Strategic Interactions Between Tax and Statutory Auditors and Different Information Regimes: Implications for Tax Audit Efficiency

We examine whether tax audits become more efficient if tax auditors have access to audited financial statements and information about statutory audit adjustments. We extend the standard tax compliance game by a statutory auditor to analyze the strategic interactions among a firm issuing financial and tax reports, a statutory auditor, and a tax auditor. For medium-powered tax auditor incentives and firms that place a high weight on book income, we show that granting the tax auditor access to information on statutory audit adjustments increases firms’ tax compliance, raises tax revenues, and decreases tax audit frequency. Thus more information sharing between statutory and tax auditors could be an important but so far overlooked policy instrument to combat tax evasion and increase tax audit efficiency. However, we also highlight the limitations of this approach. The efficiency effect of information sharing turns out to be ambiguous in many constellations and depends on the strength of tax auditor incentives and the weight that firms place on book income.


Introduction
In many countries, tax administration budgets have declined significantly in recent years, and tax administrations, therefore, reduced their workforce. For example, the United States Internal Revenue Service reduced the number of its employees from 94,711 in 201094,711 in to 73,518 in 201894,711 in (IRS, 2019. Similarly, the number of employees at the U.K. tax authority, the HMRC, fell from 91,167 in 200591,167 in to 61,370 in 201491,167 in (Slemrod, 2016. This raises the importance of improving the tax audit selection process to increase tax revenues per tax audit costs, i.e., tax audit efficiency. Using a gametheoretical model, this paper investigates whether giving tax auditors access to audited financial statements and information about statutory audit adjustments increase the efficiency of the tax audit regime.
The idea of this paper is based on the observation that financial statements are often subject to two kinds of audit: a statutory audit that is conducted by an independent private auditor and, usually with a time lag, a tax audit that is conducted by the country's tax administration. This observation raises the question of whether information sharing between statutory and tax auditors could be used to increase tax audit efficiency, i.e., to simultaneously increase tax revenues and reduce tax audit frequency.
In general, values from financial accounting are positively correlated with values in tax accounting, meaning that financial accounting information could provide an informative signal for tax auditors (Mills and Sansing, 2000). However, at first sight, it might be surprising why providing access to statutory audit adjustments could provide an additional informative signal for tax auditors because these adjustments do not contain any direct information regarding tax values. The statutory audits of financial accounting statements are conducted by independent private auditors, certified public accountants. Moreover, statutory auditors are typically perceived as conservative, i.e., they are primarily interested in preventing upward earnings management but not in preventing downward earnings management, which is what firms might do to save taxes. The statutory audit adjustments, however, reveal information on whether a firm has engaged in upward earnings management of book income. As we will show in this paper, the strategic incentives to conduct upward earnings management of book income depend to some extent on a firm's taxable income. Thus the information about statutory audit adjustments provides an additional informative signal to the tax auditor with respect to the correct tax treatment.
Tax administrations seem to have recognized the potential benefits of financial statements and statutory audit information for tax audits. Firms in many countries are obliged to file their financial accounting statements together with their tax returns, e.g., in Australia, Canada, and the United Kingdom, and in some countries, firms must also file their statutory audit report with the tax administration, e.g., in Belgium, Germany, and the United Kingdom (see Table 5, appendix B). To the best of our knowledge, however, the filed statutory audit reports do not currently contain any detailed information on the audit adjustments made by the statutory auditor. Therefore, our analysis goes one step further: We inform legislators as to whether it would be efficient to also provide the tax auditors with information on documented adjustments made by the statutory auditor. Thus we contribute to the ongoing discussion of how to improve the effectiveness and efficiency of tax administrations (OECD, 2017).
Despite its potential to increase tax compliance, the usage of information from financial accounting statements and statutory audit adjustments might also trigger opposing effects. Some firms may attempt to decrease detection risk and manipulate or avoid the signal, for example, by understating not only taxable income but also financial accounting income such that no book-tax-difference arises (conforming tax avoidance).
Other firms may have a further increased incentive to misreport only taxable income (non-conforming tax avoidance) to align book and tax income. The overall effect of using financial accounting information and statutory audit adjustments in the tax audit selection process is therefore unclear. The aim of our study is to show under which circumstances information about the statutory audit and financial statements increases tax audit efficiency.
