primary Issues

Tax evasion is a constant source of fiscal concern for most governments. The United States Internal Revenue Service [19] for instance has estimated that the federal income tax liability not reported on timely filed tax returns for 1986 was on the order of $70.1 billion. In response, the U.S. has taken steps to address the problem, including implementation of the Taxpayers Compliance Measurement Program, which is designed to provide guidance in establishing audit procedures. A longstanding belief is that a sound auditing policy accompanied by a known penalty system is necessary to insure that individuals accurately report their incomes to the taxing authority. This belief indeed forms the basis for most models of tax compliance as for instance those of Reinganum and Wilde [24; 25] and Mookherjee and Png [23].

The basic problem is this: When a taxpayer knows his own income but the revenue collector does not, the taxpayer may have an incentive to dissemble on his income report. In order to discourage this type of behavior and to insure that taxpayers comply with the tax laws, the revenue collector may selectively audit taxpayer reports and impose penalties when false reports are identified.(1) Auditing, however, is costly.

This paper presents a positive theory of tax evasion through a formal analysis of the problem just described. One important feature of the examination includes how the optimal auditing policy changes as the willingness of the tax authority to renegotiate the penalty changes. While penalty renegotiation is an observed practice, this aspect of the tax compliance problem and its implications have not yet been investigated within a formal context.

While false reporting that goes undetected increases net-of-tax consumable income for the dissembler, empirical evidence suggests that individuals suffer psychic costs from cheating [2; 16]. In this paper we show how auditing policy interacts with the individual taxpayer's perception of the psychic cost of cheating. The individuals in our model exhibit heterogenous psychic costs of dissembling which cannot be observed by the tax collector.(2) These psychic costs may be so high that some people would not cheat even if the expected marginal penalty if detected were zero.

The tax revenue collector is typically not the political body which proposes and adopts the tax structure. The tax collector is however, typically empowered to implement the means for collection. For these reasons we restrict the revenue collector's problem to the selection of the penalty for cheating, and the audit policy employed, taking the tax structure as given. The tax collector may also be empowered to renegotiate the penalty for any individual who is discovered through an audit to have falsely reported her income. We grant this privilege to the tax collector in our model after first investigating the consequences of a policy of no renegotiation.

Ex post renegotiation of the penalty has been an heretofore neglected aspect in the study of tax compliance and auditing policy. In practice, the actual penalty paid is often negotiated between the revenue collector and the audited dissembler. We specifically address the policy issue of whether penalty renegotiation is a tax revenue maximizing policy. Renegotiation is a feasible strategy for the tax collector when the net-of-penalty income of the dissembler is below her 'bankruptcy' threshold level and the tax collector will incur real costs in the process of seizing income to cover the tax and penalty liability. By reducing the penalty through negotiation the taxpayer avoids the stigma of bankruptcy and the tax authority avoids the seizure cost.(3) Thus, the revenue collector is able to raise net tax revenue. Of course, in order to induce a dissembler to accept the renegotiated penalty, the outcome must guarantee him a greater utility than his utility under bankruptcy. We examine the conditions under which ex post Pareto-improving renegotiation of the penalty is possible.(4)

When there is information asymmetry between the taxpayer and the tax collector regarding the individual's bankruptcy threshold, a policy of penalty renegotiation may however not be a revenue maximizing strategy. Once the possibility of renegotiation becomes public knowledge, audited dissemblers have an incentive to claim (falsely) that the penalty will place them in bankruptcy, and hence that they will not be able to cover the tax and penalty liability out of income. If there is no way for the revenue collector to sort out the true bankruptcy claims from those that are false, then too much renegotiation will occur and expected net tax revenue will be smaller than if the tax collector had perfect knowledge of the bankruptcy status of individual taxpayers. Furthermore, if renegotiation of the penalty is recognized before taxpayers report their income, then they will estimate that the expected cost of dissembling is below what it otherwise would be. This inference follows from the prediction that the revenue collector will reduce the initial penalty whenever the dissembler claims that bankruptcy is imminent in order to save the costs of seizing income. Thus, even taxpayers with large psychic costs of cheating may be induced to dissemble. We show however that a tax recovery strategy which combines aspects of a renegotiation policy and an asset seizure policy can reduce false bankruptcy claims while allowing the tax collector to obtain some of the benefits of renegotiation.

The tax collector also has the opportunity to select the penalty along with the optimal audit strategy. Two possible situations present themselves with respect to the choice of a penalty. First is the case of an arbitrary fixed penalty selected independently of the audit strategy. Second is the case of an optimally chosen penalty structure selected jointly with the audit strategy. In the latter case the penalty structure is not a priori restricted to any particular form. We begin our discussion by focusing on the optimal audit strategy, given a fixed, but arbitrarily selected, penalty. While on the surface, the notion of a fixed penalty of this sort is unappealing, it is not unusual to observe this type of penalty system in practice. The first part of the analysis therefore emphasizes only one dimension of the problem faced by the revenue collector, the optimal audit policy. We then turn to the issue of the optimal penalty structure.

Related Literature and Distinguishing Features of the Model

We borrow the concept of renegotiation from the recent literature on commitments as illustrated in Baron and Besanko [6] and Green and Laffont [17]. The focus of this literature has generally been on nonbinding agreements, not on Pareto-improving renegotiation moves. This paper introduces the idea of ex post Pareto-improving renegotiation within the context of tax evasion. The setting therefore combines the risk sharing and the renegotiation problems examined by Dewatripont [9] and Dewatripont and Maskin [10], Fudenberg and Tirole [13] and Ma [22]. There are important contextual differences however between our model and these others. In our model, the revenue collector is willing to renegotiate the penalty in order to maximize ex post net-of-seizure cost tax revenue.(5) Dissemblers are willing to accept the new offer because they can avoid the stigmatization associated with bankruptcy and income seizure. This stigmatization is potentially comprised of a social stigma, the stigma that follows the individual when seeking new employment opportunities, and the stigma that follows the individual when seeking new credit.(6)

The model presented here also has the following new features as compared to the existing literature on tax evasion and auditing. First, in our model, the revenue authority maximizes expected net-of-cost tax revenue.(7) This objective function is in sharp contrast to the objective in Mookherjee and Png [23] but is similar to the objective examined in Reinganum and Wilde [24]. In Mookherjee and Png [23], the government chooses the auditing and tax structure in order to maximize social welfare. In Reinganum and Wilde [24] the government (here the tax authority) is identified as a net revenue maximizer who is also a net cost minimizer. Our results, however, differ from those obtained in Reinganum and Wilde [24]. Reinganum and Wilde show that in equilibrium, every taxpayer reports falsely by a constant portion of his true income.(8) This implies that a constant auditing policy by the revenue maximizer is optimal. Given this auditing policy, taxpayers are actually indifferent about whether they cheat or not, but the stability of the auditing policy is supported only when taxpayers cheat by a certain constant proportion of true income. In our model, some taxpayers are better off by cheating (are not indifferent) because they have relatively lower psychic costs of dissembling than do others. They cheat because their expected income gain is greater than their psychic cost.

Second, our model incorporates the idea that personal psychic costs are incurred as a result of cheating. The first formal model to incorporate morality as a deterrent to tax evasion was presented by Gordon [14]. He introduces psychic costs into Allingham and Sandmo's [1] model of tax evasion to explain the empirical findings that some taxpayers never evade, that evasion increases with the tax rate, and that evasion decisions are independent. The emphasis of our paper is quite different from Gordon's. However, our model is rich enough to replicate some of his results.(9)

Finally, unlike the previous literature, this paper assumes that taxpayers have heterogenous bankruptcy levels. The bankruptcy level is assumed to be private information ex ante, known only to the taxpayer. The ability of an identified dissembler to threaten to declare bankruptcy has important implications for the benefits to the tax authority of following a policy of penalty renegotiation.(10)

Organization

The paper is organized as follows. Section II describes the basic model. Section III examines implications of the audit and penalty policies selected. Section IV demonstrates how renegotiation of the penalty can be a Pareto-improving decision. Section V then examines how renegotiation of the penalty may under certain circumstances, not be a revenue maximizing policy for the tax authority. Section VI examines a solution to the false bankruptcy claim problem. Section VII analyzes the relation between optimal audit strategy, the tax authority's willingness to renegotiate the penalty, and the consequent net tax revenue collected. Section VIII concludes the paper.

