Income tax compliance and evasion: a graphical approach.

Linster, Bruce G.

I. Introduction

Individual income tax compliance has been one of the most significant applications of Becker's [6] economic approach to criminal activity and punishment. This paper considers an approach to optimal audit policies that relies on the dIstribution of risk aversion among taxpayers and is consistent with some stylized facts about the U.S. tax system. Rather than considering the tax evasion situation as a game between the tax collector and identical taxpayers, this model has the tax collector choose an audit probability to maximize expected tax revenues net of audit costs knowing that each taxpayer has a reservation audit probability - the smallest audit probability that would evoke truthful reporting - depending on how averse to risk she is. As the heterogeneity in the population vanishes, this problem becomes the same as the tax compliance problem analyzed in Graetz, Reinganum, and Wilde [9].

This paper extends earlier contributions by analyzing income tax evasion and compliance with a population that is heterogeneous with respect to risk aversion. How other parameters of the problem influence optimal audit probabilities as well as the equilibrium proportion of tax evaders will be explored. The diversity in preferences over lotteries leads to some interesting comparative static results. For example, the impact of an increase in the penalty for lying is ambiguous in this model. The graphical approach taken here allows us to identify the relevant factors and how they interact. Additionally, this analysis makes it clear why the distribution of risk aversion in the population affects the equilibrium audit probability, but not the proportion of tax evaders.

This problem has been approached in several different ways.[1] For example, Clotfelter [7], Slemrod [15], Alm and Beck [2], and Witte and Woodbury [16] have studied this phenomenon econometrically, but the dearth of reliable data has made empirical analysis difficult. Other economists - most noteworthy, Alm, McClelland, and Schulze [5] and Alm, Jackson, and McKee [4] - have taken a different tack, employing the methods of experimental economics. While these seminal studies have shed substantial light on taxpayer behavior, a leap of faith is required to extrapolate from the laboratory to actual tax compliance decisions.

The theoretical tax compliance literature originally focused on the behavior of the taxpayer, as in Allingham and Sandmo [1]. That is, individuals were modeled as expected utility maximizers with exogenously given audit probabilities. More recent contributions to the theoretical literature, Reinganum and Wilde [12; 13], and Graetz, Reinganum, and Wilde [9] for example, have focused on the interaction between taxpayers and the tax collector. The taxpayers are still modeled as expected utility maximizers, and the tax collector is assumed to maximize expected net revenue. The probability an individual taxpayer will be audited is endogenously determined and depends on the amount of income the individual reports. Our understanding of the tax compliance problem has significantly improved because of these studies; however, their approaches to the problem treat all taxpayers as homogeneous with respect to risk aversion, avoiding the issue of heterogeneous taxpayers altogether.

The equilibria in the game theoretic models described above have properties that may seem implausible. For example, Graetz, Reinganum, and Wilde [9] require that if the tax collector audits with probability [Pi] [element of] (0, 1), then both the tax collector and taxpayers are indifferent between their pure strategy choices - "audit" and "don't audit" for the tax collector and "report truthfully" and "lie" for the taxpayer. The first-order optimization conditions insure that the tax collector will be indifferent as to whether or not he should audit another return. However, it is certainly true that only a very small group of taxpayers will be truly indifferent between lying and being truthful when reporting tax liability.

An important goal of the research reported here was to develop a model consistent with certain stylized facts about the U.S. tax system. The first observation is that the Internal Revenue Service relies on "voluntary" reporting of the amount of tax owed while having information on each taxpayer's gross income from employers and financial institutions. An audit frequently involves the verification of deductions claimed by the taxpayer. Second, individuals in similar financial situations frequently respond differently on tax returns. This may be due to different feelings of social responsibility, but certainly risk aversion is an important element in tax reporting decisions.(2) Finally, distinct income groups are audited with different probabilities. The model described here conforms to the above observations.

The next section describes the basic framework of the model. Although similar to the one used by Graetz, Reinganum, and Wilde [9], this model is substantially more general than its predecessors since it allows for populations that are not homogeneous with respect to risk aversion. In the subsequent sections, the equilibrium outcome is discussed, and some comparative static results are explored. Finally, the results are summarized.