Our analysis proceeds in three steps. In a first step, we assume that no information beyond the reported tax return is available. This is the standard inspection game between taxpayers and tax auditors (Graetz et al., 1986), which we label the reducedinformation regime. Firms (the taxpayers) face an asset valuation issue 1 and decide whether to report the correct tax value or evade taxes. Tax auditors decide, based on the reported tax returns, whether to conduct a tax audit.
In a second step, we add observable financial accounting statements. In this intermediateinformation regime, firms need to prepare a report not only for tax but also for financial accounting purposes. For the sake of simplicity, the asset's tax value can be either low or high; similarly, the asset's book value can be either low or high. To reflect existing similarities between financial accounting principles and tax law, we assume a positive correlation between book and tax values (Mills and Sansing, 2000). Thus, book values provide a (noisy) signal regarding a firm's correct tax value. We assume that firms have a preference for high book income and low tax income. Accordingly, firms will have an incentive to engage in upward (downward) management of book (tax) income.
As with tax returns, financial statements are usually subject to an audit. Thus statutory auditors determine, based on the reported book value, their effort in the statutory audit of the firm's asset valuation. In line with prior research, we assume a conservative statutory auditor who prefers conservative (income-decreasing) accounting choices to reduce his or her litigation and reputation risk (e.g., Francis and Krishnan, 1999;Kim et al., 2003;Cahan and Zhang, 2006). Thus the statutory auditor challenges firms' upward, but not downward, book income management. In the subsequent tax audit, the tax auditor observes the financial statements in the form in which they were prepared after the statutory audit. Therefore, the tax auditor has access to financial statement information but no information about statutory audit adjustments.
In a third step, we model a high-information regime by assuming that the tax auditor has access to audited financial statements and, in addition, information on required statutory audit adjustments.
Firms and both auditors alter their reporting and audit decisions with variations in the information regime. In each information regime, we determine the mixed-strategy equilibria, where the two auditors employ a probabilistic audit strategy and certain taxpayers randomize their reporting behavior. 2 In these equilibria, we compare the effects of the different information regimes on our measures of tax audit efficiency: firms' tax evasion, tax revenues and tax audit frequency. Our findings reveal that the effect of information on tax audit efficiency depends on the tax auditors' incentives and the importance that firms ascribe to book income.
For countries where tax auditors have medium-powered incentives and firms place a relatively high weight on book income, our analysis shows an unambiguously positive effect of information sharing between statutory and tax auditors on tax audit efficiency.
In this case, the additional information can induce higher tax revenues and a lower tax audit frequency. Information about statutory audit adjustments is particularly useful under these circumstances, as it allows the tax auditor to decrease the audit probability to zero for specific reports without any negative effect on tax compliance. Thus sharing information between statutory and tax auditors has the potential to significantly increase tax audit efficiency.
However, in line with prior research (Sansing, 1993) our results also demonstrate that giving more information to tax auditors is not always helpful to enhance tax compliance. In particular, we show that information sharing between the auditors is not effective for all firms. If tax auditors have medium-powered incentives, but firms place a much higher weight on tax than on book income, tax revenues decrease with information sharing. Because the tax audit frequency also decreases with information sharing, the overall efficiency effect is ambiguous in this case.
The reason for the different effects of information about statutory audit adjustments is as follows: In case of a relatively high weight on book income, taxpayers who conduct upward earnings management of book income report their tax income truthfully because this reduces the risk that the statutory auditor corrects earnings management.
Thus, information on detected earnings management signals tax honesty and the tax auditor can reduce the audit probability for these reports without losing tax revenues.
By contrast, in case of a relatively low weight on book income, the reduced tax audit probability for reports with detected earnings management comes at the cost of lost tax revenues. In this case, taxpayers combine upward earnings management with tax evasion because the benefit of the lower tax audit probability for reports with detected upward earnings management outweighs the disadvantage that the statutory auditor potentially corrects earnings management.
In countries where tax auditors have high-powered incentives, our model predicts that tax auditors do not decrease their audit frequency due to the additional information.
Increasing the information level of the tax auditors induces some firms to engage in more tax evasion, while other firms reduce their evasion. However, in equilibrium, average tax evasion remains at the same level. Thus the provision of the additional information does not affect tax audit efficiency. In countries with low-powered tax auditor incentives, we find that tax revenues increase if tax auditors have access to financial statements and statutory auditor reports. However, this comes at the cost of increased tax audit frequency. Thus the overall efficiency effect is ambiguous in this case.
