Endogenous Detection and Audit Intensity in the Tax Evasion Game

Transcription

1 Endogenous Detection and Audit Intensity in the Tax Evasion Game Mark D. Phillips Sol Price School of Public Policy, University of Southern California, Ralph and Goldy Lewis Hall 300, Los Angeles, CA Draft: February 1, Introduction In this paper I introduce an imperfect and endogenous detection technology into the sequential tax evasion game. During a tax agency s examination of a taxpayer, the fraction of evasion detected depends upon three factors: the taxpayer s true income; the taxpayer s unreported income; and the tax agency s exam-specific resources (i.e. intensity of examination). I solve for an equilibrium in which the tax agency chooses both whom to audit and at what intensity subject to an enforcement budget constraint. In contrast to simpler games, the tax agency cannot infer any given taxpayer s true income, even after an audit occurs. Instead, the tax agency need know only two more limited but realistic pieces of information for a given taxpayer: the expected amount of detected (not true) evasion, and how that amount changes with marginal increases in audit intensity. Calibrating the model with statistics on U.S. taxpayers underreporting as well as IRS Phone: Fax: mdphilli@price.usc.edu. I am grateful for comments and advice from Jim Alm, Gary Becker, Bruce Meyer, Casey Mulligan, Alan Plumley, and workshop participants at the University of Chicago, University of Southern California, Internal Revenue Service, and the 2014 International Tax Analysis Conference hosted by HM Revenue & Customs and the UK Economic and Social Research Council. All mistakes are my own. 1

2 practices, I find that the introduction of an endogenous detection technology explains several real-world phenomena that are absent from other game theories of tax evasion. First, many lower income taxpayers pool at a common self-reported income of 0, whereas higher income taxpayers self-report some, but not all, of their income. (The income in question is that which has no third-party information reporting.) Second, underreporting increases with true income, but at a decreasing rate. Third, all taxpayers face low but non-zero probabilities of audit, even when self-reported income grows unboundedly large. Fourth, audit rates vary minimally across self-reported income amounts. Fifth, the tax agency allocates more examination resources towards higher income taxpayers, but the allocation of resources increases at a decreasing rate. Sixth, the tax agency allocates its examination resources across a broad enough pool of examined taxpayers such that examinations detect far less than 100% of noncompliance. Using this calibration as a baseline, I also find that strategic interaction in the tax evasion game leads to a progressive bias in effective tax rates relative to the statutory tax schedule. This result stands in stark contrast to the standard game theoretic result that strategic interaction induces a regressive bias. These standard models focus primarily on the tax agency s audit selection choice, thus requiring that compliance be incentivized via an audit probability that decreases with self-reported income. Therefore, higher income taxpayers with higher self-reported income face lower audit probabilities and lower effective tax rates. In contrast, the introduction of an endogenous detection technology, along with a strategic tax agency choice of audit intensity, implies that compliance can also be incentivized by its effect on reducing detection rates. This in turn allows for the finding of a progressive bias. I also utilize this calibration as a baseline for comparison of equilibrium outcomes when the detection technology changes. For instance, what happens to the equilibrium when detection rates increase for all taxpayers, independent of any strategic changes by taxpayers or the tax agency? Alternatively, what happens when detection rates become increasingly responsive to changes in true income, underreported income, or examination intensity? Such 2

3 counterfactuals are particularly policy-relevant given the ongoing improvements in computational technology and big data. Of particular note, improvements in detection technology will increase government revenues, but these improvements may manifest themselves via increased self-reported tax liabilities. In fact, revenues that originate from examinations may actually decrease due to the fact that taxpayers react with endogenous increases in self-reported income. These results suggest that, as detection technology improves, lower returns to tax agency resources should not be interpreted as failures on the part of the tax agency. Rather, it is precisely the improved performance of the tax agency that compels taxpayers to self-report more income; therefore, the tax agency never has the opportunity to examine the most readily detectable noncompliance with its new and improved technology. The paper is organized as follows. Section 2 describes the detection technology function and the role it plays in the objectives of taxpayers and the tax agency. Section 3 solves for the game s equilibrium. Section 4 calibrates the game s parameters to match data on U.S. taxpayers underreporting as well as IRS tax administration practices. This calibration then serves as a baseline for the different detection technology improvement counterfactuals. Section 5 concludes with the paper s policy and tax administration implications as well as suggestions for further research. 2 Detection Technology and Audit Intensity in the Taxpayer and Tax Agency Problems Reinganum and Wilde (1986) and Erard and Feinstein (1994) are among the many studies that have examined tax compliance in a sequential game structure, and the analysis that follows uses their models as a foundation. 1 The current analysis is distinct from the existing literature in that it introduces a nonconstant and imperfect detection technology that is endogenous to factors within the control of taxpayers and the tax agency. Whereas most 1 See Andreoni, Erard, and Feinstein (1998) for a summary of game theoretic tax evasion analyses. 3