II. The Model

Since this model focuses primarily on income tax compliance and evasion, income levels, tax rates, and fines are exogenously determined. Pencavel [11] and Sandmo [14] have explored the relationship between tax evasion and labor supply, but the issue is assumed away in this paper. Treating tax rates and fines as exogenous is reasonably realistic since they are, after all, usually set by the legislative branch of government and not the tax collection agency. In order to keep the model tractable, this analysis assumes there are two possible levels of tax, [T.sub.H] and [T.sub.L]([T.sub.H] [greater than] [T.sub.L]) for a given gross income I. We can think of the tax collector receiving reports from employers and financial institutions, allowing it to sort taxpayers according to their gross income. However, based upon personal circumstance, the taxpayer with gross income I is legally obliged to pay either [T.sub.H] or [T.sub.L]. A taxpayer who is supposed to pay [T.sub.H] can be thought of as having few deductible expenses while one who owes [T.sub.L] has more. Further, the proportion of individuals actually owing the low tax is [Rho].

Each taxpayer with income I will report [T.sub.H] or [T.sub.L] to the tax collector who chooses [Pi], the probability of an audit for those reporting [T.sub.L]. Certainly, taxpayers reporting [T.sub.H] should never be audited. If a "high tax" individual is audited and discovered to be untruthful, she pays the difference between [T.sub.H] and [T.sub.L] plus a fine F. It is further assumed that the government knows the proportion of the population, [Rho], that can truthfully report [T.sub.L]. A truthful "high tax" individual will have after-tax income I - [T.sub.H]. An untruthful "high tax" taxpayer will have an after-tax income of I - [T.sub.L] if she is not audited and I - [T.sub.H] - F if she is audited. A "low tax" individual will have after-tax income I - [T.sub.L] in any case.

Taxpayers in this model are allowed to have different preferences over lotteries, but for ease of exposition they are all assumed to have expected utility functions for which their risk aversion can be parameterized. Specifically, the constant absolute risk aversion (CARA) utility function is used in this analysis.(3) Each taxpayer's preferences are represented by the von Neumann-Morgenstern utility function u(y) = -[e.sup.-ay], so the Arrow-Pratt measure of absolute risk aversion is A(y) = -u[double prime] (y)/u[prime] (y) = a. A higher value of a represents more risk aversion. The individuals in each income class are identical in every respect except their attitudes towards risk.

Measures of absolute risk aversion are distributed according to p: A [approaches] [0, 1] (where A is the set of possible risk aversion measures) so that p(a) represents the proportion of the population with risk aversion parameter a. Further, the cumulative probability distribution function, P(a) = [summation over z [less than or equal to] a] p(z), represents the proportion of the population with risk aversion measure no greater than a. A very large population is assumed so p(a) can be approximated as a continuous probability density with probability distribution function P(a) = [integral of] p(z)dz between limits a and -[infinity].

The taxpayers and the tax collector simultaneously choose from their respective strategy sets. That is, taxpayers choose to "lie" or "report truthfully." The tax collector in this model chooses an audit probability in order to maximize net tax revenue, or expected gross tax revenue less audit costs which are assumed to be constant at c per taxpayer audited. It must be true that c [less than] (1 - [Rho])[[T.sub.H] - [T.sub.L] + F], or the cost of an audit is less than the maximum possible expected return to an audit.

III. Equilibrium Tax Evasion

Equilibrium is achieved in this model when the tax collector has chosen the audit probability, [Pi], to maximize expected net revenue given the proportion of taxpayers reporting untruthfully, and each taxpayer reports a tax obligation, either [T.sub.L] or [T.sub.H], so as to maximize expected utility given the probability she will be audited. When everyone is acting optimally given what others are doing, neither the tax collector nor any taxpayer will have any regrets.

Solving for the equilibrium outcome is most easily accomplished in steps. First, notice that a "low tax" individual will always report [T.sub.L]. A "high tax" taxpayer optimizes by truthfully reporting [T.sub.H] when

-[Pi][e.sup.-a(I - [T.sub.H] - F)] - (1 - [Pi])[e.sup.-a(I - [T.sub.L])] [less than or equal to] -[e.sup.-a(I - [T.sub.H])].(1)

She will report [T.sub.L] otherwise. Using the above, we can define the function [Mathematical Expression Omitted] which assigns to each possible combination of [Pi], [T.sub.H], [T.sub.L], and F a risk aversion measure indicating which type of "high tax" individual will be just indifferent between lying and reporting truthfully. Another way of looking at this is to note that [a.sup.*] ([Pi], [T.sub.H], [T.sub.L], F) assigns to each vector ([Pi], [T.sub.H], [T.sub.L], F) the smallest level of risk aversion a "high tax" individual can have and maximize her expected utility by reporting truthfully. Specifically, [a.sup.*]([Pi], [T.sub.H], [T.sub.L], F) implicitly solves