II. The Economic Setting

Basic Features of the Model

The income of any particular individual may take the value [Y.sub.1] (high) or the value [Y.sub.2] [less than] [Y.sub.1] (low). We assume that each individual taxpayer knows his own income draw but ex ante the tax collector does not. Taxes on incomes [Y.sub.1] and [Y.sub.2] equal [T.sub.1] and [T.sub.2] respectively. We assume that taxes are graduated and hence that [T.sub.1] [greater than] [T.sub.2]. A taxpayer with income [Y.sub.1] therefore has an incentive to dissemble on her income report, ceteris paribus. Taxpayers with [Y.sub.2] income have no incentive to report falsely.

The tax collector can discover an individual's true income by conducting an audit of the individual's income tax report. The audited proportion of high income reports is always zero since the revenue collector recognizes that a reported income of [Y.sub.1] gives the greatest collectable tax bill for the individual. Therefore, only [Y.sub.2] (low income) reports are audited. Taxpayers who report [Y.sub.2] income know that the fraction [Delta] of all [Y.sub.2] reports will be audited. Auditing is costly and auditing costs are assumed to be increasing in [Delta]. We assume that the revenue collector can and must commit to an audit policy [Delta] before individuals report their incomes.(11) A penalty f is incurred by the taxpayer if an audit reveals that her income was [Y.sub.1] while she reported [Y.sub.2], and, she would also be required to pay the difference in the tax due ([T.sub.1] - [T.sub.2]). Income reporters know the penalty structure. Finally, the revenue collector is allowed to renegotiate the penalty for any identified dissembler.

Taxpayers have psychic costs of dissembling which are heterogenous individually and cannot ever be observed by the revenue collector. Let [Epsilon] denote the psychic cost of dissembling, where [Epsilon]([greater than] 0) is a continuous variable which is distributed between 0 and infinity. Let g([Epsilon]) denote the population density of [Epsilon]. The density function g([Epsilon]) is assumed to be continuous and twice differentiable and is known by the tax collector.

We consider an economy in which the cost of auditing all [Y.sub.2] income reporters is prohibitive. Hence a policy which would force all individuals to report income truthfully is not in the tax collector's feasible set of audit policies. Under these circumstances, some taxpayers with a relatively low psychic cost of dissembling are therefore likely to file false income reports.

Bankruptcy and Its Implications

Taxpayers have heterogenous bankruptcy threshold levels. We formally define bankruptcy as the inability to pay taxes due, any tax penalties and any personal obligations out of income Y. The bankruptcy threshold level for any individual is assumed to be private information that is known ex ante to the taxpayer but is not known by the revenue collector. The revenue collector perceives that an audited dissembler will declare bankruptcy if after subtracting taxes, penalties and personal obligations from income, the result is negative. Ex post, the declaration of bankruptcy in a world with no renegotiation would unequivocally identify the bankruptcy type of the individual. Suppose however, that the tax authority is willing to renegotiate the penalty and that this is public knowledge. The dissembler who is found out through an audit may in this case falsely claim that he will be bankrupted by the penalty. The tax collector however cannot verify the claim prior to the initiation of the penalty negotiation. In this case the dissembler may be able to gain from a renegotiated reduction of the penalty.

Taxpayers are classified into two bankruptcy level types. Let the bankruptcy threshold level of a type 1 individual be 0 and the bankruptcy threshold level of a type 2 be u. The threshold equals the level of personal obligations of the individual. The fractions of each type in the population, are [Alpha] and (1 - [Alpha]) respectively, and [Alpha] is assumed to be independent of the psychic cost of dissembling [Epsilon]. The tax collector knows [Alpha] and the bankruptcy thresholds. We assume that 0 [less than] u [less than] [Y.sub.2] - [T.sub.2]. This precludes the possibility that taxpayers declare bankruptcy whenever they pick the low income draw [Y.sub.2] and therefore permits us to focus our attention on the relation between bankruptcy, renegotiation and the tax evasion penalty.

If bankruptcy is declared, the revenue collector is assumed to have the authority to seize any income above the threshold bankruptcy level necessary to meet the tax liability plus the penalty. The cost of seizing income is s. Declaring bankruptcy also produces disutility for the taxpayer through stigmatization. The quantity [[Phi].sub.1] ([[Phi].sub.2]) represents the level of disutility a type 1 (type 2) incurs from the stigmatization associated with declaring bankruptcy, where we assume [[Phi].sub.1] [greater than] 0 and [[Phi].sub.2] [greater than] 0. The tax collector knows [[Phi].sub.1] and [[Phi].sub.2]. In the first part of our analysis we assume that dissemblers declare bankruptcy only when net-of-penalty and tax income is below the threshold level. Section V introduces the problem of false bankruptcy declarations.

Consider the setting in which penalty renegotiation is not permitted. If [Y.sub.1] - [T.sub.1] - f [greater than or equal to] u, no audited dissembler will declare bankruptcy since the dissembler's income is greater than or equal to the threshold level. However, if 0 [less than] [Y.sub.1] - [T.sub.1] - f [less than] u, audited bankruptcy type 2 dissemblers will declare bankruptcy since their net-of-penalty income is lower than their bankruptcy threshold, assuming they also pay the correct amount of tax due. If [Y.sub.1] - [T.sub.1] - f [less than or equal to] 0, both types of dissemblers will declare bankruptcy. These scenarios will not hold when penalty renegotiation is permitted.

Determining the Proportion of Taxpayers Who Report Falsely

Taxpayers are assumed to act like risk neutral expected utility of consumption maximizers when making a decision about whether to report income falsely. The maximum expected utility of a bankruptcy type i person (i = 1, 2) with income [Y.sub.1] who also dissembles is denoted by [V.sub.i] (i = 1, 2). Hence,

[V.sub.1]([Delta]; [Epsilon]) = [Delta][Max(0, [Y.sub.1] - [T.sub.1] - f) - [d.sub.1][[Phi].sub.1]] + (1 - [Delta])([Y.sub.1] - [T.sub.2]) - [Epsilon],

[V.sub.2]([Delta]; [Epsilon]) = [Delta][Max(u, [Y.sub.1] - [T.sub.1] - f) - [d.sub.2][[Phi].sub.2]] + (1 - [Delta])([Y.sub.1] - [T.sub.2]) - [Epsilon],

where, [d.sub.i] (i = 1, 2) is a dummy variable defined as follows:

Case A: if [Y.sub.1] - [T.sub.1] - f [less than or equal to] 0, [d.sub.1] = [d.sub.2] = 1 (both types declare bankruptcy),

Case B: if 0 [less than] [Y.sub.1] - [T.sub.1] - f [less than] u, [d.sub.1] = 0 and [d.sub.2] = 1 (only type 2 individuals declare bankruptcy),

Case C: if [Y.sub.1] - [T.sub.1] - f [greater than or equal to] u, [d.sub.1] = [d.sub.2] = 0 (no type declares bankruptcy).

While in both Cases A and B ex post renegotiation of the penalty is possible, we restrict ourselves to the more interesting of the two cases, where only type 2 dissemblers declare bankruptcy when detected (i.e., 0 [less than] [Y.sub.1] - [T.sub.1] - f [less than] u). This focus will permit us to directly examine how type 1 dissemblers will disguise themselves as type 2 dissemblers if they expect in advance that renegotiation of the penalty between type 2 dissemblers and the revenue collector is possible.

Suppose therefore that 0 [less than] [Y.sub.1] - [T.sub.1] - f [less than] u, then

[V.sub.1]([Delta]; [Epsilon]) = [Delta]([Y.sub.1] - [T.sub.1] - f) + (1 - [Delta])([Y.sub.1] - [T.sub.2]) - [Epsilon]

[V.sub.2]([Delta]; [Epsilon]) = [Delta](u - [[Phi].sub.2]) + (1 - [Delta])([Y.sub.1] - [T.sub.2]) - [Epsilon].