4 game theoretic analyses of evasion employ a constant detection rate, most often implicitly equal to 1, I posit that detection rates will hinge on three distinct factors: the amount of income underreported (E), the taxpayer s true income (W ), and the intensity of resources with which the tax agency conducts an examination (I). A noteworthy exception is Rablen (2013) which examines the tax agency s trade-off between audit probabilities and audit effectiveness, though in an environment of homogeneous taxpayers. Therefore it does not address the strategic design of audit selection and intensity rules across taxpayers. I assume the existence of a detection production function δ(e, W, I) that is additively separable in logs with respect to its three inputs: Ω(δ(E, W, I)) = Ψ 0 + Ψ E ln E + Ψ W ln W + Ψ I ln I (1) where δ(e, W, I) [0, 1] for all E,W, and I and the function Ω maps from the [0, 1] domain onto the complete real line with Ω (δ) > 0. 2 Intuition dictates that a ceteris paribus increase in underreporting increases the detection rate, while a ceteris paribus increase in true income decreases the detection rate. For instance, if some types of income are more inherently detectable than others (e.g. credit card receipts vs. cash transactions), a strategically underreporting taxpayer will underreport his income in sequence from least to most detectable. The marginal dollar of underreporting will therefore have a higher likelihood of detection, and could furthermore raise the detectability of previously unreported income. Intuition also dictates that a marginal increase in audit intensity will raise the detection conditional on a given amount of underreported and true income. In terms of the detection technology, these intuitions correspond to assumptions that δ E > 0, δ W δ < 0, and, or alternatively, that Ψ I E > 0, Ψ W < 0, and Ψ I > 0. I also ( 2 In the ) later calibration section, I will impose that the detection technology is logistic, with Ω(δ) = ln. The general description of the equilibrium does not require this additional structure however. δ 1 δ 4

5 assume that Ψ E + Ψ W + Ψ I > 0, an assumption most readily explained by rewriting (1) as ( ) ( ) E I Ω(δ(E, W, I)) = Ψ 0 + (Ψ E + Ψ W + Ψ I ) ln E Ψ W ln + Ψ I ln. (2) W E Ψ E + Ψ W + Ψ I = 0 would imply that the detection rate depends solely on the fraction of income underreported (E/W ) and the ratio of examination intensity to underreporting (I/E). However, consider two taxpayers who both underreport 100% of income, but taxpayer A has W = $1, 000 while taxpayer B has W = $1, 000, 000. Furthermore, suppose the tax agency examines both with an intensity equal to 0.1% of the underreported amount, so I = $1 for A and I = $1, 000 for B. A s light touch audit may reveal virtually none of his noncompliance, especially if his minimal W = $1, 000 stems from cash transactions that have no paper trail. In contrast, the examination of taxpayer B is likely to detect a nontrivial portion of his noncompliance, even if that portion is less than unity. In order for the detection technology given in (1) to reflect this intuition, it must be the case that Ψ E + Ψ W + Ψ I > Taxpayer Problem The taxpayer evasion decision is driven by amoral cost-benefit calculation and expands upon the Allingham and Sandmo (1972) and Yitzhaki (1974) applications of the Becker (1968) economic theory of crime. Each risk-neutral taxpayer knows his true taxable income W and chooses how much to self-report to the tax agency (X) and how much to evade (E, with W = X + E) in order to maximize his expected utility. Taxpayers take as given the tax agency s strategy with respect to both the probability and intensity of audit. Since the tax agency can only observe a taxpayer s self-reported X prior to examination, the agency s strategies are given by ˆα(X) and Î(X), where the former is the probability of audit and the latter is the intensity of audit, both conditional on a given amount of reported income. These strategies are conditioned on the class of taxpayer. Per Scotchmer (1987), a class 5

6 consists of all taxpayers that have equivalent observable characteristics (from the ex ante perspective of the tax agency) besides self-reported income. Therefore the current game is best interpreted as a class-specific subgame. 3 Conditional upon examination, the taxpayer therefore has δ(e, Î(X), W )E of detected underreporting. In that event the taxpayer must pay θτδe, where θ 1 is the multiplicative factor that scales up the detected unpaid liability to account for the evasion penalty. I assume that θ is endogenous to the amount of detected underreporting D, reflecting the fact that more egregious noncompliance is typically penalized more heavily. 4 I also assume that θ depends inversely upon the tax agency s intensity of examination, an assumption that is intended to limit the incentive for the tax agency to conduct more intense examinations solely to drive up penalty rates. 5 In net, I assume that the endogenous penalty rate is determed according to some function θ (D/I), where the argument (D/I) is endogenous to the taxpayer s choice of underreporting but the remaining penalty function parameters are statutory in nature and therefore exogenous from the taxpayer and tax agency perspectives. In sum, a taxpayer with W faces the following optimization problem max E,X { ( ) (1 τ)w + τ 1 ˆα (X) θ δ E, W, Î (X) E δ Î (X) } s.t. X + E = W and X 0 ) (E, W, Î (X) E (3) where the constraint X 0 can be interpreted in two ways. Most simply, it may reflect a statutory tax schedule that does not allow for tax rebates. Alternatively, W and E can be interpreted as the taxpayer s true and self-reported gross positive income receipts, and 3 See Rhoades (1992), Andreoni, Erard, and Feinstein (1998), and Boserup and Pinje (2013) for further discussion of taxpayer classes and their relevance to the evasion game. 4 In the U.S., the statutory penalties are 20% for substantial understatements and 75% for civil fraud. Severe noncompliance can involve prison time (which presumably corresponds to a monetized penalty rate in excess of 75%) whereas minimal noncompliance may simply involve repayment of the unpaid, detected tax liability with no additional penalty. 5 I consider it a reasonable presumption that the increasing nature of statutory penalty rates is intended to penalize taxpayers based upon their own choice of underreporting, not the extent to which the tax agency chooses to intensify its examination. 6

