Income Tax Avoidance and Evasion

Abstract

We characterize optimal individual tax evasion and avoidance when taxpayers “narrow bracket” the joint avoidance/evasion decision by exhausting all gainful methods for legal avoidance before choosing whether or not also to evade illegally. We find that (1) evasion is an increasing function of the audit probability when the latter is low enough, yet tax avoidance is always decreasing in the probability of audit; (2) an analogous finding to the so-called Yitzhaki puzzle for evasion also holds for tax avoidance—an increase in the tax rate decreases the level of avoided income and the level of avoided tax; and (3) that, holding constant the expected return to evasion, it is not always the case that the combined loss of reported income due to avoidance and evasion can be stemmed by increasing the fine rate and decreasing the audit probability.

Keywords

tax avoidance tax evasion narrow bracketing financial intermediaries

Individuals take a variety of actions to reduce their tax liabilities. The UK tax authority, for instance, distinguishes three distinct types of action (Her Majesty’s [H.M.] Treasury and H.M. Revenue and Customs 2011): those that breach tax law (tax evasion), those that “use the tax law to get a tax advantage that Parliament never intended” (tax avoidance) (p. 3), and those that “use tax allowances for the purposes intended by Parliament” (tax planning) (p. 7). By these definitions, both tax evasion and tax avoidance are responsible for significant losses in public revenue: estimates provided by the UK tax authority put the value of tax avoidance at £2.7 billion and the value of tax evasion at £4.4 billion (H.M. Revenue and Customs 2015). Given the first-order significance of tax avoidance, it is of note that the first economic studies relating to tax compliance (e.g., Allingham and Sandmo 1972; Yitzhaki 1974; Christiansen 1980) neglect the possibility of tax avoidance altogether, and the economic literature that followed has largely retained this bias.

In this article, we introduce tax avoidance into the portfolio model of tax evasion (Yitzhaki 1974). To model the joint tax avoidance/evasion decision, we build on insights developed in psychology and behavioral economics. In particular, we allow for a pervasive propensity among human decision makers facing multiple-dimension problems—that of narrow bracketing. In our context, a decision maker who narrow brackets would decompose sequentially the joint decision {avoidance, evasion} into narrow brackets, for example, {avoidance} followed by {evasion}. A key feature of narrow bracketing is that the decision maker tends to choose an option in each stage without full regard to the other decisions and circumstances that he or she faces (Rabin and Weizsäcker 2009) . Important in this context is whether the taxpayer is more likely to make the avoidance or evasion choice first. This question—as to the order in which a complex decision is mentally staged—is thought to depend heavily on mentally focal qualitative features of the choice set. We argue that a focal feature of the choice set is that avoidance is ostensibly legal whereas evasion is illegal. Indeed, judiciaries have long upheld the right of a citizen to challenge the proper interpretation of tax law and to pay only the tax they owe in law. Thus, while tax avoidance can be seen as the rightful exercise of a basic right by some lights, tax evasion lacks an equivalent interpretation. Accordingly, we suppose taxpayers focus on exhausting opportunities for legal tax avoidance before subsequently focusing on opportunities for illegal evasion.

We are by no means the first to propose that taxpayers distinguish qualitatively between legal and illegal actions, however. This distinction has previously been represented by supposing that a cost owing to social stigma and/or personal guilt is attached to the illegal act of tax evasion. This cost can be financial (e.g., Lee 2001) or psychic (e.g., Gordon 1989). Narrow bracketing offers an alternative perspective: in our model, illegal evasion is not considered until all gainful avenues for legal avoidance have been exhausted.¹ In this way, our article relates to a literature on two-stage decision-making (e.g., Blackorby et al. 1970).

In addition to the legal distinction between avoidance and evasion, we further assume that avoidance is costly whereas evasion is costless. Devising avoidance schemes that reduce a tax liability without ostensibly violating tax law invariably requires a detailed understanding of tax law, coupled with a degree of ingenuity. A classical form of avoidance scheme, for instance, involves the implementation of a circular sequence of self-cancelling option agreements that return the seller to his or her original position, but in the process create an allowable loss. As, however, few taxpayers are equipped to conceive of and implement independently such avoidance schemes, it is necessary to purchase them.² Satisfying this demand for tax avoidance is a substantial industry dedicated to the development and marketing of avoidance schemes (see, e.g., Sikka 2012; Addison and Mueller 2015). By contrast, many forms of tax evasion require no technical or legal expertise. Intentionally understating income on the tax return, for instance, may readily be performed independently.

We find that, in two respects, allowing for tax avoidance importantly changes the characteristics of optimal tax evasion. First, under plausible conditions, evasion is an increasing function of the probability of audit. Second, we reexamine the finding of Christiansen (1980, 391) that “if the fine is increased, but the efforts to detect tax evaders are adjusted so as to keep the expected gain from tax evasion unaltered, risk averters will always reduce their tax evasion.” We are again able to prove this result, yet in our model it is the total amount of lost tax (through both avoidance and evasion) that is economically pertinent. When we consider both avoidance and evasion, it is possible that taxpayers declare more income if the probability of audit is increased, and the fine decreased, holding the expected gain from tax evasion constant.

This article adds to the small, but growing, economic literature on tax avoidance. The two closest analyses to ours are Alm and McCallin (1990) and Alm (1988). The former describes avoidance and evasion as risky assets—each asset has a return characterized by a mean and variance, and the interdependence between the two returns is characterized by a covariance—while the latter characterizes avoidance as a riskless, albeit costly, asset. Whereas both of these analyses consider the simultaneous determination of evasion and avoidance, in our framework, we argue that these are chosen sequentially. Different from Alm and McCallin, we model the mean, variance, and covariance of evasion and avoidance, rather than taking these quantities as exogenous. Unlike in Alm (1988), we take avoidance to be risky, owing to the possibility of effective antiavoidance measures by the tax authority.