Type i individuals with [Y.sub.1] income dissemble when

[V.sub.i]([Delta]; [Epsilon]) [greater than] [Y.sub.1] - [T.sub.1] for i = 1, 2

or in terms of [Epsilon] when

[[Epsilon].sub.i]([Delta]) [greater than] [Epsilon] for i = 1, 2

where,

[[Epsilon].sub.1]([Delta]) [equivalent to] ([T.sub.1] - [T.sub.2]) - [Delta]([T.sub.1] - [T.sub.2] + f)

[[Epsilon].sub.2]([Delta]) [equivalent to] ([T.sub.1] - [T.sub.2]) - [Delta][[Y.sub.1] - [T.sub.2] - (u - [[Phi].sub.2])]. (1)

If the psychic cost of dissembling is less than the critical value [[Epsilon].sub.i] ([Delta]), type i taxpayers will dissemble.

Given (1) and [Y.sub.1] - [T.sub.1] - u [less than] f, the decision to file a false report depends upon the audit policy [Delta]. Three cases present themselves.

Case 1: [Delta] [greater than or equal to] ([T.sub.1] - [T.sub.2])/([T.sub.1] - [T.sub.2] + f)

No taxpayer dissembles on his income report and the incentive compatibility condition is fulfilled for both types.

Case 2: ([T.sub.1] - [T.sub.2])/[([Y.sub.1] - [T.sub.2]) - (u - [[Phi].sub.2])] [less than or equal to] [Delta] [less than] [([T.sub.1] - [T.sub.2])/([T.sub.1] - [T.sub.2] + f)]

Type 1 taxpayers with a psychic cost less than [[Epsilon].sub.1] ([Delta]) dissemble and the incentive compatibility condition is fulfilled for type 2 taxpayers.

Case 3: [Delta] [less than] ([T.sub.1] - [T.sub.2])/[([Y.sub.1] - [T.sub.2]) - (u - [[Phi].sub.2])]

Type 1 and 2 taxpayers with a psychic cost less than [[Epsilon].sub.1]([Delta]) and [[Epsilon].sub.2]([Delta]) respectively will dissemble and the incentive compatibility condition will not be fulfilled.

Define [[Delta].sub.c] as:

[[Delta].sub.c] [equivalent to] ([T.sub.1] - [T.sub.2])/[([Y.sub.1] - [T.sub.2]) - (y - [[Phi].sub.2])].

When [Delta] [less than or equal to] [[Delta].sub.c], the population proportion of taxpayers who dissemble on their income reports is P([Delta]):

P([Delta]) = [Alpha] [integral of] g([Epsilon])d[Epsilon] between limits [[Epsilon].sub.1]([Delta]) and 0 + (1 - [Alpha]) [integral of] g([Epsilon])d[Epsilon] between limits [[Epsilon].sub.2]([Delta]) and 0 (2)

where again g([Epsilon]) is the population density of the psychic cost [Epsilon], and [Alpha](1 - [Alpha]) is the fraction of type 1 (type 2) individuals in the economy.

The following comparative statics results yield four conclusions

[Delta]P([Delta])/[Delta][Delta] = -[Alpha]g([[Epsilon].sub.1])([T.sub.1] - [T.sub.2] + f) - (1 - [Alpha])g([[Epsilon].sub.2])[[Y.sub.1] - [T.sub.2] - (u - [[Phi].sub.2])] [less than] 0

[Delta]P([Delta])/[Delta]f = -[Alpha]g([[Epsilon].sub.1])[Delta] [less than] 0

[Delta]P([Delta])/[Delta][T.sub.1] = [Alpha]g([[Epsilon].sub.1])(1 - [Delta]) + (1 - [Alpha])g([[Epsilon].sub.2]) [greater than] 0

[Delta]P([Delta])/[Delta][T.sub.2] = -(1 - [Delta])[[Alpha]g([[Epsilon].sub.1]) + (1 - [Alpha])g([[Epsilon].sub.2])] [less than] 0

[Delta]P([Delta])/[Delta]u = (1 - [Alpha])g([[Epsilon].sub.2])[Delta] [greater than] 0

[Delta]P([Delta])/[Delta][[Phi].sub.2] = -(1 - [Alpha])g([[Epsilon].sub.2])[Delta] [less than] 0.

Remark 1. As either the audited proportion increases or the penalty increases, only taxpayers with low psychic costs are likely to file false reports, and overall, fewer taxpayers will dissemble.

Remark 2. With more progressive tax rates, even taxpayers with larger psychic costs will cheat and overall more taxpayers will dissemble.(12)

Remark 3. As the bankruptcy threshold level for type 2 individuals increases, more will report falsely because they expect to declare bankruptcy anyway if detected by an audit. That is, the incentive effects of the penalty are diluted.

Remark 4. If the disutility from declaring bankruptcy increases, fewer taxpayers will declare bankruptcy.

The Objective Function of the Tax Revenue Collector

We examine the implications of three forms of behavior which the tax revenue collector may adopt. These conditions directly influence the revenue collector's objective function and hence the audit policy adopted. The first case occurs when the revenue collector does not take the possibility of renegotiation into account (i.e., renegotiation is not permitted). The second case occurs when renegotiation can be concealed from public view, that is, when all renegotiation activities are "confidential". In this case the revenue collector is assumed to believe that any renegotiation action can be hidden from view. Taxpayers will in general perceive that renegotiation is not possible, and hence will act as if there is no incentive to make a false bankruptcy claim. The final case occurs when the revenue collector takes renegotiation into account and anticipates that all renegotiation activities will be public knowledge and hence will provide incentives for false bankruptcy claims to arise. In order to establish the relative effects of renegotiation we first examine the case in which the tax collector ignores the possibility of renegotiation occurring (i.e., follows a tacit policy of no renegotiation). The latter two cases are taken up in section VII.

Let [[Theta].sub.1]([Delta]) and [[Theta].sub.2]([Delta]) denote the proportion of type 1 and 2 people who dissemble on their income reports, respectively. Then we have

[[Theta].sub.1]([Delta]) = [integral of] g([Epsilon])d[Epsilon] between limits [[Epsilon].sub.1]([Delta]) and 0 and [[Theta].sub.2]([Delta]) = [integral of] g([Epsilon])d[Epsilon] between limits [[Epsilon].sub.2]([Delta]) and 0.

The optimal auditing policy when renegotiation is not permitted is characterized by [[Delta].sup.*], and is the solution to the following problem for the tax collector:

Max {[T.sub.1] - (1 - [Delta])([T.sub.1] - [T.sub.2])[[Alpha][[Theta].sub.1]([Delta]) + (1 - [Alpha])[[Theta].sub.2]([Delta])]

+ [Delta][[Alpha][[Theta].sub.1]([Delta])f + (1 - [Alpha])[[Theta].sub.2]([Delta])([Y.sub.1] - [T.sub.1] - u - s)] - c([Delta])}. (3)

In (3), [Alpha](1 - [Alpha]) is the fraction of type 1 (type 2) taxpayers, f [greater than] ([Y.sub.1] - [T.sub.1] - u) is the penalty imposed on the detected [Y.sub.1] income dissembler, ([Y.sub.1] - [T.sub.1] - u) is the maximum that can be seized from the audited type 2 dissembler, s is the cost incurred by the tax collector when seizing taxpayer income, and finally c([Delta]) is the auditing cost which depends on the audited proportion [Delta].(13) We assume that 0 [less than] s [less than] [Y.sub.1] - [T.sub.1] - u, otherwise, the revenue collector will not pursue the collection of penalties. Expression (3) states that the revenue collector maximizes the tax revenue that would be collected from a high income person ([T.sub.1]) that reported truthfully, minus the expected uncollected tax revenue from taxpayers who dissemble, plus the expected net-of-seizure cost penalty revenue, minus the auditing cost c([Delta]) (where c(0) = 0, c[prime]([Delta]) [greater than] 0 and c[double prime]([Delta]) [greater than] 0).(14)