7 by construction the taxpayer can self-report no less than $0 of gross positive income. The burden of proof for reporting gross income losses is entirely different from that for reporting gross income gains, as taxpayers must substantiate gross losses with documentation. If gross losses are overreported, the false documentation that substantiates such losses is likely to be subject to an inherently different (and better) detection technology compared to that which applies to the omission of gross income gains. Assuming that ˆα(X) and Î(X) are differentiable, an interior solution to the taxpayer problem is given by 6 1 ˆαθδ = ˆαθδ [ d ln ˆα d ln X E X ( + d ln θ d ln(d/i) ( + ln δ ln E d ln Î d ln X E X ) ] ln δ ln E ln δ ln I d ln Î d ln X E X ( 1 ln δ ) ) ln I. (4) The left side of the equality is the traditional marginal benefit of evasion, the expected increase in consumption conditional on fixed audit, penalty, and detection rates. The three lines on the right side of the equality reflect the marginal costs of evasion: first, the cost associated with endogenous changes in the probability the audit; second, the cost associated with endogenous changes in the penalty rate; and third, the cost associated with endogenous changes in the rate of detection. Classical analyses of the tax evasion game treat penalty and detection rates as exogenous, in which case (4) shows that taxpayers can only be disincentivized from additional evasion by a tax agency strategy with dˆα dx < 0. In the presence of endogenous detection however, changes in ˆα(X) do not have to shoulder the entire burden of inducing compliance. 6 Non-differentiable tax agency strategies and corner solutions to the taxpayer optimization problem are discussed in Section

8 2.2 Tax Agency Problem I assume that the tax agency s objective is to maximize gross tax revenues (the sum of taxpayers self-reported liability as well as the revenues from audits) subject to a exogenous enforcement budget constraint. This objective follows Reinganum and Wilde (1985) and Erard and Feinstein (1994) and is intended to reflect that a tax agency s budget is typically fixed by legislators. The validity of such an assumption, in contrast to an objective that maximizes revenues net of enforcement costs, is strengthened by estimates of the marginal return to IRS resources. In a March 31, 2011 statement to Congress, IRS Commissioner Doug Shulman estimated that a $600 million IRS budget cut would lead to an enforcement revenue reduction of $4 billion. 7 When the tax agency is instead assumed to maximize net revenue, the marginal effect of budget on revenues would be 1:1. The tax agency observes X for each taxpayer and the corresponding cumulative distribution of X s given by F (X). The agency then jointly decides the probability and intensity at which it will examine an X report. From the tax agency s perspective, it faces a detected underreporting function D(I, X) that identifies the amount of detected underreporting for a taxpayer with self-reported income X that is audited with intensity I. The relationship between this D function and the detection technology function is described in Section 3, but in the meantime I simply assert its existence along with the property that D I > See US tax head warns against Republican IRS cuts, Reuters: March 31, In the game of incomplete information, it is required that all players have common priors over the distribution of types, in this case unmatched income. According to Jehle and Reny (2001), the common prior assumption can be understood in at least two ways. The first is that [the distribution of is types]...is simply an objective empirical distribution over the players types, one that has been borne out through many past observations...[the second is that] before players are aware of their own types and are therefore in a symmetric position each player s beliefs about the vector of player types must be identical, and equal... Given that the tax agency has expectations only over the detected evasion for a given income report, and not the actual unmatched income, the former interpretation is more applicable with expectations regarding detected evasion arising from prior audit experience. 8

9 In sum, the agency faces the optimization problem max {ˆα(X),Î(X)} { ( ( ) ) D(Î(X), X) τ X + ˆα(X)θ 0 Î(X) D(Î(X), X) df (X) } s.t ˆα(X)Î(X)dF (X) B 0 (5) in which it chooses ˆα(X) and Î(X) for all observed X s, audit probabilities are constrained to fall between 0 and 1, and B is the exogenous agency budget. The agency s objective is linear in its ˆα(X) choice variables; therefore, any mixed equilibrium tax agency strategy requires that ( ) D(Î(X), X) θ D(Î(X), X) = µî(x) (6) Î(X) for all observed X, where µ is the Lagrangian multiplier on the agency s budget constraint. The first-order condition for examination intensity is instead that θd Î ( ln D ln I + d ln θ ( )) ln D d ln(d/i) ln I 1 = µ (7) for all observed X s, where the first term in parentheses reflects the marginal benefit of I associated with an increase in detected underreporting and the second term is the marginal benefit of increases in θ that arise from a marginal increase in I. In conjunction, (6) and (7) imply that ln D ln I = 1 (8) for all observed X s in an equilibrium where the tax agency employs a mixed strategy. This in turn implies that (7) reduces to θḓ I ln D ln I = µ and that the tax agency has no marginal incentive to increase I in order to increase the equilibrium penalty rate. This is precisely the property that was desired when assuming that θ depend on the ratio of D to I. 9

10 3 Equilibrium Description Using the results from the taxpayer and tax agency optimization problems, this section will now describes an equilibrium to the tax evasion game. Though I do not claim that the equilibrium is generically unique, it is in fact unique when conditioning upon the desirable property that taxpayers face a strictly positive, non-unity probability of audit even in the limit as self-reported incomes approach infinity. The strategy underlying this analysis is to first guess that a fully separating equilibrium exists, and then verify the conditions under which this is true. As it turns out, this equilibrium cannot exist if there exists a density of taxpayers with sufficiently low true incomes, in which case I will expand the structure of the guessed equilibrium to one in which lower income taxpayers pool at a common equilibrium self-reported income of $0 while higher income taxpayers separate. 3.1 Separating Equilibrium Let the solutions to the taxpayer s first-order condition (4) be defined as E (W ) for underreported income and X (W ) for self-reported income. A separating equilibrium requires that a strictly monotonic relationship between X and W, though I will posit the more restrictive but realistic restriction that dx dw de (0, 1] (or alternatively, dw [0, 1)). If these restrictions hold, then there exist functions that relate an observed X to a corresponding E and W. I denote these functions Ê(X) and Ŵ (X), and they are implicitly and jointly defined by Ê(X) = E (Ŵ (X)) and Ŵ (X) = X + Ê(X). In that case, the tax agency s perceived D function is defined by the relationship D(I, X) = δ(ê(x), Ŵ (X), I)Ê(X) (9) and D I δ > 0 holds if I > 0 and Ê(X) > 0. 10