[[e.sup.-[a.sup.*](I - [T.sub.H])]] - [e.sup.-[a.sup.*](I - [T.sub.L])]]/[[e.sup.-[a.sup.*](I - [T.sub.H] - F)] - [e.sup.-[a.sup.*](I - [T.sub.L])]] - [Pi] = 0

([e.sup.[a.sup.*][T.sub.H]] - [e.sup.[a.sup.*][T.sub.L])/[[e.sup.[a.sup.*]([T.sub.H] + F)] - [e.sup.[a.sup.*][T.sub.L]]] - [Pi] = 0. (2)

It is not difficult to show that (a) [Delta][a.sup.*]/[Delta][Pi] [less than] 0 and [Delta][a.sup.*]/[Delta]F [less than] 0, or as audit probability or the fine increases, less risk averse individuals will begin reporting truthfully; and (b) [Delta][a.sup.*]/[Delta][T.sub.H] [greater than] 0 and [Delta][a.sup.*]/[Delta][T.sub.L] [less than] 0, or as the difference between the two taxes increases, "high tax" individuals who are more risk averse optimize by lying. The intuition for the above results is straightforward and reflects that individuals will respond in a predictable way if the expected benefit from lying increases or decreases. It is worth noting that the actual values of the above partial derivatives depend on the values of [Pi], [T.sub.H], [T.sub.L], and F.

For notational convenience, [a.sup.*] ([Pi], [T.sub.H], [T.sub.L], F) will be denoted as [a.sup.*] ([Pi]) until considering comparative statics. This notation emphasizes that the only choice variable for the tax collector is [Pi]. Hence, if the tax collector audits with probability [Mathematical Expression Omitted], any "high tax" individuals with risk aversion measures of at least [Mathematical Expression Omitted] will optimize by reporting [T.sub.H], and those less risk averse will report [T.sub.L].

Analyzing this income tax compliance problem graphically is straightforward. The tax collector solves the following program

[Mathematical Expression Omitted].

In the above, [Gamma] represents the proportion of high tax individuals who report untruthfully.

Some simple algebra reveals that the tax collector's optimal audit probability ([[Pi].sup.*]) is zero if [Gamma] [less than or equal to] [Rho]c/[(1 - [Rho])([T.sub.H] - [T.sub.L]+ F - c)] and [[Pi].sup.*] = 1 if [Gamma] [greater than or equal to] [Rho]c/[(1 - [Rho])([T.sub.H] - [T.sub.L] + F - c)]. In other words, the best reply for the tax collector is to audit nobody if the proportion of the high tax individuals reporting [T.sub.L] is smaller than [[Gamma].sup.*] = [Rho]c/[(1 - [Rho])([T.sub.H] - [T.sub.L] + F - c)]. If the proportion is greater, the tax collector optimizes by auditing every taxpayer reporting [T.sub.L], and auditing with any probability [Pi] [element of] [0, 1] is optimal if the proportion of high tax individuals reporting [T.sub.L] is [[Gamma].sup.*] = [Rho]c/(1 - [Rho])([T.sub.H] - [T.sub.L] + F - c), which does not depend on P(a). To analyze this portion of the problem graphically, consider the tax collector's best reply to the proportion of "high tax" individuals reporting [T.sub.L]. Based on the above, the optimal response by the tax collector is represented by [BR.sub.TC] in Figure 1.

The taxpayers' aggregate response, [R.sub.TP], can also be summarized graphically, but a four quadrant diagram is required. The function [a.sup.*]([Pi]) is plotted in the northwest quadrant. Below that, in the southwest quadrant, the probability distribution function P(a) is represented. Through the use of a 45 degree line, the taxpayers' aggregate response, [R.sub.TP] = [Gamma]([Pi]) = P[[a.sup.*]([Pi])], is reflected into the northeast quadrant. It is easily shown that [Gamma][prime]([Pi]) = p([a.sup.*])[Delta][a.sup.*]/[Delta][Pi] [less than] 0, or [R.sub.TP] is downward sloping.

The equilibrium audit probability ([[Pi].sup.*]) and proportion of the "high tax" individuals reporting untruthfully ([[Gamma].sup.*]) is found at the intersection of the tax collector's best reply and the aggregate taxpayers' response curves in the northeast quadrant. It is interesting to note that the equilibrium proportion of tax evaders does not depend on the distribution of the risk aversion parameter in the population. Instead, it depends only on [Rho], c, [T.sub.H], [T.sub.L], and F. The optimal audit probability, on the other hand, depends on all the parameter values.(4) In the next section, the comparative statics of the equilibrium will be discussed.