III. Implications of the Audit and Penalty Policies Selected

The Audit Policy

We restrict our attention to the case where the total auditing cost for the otherwise optimal [[Delta].sup.*] is so large that truthful revelation by all taxpayers will not be observed. We therefore focus on the solution [[Delta].sup.*] [less than] [[Delta].sup.c]. The first order condition for [[Delta].sup.*] [less than] [[Delta].sub.c] gives:

c[prime]([[Delta].sup.*]) = ([T.sub.1] - [T.sub.2])[[Alpha][[Theta].sub.1]([[Delta].sup.*]) + (1 - [Alpha])[[Theta].sub.2]([[Delta].sup.*])]

+ [[Alpha][[Theta].sub.1]([[Delta].sup.*])f + (1 - [Alpha])[[Theta].sub.2]([[Delta].sup.*])([Y.sub.1] - [T.sub.1] - u - s)]

- (1 - [[Delta].sup.*])([T.sub.1] - [T.sub.2])[[Alpha][[Theta][prime].sub.1]([[Delta].sup.*]) + (1 - [Alpha])[[Theta][prime].sub.2]([[Delta].sup.*])]

+ [Delta][[Alpha][[Theta][prime].sub.1]([[Delta].sup.*])f + (1 - [Alpha])[[Theta][prime].sub.2]([[Delta].sup.*])([Y.sub.1] - [T.sub.1] - u - s)] (4)

where [[Theta][prime].sub.1] ([[Delta].sup.*]) = -[([T.sub.1] - [T.sub.2] + f)g([[Epsilon].sub.1])] [less than] 0 and [[Theta][prime].sub.2]([[Delta].sup.*]) = -[([Y.sub.1] - [T.sub.2] - u + [[Phi].sub.2])g([[Epsilon].sub.2])] [less than] 0.

Equation (4) indicates that the marginal cost of auditing equals the expected marginal revenue of auditing at [[Delta].sup.*]. The right hand side of (4) can be explained as follows. There are four components which cause the change in tax and penalty revenues. With a given pool of dissemblers, as [[Delta].sup.*] increases a dissembler faces a lower probability of evading the tax and penalty, which from the first and second components of the r.h.s. of (4) results in

([T.sub.1] - [T.sub.2])[[Alpha][[Theta].sub.1]([[Delta].sup.*]) + (1 - [Alpha]) [[Theta].sub.2]([[Delta].sup.*])] + [[Alpha][[Theta].sub.1]([[Delta].sup.*])f + (1 - [Alpha])[[Theta].sub.2]([[Delta].sup.*])([Y.sub.1] - [T.sub.1] - u - s)] [greater than] 0. (5)

Expression (5) is the expected marginal tax and penalty revenue of auditing for a given pool of dissemblers. The third and fourth components of (4) give the effect of individuals' self-selections to dissemble. As the threat of an audit increases, taxpayers with the greatest psychic costs among dissemblers are discouraged from dissembling on their income reports (Remark 1). Equation (6) is the expected marginal tax revenue from auditing.

- (1 - [[Delta].sup.*])([T.sub.1] - [T.sub.2])[[Alpha][[Theta][prime].sub.1]([[Delta].sup.*]) + (1 - [Alpha])[[Theta][prime].sub.2]([[Delta].sup.*])] [greater than] 0. (6)

The sign of (6) is positive because the expected uncollected tax revenue decreases as taxpayers with the greatest psychic cost reduce their dissembling.

Equation (7) is the expected marginal penalty revenue of auditing,

[[Delta].sup.*][[Alpha][[Theta][prime].sub.1]([[Delta].sup.*])f + (1 - [Alpha])[[Theta][prime].sub.2]([[Delta].sup.*])([Y.sub.1] - [T.sub.1] - u - s)] [less than] 0. (7)

The sign of (7) is negative because the expected penalty revenue decreases as taxpayers with the greatest psychic cost reduce their dissembling. The sum of (5), (6) and (7) (i.e., the r.h.s. of (4)) must be positive for a positive value of [[Delta].sup.*] to be an optimal solution.

The second order condition for the maximization problem (3) requires that H [less than] 0 where

H = 2([T.sub.1] - [T.sub.2])[[Alpha][[Theta][prime].sub.1]([[Delta].sup.*]) + (1 - [Alpha])[[Theta][prime].sub.2]([[Delta].sup.*])]

+ 2[[Alpha][[Theta][prime].sub.1]([[Delta].sup.*])f + (1 - [Alpha])[[Theta][prime].sub.2]([[Delta].sup.*])([Y.sub.1] - [T.sub.1] - u - s)]

- (1 - [[Delta].sup.*])([T.sub.1] - [T.sub.2])[[Alpha][[Theta][double prime].sub.1]([[Delta].sup.*]) + (1 - [Alpha])[Theta][double prime]([[Delta].sup.*])]

+ [[Delta].sup.*][[Alpha][[Theta][double prime].sub.1]([[Delta].sup.*])f + (1 - [Alpha])[[Theta][double prime].sub.2]([[Delta].sup.*])([Y.sub.1] - [T.sub.1] - u - s)] - c[double prime]([[Delta].sup.*]),

and,

[[Theta][double prime].sub.1]([[Delta].sup.*]) = [([T.sub.1] - [T.sub.2] + f).sup.2]g[prime]([[Epsilon].sub.1]) and [[Theta][double prime].sub.2]([[Delta].sup.*]) = [([Y.sub.1] - [T.sub.1] - u + [[Phi].sub.2]).sup.2]g[prime]([[Epsilon].sub.2]).

We assume this condition is satisfied. Further, from (4), we have:

[Delta][[Delta].sup.*]/[Delta]s = (1 - [Alpha])[[[Theta].sub.2]([[Delta].sup.*]) + [[Delta].sub.2][[Theta][prime].sub.2]([[Delta].sup.*])]/H (8)

[Delta][[Delta].sup.*]/[Delta][[Phi].sub.2] = [[Delta].sup.*](1 - [Alpha])([T.sub.1] - [T.sub.2])g([[Epsilon].sub.2]) + [[Delta].sup.*](1 - [Alpha])([Y.sub.1] - [T.sub.1] - u - s)g([[Epsilon].sub.2])/H [less than] 0 (9)

[Delta][[Delta].sup.*]/[Delta]f = {([T.sub.1] - [T.sub.2])[Alpha][[Delta].sup.*]g([[Epsilon].sub.1]) + [Alpha]f[[Delta].sup.*]g([[Epsilon].sub.1]) - [[Alpha][[Theta].sub.1]([[Delta].sup.*]) + [Alpha][[Delta].sup.*][[Theta][prime].sub.1]([[Delta].sup.*])]}/H. (10)

The sign of (8) is not uniformly determined because [[Theta][prime].sub.2]([[Delta].sup.*]) [less than] 0. While the sign of (9) is uniformly negative, the sign of (10) is not uniformly determined (but see Remark 7 below). Equations (8)-(10) give the following intuitive results:

Remark 5. If the seizure cost (s) increases, the expected marginal revenue of auditing is changed by two forces. First, for a given pool of dissemblers, when type 2 dissemblers are audited, the revenue collector incurs a larger seizure cost. This decreases the expected marginal penalty revenue of auditing for a given pool of dissemblers (i.e., (1 - [Alpha])[[Theta].sub.2]([[Delta].sup.*])/H [less than] 0 in (8). Second, with a greater seizure cost, the loss of expected marginal penalty revenue that arises from discouraging taxpayers with the greatest psychic costs among dissemblers from reporting falsely (that is, (1 - [Alpha])[[Theta][prime].sub.2]([[Delta].sup.*])/H [greater than] 0 in (8)) decreases. Since these two marginal forces work in the opposite direction, the overall change is ambiguous. However, it seems reasonable to believe that the population proportion of type 2 dissemblers (i.e., [[Theta].sub.2] ([[Delta].sup.*])) is always greater than the marginal change of the proportion caused by an increment of [[Delta].sup.*] (i.e., [absolute value of [[Theta][prime].sub.2]([[Delta].sup.*])]). If this is the case, the first force dominates the second force, and the expected marginal revenue of auditing decreases with seizure cost. Therefore, the revenue collector optimally decreases the audited proportion of [Y.sub.2] income reports.