Most previous tax compliance research explains tax evasion as taxpayers maximizing their expected utility given the detection probability and the size of the penalty (Allingham and Sandmo, 1972). However, this fails to recognize that the detection probability is the result of a game between taxpayers and the government and thus endogenous. Graetz et al. (1986) and Reinganum and Wilde (1986) were the first to study tax com-pliance as a game in which the revenue agency could not credibly commit to an audit strategy. Reinganum and Wilde (1988) introduce taxpayer uncertainty regarding the government's audit costs, and Beck and Jung (1989) consider tax liability uncertainty.
In Beck et al. (1996), taxpayers are allowed to purchase tax advice to resolve all uncertainty. Erard and Feinstein (1994) modify the approach of Reinganum and Wilde (1986) to consider nonstrategic taxpayers. Rhoades (1999) analyzes the impact of component reporting on the tax compliance game. Bayer (2006) considers taxpayers' concealment costs and demonstrates that higher tax rates lead to higher tax evasion and higher concealment and audit costs. Other studies analyze the effect of a signal regarding the taxpayer's income on the audit process. Sansing (1993) examines a signal regarding the correct tax treatment of a loss. He demonstrates that this signal can result in an increased amount of tax evasion while tax revenues (before audit costs) remain unaffected. Beck et al. (2000) study the effect of voluntary disclosures regarding an uncertain tax benefit to avoid a substantial underpayment penalty if the tax return is audited. Related to the study of Beck et al. (2000), De Simone et al. (2013 investigate the benefits of "Enhanced Relationship Tax Compliance Programs". Niggemann (2018) finds that book-tax conformity restricts misreporting positive book-tax differences but may impair the accuracy of financial statements.
The two studies most related to our paper are Mills and Sansing (2000) and Mills et al. (2010). In Mills and Sansing (2000), the tax auditor observes the (correct) financial statement valuation, which results in a higher audit probability for positive book-tax differences. Mills et al. (2010) investigate the effects of disclosed uncertain tax positions according to FIN 48. Contradicting public opinion, they demonstrate that some taxpayers will actually benefit from mandatory disclosure requirements. Empirical research has confirmed that public financial accounting information can be useful for tax auditors to infer the tax aggressiveness of companies (e.g. Lisowsky et al. (2013), Bozanic et al. (2017)).
We contribute to this research by adding the interaction between strategic statutory auditors and tax auditors while allowing taxpayers to misreport both financial and tax reporting positions. The strategic interaction between statutory auditors, tax auditors, and taxpayers reveals that information sharing between statutory and tax auditors has the potential to increase tax audit efficiency. In times of decreasing resources for many tax administrations (Nessa et al., 2019), this is an important policy result. However, our model also shows that firms and auditors respond to the changing information environments and that the desired positive effects may depend on the specific countries' institutional environments that determine, for example, the incentives of statutory and tax auditors. Thus the optimal tax policy will vary across countries, which might explain the large empirical variation that we observe in the obligations to file financial statements and statutory audit reports with the tax administration (Table 5, appendix B).
The remainder of this paper is organized as follows: In the second section, we present the model. In the third section, we discuss some important aspects of the equilibrium analysis of the game and introduce our efficiency measures. Sections 4, 5 and 6 investigate how the three different information environments affect the efficiency of the tax audit regime. The last section discusses the results and implications for future research.

The model
We consider a multi-stage game involving three players: the manager of a firm, a statutory auditor, and a tax auditor. The manager has to release a report about both the tax and the financial statement valuation of an issue. 3 Subsequently, the statutory auditor determines her audit effort and the tax auditor has to decide whether to audit 3 To avoid having an excessively complex model, we exclude agency conflicts from the analysis. The effect of agency problems (among shareholders or between shareholders and managers) on corporate tax avoidance is analyzed, for example, in Chen and Chu (2005) the tax report. All players are assumed to be risk neutral. A definition of all variables is presented in appendix A.