Much of the remaining literature on tax avoidance, however, is concerned with whether income tax has “real” effects upon labor supply or simply leads to changes in the “form” of compensation (e.g., Slemrod and Kopczuk 2002; Piketty, Saez, and Stantcheva 2014; Slemrod 1995). Accordingly, in these studies, the term “tax avoidance” typically refers to all form-changing actions that reduce a tax liability.³ This definition overlaps with ours but is broader in the sense that it also includes actions that fall in to our notion of tax planning. By this broader definition, Lang, Nöhrbaß, and Stahl (1997) estimate that tax avoidance costs the German exchequer an amount equal to around 34 percent of income taxes paid.

The plan of the article is as follows: in the second section, we motivate the key behavioral assumptions behind our analysis, from which the third section develops a formal model. The fourth section performs the main analysis and the fifth section compares our findings to the literature. We extend the model in the sixth section to allow for risk in the tax authority’s efforts to illegalize avoidance schemes, and the seventh section concludes. All proofs are in the Appendix.

Deciding to Avoid and/or Evade Tax

Two key features of our modeling of the joint decision to avoid and/or evade are that (1) the taxpayer makes the avoidance and evasion decisions sequentially and (2) that the avoidance decision is made first. The first feature—that complex decisions are routinely broken down into smaller ones—is often termed narrow bracketing in the behavioral literature. The second feature—the choice of how to stage the subdecisions within the larger composite decision—is sometimes termed decision staging (Johnson et al. 2012). We discuss each of these features in turn.

Narrow Bracketing

A mass of evidence suggests that people narrowly bracket: a decision maker who faces a multidimensional decision tends to break the decision down sequentially, proceeding at each stage to isolate a single dimension of the problem without full regard to the other dimensions of the problem. In the context of monetary risk, Tversky and Kahneman (1981) present an experiment that demonstrates how powerful this propensity is. In their experiment, people narrowly bracket even when faced with only a pair of independent simple binary decisions that are presented on the same sheet of paper. Narrow bracketing lies at the heart of current explanations of phenomena such as the stock market participation puzzle (Barberis, Huang, and Thaler 2006), the equity premium puzzle (Benartzi and Thaler 1995; Gneezy and Potters 1997), and choice among lotteries (Battalio, Kagel, and Jiranyakul 1990; Langer and Weber 2001).

Cognitive limitations—in perception, attention, memory, and analytical processing—are thought to be an important reason for narrow bracketing (Read, Loewenstein, and Rabin 1999). Accordingly, in the experiment of Read et al. (2001), subjects who were required to resolve a complex choice problem sequentially actually made better choices than those subjects who were required to proceed simultaneously. Clearly, however, decision-making outcomes under narrow bracketing can, in other contexts, appear worse than those arrived at from a wide bracketing perspective. For instance, decision-making under narrow bracketing may violate first-order stochastic dominance (Rabin and Weizsäcker 2009) and the “one day at a time” bracketing observed among New York cab drivers fails to maximize earnings per hour across days (Camerer et al. 1997). Thus, while the desirability of narrow bracketing is still debated (see, e.g., Köszegi and Rabin 2009), the pervasiveness of the phenomenon is not. It is thus of relevance to understand the nature of the joint avoidance and evasion decision under narrow bracketing.

Decision Staging

Having established that taxpayers may well mentally separate the joint avoidance and evasion decision, there remains the question as to how this is done. According to Kahneman (2003), the way individuals will choose to stage or frame a decision is heavily shaped by the features of the situation at hand that come to mind most easily. This notion is supported in the context of tax-related decision-making by McCaffery and Baron (2004). These authors employ a slightly different terminology—the isolation effect—which, however, refers to the tendency of respondents in their study to “decide complex matters by responding to the most salient or obvious aspect of a choice set or decision problem.”

In the context of the joint avoidance and evasion decision, we argue that the most accessible feature of the choice set is that tax avoidance and evasion are qualitatively distinct: one is legal, and the right to practice it has often been defended by the judiciary, whereas the other is a crime. Courts on both sides of the Atlantic have for many years upheld the right of citizens to challenge the interpretation of tax law (Barker 2009; Prebble and Prebble 2010). In 1936, Lord Tomlin surmised that “[e]very man is entitled, if he can, to order his affairs so that the tax attaching under the appropriate Acts is less than it otherwise would be. If he succeeds in ordering them so as to secure this result, then, however unappreciative the Commissioners of Inland Revenue or his fellow taxpayers may be of his ingenuity, he cannot be compelled to pay an increased tax.” Similarly, in the United States, Judge Learned Hand stated in 1947 (in Commissioner vs. Newman) that “[t]here is nothing sinister in so arranging one’s affairs as to keep taxes as low as possible. Everybody does so, rich or poor; and all do right, for nobody owes any public duty to pay more than the law demands.”

There is evidence that these traditional legal arguments continue to affect public sentiment toward tax avoidance. In the qualitative study of Kirchler, Maciejovsky, and Schneider (2003), participants relate tax avoidance to lawful acts enabling tax reduction, to cleverness, and to costs. Tax evasion, by contrast, is associated with illegal acts such as fraud, criminal prosecution, risk, tax audits, punishment, penalty, and the risk of detection. In this sense, we argue that, for many taxpayers, tax avoidance is qualitatively preferred to evasion. Accordingly, we argue that taxpayers would exhaust the scope for legal avoidance before subsequently deciding whether or not they additionally wish to evade illegally (rather than the other way round).

Model

A taxpayer has an income (wealth) w and faces a tax on income given by tw, where t ∈ (0, 1). Taxpayers behave as if they maximize expected utility, where utility is denoted by U(z) = log z.⁴ The taxpayer’s true income is not observed by the tax authority, but the taxpayer must declare an amount x ∈ [0, w]. The taxpayer can choose to avoid paying tax on an amount of income A ∈ [0, w], and subsequently to evade illegally an amount of income E ∈ [0, w − A], so x = w − A − E.