11 (9), along with the tax agency s optimality condition (8), imply that ln δ(ê(x), Ŵ (X), Î(X)) ln I = 1 (10) for all X. This result and the log separable form of δ in (1) in turn imply that each separating taxpayer attains the same equilibrium detection rate δ defined by δ Ω (δ ) = Ψ I. (11) Furthermore, the tax agency s optimality condition (6), along with the θ function s assumed dependence on D/I, imply that each separating taxpayer attains the same equilibrium penalty rate θ defined by ( µ ) θ = θ, (12) τθ and also attains the same elasticity of θ with respect to D/I, where said elasticity is denoted ɛ and defined by ɛ = d ln θ ( µ τθ ) d ln(d/i). (13) Taxpayer Strategy The tax agency s first-order conditions are sufficient to determine taxpayers equilibrium underreporting strategy. The constancy of δ across all taxpayers implies (using (1) and (6)) that ln(δ /(1 δ )) = Ψ 0 + Ψ E ln E (W ) + Ψ W ln W + Ψ I ln(θ τδ E (W )/µ) for all W. Differentiating this identity with respect to W therefore reveals that in equilibrium taxpayers must underreport income such that E (W ) has constant elasticity with respect to W. In particular, the equilibrium taxpayer strategy is given by E (W ) = kw β, (14a) 11

12 where the shifter k and elasticity β are defined according to β = Ψ W Ψ E + Ψ I and (14b) ( ) δ ln = Ψ 1 δ 0 + (Ψ E + Ψ I ) ln k + Ψ I ln ( θ τδ µ ). (14c) β > 0 is guaranteed by Ψ E > 0, Ψ W < 0, and Ψ I > 0, whereas β < 1 is guaranteed by Ψ E + Ψ W + Ψ I > 0. Therefore taxpayers underreporting increases with income, but underreporting as a fraction of true income decreases with income, a joint equilibrium property that aligns well with most empirical analyses of tax compliance. 9 The constraint that taxpayers must self-report non-negative amounts of income reveals a necessary condition for the existence of the separating equilibrium, namely that the minimum true income, W min, satisfy the inequality W min k 1/(1 β). This condition cannot be satisfied if W min = 0; however, let us assume for the time being that W min is sufficiently large that the separating equilibrium exists, while the next subsection will discuss an alternative equilibrium when this necessary condition is unmet. W min k 1/(1 β) is also a sufficient (but unnecessary) condition for the monotonicity of X (W ), a requirement for the existence of a separating equilibrium. The necessary condition for monotonicity is W min (βk) 1/(1 β), which is necessarily satisfied if β (0, 1) and W min k 1/(1 β). The equilibrium taxpayer strategy also dictates the relationship between F (X), the tax agency s observed cumulative distribution of X, and G(W ), the cumulative distribution of true incomes: F (X (W )) = G(W ) Tax Agency Strategies With our discussion of the equilibrium taxpayer strategies concluded, we now turn to the tax agency s equilibrium strategies, ˆα(X) and Î(X). The agency s indifference condition (6) 9 See the Andreoni, Erard, and Feinstein (1998) survey. 12

13 and the equilibrium relationship in (9) implies that Î(X) = θ τδ µ Ê(X) (15a) where our prior solutions for taxpayers underreporting strategy enable implicit defining of the Ê(X) and Ŵ (X) functions according to Ê(X) = kŵ (X)β and (15b) Ŵ (X) = X + Ê(X). (15c) This formulation implies that that the tax agency s intensity of exam increases with X but at a decreasing rate. 10 Defining I (W ) to be the equilibrium audit intensity faced by a taxpayer with true income W, I (W ) = Î(X (W )) and therefore I (W ) = θ τδ µ E (W ). (16) In equilibrium, the intensity of audit grows in direct proportion to a taxpayer s (unobserved) total underreporting. Therefore equilibrium audit intensity increases with true income W, but at a decreasing rate. I now turn attention to the derivation of the tax agency s unique ˆα(X) strategy in a separating equilibrium. The solution strategy involves solving for the ˆα(X) function that induces taxpayers to use the E (W ) strategy previously recovered. It will therefore prove useful to rewrite the taxpayer first-order condition in a simplified manner that takes advantage of the equilibrium characteristics discussed so far (along with the equilibrium identity 10 dî(x) dx β < 1. = θ τδ µ d 2 Î(X) dx 2 d(ê(x)/ŵ (X)) dx < 0. βê(x)/ŵ (X) 1 βê(x)/ŵ = θ τδ µ (X) > 0 because Ê(X)/Ŵ (X) 1 for all X in a separating equilibrium with β (1 βê(x)/ŵ (X))2 d(ê(x)/ŵ (X)) dx < 0 follows from the additional observation that 13