In the first stage of the game, the firm's manager (in the following denoted as the "taxpayer") faces an asset valuation issue. The asset's correct book value in the financial statements is b ∈ {b, b}. A similar valuation issue -with a potentially different outcome -arises for tax purposes, where the proper valuation is t ∈ {t,t}. We define tions. For both financial statement and tax issues, a low (high) valuation also implies a low (high) income. Therefore, we use the terms book (tax) valuation and book (tax) income interchangeably.
We refer to the true valuations (b,t) as the taxpayer's type. Accordingly, in the game, the taxpayer can be any of the following four types: (b,t) , b,t , (b,t) , b,t . At the beginning of the game, nature chooses a type for the taxpayer. Nature's move is observed only by the taxpayer such that the correct valuations (b,t) are the taxpayer's private information. 4 We assume the following common prior probability distribution over the taxpayer's type: 1. The ex ante probabilities Prob{t} and Prob{t} are 0.5.
The joint probabilities are Prob{b,t} = 0.5(1 − p), Prob{b,t} = 0.5p, Prob{b,t} = 0.5p and Prob{b,t} = 0.5(1 − p). Furthermore, Prob{b} = Prob{b} = 0.5. 4 We do not address taxpayer's uncertainty about the correct valuations in this paper. An analysis of tax uncertainty can be found, for example, in Graetz et al. (1986), Beck and Jung (1989), Beck et al. (1996) The parameter p is a measure of the conformity between tax and financial statement valuation. A strong conformity means p → 1, and a low conformity implies p → 0.5.
In this sense, the assumption that p > 0.5 ensures that the correlation between tax and financial statement valuation is positive.
At the beginning of the game, after observing the true values b and t, the taxpayer decides the valuationsb andt to be reported. We assume that the taxpayer makes simultaneous reports of book and tax income whereas in practice there is usually a time gap between the filing of annual statements and tax returns. Our simplifying assumption of simultaneous reports is based on the fact that taxpayers have to calculate the tax liability when preparing the annual accounts because otherwise they wouldn't be able to report the correct amount of deferred taxes. The taxpayer may bias the reported valuation for example because she is interested in a low asset value for tax assessment; we specify the taxpayer's objective function after we have introduced the other players of the game.
In the second stage of the game the statutory auditor conducts the annual financial statement audit after having observed the taxpayer's reported valuations b ,t . The statutory auditor determines the audit effort that can be either high (a S = 1) or low (a S = 0). The costs of the statutory audit are given by C S (a S = 1) = C S > 0 and C S (a S = 0) = 0. For the sake of simplicity, we assume a perfect audit technology for the statutory auditor with regard to the financial statement valuation. That means that for a S = 1, the statutory auditor with certainty detects an inaccurate financial statement valuation by the taxpayer. For a S = 0 the statutory auditor will not detect any misreporting. The statutory auditor's benefits come from reducing litigation and reputation risk. She benefits from detecting an overstatement of the asset value b = b although b = b but accepts conservative (income-decreasing) accounting choices because they have no impact on litigation and reputation risk. We assume that the benefit is proportional to the overstatement, λ b − b . Thus, the preferences where b (a S ) is the final financial statement valuation after the statutory auditor's decision a S , with b (0) =b and b (1) as in table 1.
We need to assume that λ −C S > 0 to ensure that high effort can be induced as equilibrium behavior. The statutory auditor's audit effort decision a S is not observable to the tax auditor who enters the game at its final stage.
The tax auditor is modeled in a very similar fashion to the statutory auditor. She also possesses a perfect audit technology with regard tot and has to decide whether to audit (a T = 1) the tax valuationt or not (a T = 0). The tax auditor benefits from detecting an understated tax valuation (t = t although t = t) but earns no benefit from detecting an overstatement. 5 Denoting the tax auditor's preference parameter for detecting an understatement by δ and the personal cost of auditing actions a T by C T (a T ), the tax auditor's objective function can be written as Here t (a T ) denotes the final tax valuation after the tax auditor's decision a T , with t (0) =t, t (1) = t. We further assume C T (0) = 0 and C T (1) = C T > 0.