Evasion is financially costless but avoidance technology must be bought in a market in which “promoters” sell avoidance schemes to “users.”⁵ A common feature of this market is the “no saving, no fee” arrangement under which the price received by a promoter is linked to the amount by which their scheme stands to reduce the user’s tax liability. Although systematic information regarding the contractual terms of avoidance schemes is scarce, we understand from a detailed investigation in the UK that, for the majority of mass-marketed schemes, the fee is related to the reduction in the annual theoretical tax liability of the user, not the expost realization of the tax saved (Committee of Public Accounts 2013). Thus, the monetary risks associated with the possible subsequent detection and termination of a tax avoidance scheme are borne by the user.⁶ Accordingly, we assume that the promoter’s fee is a proportion φ ∈ (0, 1) of the amount by which the taxpayer’s tax liability stands to be reduced, tA. In this way, φ may be interpreted as measuring the degree of competition in the market for tax avoidance schemes, with lower values of φ indicating the presence of stronger competitive forces.

Although the older judicial arguments asserting the morality of tax avoidance continue to affect importantly public sentiment, there has nonetheless been a discernible shift in the attitudes of the judiciary, beginning in the 1980s (Stevens 2013). Increasingly, courts apply a purposive interpretation, as summarized by Judge Ribeiro, who states (in Collector of Stamp Revenue v. Arrowtown Assets Ltd.) that “the ultimate question is whether the relevant statutory provisions, construed purposively, were intended to apply to the transaction, viewed realistically.” Armed with this purposive interpretation of the law, tax authorities now routinely seek to have avoidance schemes ruled illegal. Yet taxpayers may legitimately continue to use an avoidance scheme while the (often lengthy) process of shutting it down is ongoing. Moreover, if the scheme is eventually declared illegal, the tax authority can only seek the amount of tax that was properly due (it cannot levy fines retrospectively). Inherent in our definition of tax avoidance (as distinct from tax planning) is that—should the tax authority learn of the scheme—it will consider it illegal.

The taxpayer’s income declaration is audited with probability p ∈ (0, 1). If audited, A and E are observed and the taxpayer has to pay [1 + f]tE on account of the amount of evaded tax, where f > 0 is the fine rate. The tax authority mounts a legal challenge to the avoidance scheme, which is successful with probability p_L . If the legal challenge is successful, the tax authority obtains the right to reclaim the tax owed (but cannot levy a fine). In this case, instead of paying tx in tax, the taxpayer must instead pay t[x + A].

The taxpayer’s expected utility is therefore given by

E U (A, E) = [1 - p] U (w^{n}) + p p_{L} U (w^{a_{s}}) + p [1 - p_{L}] U (w^{a_{u}}),

where wⁿ is the taxpayer’s wealth in the state in which they are not audited,

w^{a_{s}}

is the taxpayer’s wealth in the state in which they are audited and the tax authority’s legal challenge is successful, and

w^{a_{u}}

is the taxpayer’s wealth in the state in which they are audited and the tax authority’s legal challenge is unsuccessful:

w^{a_{s}} (A, E) = w - t [w - E] - [1 + f] t E - φ t A;

w^{a_{u}} (A, E) = w - t [w - A - E] - [1 + f] t E - φ t A;

w^{n} (A, E) = w - t [w - A - E] - φ t A .

A key distinguishing factor between evasion and avoidance in this context is that avoidance entails a cost φtA in all states of the world. Thus, if avoidance is detected and the scheme closed down, a taxpayer is worse off for having chosen to avoid, even though they are not fined on avoided income. To ensure that the amount of taxes, fines, and fees never exceeds a taxpayer’s wealth for any x, we must assume [1 − t]/t > max{φ,f}.

We suppose that taxpayers choose their preferred level of avoidance and evasion sequentially: gainful opportunities for tax avoidance are exhausted before the taxpayer decides whether to engage additionally in evasion. Thus, taxpayers first choose avoided income as:

A^{*} = \arg \max_{A} E U (A,0);

and then evaded income as:

E^{*} = \arg \max_{E} E U (A^{*}, E) .

Analysis

We now present an analysis of the model of the previous section. For analytic tractability, we shall consider the special case of the model with p_L = 1, such that legal challenges by the tax authority are always successful. In a later section, we shall demonstrate numerically how the results with p_L = 1 relate to the results for the more general case with p_L < 1.

To begin, it is helpful to define the function R(z) = [1 − z]/z, such that, for example, R(p) is the classical odds ratio found in decision theory. We may then state our first proposition:

Proposition 1: An interior optimum for avoidance and evasion satisfies

A^{*} = \frac{p R (t)}{1 - φ} [R (p) R (φ) - 1] w; E^{*} = \frac{p R (t)}{1 - φ} \frac{[1 - p] [1 - f R (φ)]}{f} w

where

R (p) R (φ) > 1 > f R (φ); \frac{p R (t)}{1 - φ} \frac{[1 - p] [1 - f R (φ)] + f [R (p) R (φ) - 1]}{f} < 1.

Proposition 1 gives closed-form expressions for optimal avoidance and evasion when both are at an interior maximum and the conditions needed for a such an interior maximum to arise. The first sequence of inequalities at the bottom of the proposition guarantees that A*, E* > 0. The left-side inequality, R(p) R(φ) > 1, is the condition that the avoidance gamble be better than fair. As a necessary condition for both inequalities to hold, it must be that R(p) R(φ) > fR (φ), which implies R(p) > f. This is the standard restriction in the portfolio model of tax evasion that the evasion gamble be better than fair. The right-side inequality at the bottom of the proposition ensures that A* + E* < w. To gain insight into how A* and E* are related, note that we may write one as a (linear) function of the other:

E^{*} (A^{*}) = \frac{p [w R (t) - φ A^{*}] [R (p) - f]}{f} - p A^{*} .