IV. Comparative Statics

Some simple comparative static results can be derived from the graphical solution. First, consider what happens to the optimal audit probability and proportion of tax evaders if the proportion of "low tax" taxpayers ([Rho]) or the cost of an audit (c) increases. Changes in [Rho] or c have unambiguous implications for equilibrium [Pi] and [Gamma] because they only change the tax collector's best reply. Specifically, the vertical portion is shifted rightward if either [Rho] or c increases. Since the taxpayers' aggregate response is downward sloping, the result is a higher proportion of tax evaders and a lower audit probability. The underlying intuition is clear.

Now consider changes in any of the other parameters of the problem, [T.sub.H], [T.sub.L], and F. As before, the graphical approach will provide some insight that is much more difficult to obtain through standard calculus techniques. Suppose the fine changes by a small amount [Delta]F. The vertical portion of [BR.sub.TC] then shifts by approximately [Delta][[Gamma].sup.*] [congruent] - [Rho]c/[(1 - [Rho])[([T.sub.H] - [T.sub.L] + F - c).sup.2]][Delta]F which will have a sign opposite that of [Delta]F. If the fine increases, for example, the vertical portion of [BR.sub.TC] shifts to the left. The intuition here is that if the fine increases, the tax collector will be indifferent between auditing and not auditing at a lower proportion of tax cheaters.

The difference between the comparative statics in this situation and when only [Rho] or c changes is that the taxpayers alter their behavior when [T.sub.H], [T.sub.L], or F vary. To complete the comparative static analysis of the above situation we must examine how taxpayer behavior will be affected by a change in F. First, notice that if the fine increases by [Delta]F, the level of risk aversion at which "high tax" taxpayers become indifferent to reporting truthfully and lying when the audit probability is [[Pi].sup.*] changes by

[Delta][a.sup.*] = [a.sup.*]([[Pi].sup.*], [T.sub.H], [T.sub.L], F) - [a.sup.*]([[Pi].sup.*], [T.sub.H], [T.sub.L], F + [Delta]F) [congruent] [[Delta][a.sup.*]([[Pi].sup.*], [T.sub.H], [T.sub.L], F)/[Delta]F][Delta]F.

This change is represented in Figure 2 and has a sign opposite that of [Delta]F.(5) This change, [Delta][a.sup.*], is translated into the proportion of taxpayers through the cumulative probability distribution function in the southwest quadrant. This leaves the change in the proportion of "high tax" individuals who will cheat as

[Delta][Gamma]([[Pi].sup.*]) [congruent] p[[a.sup.*] ([[Pi].sup.*], [T.sub.H], [T.sub.L], F)][Delta][a.sup.*] [congruent] p[[a.sup.*] ([[Pi].sup.*], [T.sub.H], [T.sub.L], F)][[Delta][a.sup.*]([[Pi].sup.*], [T.sub.H], [T.sub.L], F)/[Delta]F][Delta]F.

Figure 2 shows us that the equilibrium audit probability will decrease if

[Mathematical Expression Omitted],

and the equilibrium audit probability will increase if the inequality is reversed. This result is somewhat counterintuitive since the previous literature on this subject held that an increase in the fine would cause a decrease in the equilibrium audit probability. Here an increase in the fine can cause an increase in equilibrium audit probability if the conditions are right. Specifically, if the slope of the probability distribution function (probability density) is small enough or the reservation level of risk aversion is sufficiently unresponsive, an increase in the fine can lead to an increase in equilibrium audit probability.

A similar exercise can be used to examine the comparative statics associated with changes in [T.sub.H] and [T.sub.L]. It can easily be verified that the equilibrium audit probability will decrease if

[Mathematical Expression Omitted]

or

[Mathematical Expression Omitted]

and the equilibrium audit probability will increase if the inequalities are reversed.