Remark 6. If declaring bankruptcy causes greater disutility from the associated stigmatization ([[Phi].sub.2]), fewer type 2 taxpayers dissemble on their income report. Further, the expected marginal tax and penalty revenue of auditing that arises from discouraging type 2 taxpayers with the greatest psychic costs among dissemblers from reporting falsely decreases. As a result, the revenue collector optimally decreases the audited proportion of [Y.sub.2] income reports.

Remark 7. If the penalty (f) is increased, the expected marginal revenue of auditing is influenced by two forces. First, as explained above, the net expected marginal penalty revenue of auditing for a given pool of dissemblers will increase (i.e., from eq. (10), -[[Alpha][[Theta].sub.1] ([[Delta].sup.*]) + [Alpha][[Delta].sup.*] [[Theta][prime].sub.1] ([[Delta].sup.*])]/H [greater than] 0). Second, with a greater penalty, fewer bankruptcy type 1 individuals will dissemble on their income reports. The expected marginal (tax plus penalty) revenue that results from discouraging type 1 individuals with the greatest psychic costs from reporting falsely will be negative (i.e., from eq. (10), [([T.sub.1] - [T.sub.2])[Alpha][[Delta].sup.*]g([[Epsilon].sub.1]) + [Alpha]f[[Delta].sup.*]g([[Epsilon].sub.1])]/H [less than] 0). Because these changes are in opposite directions, the overall change is ambiguous.

The Penalty

The penalty can also be evaluated using equation (3). Taking the first order derivative of (3) with respect to f gives the marginal net tax revenue from a change in the penalty

(1 - [Delta])([T.sub.1] - [T.sub.2])[[Alpha]g([[Epsilon].sub.1)[Delta]] + [Delta][[Alpha][[Theta].sub.1]([Delta]) - [Alpha]g([[Epsilon].sub.1])[Delta]f].

Let [f.sup.*] be the penalty that maximizes net tax revenues and let the condition's value at [f.sup.*] be q. Evaluating the sum at [f.sup.*] shows that the first term is positive. Most penalties for noncompliance appear to be bounded, therefore we assume that [f.sup.*] lies in the interval [a, b] inclusive. The second bracketed term [[Alpha][[Theta].sub.1]([Delta]) - [Alpha]g([[Epsilon].sub.1])[Delta]f], evaluated at [f.sup.*], is the net consequence of a direct and indirect effect from a change in the penalty. The direct effect [Alpha][[Theta].sub.1]([Delta]) is the marginal revenue from an increase in the penalty for the given pool of audited dissemblers. The indirect effect [Alpha]g([[Epsilon].sub.1])[Delta]f is the marginal revenue from the change in the pool that occurs as the penalty increases. We assume that the direct effect exceeds the indirect effect. This is likely to be true for most reasonable parameterizations of the economy being considered. In this case the second bracketed term, when evaluated at [f.sup.*], is positive. Therefore, the condition would yield q [greater than] 0 at [f.sup.*], and hence we would conclude that [f.sup.*] = b, that is, the optimal penalty from the tax authority's perspective is the upper bound. Henceforth f will denote this upper bound.(15)

IV. Pareto-Improving Renegotiation of the Penalty

A type 2 taxpayer has the bankruptcy threshold u. If such a taxpayer files a false income report and is discovered through an audit, he will have insufficient income to cover the tax due plus the penalty (i.e. f [greater than] [Y.sub.1] - [T.sub.1] - u) because of the bankruptcy threshold. Before declaring bankruptcy publicly, it is assumed that these individuals must report their near bankruptcy status to the revenue collector. Unless the revenue collector offers a one-time penalty reduction which makes the penalty payable, the type 2 dissemblers will declare bankruptcy.

However, the revenue collector has an incentive to renegotiate the penalty. By mitigating the penalty (i.e., by making the penalty payable), the revenue collector may be able to raise net tax revenue. Of course, in order to make type 2 dissemblers take the offer, the new offer must guarantee the dissembler a greater utility than the utility she will receive under bankruptcy. In this case, a possibility exists for ex post Pareto-improving renegotiation of the penalty between type 2 dissemblers and the revenue collector.

PROPOSITION. If (i) the renegotiation of the penalty can be executed privately between the revenue collector and the audited type 2 dissemblers (i.e. renegotiation can be concealed from view), and (ii) it is costly for the revenue collector to seize the income of type 2 dissemblers who declare bankruptcy (i.e., s [greater than] 0), then there exists an ex post Pareto-improving renegotiation plan between audited type 2 workers and the revenue collector.

An intuitive explanation of this proposition is as follows. By the assumption that the revenue collector has the authority to initiate the renegotiation of the penalty, the revenue collector will initially offer type 2 dissemblers the maximum payable penalty in order to maximize ex post revenue. The maximum payable penalty of type 2 dissemblers is [Y.sub.1] - [T.sub.1] - u. By offering a reduced penalty, the revenue collector induces type 2 dissemblers to make a voluntary payment and thus increases net tax revenue by saving the seizure cost s. Likewise, type 2 dissemblers may accept the alternative offer because they can thereby avoid the stigmatization associated with declaring bankruptcy. That is, after accepting the renegotiated penalty, the utility of type 2 audited dissemblers is u, which is greater than their utility when bankruptcy is declared (u - [[Phi].sub.2]). Therefore, both the type 2 dissemblers and the revenue collector are better off and are thus willing to agree to the renegotiated offer. In general, the revenue collector's incentive to offer a revised penalty increases with s, and the type 2 dissembler's incentive to accept the offer increases with [[Phi].sub.2].

Even when the seizure cost is zero, a renegotiated offer may still make type 2 dissemblers better off because they avoid the stigmatization of bankruptcy. However, if s = 0, the revenue collector has no incentive to offer the renegotiation opportunity because the maximum penalty can always be collected.

V. Problems with Renegotiation as a Revenue Maximizing Policy

The proposition just described imposed the conditions that any renegotiation be concealed from public view - in essence, individual taxpayers behave as if renegotiation is not possible because it is not a declared policy. When renegotiation is either a declared policy or an observed practice, the renegotiation gains to the tax authority shown above may vanish.

There are two problems which can arise. The first problem is false claims of potential bankruptcy by taxpayers, and the second problem is that more taxpayers will dissemble if the opportunity to renegotiate the penalty is recognized in advance (a form of moral hazard problem).

False Bankruptcy Declarations

Once the possibility of renegotiation is recognized publicly, type 1 audited dissemblers have incentives to claim that the penalty will place them in bankruptcy, which would be false. If the revenue collector knew the personal bankruptcy levels of all taxpayers, then type 1 dissemblers would be unable to make this false claim. However, the problem of a false bankruptcy claim manifests itself when the revenue collector cannot sort out whether the claim is true or false prior to initiating renegotiation. In this case, the expected maximum revenue gain with renegotiation becomes:

(1 - [Alpha])s - [Alpha][f - ([Y.sub.1] - [T.sub.1] - u)].

The second term of the above is the loss of the tax collector's penalty revenue caused by renegotiation with reporters who claim falsely that they will be forced to declare bankruptcy. Of course, the sign of the above may be negative. With a large fraction of type 1 people and with a greater initial penalty f, the false bankruptcy claim problem is more likely to decrease net tax revenue after the renegotiation. Therefore, if renegotiation is public policy and taxpayers anticipate that it is an option, then it is less likely to be effective as a revenue maximizing tool.

Dissembling When Renegotiation Is Expected in Advance

Suppose that taxpayers know in advance that there is always an opportunity for renegotiating the penalty if they report income falsely and are detected. They will calculate an expected penalty which is less than the statutory penalty. As a result, even taxpayers with large psychic costs may be induced to dissemble.