From equation (7), we note that $\partial E^{*} / \partial A^{*} = - [f R (φ) + R (p)] / {f [1 + R (p)] [1 + R (φ)]} < 0$ , so the amount of evaded income E* is negatively related to the amount of avoided income A*. This finding matches that of Alm, Bahl, and Murray (1990) using data from Jamaica, who find that evasion and avoidance are substitutes. We now consider the comparative statics of optimal avoidance and evasion:

Proposition 2: At an interior optimum for avoidance and evasion, it holds for A* that

\begin{array}{l} \frac{\partial A^{*}}{\partial w} = \frac{A^{*}}{w} > 0; \frac{\partial A^{*}}{\partial t} = - \frac{A^{*}}{t [1 - t]} < 0; \frac{\partial A^{*}}{\partial f} = 0; \\ \frac{\partial A^{*}}{\partial φ} = - \frac{w R (t) [p + (1 - p) R {(φ)}^{2}] {[1 + R (φ)]}^{2}}{R {(φ)}^{2}} < 0; \\ \frac{\partial A^{*}}{\partial p} = - \frac{R (t) w}{φ [1 - φ]} < 0; \end{array}

and for E* that

\begin{array}{l} \frac{\partial E^{*}}{\partial w} = \frac{E^{*}}{w} > 0; \frac{\partial E^{*}}{\partial t} = - \frac{E^{*}}{t [1 - t]} < 0; \frac{\partial E^{*}}{\partial f} = - \frac{E^{*}}{[1 - f R (φ)] f} < 0; \\ \frac{\partial E^{*}}{\partial φ} = \frac{f {[R (φ)]}^{2} + 1}{[1 - φ] [1 - f R (φ)]} E^{*} > 0; \\ \frac{\partial E^{*}}{\partial p} = \frac{R (p) - 1}{1 - p} E^{*} ≷ 0 \Leftrightarrow R (p) ≷ 1. \end{array}

Proposition 2 is derived via straightforward differentiation of the expressions for A* and E* in Proposition 1, so we omit the proof. Beginning with the comparative statics of A*, we see that wealthier people are predicted to avoid more income than less wealthy people. A second result is an extension of the well-known Yitzhaki paradox for evasion to the case of avoidance—avoided income falls as the tax rate is increased. The intuition for this result is analogous to that for evasion: a higher marginal tax rate makes the taxpayer feel poorer, and thereby more risk averse. An increase in the competitiveness of the market for avoidance schemes (a decrease in φ) increases avoided income, and an increase in the probability of audit decreases avoided income. Knowing ∂A*/∂t < 0 does not warrant that the total tax avoided, tA, also falls. It is straightforward to show, however, that $\partial [t A^{*}] / \partial t = - {t^{- 1} {[1 - t]}^{- 1} - 1} A^{*} < 0.$

Turning to evasion, the logic of the chain rule implies that, for an arbitrary exogenous variable z, it must hold that

\frac{\partial E^{*}}{\partial z} = \frac{\partial E^{*}}{\partial z} |_{A^{*} = cons .} + \frac{\partial E^{*}}{\partial A^{*}} \frac{\partial A^{*}}{\partial z},

where the first term on the right side is the direct effect of z on evasion, and the second term captures the indirect effect on evasion arising from the effects of z upon avoidance. Intuitively, the indirect effect is the income effect imparted upon the evasion choice by movements in avoidance. Noting that ∂E*/∂A* < 0, it follows that if ∂A*/∂z and ${\partial E^{*} / \partial z |}_{A^{*} = cons .}$ are of the same sign—as turns out to be the case for each of the variables {w, t, f, p}—then the direct and indirect effects in equation (8) oppose each other. This observation notwithstanding, the first three results in Proposition 2—those for {w, t, f}—are each unambiguous and consistent with Yitzhaki (1974). Unambiguity in this context arises as the direct effect can be shown to always dominate the indirect effect. To take the tax rate as an example, we find the direct effect—using equation (7)—as

{\frac{\partial E^{*}}{\partial t} |}_{A^{*} = cons .} = - \frac{w {[1 + R (t)]}^{2} [R (p) - f]}{f [1 + R (p)]} < 0,

Combining equation (9) with ∂A*/∂t in Proposition 1 and ∂E*/∂A*, we can rewrite the direct effect in terms of the indirect effect,

{\frac{\partial E^{*}}{\partial t} |}_{A^{*} = cons .} = - \frac{[R (p) - f] [1 + R (p)] R (φ)}{[R (p) + f R (φ)] [R (p) R (φ) - 1]} \frac{\partial E^{*}}{\partial A^{*}} \frac{\partial A^{*}}{\partial t},

such that, by equation (8), we obtain an alternative form for ∂E*/∂t to that given in Proposition 2:

\frac{\partial E^{*}}{\partial t} = - \frac{R (p) [1 + R (φ)] [1 - f R (φ)]}{[R (p) + f R (φ)] [R (p) R (φ) - 1]} \frac{\partial E^{*}}{\partial A^{*}} \frac{\partial A^{*}}{\partial t} < 0.

As well as evaded income being decreasing in the tax rate, it is straightforward to show that evaded tax, tE*, is decreasing in the tax rate, too. As it has no direct effect on evasion, the effect of competition in the market for avoidance (as captured by φ) is given by equation (8) as simply $\partial E^{*} / \partial φ = [\partial E^{*} / \partial A^{*}] [\partial A^{*} / \partial φ] > 0$ . Thus, a decrease in the competitiveness of the avoidance industry increases evasion.

The final finding is that tax evasion is increasing in the probability of audit if R(p) > 1 (equivalently, p < .5) and decreasing otherwise. In this case, there are again competing direct and indirect effects upon evasion, but now the direct effect does not always dominate the indirect effect. Following the same steps as in the tax rate example above, we obtain

\frac{\partial E^{*}}{\partial p} = \frac{[R (p) - 1] [1 - f R (φ)]}{R (p) + f R (φ)} \frac{\partial E^{*}}{\partial A^{*}} \frac{\partial A^{*}}{\partial p} .

From equation (12), it is immediate that the direct effect dominates when R(p) < 1 and the indirect dominates when R(p) > 1.

How plausible is the condition p < .5 required for evasion to be increasing in the probability of audit? A priori it appears highly plausible given that only around 0.96 percent of US individual tax returns filed in calendar year 2012 were examined (Internal Revenue Service 2014). If audits are concentrated on the 20 percent or so of people in the United States who are self-employed, the probability for this group would rise to 4.8 percent, still well below the 50 percent level.