14 ln δ = Ψ ln E E/Ψ I for all taxpayers 11 ): 1 ˆα(X)θ δ = d α(x) ˆ ˆα(X)θ δ dx Ê(X) ˆα(X) ( + ɛ 1 + Ψ ) E + ( Ψ E Ψ I Ψ I dê(x) dx ) (17) for all observed X. I assume that the true income distribution has a semi-infinite support that extends to positive infinity, which implies that the tax agency s observed reported income distribution also has a semi-infinite support that extends to positive infinity. 12 In order for lim X ˆα(X) to be finite, it must be the case that lim X dˆα(x) dx dˆα(x) = 0. Plugging dx dê(x) = 0 and = 0 dx (as is true in the limit) into (17), the separating equilibrium therefore has the property ˆα(X) X lim ˆα(X) = 1 X θ δ (1 + Ψ E Ψ I )(1 + ɛ ). (18) The equilibrium audit selection strategy must also decrease in self-reported income: ˆα(X) < 0. For contradiction, assume first that X = 0. In that case all taxpayers face the same equilibrium audit rate, and (17) reveals that all taxpayers have the same marginal benefit of evasion (the left side of the equality). All taxpayers also have identical costs associated with endogenous α (the first line of the right side, and equal to 0 by assumption) and θ (the second line). However, higher income taxpayers will face a larger marginal cost associated with endogenous δ (the third line) because of the fact that d2ê(x) dx 2 < 0. Therefore (17) cannot hold for all observed X because lower income taxpayers would have marginal benefits in excess of marginal costs while the reverse would hold for high income taxpayers. This contradiction suggests then that the equilibrium can only be sustained if dˆα(x) dx < 0 and 11 The additively separable expression for δ in (1) provides that ln δ ln E = Ψ E/(δΩ (δ)), which combined with the equilibrium expression for δ in (11) yields ln δ ln E = Ψ E/Ψ I for all taxpayers. 12 β (0, 1) guarantees the semi-infinite support for X. 14

15 d 2 ˆα(X) dx 2 > 0 so that lower income taxpayers face a relatively larger marginal cost associated with endogenous audit rates. 13 Finally, the game has sufficient structure such that the an explicit expression can be obtained for ˆα(X), though I leave the derivation of the formula for an appendix: ˆα(X) = 1 θ δ (1 + Ψ E Ψ I ) (1 + ɛ ) 1 + β 1 β ( )) Γ s, t (Ê(X) ) s ( )) t (Ê(X) exp t (Ê(X) (19a) where Γ(s, t) is the upper incomplete gamma function (Γ(s, t) = x s 1 exp( x)dx) and t its arguments are endogenously determined according to s = β ( ( 1 β Ψ ) ) E (1 + ɛ ) Ψ I and (19b) t(e) = ( 1 + Ψ E Ψ I ) (1 + ɛ ) k 1/β (1 β) E ( 1 β β ). (19c) Each taxpayer s equilibrium audit probability as a function of his true income is therefore given by α (W ) = ˆα(X (W )). (20) In summary, the unique taxpayer and tax agency strategies for a separating equilibrium have been described in full. They are written in terms of the endogenous parameters β, δ, θ, ɛ, k, s, and µ, which are jointly determined by solving the system of seven equations given by (11), (12), (13), (14b), (14c), (19b), and the tax agency budget constraint W min α (W )I (W )dg(w ) = B. (21) 13 The result can be proven more formally proven by also assuming (for contradiction) that dˆα(x) dx > 0. In that case ˆα(X) < ˆα( ) for all X, which in turn implies that 1 ˆα(X)θ δ > (1 + Ψ E /Ψ I )(1 + ɛ ) 1 for all X. Combining this result with (17) in turn implies that dˆα(x) dx be true. Therefore, it must be the case that dˆα(x) ˆα(X) asymptote to ˆα( ). dx ˆα(X)θ δ Ê(X) ˆα(X) + dê(x) dx < 0 for all X, which cannot < 0, which in turn requires that d2 ˆα(X) dx 2 > 0 in order that 15

16 3.2 Partially Pooling Equilibrium The fully separating equilibrium is attainable only if X (W ) 0 for all taxpayers, which in turn implies that the lower bound of the true income distribution must satisfy W min k 1/(1 β). While this condition depends on the endogenous parameters k and β, it is clear that it cannot be satisfied if the support of the W distribution extends down to 0. In contrast to games such as those in Reinganum and Wilde (1986) and Feinstein and Erard (1994), a separating equilibrium is unattainable even if taxpayers face no lower bound on self-reported income but the support of the W distribution still goes to 0. This arises from the fact that dx (W ) < 0 for W [0, (βk) 1/(1 β) ), dx (W ) = 0 for W = (βk) 1/(1 β), and dx (W ) dw dw dw > 0 for W > (βk) 1/(1 β), which jointly imply that all observed non-positive X s would correspond to two, not one, values of W. I therefore consider an attainable equilibrium in which taxpayers below a critical level of income, W, pool at the common self-reported income of X = 0, while taxpayers above W separate according the strategies derived in the previous subsection. This equilibrium is similar to the bright line equilibrium in Erard and Feinstein (1994). In this article, the authors similarly define a critical level of true income below which all taxpayers pool at a common self-reported income amount. This self-reported amount represents the tax agency s bright line, as the equilibrium requires that the agency promise to audit with high probability any taxpayer who reports below this level. However, the authors also note that the bright line equilibrium is eliminated by the divinity refinement of Banks and Sobel (1987). The bright line equilibrium with pooling at X = 0 for instance requires strictly positive audit probabilities for all off-equilibrium reports below 0. However, the set of agency actions that would make a given W willing to deviate to some X < 0 is greatest for the W = 0 taxpayer. 14 Therefore, the divinity refinement forces the government to expect that a hypothetical report of X < 0 is being made by a W = 0 taxpayer, which in turn leads to a low expected amount of detection and the government is not in fact willing 14 All of those taxpayers pooling at 0 gain the same reduction in self-reported tax liability by switching to a given X < 0, but the lower one s W, the lower the cost of facing the new audit probability. 16