Similar to the statutory auditor we impose the regularity condition δ −C T > 0. Figure   5 There are typically implicit incentives for assessing additional taxes during tax audits because the effectiveness of the tax audit staff is evaluated with respect to additional taxes 'earned' from tax audits (Alissa et al., 2014). In some countries, explicit incentives also exist (Kahn et al., 2001  We are now able to specify the taxpayer's objective function. Assuming that the taxpayer is interested in a low asset value for tax assessment and in a high asset value in the financial statements, a taxpayer of type (b,t) maximizes the following function F T and F B denote disutility from penalties due when an incorrect valuation is detected either in the tax (F T ) or in the financial statements (F B ), respectively. They are defined by The parameters ω and γ are positive weights indicating the taxpayer's preferences for financial statement and tax valuation. In the following, we consider the case of γ > ω, implying that the taxpayer's preferences are more sensitive to the tax than to the financial statement valuation. This is typical for firms characterized by low capital market exposure, by managers not predominantly driven by financial reporting measures and by less important outside debt financing. Thus, taxpayers with γ > ω can be regarded as managers of mainly private firms. Our focus on firms that give at least a small prioritization of tax outcomes over book outcomes is based on the fact that only these firms have an incentive to conduct conforming tax avoidance. However, we will discuss the implications for firms that prioritize book over tax outcomes in section 7.
Analyzing the effect of different information environments on tax compliance, we con- sider three different regimes. Across the regimes, we vary the information available to the tax auditor when he or she has to select an audit strategy, see table 2.
We begin with a benchmark setting (reduced-information regime) in which neither financial statements nor statutory auditor are present. The game is only between the taxpayer and the tax auditor, who observes the tax report. Thus our benchmark is a standard tax compliance inspection game (Graetz et al., 1986). The second regime (the intermediate-information regime) simultaneously introduces the need for financial statement valuation and the statutory auditor. We assume that the statutory auditor observes both the report on financial statementsb and that on tax valuationt. The tax auditor, in contrast, observes the report on tax valuationt and the final financial statement valuation b after the statutory auditor's effort choice. In regime 3 (the highinformation regime), both auditors observe the same information as in regime 2, but the tax auditor can also observe whether the statutory auditor has corrected the taxpayer's financial statement valuation. Formally, this is equivalent to a setting where the tax auditor observes the tax reportt, the final valuation b and the financial statement reportb.

Preliminary analysis
The game described in the previous section is an inspection game with two auditors.
Inspection games only have pure-strategy equilibria for extremely low or high inspection costs. Thus the inspectors never or always audit, and the inspectee always or never reports falsely. We do not consider these equilibria because they are rarely descriptive of real-world audit and reporting behavior. Therefore, we concentrate on mixed-strategy equilibria, in which the two auditors employ a probabilistic audit strategy and some taxpayer types randomize their reporting behavior. In these equilibria, we compare the effects of the different information regimes on the efficiency measures of tax evasion, tax revenue and tax audit frequency. This section (1)

Equilibrium in the reduced-information regime
As a first step, we compute the equilibrium in the reduced information setting. In this case, the game reduces to a simple variant of the inspection game. Thus proposition 1 replicates equilibrium strategies well known in the literature.

Proposition 1 (Reduced information regime)
If there is no financial statement valuation, the equilibrium strategies of the taxpayer and tax auditor are as follows: • Taxpayer: Type t reports truthfully. Type t reports t with probability θ T := C T δ −C T as long as θ T ≤ 1. Type t always reports t for θ T > 1.
• Tax auditor: Report t is never verified. Report t is verified with probability α = γ γ+F T for θ T ≤ 1 and never for θ T > 1.

Proof: See appendix D.1
The term θ T = C T δ −C T can be interpreted as a reverse measure of the tax auditor's incentives. Low values of θ T represent low audit costs and high rewards for detecting underreporting, implying high-powered incentives. A combination of high audit costs and low rewards yields low-powered incentives for the tax auditor and a large θ T .
Proposition 1 states that the taxpayer's misreporting probability decreases with the tax auditor's incentives. By choosing a misreporting probability amounting to θ T , the taxpayers ensure that the tax auditor is indifferent between auditing and not auditing a reportt = t, and by choosing the tax audit probability α, the tax auditor makes the taxpayers indifferent between tax evasion and honest tax reporting. The unique mixedstrategy equilibrium vanishes if incentives become very low. In this case, the taxpayer always misreports, and the tax auditor never challenges the reported tax valuation.

Dominated strategies in regimes 2 and 3 and notation
We proceed with the intermediate-and the high-information regime. In these cases, financial statements and the statutory auditor enter the scene and things become more complicated. To simplify the following analysis, it is useful to rule out actions that will never be part of any equilibrium because they are dominated by other actions.