We now characterize the total level of undeclared income, w − x (= A* + E*):

Proposition 3: An interior optimum for avoidance and evasion, it holds for w − x that

\begin{matrix} \frac{\partial [w - x]}{\partial w} = \frac{w - x}{w} > 0; \frac{\partial [w - x]}{\partial t} = - \frac{w - x}{t [1 - t]} < 0; \frac{\partial [w - x]}{\partial f} = \frac{\partial E^{*}}{\partial f} < 0; \\ \frac{\partial [w - x]}{\partial φ} = \frac{p^{2} w R (t) {R (p) {1 - f - f R (p) {[R (φ)]}^{2}} - f}}{{[1 - φ]}^{2} f} ≷ 0 \\ \Leftrightarrow R (p) {1 - f - f R (p) {[R (φ)]}^{2}} - f ≷ 0; \\ \frac{\partial [w - x]}{\partial p} = \frac{w R (t) [1 + R (φ)] {R (p) [1 - f - 2 f R (φ)] - 1 - f}}{f [1 + R (p)] R (φ)} ≷ 0 \\ \Leftrightarrow R (p) [1 - f - 2 f R (φ)] - [1 + f] ≷ 0. \end{matrix}

The first three results of Proposition 3 follow immediately from Proposition 2, for the comparative static effects for both A* and E* go in the same direction. The remaining two effects—those for φ and p—may go in either direction, however. Unlike the condition for E* to increase in p, though, the condition needed for w − x to increase in p seems far from being satisfied empirically. In particular, it requires an especially low f, which, in turn, forces the tax rate to be implausibly high. A similar remark applies to the condition needed for w − x to be increasing in φ. The results of Proposition 3 allow us to characterize readily the comparative statics of declared income x. For all exogenous variables except w, we obtain that the effect for declared income will take the opposite sign to the effect for total undeclared income, that is, ∂x*/∂(⋅) = −∂[w − x]/∂(⋅). For w, however, we obtain ∂x/∂w = x/w > 0.

As a final perspective on the properties of optimal avoidance and evasion, we may characterize the properties of the share of unreported income that is avoided: s_A ≡ A/[A + E]. This shall be instructive when we come to compare our results with the existing literature.

Proposition 4: At an interior optimum for avoidance and evasion, it holds that

\begin{matrix} \frac{\partial s_{A}}{\partial t} = \frac{\partial s_{A}}{\partial w} = 0; \frac{\partial s_{A}}{\partial f} = φ p^{2} [1 - p - φ] R (p) ϑ > 0; \\ \frac{\partial s_{A}}{\partial φ} = - p^{2} [1 - p - f p] R (p) ϑ < 0; \\ \frac{\partial s_{A}}{\partial p} = - f {[φ p]}^{2} [1 - f R (φ)] [1 + R {(p)}^{2} R (φ)] ϑ < 0; \end{matrix}

where

ϑ \equiv {\frac{A^{*}}{f [A^{*} + E^{*}] [1 - p - φ]}}^{2} > 0

Proposition 4 clarifies that the share of undeclared income that is avoided is independent of the tax rate (t) and the taxpayer’s wealth (w). This follows from the observation in Proposition 1 that w and R(t) enter both avoidance and evasion as multiplicative factors. We find that the probability of audit unambiguously reduces the share of undeclared income that is avoided, even though the effect of p on evasion can be of either sign. The results for the effects of the fine rate and the cost of avoidance follow directly from Proposition 2.

Comparison with the Literature

We now compare the findings of the previous section to the existing literature. First, we consider Alm, Bahl, and Murray (1990). These authors find that the quantity [1 − f^t]⁻¹ is negatively related to evaded income, implying that an increase in either f or t reduces evasion (consistent with Proposition 2). They also find, like us, that avoided income is decreasing in the cost of avoidance (as measured in our model by the parameter φ). A caveat, however, is that Alm, Bahl, and Murray identify avoidance as a riskless asset, somewhat different from the definition of avoidance as a risky asset we employ here. Second, we may compare our findings for optimal evasion to those of Yitzhaki’s (1974) canonical model of tax evasion. Our findings for the effect of wealth, the tax rate, and the fine rate on evasion are consistent with Yitzhaki, but the finding that evasion may increase in the probability of audit is different from that in Yitzhaki (where evasion is always decreasing in the probability of audit).

Third, we may compare our findings to those of Alm and McCallin (1990), who report comparative statics results for reported income (x) and for the share of undeclared income that is avoided (s_A ). Like these authors, we find that higher fines for evasion increase reported income and increase the share of undeclared income that is avoided. Different from these authors, however, we retain the well-known result of Yitzhaki (1974) that a tax rate rise will increase reported income (whereas Alm and McCallin report the opposite relationship) and, whereas Alm and McCallin find that a tax rate rise increases the share of undeclared income that is avoided, we find that this share is independent of the tax rate. We are unable, however, to follow Alm and McCallin in examining the comparative statics effects of quantities such as the mean and variance of the return to evasion and avoidance as, in our model, these quantities are determined endogenously.

Last, we may compare our findings to those of the theoretical model of Alm (1988), albeit an important difference between his model and ours is that he models avoidance as a riskless asset. Alm presents comparative statics results for the quantities w − E and the share s_x ≡ x/[w − E], that is, the fraction of income net of evasion that is avoided. In his very general framework, Alm finds all comparative statics for the share s_x to be ambiguous in sign. We similarly find that the effects of φ and p on s_x are ambiguous, but we find that ∂s_x/∂f > 0 and ∂s_x/∂t > 0. We find that s_x is independent of a taxpayer’s wealth: ∂s_x/∂w = 0. The only other two clear-cut results in Alm (1988) are that evasion is decreasing in the fine rate and in the audit probability. In our model, the first of these results is preserved, but we find that evasion can be increasing in the probability of audit.