When the opportunity for renegotiation is recognized in advance, expected uncollected tax revenue increases since more tax payers dissemble. When the opportunity for renegotiation is not recognized in advance, type i taxpayers with [Epsilon] less than [[Epsilon].sub.1]([[Delta].sup.*])(i = 1, 2) where

[[Epsilon].sub.1]([[Delta].sup.*]) = ([T.sub.1] - [T.sub.2]) - [[Delta].sup.*]*([T.sub.1] - [T.sub.2] + f)

[[Epsilon].sub.2]([[Delta].sup.*]) = ([T.sub.1] - [T.sub.2]) - [[Delta].sup.*][[Y.sub.1] - [T.sub.2] - (u - [[Phi].sub.2])]

will dissemble. However, when renegotiation is expected to reduce the penalty, type i taxpayers with [Epsilon] less than [Mathematical Expression Omitted] where

[Mathematical Expression Omitted]

will dissemble. As f [greater than] [Y.sub.1] - [T.sub.1] - u, it follows that ([T.sub.1] - [T.sub.2] + f) [greater than] ([Y.sub.1] - [T.sub.2] - u) and [Mathematical Expression Omitted] for i = 1, 2, which means more taxpayers dissemble and net tax revenue will be adversely affected.

VI. Reducing False Bankruptcy Declarations

False bankruptcy claims can be reduced but this calls for the tax authority to follow a plan of randomizing between two tax recovery strategies. The first strategy involves renegotiation of the penalty for audited dissemblers who make claims that they will be put in bankruptcy by the penalty. The second strategy involves seizing the assets of dissemblers to cover the penalty due. A declared policy of randomizing between these two strategies will reduce the incentives for type 1 dissemblers to declare bankruptcy falsely and will minimize the tax collector's expected seizure cost.

Type 1 dissemblers always have the ability to pay the penalty due, but type 2 dissemblers do not. The tax collector therefore has several options. First, minimize seizure costs by renegotiating the penalty of all audited dissemblers who make a bankruptcy claim, allowing each to pay ([Y.sub.1] - [T.sub.1] - u). This option (as already suggested) increases the incentives for type 1 dissemblers to make false bankruptcy declarations when they are audited. Second, seize the incomes of every audited dissembler. If the tax authority happens to face type 2 dissemblers who really are bankrupt, the maximum income which can be seized is [Y.sub.1] - [T.sub.1] - u. However, if they face type 1 dissemblers who falsely declare bankruptcy, the tax collector can seize all of the tax due plus the penalty f. This policy would permit the tax collector to maximize before seizure cost tax revenue, but seizure costs would also be maximized. The third option is the mixed strategy already mentioned. Specifically, let the tax collector select a randomization policy between the renegotiation and seizure strategies which minimizes "expected" seizure cost where the expectation is taken over the optimal randomization, subject to a self-selection constraint which would force dissemblers to only declare that bankruptcy was imminent when this was in fact the case.

Define the pool of individuals who report [Y.sub.2] (low income), who are audited and discovered to have actual income [Y.sub.1], and who declare that the penalty will place them in bankruptcy. Let [Tau] be the fraction of this pool who are offered the renegotiated penalty. Then, (1 - [Tau]) is the fraction whose income is seized by the tax collector. Recall that the tax collector does not know the bankruptcy threshold for any particular individual. The optimal value of [Tau] is given by the solution to

[Mathematical Expression Omitted]

where [Rho] is the nonpecuniary and pecuniary stigma cost associated with being detected as one making a false bankruptcy claim. Thus, [Rho] may include an imprisonment cost and any reputation cost which is imposed by the capital and labor markets. When income seizure occurs, the bankruptcy status of the dissembler is revealed. This occurs because the bankruptcy thresholds u and 0 explicitly determine the total level of income that is available for seizure. The solution to the problem is

[[Tau].sub.*] = [Rho]/{[f - ([Y.sub.1] - [T.sub.1] - u)] + [Rho]}. (11)

Notice that in the self-selection constraint the type 1 dissembler whose income report was audited recognizes that renegotiation is possible with probability [[Tau].sup.*] but seizure (and hence detection of true bankruptcy type) occurs with probability (1 - [[Tau].sup.*]). The type 1 dissembler trades off the gain from renegotiation [f - ([Y.sub.1] - [T.sub.1] - u)] against the cost of false bankruptcy declaration. The solution to the tax collector's problem produces the strategy [[Tau].sup.*] which is optimal in terms of minimizing expected cost. This leads to the following results:

Remark 8. As the bankruptcy threshold becomes more heterogenous (i.e., u increases), some taxpayers (i.e., the type 1 dissemblers) have a greater incentive to declare bankruptcy falsely. In order to reduce this incentive, the tax collector must raise the fraction of audited dissemblers whose income is seized.

Remark 9. As the cost associated with being discovered to have made a false bankruptcy claim increases (i.e., [Rho] increases), the false bankruptcy problem can be solved by seizing the incomes of only a fraction of those making a bankruptcy claim.

VII. Renegotiation, the Optimal Audit Strategy, and Net Tax Collections

This section compares different auditing strategies that are contingent on the revenue collector's willingness to renegotiate the penalty for dissemblers. We then examine net tax collections under these alternative circumstances. We find that the tax authority is better off following a policy of randomizing between strategies of renegotiation of the penalty and asset seizure (i.e., a mixed strategy) as compared to following a policy of no renegotiation at all.

The objective function of the tax collector when penalty renegotiation is not permitted is, from equation (3)

[T.sub.1] - (1 - [Delta])([T.sub.1] - [T.sub.2])[[Alpha][[Theta].sub.1]([Delta]) + (1 - [Alpha])[[Theta].sub.2]([Delta])] + [Delta][[Alpha][[Theta].sub.1]([Delta])f + (1 - [Alpha])[[Theta].sub.2]([Delta])([Y.sub.1] - [T.sub.1] - u - s)] - c([Delta]) [equivalent] [R.sup.n]. (3[prime])

As defined earlier s [greater than] 0 is the seizure cost. Now suppose the tax collector behaves as if private (concealed) renegotiation is possible. Recall that under these conditions renegotiation will not lead to an increase in false bankruptcy claims nor will there be an increase in the number of dissemblers. In this case, the objective function of the tax authority is

[T.sub.1] - (1 - [Delta])([T.sub.1] - [T.sub.2])[[Alpha][[Theta].sub.1]([Delta]) + (1 - [Alpha])[[Theta].sub.2]([Delta])] + [Delta][[Alpha][[Theta].sub.1]([Delta])f + (1 - [Alpha])[[Theta].sub.2]([Delta])([Y.sub.1] - [T.sub.1] - u)] - c([Delta]) [equivalent] [R.sup.r]. (12)

which is equation (3[prime]) except that the seizure cost is now zero (s = 0).

We know that [Delta][[Delta].sup.*]/[Delta]s is likely to be negative (Remark 5). Therefore, it can easily be seen that the optimal audit proportion is larger when the tax collector takes renegotiation into account (i.e., s = 0) than when renegotiation is not permitted (i.e., s [greater than] 0). This follows because the expected net penalty revenue with renegotiation is higher than without it (see also equation (19) below).

The conclusion that the optimal audit proportion is larger under s = 0 versus s [greater than] 0 may not be true when the tax authority expects false bankruptcy claims and a higher degree of dissembling, events which would accompany a public policy of renegotiation. That is, suppose taxpayers know that renegotiation is public policy. When the tax collector takes these facts into account, the objective function becomes

[T.sub.1] - (1 - [Delta])([T.sub.1] - [T.sub.2])[[Alpha][[Theta].sub.1]([Delta]) + (1 - [Alpha])[[Theta].sub.2]([Delta])] + [Delta]([Y.sub.1] - [T.sub.1] - u)[[Alpha][[Theta].sub.1]([Delta]) + (1 - [Alpha])[[Theta].sub.2]([Delta])]. (13)

Note that in this case, the tax collector can save the seizure cost by offering a renegotiated penalty to income reporters who claim bankruptcy but the penalty revenue is lower due to the fact that the incidence of false bankruptcy claims by the type 1's increases. In comparison with the objective function (3[prime]), the actual penalty in (13) to the type 1 dissemblers is ([Y.sub.1] - [T.sub.1] - u)([less than] f). From Remark 7, the optimal audit proportion varies ambiguously with the penalty. That is, the sign of [Delta][[Delta]*]/[Delta]f is ambiguous for any value of s and, in particular, when s = 0. Therefore, in comparison with the audit policy without the possibility of renegotiation, the optimal audit policy cannot be determined uniformly when renegotiation is publicly acknowledged and the resulting false bankruptcy problem arises.