Another interesting question is what will happen to the equilibrium audit probability if there is a change in the distribution of risk aversion in the population. For example, suppose there is a median preserving spread in the distribution. It can be seen that the effect such a change will have on the equilibrium audit probability depends on the equilibrium proportion of high tax individuals who are lying. Notice in Figure 3 that [BR.sub.TC] is unaffected by a change in the distribution of risk aversion. However, [R.sub.TP], changes so that a median preserving (this is also mean preserving if the distribution is symmetric) increase in the variance of the distribution will lead to an increase in the audit probability if [[Gamma].sup.*] [less than] 0.5. However, if [[Gamma].sup.*] [greater than] 0.5, the opposite is true. The implication is that if [Rho]c/[(1 - [Rho])([T.sub.H] - [T.sub.L] + F - c)] [less than] 0.5, a median preserving increase in the variance will lead to an increase in the equilibrium audit probability. The equilibrium audit probability will decrease if the inequality is reversed.

Finally, suppose only the mean of the risk aversion parameter distribution changes. Specifically, suppose the mean increases while all other parameters of the distribution stay the same. It is clear that such a change will lower the aggregate taxpayer response without changing the tax collector's best reply. The result will be an unambiguous decrease in equilibrium audit probability and the proportion of high tax individuals who lie will remain constant. A change of this type is represented with dashed lines in Figure 3.

V. Conclusions

This analysis sheds new light on the issue of tax evasion. The graphical approach presented here allows us to analyze what happens as parameter values vary. It is possible to make reasonable assumptions about the distribution of risk aversion in a population, allowing an analysis of how changes in audit probability will alter net expected revenue given the proportion of "low tax" individuals, the cost per audit, and the levels of taxes and fines.

One seemingly restrictive assumption in this model is that each individual has a CARA utility function. The importance of this assumption can at least be partly diminished when we consider that we are looking at individuals in a particular income group. As long as the difference between the after-tax incomes realized for each contingency is relatively small for an income group, the results will not be significantly altered. The meaningful results from this paper do not depend on the CARA utility function. The important issue is that disparate individuals feel differently about risk, and any tax compliance model should take that into account. How risk aversion is distributed throughout the population will have an important impact on optimal audit policies. Additionally, it should be remembered that the tax collector can solve many similar problems simultaneously. If individuals are categorized by gross income, each group will pose a separate problem for the tax collector, resulting in different optimal audit probabilities depending on income.

Another strong assumption in this model is the restriction to only two possible levels of tax for each income group. This simplification was made primarily for tractability. A similar analysis could be accomplished with individuals choosing an amount of tax to report between [T.sub.L] and [T.sub.H]. However, this would complicate the problem while obscuring the central idea in this analysis, which is that the distribution of risk aversion is important in choosing optimal tax compliance policies.

The model described in this paper conforms with certain stylized facts about tax compliance with information requirements that are similar to those we observe in the US. That is, the IRS has information on gross income from employers and financial institutions, but it relies on "voluntary" reporting for the actual amount of tax owed. The model stresses the fact that individuals feel differently about risk, and some taxpayers will be more compliant than others in similar circumstances. Additionally, it allows different audit rates for different income groups, letting higher income groups be subject to different audit probabilities than lower income groups. Moreover, different distributions of risk aversion among various income groups are possible. For example, if higher income groups are believed to be less risk averse than others, this can be modeled.

Individual tax compliance has been the topic of many studies. How taxpayers feel about risk, though, must be an important element of any model that attempts to explain compliance behavior. That populations respond to higher fines and audit probabilities in the expected way has been demonstrated in the studies noted earlier. This model forms a theoretic basis for analyzing this problem.

1. An excellent survey of the tax compliance literature is given by Frank A. Cowell [8].

2. Clearly, risk aversion cannot completely explain tax compliance behavior, but those whose feelings of social responsibility compel them to pay their full taxes regardless of audit probabilities can be thought of as infinitely risk averse for the purposes of this analysis.

3. Another possibility for the kind of utility function desired for this analysis is the constant relative risk aversion utility function, u(y) = [y.sup.1-b]/(1 - b). Here the coefficient of relative risk aversion, R(y) = A(y)y = b, is a constant. In either case, the level of risk aversion is easily parameterized. For an excellent discussion of useful utility functions, see Ingersoll [10, 39-40].

4. The analysis here doesn't depend on income only because CARA utility functions are assumed. Adding gross income to the analysis is straightforward. For example, if constant relative risk aversion utility functions are used, income is easily incorporated into the model.

5. An affine relationship is assumed between [Pi] and [a.sup.*] for simplicity. Although the slope ([Delta][Pi]/[Delta][a.sup.*]) can vary as [T.sub.H], [T.sub.L], or F changes, such shifts will only affect the magnitude of the audit probability change, not its sign.

References

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