Audit Probability versus Fine Rate

As a final comparison to the literature, we consider the finding of Christiansen (1980) that, for a constant expected return to evasion, the amount evaded is always reduced by increasing the fine rate and by decreasing the audit probability. Following Christiansen (1980), we first restrict analysis solely to evasion. For a given level of avoidance, the expected return to evasion is given by μ_E ≡ p[1 + f] − 1. Holding this constant by appropriate variation of f, and differentiating E* with respect to p, we obtain:

Proposition 5: An interior optimum for avoidance and evasion it holds that

{\frac{\partial E^{*}}{\partial p} |}_{μ_{E} = cons .} = \frac{w R (t) {[1 - R (p)] f^{2} R (φ) + [1 + 2 f] R (p) - f}}{f^{2} [1 - φ] [1 + R (p)]} > 0.

According to Proposition 5, we are able to replicate Christiansen’s finding: it always worsens evasion to raise the audit probability and lower the fine rate, holding the expected return to evasion fixed. In the context of a model containing both avoidance and evasion, however, what is relevant to a tax authority seeking to maximize tax revenue is the effect of varying p and f on the total level of income that does not get taxed. On this point, we have that:

{\frac{\partial [A^{*} + E^{*}]}{\partial p} |}_{μ_{E} = cons .} = \frac{w R (t) {[1 - p] {1 + f [3 - 2 f R (φ)]} - f [1 + f]}}{[1 - φ] f^{2}} ≷ 0.

As the right side of equation (13) can take either sign, depending upon parameter values, we now no longer find that raising fines is always superior to raising audit probability. Intuitively, this finding stems from the observation that increasing the fine rate only affects the evasion decision, whereas increasing the audit probability affects both avoidance and evasion.

Probabilistic Antiavoidance Outcomes

Up until this point, the analysis has been undertaken with the simplifying assumption that, if the tax authority mounts a legal challenge to the avoidance scheme, its challenge is always successful. While important in securing a tractable model, clearly tax authorities are not always successful in their attempts to shut down avoidance schemes, so it is of interest to understand how this consideration affects our findings.

Solving for A* using the definition in equation (5) and the full expression for expected utility given in equation (1), we obtain

A^{*} = \frac{p p_{L} R (t)}{1 - φ} [R (p p_{L}) R (φ) - 1] w,

which can be obtained from the solution for A* given in Proposition 1 (for the case of p_L = 1) simply by replacing p with pp_L .⁷ It follows that the comparative statics results for A* given in Proposition 2 continue to hold and that the effects of p_L upon avoidance are analogous to those of p. The solution for E* coming from equation (6) is complex, however. We note, though, that the taxpayer’s wealth and the tax rate both still enter the solution multiplicatively—as they do also for A* in equation (14)—so these two variables stay independent of the share of undeclared income that is avoided, s_A .

To make further progress, we assess the properties of optimal evasion via a numerical optimization procedure. Figure 1 depicts optimal avoidance and evasion as p_L is allowed to vary on the unit interval.⁸ For very low values of p_L in the interval denoted $[{0, {\hat{p}}_{L} |}_{E = 0}]$ , avoidance is seen to be maximal, and the taxpayer does not evade. In a second interval, denoted in the figure by $({\hat{p}}_{L} |_{E = 0}, {\hat{p}}_{L} |_{A + E = w}]$ , the taxpayer both avoids and evades and reports no income (x ≥ 0 is binding). In a third interval, denoted $({{\hat{p}}_{L} |}_{A + E = w},1]$ , the taxpayer again both avoids and evades, but now x > 0 (this is the case to which our comparative statics analysis applies). Within this interval, we observe that optimal evasion increases as the probability of a successful legal challenge increases. This is as expected, for an increase in p_L leaves the returns to evasion unaffected, but reduces the returns to avoidance, making evasion more attractive relative to avoidance.

Figure 1.

Optimal avoidance and evasion for p_L ∈ [0, 1].

In figure 2, we explore the effect of varying p_L on our earlier finding that evasion is increasing in the probability of audit.⁹ On the interval of figure 2 where both optimal evasion and avoidance are interior, we see that reducing p_L below unity reduces to a value below one-half the threshold audit probability above which evasion is decreasing in p. Thus, lower values of p_L imply a smaller set of parameter values for which evasion is observed to be increasing in audit probability.

Figure 2.

Optimal avoidance and evasion for p_L < 1 and p_L = 1.

Other numerically generated results we have analyzed—which we do not report here for brevity—indicate that the qualitative nature of the results given in Propositions 2 to 4 continue to hold. In particular, the taxpayer’s wealth and the tax rate continue to act as multipliers in the expressions for optimal avoidance and evasion, and w − x may be either an increasing or decreasing function in φ and p.

Conclusion

Although the economic literature has largely limited itself to the study of tax evasion, tax avoidance is empirically observed alongside tax evasion. We therefore examine the choice of a taxpayer of how much tax to avoid and how much to evade, under the assumptions that (1) a taxpayer narrow brackets the joint avoidance/evasion decision—breaking the decision down into separate avoidance and evasion subdecisions, and taking the first of these two subdecisions in isolation from the second and (2) that a taxpayer will decide first on whether and how much tax to avoid legally before deciding whether and how much tax to evade illegally. Among our results are, first, that an analogous finding to the so-called Yitzhaki puzzle for evasion also holds for tax avoidance—an increase in the tax rate decreases the level of avoided income and the avoided tax. Second, for a small enough audit probability, evasion is an increasing function of the audit probability. Although tax avoidance is always decreasing in the probability of audit, in some circumstances even the total amount of income lost to evasion and avoidance can be increasing in the probability of audit. Last, holding constant the expected return to evasion, it is not always the case that combined loss of reported income due to avoidance and evasion can be stemmed by increasing the fine rate and decreasing the audit probability.

We finish with some possible avenues for future research. First, it would be of interest to allow for imperfect audit effectiveness, as in Rablen (2014) and Snow and Warren (2005), for it might be that evasion and avoidance differ in the amount of tax inspector time required to detect them. A last suggestion is to embed the model within a general equilibrium framework (see, e.g., Alm and Finlay 2013), for the partial equilibrium setting explored here may miss some important wider interactions between avoidance and evasion that should properly be accounted for.