When the tax collector follows a publicly known policy of renegotiation more dissembling may occur. In this case, the fraction of each type who dissembles on their income report becomes

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted]. Note that [Mathematical Expression Omitted]. Replacing [[Theta].sub.1]([Delta]) and [[Theta].sub.2]([Delta]) by [Mathematical Expression Omitted] in (3[prime]), the tax collector's objective function becomes:

[Mathematical Expression Omitted].

The first-order condition that maximizes (14) is

[Mathematical Expression Omitted].

In order to compare the optimal audit policy when it does not cause more dissembling with the one that does, one must compare the above condition (15) with condition (4) in section III. It is clear that the first two terms in (15) are always greater than the first two in (4). However, the comparisons between the third and fourth terms are ambiguous and as a result no general conclusions can be drawn.

The tax authority's objective function taking the mixed strategy into consideration, and that this strategy reduces false bankruptcy claims, is

[T.sub.1] - (1 - [Delta])([T.sub.1] - [T.sub.2])[[Alpha][[Theta].sub.i]([Delta]) + (1 - [Alpha])[[Theta].sub.2]([Delta])] + [Delta]{[Alpha][[Theta].sub.1]([Delta])f + (1 - [Alpha])[[Theta].sub.2]([Delta])[[[Tau].sup.*]([Y.sub.1] - [T.sub.1] - u) + (1 - [[Tau].sup.*])([Y.sub.1] - [T.sub.1] - u - s)]} - c([Delta]) (16)

which can be written as

{[T.sub.1] - (1 - [Delta])([T.sub.1] - [T.sub.2])[[Alpha][[Theta].sub.1]([Delta]) + (1 - [Alpha])[[Theta].sub.2]([Delta])] + [Delta][Alpha][[Theta].sub.1]([Delta])f + [Delta](1 - [Alpha])[[Theta].sub.2]([Delta])[([Y.sub.1] - [T.sub.1] - u) - (1 - [[Tau].sup.*])s] - c([Delta])} [equivalent] [R.sup.m] (17)

where [[Tau].sup.*] = [Rho]/{[f - ([Y.sub.1] - [T.sub.1] - u)] + [Rho]}. In Remark 5, we argued that [Delta][[Delta].sup.*]/[Delta]s [less than] 0. Note also that [[Tau].sup.*] does not depend upon s. These facts imply that the audit proportion under the mixed strategy will lie between the audit proportions with s = 0 and s [greater than] 0. Let [[Delta].sup.n], [[Delta].sup.r], and [[Delta].sup.m] denote the optimal audit proportions with no renegotiation (s [greater than] 0), successful but concealed renegotiation (s = 0), and under the mixed strategy. Then the above discussion suggests that [[Delta].sup.r] [greater than] [[Delta].sup.m] [greater than] [[Delta].sup.n].

Equations (3[prime]) and (12) can be used to identify the relation between net tax revenue collections under no renegotiation (n) and concealed renegotiation (r),

[R.sup.n] = [R.sup.r] - [Delta](1 - [Alpha])[[Theta].sub.2]([Delta])s. (18)

Let [[Delta].sup.n] maximize (3[prime]) with the maximized value equal to [R.sup.n]([[Delta].sup.n]). Inserting [[Delta].sup.n] in (12), we can write (18) as

[R.sup.n]([[Delta].sup.n]) = [R.sup.r]([[Delta].sup.n] - [[Delta].sup.n](1 - [Alpha])[[Theta].sub.2]([[Delta].sup.n])s. (19)

The second right-hand side term in (19) is positive, which implies [R.sup.n]([[Delta].sup.n]) [less than] [R.sup.r] ([[Delta].sup.n]). Note also that [[Delta].sup.n] does not necessarily maximize (12). This means the tax collector's payoff under no renegotiation is lower than the payoff under concealed renegotiation (i.e., no false bankruptcy claims). Now using (3[prime]) and (17), we can compare net tax revenue under the mixed strategy and under concealed renegotiation

[R.sup.m] = [R.sup.r] - [Delta](1 - [Alpha])[[Theta].sub.2]([Delta])(1 - [[Tau].sup.*])s. (20)

Let [[Delta].sup.m] denote the optimal audit proportion under the mixed strategy. That is, [[Delta].sup.m] maximizes (17) and let the maximized value equal [R.sup.m]([[Delta].sup.m]). Inserting [[Delta].sup.m] in (12), we can write (20) as

[R.sup.m]([[Delta].sup.m]) = [R.sup.r]([[Delta].sup.m]) - [[Delta].sup.m] (1 - [Alpha])[[Theta].sub.2]([[Delta].sup.m])(1 - [[Tau].sup.*])s.

It is clear that [R.sup.m]([[Delta].sup.m]) [less than] [R.sup.r]([[Delta].sup.m]). Again, [[Delta].sup.m] does not necessarily maximize (12). Thus the tax collector's payoff under the mixed strategy is lower than the payoff under concealed renegotiation. From (3[prime]), (12) and (17) we have

[R.sup.n] = [R.sup.m] - [Delta](1 - [Alpha])[[Theta].sub.2]([Delta])[[Tau].sup.*]s = [R.sup.r] - [Delta](1 - [Alpha])[[Theta].sub.2]([Alpha])s. (21)

Given the optimal audit policy [[Delta].sup.n] = [[Delta].sup.*], we can therefore rank the payoffs of the tax collector as follows:

[R.sup.r]([[Delta].sup.*]) [greater than] [R.sup.m]([[Delta].sup.*]) [greater than] [R.sup.n] ([[Delta].sup.*]).

This relation indicates that expected net tax revenue is greatest when concealed renegotiation can be practiced. When this is not the case, the mixed strategy of renegotiation and seizure dominates a strategy of no renegotiation.

Finally, as can be seen from equation (11), the parameter [[Tau].sup.*] depends on f. As f increases [[Tau].sup.*] falls, ceteris paribus. This has an effect on (17) similar to an increase in s. Since we have argued that [Delta][[Delta].sup.*]/[Delta]s [less than] 0, it follows that [[Delta].sup.*] falls as f is increased.

VIII. Summary and Conclusions

This paper shows that with progressive tax rates more taxpayers will dissemble. This is true even for taxpayers with large psychic costs of cheating. However as the audited proportion of taxpayers is increased, or, as the penalty is increased, only taxpayers with low psychic costs are likely to cheat. In this case, fewer taxpayers will dissemble. Increasing the audited proportion of income reports discourages taxpayers with the greatest psychic costs (among dissemblers) from cheating. This increases expected marginal tax revenue.

We also evaluate whether a policy of renegotiating the tax evasion penalty is a revenue maximizing policy. The reason behind ex post renegotiation between the revenue collector and the audited dissembler is clear. By mitigating the penalty on someone who is incapable of paying his tax bill plus the penalty for filing a false report, the revenue collector is able to raise its net revenue because seizure costs are saved. Audited dissemblers are willing to accept the revenue collector's offer because it eliminates the stigmatization associated with declaring bankruptcy. The possibility therefore exists for ex post Pareto-improving renegotiation of the penalty because both dissemblers and the revenue collector are better off. However, a policy of penalty renegotiation is associated with at least two important problems. Once the renegotiation possibility is recognized, audited dissemblers have incentives to claim that the penalty will force them into bankruptcy. If the revenue collector cannot sort out true bankruptcy claims from those that are false, ex ante, then expected net tax revenue will be smaller than when the revenue collector has perfect knowledge. Further, if renegotiation of the penalty is recognized before taxpayers report their incomes, then taxpayers will estimate that the expected penalty for dissembling is less severe. This occurs because the revenue collector will reduce the initial penalty whenever the dissembler claims that the penalty will force them into bankruptcy in order to save on seizure costs. In this case, even taxpayers with large psychic costs associated with cheating are induced to dissemble, which may lead to smaller tax revenues than when renegotiation is not possible.