Lemma 1 In regimes 2 and 3, the following actions will never be played in equilibrium: • The tax auditor will never audit a tax report t.
• The statutory auditor will never audit a financial statement report b with high effort.
• A taxpayer of type b,t will always report truthfully.
• A taxpayer of type b,t will never report b,t.
• A taxpayer of type b,t will never report b,t and b,t.
Intuitively, the tax and statutory auditors will never audit reports that do not offer them any potential benefits. The taxpayer will never issue reports that imply a lower payoff, even absent any detection risk, than the payoff under truthful reporting. Table   3 displays the reporting strategies for the taxpayer reports that are not dominated (see  Table 4 contains the definitions of the tax and statutory auditor's audit probabilities. The tax auditor's audit probabilities depend on whether we consider regime 2 or regime 3. While in regime 2 the tax auditor can only observe the final financial statement valuation in addition to the tax report, in regime 3 the tax auditor can make the audit probabilities also conditional on the (possible) correction by the statutory auditor.
x 1 Statutory auditor audits report b with high effort given tax report t x 2 Statutory auditor audits report b with high effort given tax report t α Tax auditor audits t given report b,t (regimes 2 and 3) β Tax auditor audits t given report b,t (regime 2) β 1 Tax auditor audits t given uncorrected report b,t (regime 3) β 2 Tax auditor audits t given corrected report b,t (regime 3) In the following, we use superscripts "R1", "R2" and "R3" to identify the three different regimes. For example, α R2 denotes the tax audit probability for report b,t in regime 2.

Efficiency measures
The tax evasion probability (TE) is generally defined as the unconditional probability of a low tax valuation report by the high-tax-valuation types, i.e., The following lemma shows that different information regimes do not necessarily imply a different tax evasion probability.
Lemma 2 (Constant tax evasion property) 1. Suppose that an equilibrium is char- in the high-information regime 3. This implies that TE = 1 2 θ T independent of the information regime.
2. Suppose that one of the audit probabilities in regime 1, 2 or 3 is equal to zero.
Lemma 2 states that additional information does not affect tax evasion if all reports are subject to a positive tax audit probability. The reason for this result is that in equilibrium the tax auditor is indifferent between auditing and not auditing the report.
Indifference on the part of the tax auditor, however, requires that the ratio of false to correct reports is identical for each type of report and set such that the tax auditor expects on average "no profit". Therefore, as long as each report with low tax values is audited by the tax auditor with a positive probability, firms' average tax evasion must remain the same in all information regimes.
In addition to tax evasion, we analyze two other measures that characterize the tax audit efficiency. These measures are defined as follows: 1. Frequency of tax audits: The audit frequency can be measured by the ex ante probability of a tax audit. In the reduced-information regime, the tax audit frequency can simply be written as In the intermediate-and the high-information regime we define in regime 2 and in regime 3.

Lost tax revenue (LTR):
The probability of taxing a type-t taxpayer who reports t based on a reported tax valuation t can be interpreted as "lost tax revenue". LTR measures the fraction of tax evasion that is not detected by the tax auditor and can be seen as an inverse measure of tax revenue. A low value of LTR implies a high tax revenue, and vice versa. In the reduced-information regime we obtain In the intermediate-and the high-information regimes, we define the measure as and in regime 3.

Tax auditors with high-powered incentives: no information effect
In the following three sections, we will show how the changes in the measures of the tax audit efficiency implied by variations in the information regimes critically depend on the tax and statutory auditors' incentives to provide an audit. We differentiate between high-powered incentives, medium incentives and low incentives. In our model, the incentives for both auditors are represented (1) by the rewards for a successful audit (represented by λ for the statutory auditor and δ for the tax auditor) and (2) by the audit costs (C S and C T ). In addition to θ T = C T δ −C T , it is useful to define as a measure of the statutory auditor's incentives. Values of θ S,T close to zero are equivalent to high-powered incentives, increasing values imply medium and even lower incentives. The following analysis reveals that θ T and θ S determine the equilibrium audit and reporting decisions. As a consequence, the impact of different information regimes on the efficiency measures also depends on these variables. Our analysis concentrates on the tax auditor and, therefore, we focus on variations in θ T . We obtain different results for (1) high-powered incentives (0 < θ T ≤ θ T (θ S ), this sec-tion), (2) medium incentives (θ T (θ S ) < θ T ≤ 1, section 5) and (3) lower incentives (1 < θ T ≤ θ T (θ S ), section 6), where the upper and lower bounds on θ T depend on θ S .