17 to audit the taxpayer at the aforementioned sufficiently large probability. While Erard and Feinstein s bright line equilibrium is eliminated by the divinity refinement, this is not the case in the current analysis that distinguishes between the differences inherent in the underreporting of gross positive income versus the overreporting of gross losses. Why should the agency allow claims of gross positive income less than 0 to go unaudited when the agency knows that 0 is the minimum possible value? My inclusion of the X 0 constraint in the taxpayer s optimization problem is therefore interpretable as an endogenous outcome of the partially pooling equilibrium with a bright line at X = 0. This bright line equilibrium still requires sufficiently high off-equilibrium audit probabilities should any taxpayer report negative income, but given the virtual costlessness of proving that true income is at least 0, the agency would in fact be willing to make good on its commitment. Besides the intuitive appeal of this equilibrium type, its predictions also align closely with the empirical findings of recent articles such as Kleven et al. (2011) and Phillips (2013) which report significant pooling of noncompliant taxpayers at X = Taxpayers with W > W separate in this new equilibrium and follow strategies identical to those derived in the previous subsection. These taxpayers have self-reported incomes greater than or equal to a critical amount X (W ). The tax agency therefore employs strategies identical to those derived in the previous subsection for all X X (W ). However, the introduction of pooling at X = 0 introduces three additional endogenous parameters: the critical income level that separates pooling and separating taxpayers (W ); the probability that pooling taxpayers are audited (α); and the intensity of audit for pooling taxpayers (I). These three values are determined by three additional equilibrium conditions that I now discuss. First, the taxpayer who possesses true income of W must be indifferent between pooling and separating. Given that he is risk-neutral, this is the same as saying that his expected 15 Kleven et al. (2011) and Phillips (2013) examine the compliance behavior of taxpayers in Denmark and the U.S., respectively. In both instances, the self-report of X = 0 applies to income that is unmatched by third-party information reporting, the type of income that is the focus of the current game that is conditioned on taxpayer characteristics like third-party-reported income that are ex ante observable to the tax agency. 17

18 tax payment must be identical under either option: ( ) δ(w, W, I)W τx (W ) + τα (W )θ δ E (W ) = ταθ δ(w, W, I)W. (22) I Second, the tax agency must still be indifferent between auditing any observed X, including X = 0, in order for it to employ a mixed strategy. Unlike those X s that correspond to pooling taxpayers however, X = 0 poolers selected for audit may experience different detection amounts and penalties, such that the tax agency s first-order condition must take into account expected outcomes: W 0 ( ) δ(w, W, I)W τθ δ(w, W, I)W dg(w ) I G(W ) = µi, (23) the α analogue to first-order condition (6). Third, agency optimality requires that the return to a marginal dollar of intensity in X = 0 reports is identical to the return to a marginal dollar of intensity in all other reports. Once again however, the presence of many different true incomes among pooling taxpayers implies that this first-order condition must take into account the expected marginal return: ( W d τθ 0 ( δ(w,w,i)w I ) di ) δ(w, W, I)W dg(w ) G(W ) = µ, (24) the I analogue to the first-order condition (7). Finally, the introduction of pooling taxpayers changes the tax agency s budget constraint from (21) to α I G(W ) + W α (W )I (W )dg(w ) = B. (25) In summary then, the partially separating equilibrium results in a mass of taxpayers at X = 0 and a density of taxpayers at all X > X (W ). These results correspond remarkably well to empirical observations of taxpayer behavior. However, the final component of the 18

19 partially pooling equilibrium is that the tax agency commit to examining off-equilibrium income reports between 0 and X (W ) at a sufficiently large probability to discourage taxpayers from reporting those amounts. In reality, there is no gap in the self-reported income distribution just above X = 0 and the tax agency does not audit such reports with anywhere near a 100% audit probability. While unsatisfying, this result could be eliminated by allowing for errors in taxpayer optimization or for taxpayer honesty akin to that in Erard and Feinstein (1994). 16 In spite of this one peculiar prediction, it is a byproduct of the desirable and realistic prediction of pooling at X = 0. I opt to maintain the current game s structure in order to hone in on the primary effects of detection technology and strategic examination intensity on the tax evasion game. 4 Simulations In order to better understand the nature of the game s equilibrium, I now calibrate the model to match observations on U.S. taxpayers noncompliance and IRS administrative practices. The calibrated model also provides an opportunity to test how the equilibrium changes under counterfactuals such as changes in the detection technology. 4.1 Calibrated Base Line In order to calibrate the model I must place additional structure on the general detection and penalty rate functions from Section 2. I assume that Ω(δ) = ln ( δ 1 δ ) which in turn implies that δ is logistic in E, W, and I: δ(e, W, I) = exp( Ψ 0 )E Ψ E W Ψ W I Ψ I. (26) 16 Erard and Feinstein (1994) allows for the existence of honest taxpayers who self-report true income independent of financial incentives. In this case, an equilibrium audit schedule cannot jump due to the existence of honest taxpayers (who are only revealed to be honest upon audit) at all self-reported income amounts. If small but strictly positive reported incomes were subject to a 100% audit rate, then only honest taxpayers would in fact choose to self-report these small amounts, but in that case the agency would not expect to detect any evasion on these reports and in turn would deviate from the strategy of 100% auditing. 19