Footnotes

Appendix

Proof of Proposition 1: From equation (1), we have that A1

\frac{\partial E U (A,0)}{\partial A} = φ p {\frac{R (p) R (φ)}{A [1 - φ] + w R (t)} - \frac{1}{w R (t) - φ A}} .

Solving for the point ∂EU(A, 0)/∂A = 0 gives A2

A^{*} = \arg \max_{A} E U (A,0) = \frac{p R (t)}{1 - φ} [R (p) R (φ) - 1] w .

Rewriting A* in equation (A2) as $A^{*} = w R (t) [1 - p - φ] φ^{- 1} {[1 - φ]}^{- 1}$ , substituting this expression for A* into equation (1), and differentiating with respect to E gives A3

\frac{\partial E U (A^{*}, E)}{\partial E} = φ p {\frac{R (p)}{[1 - p] w R (t) + E φ} - \frac{f R (φ)}{p w R (t) - E f [1 - φ]}} .

Evaluating at ∂EU(A*, E)/∂E = 0 and solving for E gives A4

E^{*} = \arg \max_{E} E U (A^{*}, E) = \frac{p R (t)}{1 - φ} [\frac{[1 - p] [1 - f R (φ)]}{f}] w .

From equations (2) and (4), we see that A5

A^{*} > 0 \Leftrightarrow R (p) R (φ) > 1; E^{*} > 0 \Leftrightarrow f R (φ) < 1.

From equations (2) and (4), we compute A6

A^{*} + E^{*} = \frac{p R (t)}{1 - φ} [\frac{[1 - p] [1 - f R (φ)] + f [R (p) R (φ) - 1]}{f}] w,

so A7

A^{*} + E^{*} < w \Leftrightarrow \frac{p R (t)}{1 - φ} [\frac{R (p) - f + f R (p) [R (p) R (φ) - 1]}{f}] < 1.

Proof of Proposition 5: Noting that A8

{\frac{\partial^{2} E^{*}}{\partial p \partial f} |}_{μ_{E} = cons .} = - \frac{w R (t) {2 [1 + f] R (p) - f}}{f^{3} [1 - φ] [1 + R (p)]} < - \frac{w R (t) [1 + 2 f]}{f^{2} [1 - φ] [1 + f]} < 0,

it follows that if

{\partial E^{*} / \partial p |}_{μ_{E} = cons .} > 0

when f approaches its maximum possible value, that is, f → R(φ)⁻¹, then

{\partial E^{*} / \partial p |}_{μ_{E} = cons .} > 0

for all f < R(φ)⁻¹. At f → R(φ)⁻¹, we indeed have A9

{\frac{\partial E^{*}}{\partial p} |}_{μ_{E} = cons .} = \frac{w R (p) R (t) R (φ) [1 + R (φ)]}{[1 - φ] [1 + R (p)]} > 0.

Acknowledgments

We thank the editor, James Alm, and two anonymous referees for helpful comments.

Declaration of Conflicting Interests

The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and publication of this article: Gamannossi degl’Innocenti gratefully acknowledges financial support from the Ministero dell’Istruzione, dell’Università e della Ricerca (cycle XXVIII) and from the European Commission (Erasmus mobility grant 2015-1-IT02-KA103-013713/5).

Notes

References

Addison

Susan

Mueller

Frank

. 2015. “The Dark Side of Professions: The Big Four and Tax Avoidance.” Accounting, Auditing & Accountability Journal 28:1263–90.

Allingham

Michael G.

Sandmo

Agnar

. 1972. “Income Tax Evasion: A Theoretical Analysis.” Journal of Public Economics 1:323–38.

Alm

James

. 1988. “Compliance Costs and the Tax Avoidance Tax Evasion Decision.” Public Finance Quarterly 16:31–66.

Alm

James

Bahl

Roy

Murray

Matthew N.

. 1990. “Tax Structure and Tax Compliance.” Review of Economics and Statistics 72:603–13.

Alm

James

Finlay

Keith

. 2013. “Who Benefits from Tax Evasion?” Economic Analysis and Policy 43:139–54.

Alm

James

McCallin

Nancy J.

. 1990. “Tax Avoidance and Tax Evasion as a Joint Portfolio Choice.” Public Finance / Finances Publiques 45:193–200.

Alm

James

McClelland

Gary H.

Schulze

William D.

. 1992. “Why Do People Pay Taxes?” Journal of Public Economics 48:21–38.

Barberis

Nicholas

Huang

Ming

Thaler

Richard H.

. 2006. “Individual Preferences, Monetary Gambles, and Stock Market Participation: A Case for Narrow Framing.” American Economic Review 96:1069–90.

Barker

William B.

2009. “The Ideology of Tax Avoidance.” Loyola University Chicago Law Journal 40:229–51.

10.

Battalio

Raymond C.

Kagel

John H.

Jiranyakul

Komain

. 1990. “Testing between Alternative Models of Choice Under Uncertainty: Some Initial Results.” Journal of Risk and Uncertainty 3:25–50.

11.

Benartzi

Shlomo

Thaler

Richard H.

. 1995. “Myopic Loss Aversion and the Equity Premium Puzzle.” Quarterly Journal of Economics 110:73–92.

12.

Blackorby

Charles

Lady

George

Nissen

David

Russell

Robert

. 1970. “Homothetic Separability and Consumer Budgeting.” Econometrica 38:468–72.

13.

Camerer

Colin

Babcock

Linda

Loewenstein

George

Thaler

Richard

. 1997. “Labor Supply of New York City Cabdrivers: One Day at a Time.” Quarterly Journal of Economics 112:407–41.

14.

Chiappori

Pierre-André

Paiella

Monica

. 2011. “Relative Risk Aversion is Constant: Evidence from Panel Data.” Journal of the European Economic Association 9:1021–52.

15.

Christiansen

Vidar

. 1980. “Two Comments on Tax Evasion.” Journal of Public Economics 13:389–93.