We restrict θ S such that the statutory auditor exerts high effort (a S = 1) with positive probability for at least report b,t or b,t.
Our discussion starts with a setting in which both auditors have high-powered incentives. Appendices C.1.1 and C.2.1 show that for high-powered incentives the tax auditor uses the same audit probability for all reports in all information regimes -that With positive tax audit probabilities the tax evasion probability is the same in all three information regimes by lemma 2. Also the statutory auditor exerts high effort with equal probabilities Thus changes in the information regime have no effect on tax revenue and audit probability if tax and statutory auditors have strong incentives. The following proposition 2 states the result.
Proposition 2 (No information effect) There exist critical values θ T , θ S (θ T ) such that for 0 < θ S < θ S and 0 < θ T < θ T (θ S ), the following relations hold: Note that the change from information regime 2 to regime 3 has no consequences for the composition of the taxpayer's equilibrium reporting. For example, taxpayer type b,t will report b,t and b,t with positive probability in both regimes. However, the concrete reporting probabilities will not necessarily be the same in the two regimes. Firms with b and t increase their tax compliance but firms with b and t will reduce compliant reporting -see appendix C.1.1 and C.2.1 for the details. The differences stem from the enriched tax auditor strategy: In the intermediate-information regime, the tax auditor can condition his audit on low and high financial statement valuations. The high-information regime additionally allows for a distinction between a low financial statement valuation that is corrected or uncorrected by the statutory auditor.

Tax auditors with medium incentives: mixed information effects
What happens if the tax auditor's incentives become weaker? First, the tax auditor uses different audit strategies for different reports. Second, the taxpayer adjusts her equilibrium reporting with respect to both financial statements and tax valuation. Finally, these adjustments also affect the statutory auditor's audit effort choice.
Tax auditors have intermediate incentives if the preference parameter δ together with the personal audit cost C T result in θ T such that θ T ≤ θ T < 1. Due to the existence of different equilibria, we consider important financial statement valuation (γ < ω + F B , regime 3a) and less important financial statement valuation (γ > ω + F B , regime 3b). The reduced incentives for the tax auditor have the following effects in the two regimes: 6 (1) Compared to a situation with high-powered incentives, the probability of false reporting increases for all types except b,t.
(2) Type b,t stops reporting truthfully and type b,t starts to understate both valuations by reporting b,t with a positive probability.
Both types reduce the probability of reporting b,t.
(3) The change in the reporting profiles also induces changes in the audit probabilities. In both information regimes the statutory auditor audits reports b,t with high effort (x R2 1 = x R3 1 ) but stops auditing reports b,t with high effort (x R2 2 = x R3 2 = 0) because type b,t reduces the probability for report b,t more than type b,t. The tax auditor decreases the audit probability of report b,t in both information regimes (β R2 = β R3 1 ) while the audit probability for reports b,t remains the same as in the case with high auditor incentives (α R1 = α R2 = α R3 ).
Moreover, if information on statutory audit adjustments is available (regime 3) and upward earnings management is detected the tax auditor further decreases the audit probability for these reports (β R3 2 < β R3 1 ).
The effect of lower tax auditor incentives on the efficiency measures in the two information regimes depends on the relative importance of the financial statement valuation for the taxpayer: (1) Important financial statement valuation (γ < ω + F B , regime 3a): In regimes 2 and 3a, type b,t finds the report b,t more attractive than report b,t because the statutory auditor may downgrade b,t to b,t. Thus she sacrifices the chance of a low tax valuation to obtain a high financial statement valuation with certainty. As a consequence, the probabilities for a report b,t sent by types b,t and b,t do not change from regime 2 to regime 3a because the potential senders of report b,t do not change. However, the probability of an uncorrected report b,t with false tax values has to decrease in regime 3a. This is due to the observable correction activity of the statutory auditor: The tax auditor can better conclude on types b,t and b,t. Therefore, both type t -taxpayers decrease the probability of report b,t in regime 3a. Taken together, more information induces lower tax evasion. Moreover, a report b,t corrected by the statutory auditor can only stem from type b,t. Thus, the tax auditor can reduce the probability of auditing a corrected report b,t to zero without any negative consequences regarding tax revenues.