20 I also assume that the net penalty rate (θ 1) has constant elasticity with respect to D/I: θ ( ) ( ) γ1 D D = 1 + γ 0. (27) I I This functional form allows that θ approach 1 (i.e. no penalty is assessed) as the detected underreporting approaches 0. Finally, I assume that the distribution of true income is lognormal with log-scale location parameter λ and scale parameter σ The fully specified model has ten exogenous parameters: four from the detection technology function (Ψ 0, Ψ E, Ψ W, Ψ I ), two from the penalty function (γ 0, γ 1 ), two from the W distribution (λ and σ), the tax rate (τ), and the tax agency budget (B). Due to the assumption that all players objectives exhibit risk neutrality, the value of τ affects only the endogenous value of µ, the marginal revenue of tax agency funds. I assume that τ = 0.3 for calibration purposes. Calibrating the remaining nine parameters requires nine pieces of identifying information. I use an equilibrium detection rate (i.e. δ ) of 34%. 17 The equilibrium gross penalty multiplier (i.e. θ ) is The limiting audit rate (i.e. ˆα( )) is 13.9%. 19 The elasticity of underreporting with respect to true income (i.e. β) is The marginal examination revenue of tax agency funds (i.e. µ) is $ The maximum true income at which 17 Erard and Feinstein (2010) estimate that approximately one-third of Schedule C (sole proprietor) income is detected during IRS National Research Program examinations. 18 As previously mentioned, the statutory penalty rate is 20% for substantial understatements and 75% for civil fraud. The 50% penalty used in the calibration is intended to split the difference between the two. 19 According to the IRS Data Books, 881,638 examinations were conducted on 47,457,224 Schedule C tax returns, for an average annual audit rate of 1.86%. Also based on data from these IRS Data Books, 167,630 out of 871,883 examinations resulted in no change to tax liability, implying a detectable noncompliance probability conditional on examination of 80.8%. Finally, Phillips (2013) estimates that 61% of the Schedule C taxpayers in the NRP sample have detectable noncompliance. Applying Bayes Rule to these values, the annual probability of audit conditional on having detectable noncompliance is 2.5%. 13.9% is the cumulative probability of audit over six years (13.9% 1 (1 2.5%) 6 ) and is used because U.S. law allows for an audit to examine the previous six years of tax returns. 20 Phillips (2013) estimates that U.S. taxpayers have an elasticity of detected unreported income with respect to detected total income of This parameter, denoted β to signify its reliance on detected outcomes, is related to the elasticity parameter β according to β = (δ β)/(δ β + 1 β) when E (W ) = W. Solving this expression for β, then plugging in the observed value of β yields the calibration s employed β. 21 I calibrate the model based on external estimates of µ rather than external estimates of B due to the fact that the game models strategic interaction for a given class of taxpayers. Theory predicts that the tax agency should strategically equate the marginal return to agency resources across all classes, but this result does not directly imply anything about the level resources allocated to each class. The IRS Oversight Board s FY2014 IRS Budget Recommendation Special Recommendation provides estimated return-on-investment 20

21 taxpayers pool (i.e. W ) is $7, The fraction of taxpayers who pool (i.e. G(W )) is 40%. 23 The net misreporting percentage (i.e. ( W 0 W dg(w ) + W E (W )dg(w ))/( 0 W dg(w ))) is 53.9%. 24 The final identifying condition is Ψ E + Ψ W = The calibration exercise yields paramater values of Ψ 0 = 5.45, Ψ E = 13.16, Ψ W = 13.16, Ψ I = 1.52, γ 0 = , γ 1 = 1.37, λ = 9.46, σ = 2.19, and a tax agency budget that is 0.063% of aggregate income. Figure 1 plots the tax agency s equilibrium audit strategies as a function of self-reported income. The audit rate for pooling taxpayers is 19.4% while the audit rates for separating taxpayers ranges from 19.0% for the critical X (W ) = $630 income report down to 13.9% in the limit as self-reported income approaches infinity. 26 The minimal variation in audit rates aligns well with the empirical observation of similar audit rates across different classes of income. It also stands in stark contrast to other game theoretic models of tax evasion in which audit rates for low-income taxpayers are significantly higher than those for high-income taxpayers. In these games changes in audit rates alone must disincentivize noncompliance among high income taxpayers, whereas the current game allows for changes in examination intensity and endogenous detection rates to shoulder some of the burden. Other researchers have interpreted the lack of real-world variation in audit rates as evidence of non-strategic tax agency behavior. The current game s results instead suggest that the discrepency between real-world and theoretical audit strategies is instead the result of a misspecified theory. The current game shows that (roughly) constant audit rates across incomes is in fact consistent with strategic tax agency behavior. (ROI) to various IRS activities. These estimates are typically far greater than one but less than 20. The use of µ = 20 is intend to provide a conservative estimate of the level of tax agency funding. 22 Phillips (2013) estimates that the median U.S. taxpayer who breaks free of the X = 0 underreporting strategy has detected income of $2,500. Dividing by δ yields the taxpayer s actual income amount. 23 Per Phillips (2013). 24 This is the IRS s estimate of net misreporting percentage for income amounts subject to little or no information reporting in TY01. See 25 This restriction implies that underreporting depends on audit intensity and the fraction of income underreported. This implies that all pooling taxpayers will therefore have the same equilibrium detection rate, and furthermore, this detection rate must be equal to the equilibrium detection rate among separating taxpayers. The restriction is also necessary to guarantee that the tax agency has no marginal incentive to raise poolers audit intensity in order to raise the expected penalty rate among poolers. 26 On an annualized basis (assuming α corresponds to the cumulative 6-year probability of audit, the annual audit rate is 1 (1 α) 1/6 ), these probabilities are 3.53%, 3.45%, and 2.46%, respectively. 21