16.

Committee of Public Accounts. 2013. Tax Avoidance: Tackling Marketed Avoidance Schemes. London, UK: The Stationery Office.

17.

Damjanovic

Tatiana

Ulph

David

. 2010. “Tax Progressivity, Income Distribution and Tax Non-compliance.” European Economic Review 54:594–607.

18.

Gideon

Michael

. 2015. “Do Individuals Perceive Income Tax Rates Correctly?” Public Finance Review. 45:97–117.

19.

Gneezy

Uri

Potters

Jan

. 1997. “An Experiment on Risk Taking and Evaluation Periods.” Quarterly Journal of Economics 112:631–45.

20.

Gordon

James P. P.

1989. “Individual Morality and Reputation Costs as Deterrents to Tax Evasion.” European Economic Review 33:797–805.

21.

H.M. Revenue and Customs. 2015. Measuring Tax Gaps 2015 Edition: Tax Gap Estimates for 2013-14. London, UK: H.M. Revenue and Customs.

22.

H.M. Treasury and H.M. Revenue and Customs. 2011. Tackling Tax Avoidance. London, UK: H.M. Treasury and H.M. Revenue and Customs.

23.

Internal Revenue Service. 2014. Internal Revenue Service Data Book 2013. Washington, DC: IRS.

24.

Johnson

Eric J.

Shu

Suzanne B.

Dellaert

Benedict G. C.

Fox

Craig

Goldstein

Daniel G.

Häubl

Gerald

Larrick

Richard P.

Payne

John W.

Peters

Ellen

Schkade

David

Wansink

Brian

Weber

Elke U.

. 2012. “Beyond Nudges: Tools of a Choice Architecture.” Marketing Letters 23:487–504.

25.

Kahneman

Daniel

. 2003. “Maps of Bounded Rationality: Psychology for Behavioral Economics.” American Economic Review 93:1449–75.

26.

Kirchler

Erich

Maciejovsky

Boris

Schneider

Friedrich

. 2003. “Everyday Representations of Tax Avoidance, Tax Evasion, and Tax Flight: Do Legal Differences Matter?” Journal of Economic Psychology 24:535–53.

27.

Köszegi

Botond

Rabin

Matthew

. 2009. “Reference-dependent Consumption Plans.” American Economic Review 99:909–36.

28.

Lang

Oliver

Nöhrbaß

Karl-Heinz

Stahl

Konrad

. 1997. “On Income Tax Avoidance: The Case of Germany.” Journal of Public Economics 66:327–47.

29.

Langer

Thomas

Weber

Martin

. 2001. “Prospect Theory, Mental Accounting, and Differences in Aggregated and Segregated Evaluation of Lottery Portfolios.” Management Science 47:716–33.

30.

Lee

Kangoh

. 2001. “Tax Evasion and Self-insurance.” Journal of Public Economics 81:73–81.

31.

McCaffery

Edward J.

Baron

Jonathan

. 2004. “Framing and Taxation: Evaluation of Tax Policies Involving Household Composition.” Journal of Economic Psychology 25:679–705.

32.

Piketty

Thomas

Saez

Emmanuel

Stantcheva

Stefanie

. 2014. “Optimal Taxation of Top Labor Incomes: A Tale of Three Elasticities.” American Economic Journal: Economic Policy 6:230–71.

33.

Prebble

Zoë M.

Prebble

John

. 2010. “The Morality of Tax Avoidance.” Creighton Law Review 43:693–745.

34.

Rabin

Matthew

Weizsäcker

Georg

. 2009. “Narrow Bracketing and Dominated Choices.” American Economic Review 99:1508–43.

35.

Rablen

Matthew D.

2014. “Audit Probability versus Effectiveness: The Beckerian Approach Revisited.” Journal of Public Economic Theory 16:322–42.

36.

Read

Daniel

Antonides

Gerrit

van den Ouden

Laura

Trienekens

Harry

. 2001. “Which Is Better: Simultaneous or Sequential Choice?” Organizational Behavior and Human Decision Processes 84:54–70.

37.

Read

Daniel

Loewenstein

George

Rabin

Matthew

. 1999. “Choice Bracketing.” Journal of Risk and Uncertainty 19:171–97.

38.

Reinganum

Jennifer F.

Wilde

Louis L.

. 1991. “Equilibrium Enforcement and Compliance in the Presence of Tax Practitioners.” Journal of Law, Economics, & Organization 7:163–81.

39.

Sikka

Prem

. 2012. “The Tax Avoidance Industry.” Radical Statistics 107:15–30.

40.

Slemrod

Joel

. 1995. “Income Creation or Income Shifting? Behavioral Responses to the Tax Reform Act of 1986.” American Economic Review 85:175–80.

41.

Slemrod

Joel

Kopczuk

Wojciech

. 2002. “The Optimal Elasticity of Taxable Income.” Journal of Public Economics 84:91–112.

42.

Slemrod

Joel

Yitzhaki

Shlomo

. 2002. “Tax Avoidance, Evasion and Administration.” In Handbook of Public Economics, Vol. 5, 1st ed., edited by Auerbach

Alan

Feldstein

Martin

, 1423–70. Amsterdam, Netherlands: North-Holland.

43.

Snow

Arthur

Warren

Ronald S.

Jr . 2005. “Ambiguity about Audit Probability, Tax Compliance, and Taxpayer Welfare.” Economic Inquiry 43:865–71.

44.

Stevens

Peter

. 2013. “The Changing Limits of Acceptable Tax Avoidance.” Tax Journal 1152:17–21.

45.

Stiglitz

Joseph E.

1985. “The General Theory of Tax Avoidance.” National Tax Journal 38:325–37.

46.

Tversky

Amos

Kahneman

Daniel

. 1981. “The Framing of Decisions and the Psychology of Choice.” Science 211:453–58.

47.

Yitzhaki

Shlomo

. 1974. “A Note on Income Tax Evasion: A Theoretical Analysis.” Journal of Public Economics 3:201–02.