We provide a solution to the false bankruptcy claim problem. We show that the problem can be reduced if the tax collector declares a policy randomizing between two strategies of renegotiation and asset seizure. Finally, our examination of the relation between the audit policy, net tax collections and the tax authority's willingness to renegotiate the penalty indicates that a policy of concealed renegotiation generates the greatest net tax revenues followed by the mixed strategy just described. A policy of no renegotiation produces the least net tax revenues.

1. It appears that tax evasion is a major problem in virtually all economies. For example, in the United States it is estimated that about 6 to 8 percent of Gross National Product is hidden from the tax authorities [19]. Cowell [8] has suggested that the number is 5 percent in the United Kingdom and 8 to 15 percent has been suggested for Sweden.

2. The psychic cost of cheating has two components; first is the cost of engaging in a risky endeavor and the second is the cost of violating a moral commitment. The first component obviously depends upon the degree of risk aversion, and the second one comes from anxiety, guilt or a reduction in self-image. Without the second type of psychic cost, risk neutral people will always take tax evasion gambles since a small audit probability makes the gamble actuarially favorable. However, with the presence of the second type of cost, they may not engage in such a gamble. Empirical evidence related to the latter has been reported by Baldry [5].

3. The stigmatization of bankruptcy is potentially comprised of a social stigma, the stigma that follows the individual when seeking new employment opportunities, and the stigma that follows the individual when seeking new credit.

4. Renegotiation of the penalty is conceptually similar to tax amnesty as it represents a policy for increasing tax revenues, Stella [27] and Andreoni [4]. It is important to remember however that renegotiation of a tax penalty is offered only after a dissembler is discovered. It is not the result of voluntary compliance. Every year for instance the United States Internal Revenue System abates assessed penalties resulting from audited individual income reports. The 1991 Annual Report of the U.S. Internal Revenue Service [20] indicates that in 1990 these abatements equalled 24.9% of penalties from delinquency, 9.5% of penalties from the failure to pay the tax due, 9.9% of penalties from bad checks, and 22.3% of penalties from fraud. Of course, these abatements are not associated with tax amnesty. We infer that these abatements are due to renegotiation.

5. If the tax authority decides not to renegotiate the penalty with an audited dissembler whose income is insufficient to cover the tax, the penalty and any additional personal liabilities, there are several possible outcomes:

(i) The audited dissemblers are likely to go into default and the tax collector takes self-help measures,

(ii) Default occurs and the tax collector takes self-help measures,

(iii) Default occurs and the audited dissemblers initiate legal action against the tax collector (i.e. the dispute is adjudicated in tax court).

In connection with (i), self-help can take the form of threatening to a) repossess assets or garnish wages, or b) initiate legal action. The ensuing cost of these actions can range from efforts at persuasion to legal fees. Once default occurs, the tax collector also relies on self-help actions. For example, after due notice is given, the tax collector can repossess personal assets or garnish the individual's wages. Seizure costs are incurred in the process. Moreover, the tax collector may recover a value that is smaller than the outstanding tax and penalty due plus seizure costs. Outcome (iii) may not occur because legal fees may be too large. However, if the case does reach the courts, the tax collector also faces legal fees when defending its actions (such as seizure). In 1990, 35.9% of U.S. tax court cases were determined in favor of the IRS, 3.7% were decided in favor of taxpayers, 47.6% were decided neutrally (the remaining 12.8% were classified as miscellaneous) [20].

6. Typically, declaring bankruptcy results in the garnishment of wages which imposes transaction costs on the employer. Therefore, employers tend to be outraged at their worker's apparent improvidence and seek steps to disassociate themselves from the complicated affair, tending to fire workers and to write bad references for workers whose wages have been garnished.

7. The tax authority incurs auditing and income seizure costs. Seizure can take one of the following three forms; filing a federal tax lien, levies served upon a third party, and seizures of property. For example, in 1989, filing federal tax liens generated 28% of the IRS's total seizure revenue, levies served upon third parties generated 67%, and seizures of property generated 5% [20].

8. This result is for the case in which taxpayers are risk neutral and the audit cost is constant.

9. The issue of taxpayer honesty in general has been investigated by Graetz, Reinganum and Wilde [15] Alm, McClelland and Schultze [3] and Erard and Feinstein [11]. These models do not investigate the implications of renegotiation.

10. Renegotiation is similar to plea bargaining. It is known that guilty pleas save resources which would otherwise be consumed during the trial process. For example, Landes [21] argues that the likelihood of disposition by negotiated plea should be higher the greater is the resource cost to the defendant of a trial versus a negotiated plea. Grossman and Katz [18] and Chu [7] also investigate this issue.

11. The alternative formulation is that the tax revenue collector cannot commit itself to the audited proportion. If the tax revenue collector cannot commit itself, the individual taxpayer cannot subsequently determine whether the revenue collector has actually used the preannounced proportion. In that case there is little reason to believe that the revenue collector follows the preannounced audit policy. This alternative formulation is useful in dealing with the sequential volatility or stability of the optimal audited proportion issue as the analysis of Reinganum and Wilde [24] shows. We restrict the revenue collector's choices by requiring that it precommit to an auditing policy. This restriction brings into sharp focus the central implications of penalty renegotiation.

12. We define the progressivity of the tax structure in terms of average tax rates. That is, the progressivity is measured by [Sigma] = [T.sub.1]/[Y.sub.1] - [T.sub.2]/[Y.sub.2]. If [Sigma] [greater than] 0, the tax structure is progressive, and if [Sigma] [less than] 0, the tax structure is regressive. Formby, Smith and Sykes [12] discuss alternative definitions of progressivity. The definition of progressivity employed is not crucial. Our basic results are insensitive to the definition of progressivity and so the simplest case is employed. Here, we simply say that as [T.sub.1] increases or [T.sub.2] decreases, that tax rates becomes more or less progressive.

13. We believe that seizure cost is an important part of any tax auditing system. Internal audit recommendations of the U.S. Internal Revenue Service indicate that compliance and enforcement are the prime factors to be considered in the administration of the process, and, that the decision to seize should not depend on the cost-benefit ratio of the action. However, according to a Report of the Comptroller General of the United States [26], this recommendation has been frequently violated. That document reports that taxpayer held property is seized infrequently and that the sale of seized property is also infrequent. Of the estimated 18,000 seizures made in 1975, only 3132 (17 percent) eventually resulted in property being sold. The report points out that seizures are not always cost effective when the cost of seizure is compared to the benefits of seizure. Further, the December 1972 U.S. IRS internal audit report recommended that preseizure reports, which set forth the reason why seizure is required, the estimated costs of the seizure and the estimated sales proceeds likely to be realized, be required prior to any action being taken [26]. All this suggests that seizure cost is an important factor in collecting tax and penalty liabilities owed.

14. The objective function (3) is similar to the objective defined in Reinganum and Wilde [24]. In Reinganum and Wilde the government (here the tax authority) is identified as a net revenue maximizer who is also a net cost minimizer. Note that for some audited [Y.sub.2] reports the tax authority will discover that income was actually equal to [Y.sub.2] and hence no additional tax or penalties will be collected. Let the true fraction of the population with [Y.sub.2] incomes equal p[prime], and the true fraction with [Y.sub.1] incomes equal (1 - p[prime]). Also define the expression in {} in equation (3) as the quantity q. The maximization problem for the tax authority is therefore Max(1 - p[prime])(q) + (p[prime])(0) = (1 - p[prime]) Max(q). Given a fixed p[prime], the results for the optimal audit policy shown below would be adjusted by a constant factor of proportionality. The adjustment should be regarded as being implicit.

15. From our earlier discussion in section II, in order to preserve the assumption that high income type earners are not driven into bankruptcy by the penalty, we maintain the assumption 0 [less than] [Y.sub.1] - [T.sub.1] - b.

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