22 Figure 2 plots the tax agency s equilibrium intensity strategy as a function of self-reported income. In contrast to audit probabilities, audit intensity increases with self-reported income (and dramatically so), though the increase occurs at a decreasing rate. Pooling taxpayers are allocated $24 in examination resources (i.e. I) whereas the lowest separating report (X = $630) is examined with $51 in resources. Figure 3 plots taxpayers underreporting as a fraction of true income (E (W )/W ). (The x-axis is in log-scale in order to better illustrate the outcomes for higher income taxpayers. Taxpayers with income $1,000 also pool at 100% underreporting.) The figure shows those low incomes that pool at X = 0 (i.e. E (W )/W = 1), whereas the share of income underreported decreases with income among separating taxpayers. In the limit, the share of income underreported goes to 0, though income underreporting is large even for incomes that we would typically consider high. For example, taxpayers with $1 million in true income are estimated to underreport 55% of income. However, only 34% (i.e. δ ) of this taxpayer s underreporting is detected, implying that this taxpayer will appear to have underreported only 29% of his detected true income (the sum of the taxpayer s self-reported income ($449,476) and the detected portion (34%) of his unreported income ($550,524)). Figure 4 plots taxpayers effective average tax rates, the expected tax payments inclusive of penalties divided by true income. No matter taxpayers true income, the average effective tax rate is less than the statutory tax rate of 30%. This must obviously be the case since taxpayer s have the option to self-report all income and face an effective tax rate of 30%. If they underreport, it is only because the expected tax liability is lower than it would be under an honest self-report. The most interesting result is that the effective tax rates increase with true income. Therefore, the strategic interaction in the tax evasion leads to a progressive bias relative to the flat statutory rate. For instance, the critical W = $7, 353 taxpayer pays an average rate of 5.2%, a W = $100, 000 taxpayer pays 10.7%, and a W = $1 million taxpayer pays 14.7%. The progressive bias result stands in stark contrast to most other tax evasion game anal- 22

23 yses. Progressivity arises from the (realistic) equilibrium outcome that self-reported income increases with true income, but at a decreasing rate. Among separating taxpayers, the share of true income self-reported increases and actually approaches 100% in the limit as true income approaches infinity. Therefore this limiting taxpayer has an effective marginal tax rate equal to the statutory rate, and the average effective tax rate also approaches the statutory rate of 30%. It is worth emphasizing however that the progressive bias applies within a given class of taxpayers. A given class of taxpayer has the same ex ante observable characteristics, with third-party-reported (i.e. matched ) income being one such characteristic. Taxpayers generally have to pay the statutory tax rate on matched income. To the extent that high total income taxpayers possess a larger share of unmatched income (see Johns and Slemrod (2010) and Phillips (2013)), the across-class tax bias may still be regressive. 4.2 Improvements in Detection Technology Several interesting phenomena also reveal themselves by considering exogenous changes in the detection technology. Such changes seem particularly relevant to consider in the context of rapid transitions in information and other big data technologies that will fundamentally affect tax agencies ability to detect taxpayers noncompliance. I consider three different technology improvement scenarios: an increase in Ψ 0, an increase in Ψ I, and an increase in Ψ W with compensating decrease in Ψ E that maintains the Ψ E + Ψ W = 0 equality. Referring back to the detection technology expression in (1), a ceteris paribus increase in Ψ 0 increases Ω(δ) (and therefore δ) without directly affecting the function s responsiveness to its inputs E, W, and I. A ceteris paribus increase in Ψ I both increases Ω(δ) but also increases the degree to which detection responds to increases in audit intensity. A ceteris paribus increase in Ψ W (recall that Ψ W is negative) and compensating decrease in Ψ E increases Ω(δ) and changes the degree to which detection increases in response to increases in the portion of income underreported. For this counterfactual exercise, I calculate the change in each parameter that would lead 23

24 to a 50% increase in the aggregate detection rate (total detected underreporting divided by total actual underreporting) in the absence of changes in taxpayer and tax agency strategies. As a reminder, aggregate detection is 34% in the base case (and in fact, equals 34% for each taxpayer individually), so the counterfactuals would raise aggregate detectin to 51% in the absence of changes in strategy. Referring back to the analytical results however, it is clear that significant behavioral responses will occur in the course of establishing a new equilibrium. For instance, note that (11) expresses taxpayers common equilibrium detection rate δ as a function of Ψ I solely; therefore, equilibrium detection rates cannot possibly stay as high as 51% in the scenarios where Ψ I remains unchanged. Instead, (11) reveals that equilibrium detection rates will have to drop all the way back to the base case s 34%, a phenomenon that can occur only via endogenous increases in examination intensity and/or decreases in taxpayers underreporting. Figure 5 plots taxpayers equilibrium audit probabilities under each scenario relative to those under the base case, while Figure 6 plots the relative intensity of audit. The tax agency s budget does not change under the alternative scenarios, so increases in audit probability must be roughly offset by decreases in audit intensity (and vice versa). Roughly speaking, the increase in Ψ 0 leaves equilibrium probabilities and intensities unchanged. In contrast, the increase in Ψ I generally results in a shift towards fewer, but higher intensity audits, while the increase in Ψ W (and corresponding decrease in Ψ E ) results in a shift towards more, but lower intensity audits. Changes in Ψ 0 do not affect the detection technology s responsiveness to its inputs, so the quasi-stability of equilibrium probabilities and intensities relative to base is unsurprising. The increase in Ψ I instead raises δ s responsiveness to audit intensity, thus raising the marginal benefits to audit intensity from the tax agency perspective. The agency s response is therefore to allocate more resources to fewer examinations, which is of course offset by taxpayer responses to the higher resources. The net effect is nonetheless an increase in examination intensities. The increase in Ψ W instead raises δ s responsiveness to increases in the fraction of income underreported, which in turn raises the 24