Intuitively, the high information content of the statutory auditor's observable corrections drives the result. In case of important financial statement valuation, corrected financial statements together with a reported low tax valuation are a perfect indicator for a truthful tax report. Therefore, the tax auditor will not audit these reports. However, uncorrected financial statements together with a reported low tax valuation also become a much better indicator for misreporting. The tax auditor audits these reports, and taxpayers with a high tax valuation reduce their probability of misreporting in equilibrium. In sum, providing information about the statutory auditors corrections to the tax auditor is efficient (compared to regime 2). The additional information will induce both higher tax revenue and lower audit frequency.
In the reduced-information regime 1, both tax evasion and tax audit frequency are higher than in regime 3a. However, tax audit frequency may dominate and induce lower lost tax revenue compared to regime 3a. This explains why regime 3a is not necessarily efficient with respect to the no information regime 1.
(2) Less important financial statement valuation (γ > ω + F B , regime 3b): In this case the taxpayer has relatively strong preferences for a low tax valuation. Therefore type b,t reports b,t and takes the risk of being corrected to b,t. This is the key difference from regimes 2 and 3a. The possibility of type b,t as a sender of report b,t dampens the reporting probabilities for a report b,t sent by types b,t and b,t. Moreover, in contrast to regime 3a, the tax auditor will also audit the corrected report b,t because a high tax valuation type may be the sender. Because all tax audit probabilities for reports t are positive, lemma 2 applies. Thus in contrast to regime 3a, tax evasion is not affected by providing access to the statutory audit report. However, tax revenues decrease because the report b,t of type b,t is corrected to b,t by the statutory auditor with positive probability and the taxpayer then benefits from the lower tax audit probability (compared to report b,t).
The following proposition states the effects on the efficiency measures when financial statement valuation is relatively unimportant to the taxpayer.
Proof: See appendix D.6 The lower information content of the statutory auditor's observable corrections is responsible for the difference between the efficiency result in proposition 3 and the negative tax revenue effect in proposition 4. Because of the less important financial statement valuation, the taxpayer does not report a truthful high tax valuation to ensure a favorable financial statement valuation. Therefore, a reported low tax valuation together with a corrected financial statement valuation will not perfectly indicate a truthful tax report. As a consequence, the tax auditor audits with a lower, but positive audit probability. Moreover, also a report with uncorrected low financial statement and low tax valuation is less informative. As a consequence, tax evasion is higher in equilibrium compared to regime 3a. Thus more information decreases tax revenue and audit frequency.
6 Low powered tax auditor incentives: positive tax revenue effect We now consider lower incentives for the tax auditor that are characterized by 1 < θ T < θ T . Compared to medium incentives, lower incentives have the following effects: 7 (1) They increase the probability of false reporting with respect to tax and financial statement valuation.
(2) We observe changes in the reporting profiles: Taxpayer type b,t stops truthful reporting in regimes 2 and 3b. In regime 3a truthful reporting is maintained due to the relative importance of financial statement valuation. Taxpayer type b,t will no longer issue the report b,t in regime 2 and 3b. Again, the profile in regime 3a does not change due to the important financial statement valuation.
(3) Changes in the reporting profiles entail changes in the equilibrium tax audit probabilities, while the statutory auditor's behavior remains the same as in the case with medium incentives for tax auditors. In regime 2, the tax auditor never audits any report b,t. Although this report can be filed by two types with high tax valuation, the possibility of a true or corrected report originated by type b,t is sufficient to justify the no audit decision.
This changes in both regimes 3a and 3b. In those cases, the tax auditor can observe the statutory auditor's correction of an overstated financial statement valuation. In the event of a correction, the reporting probabilities for b,t and the statutory auditor's correction probability x 1 are determined such that the tax auditor has no incentive to audit a corrected report. However, the tax auditor still prefers to audit an uncorrected report b,t.
Proposition 5 demonstrates how the tax system's efficiency measures vary with the changes in the information environment.
The reduced-information scenario (regime 1) induces the highest lost tax revenue and the lowest audit frequency because the tax auditor terminates his audit activities and the taxpayer always understates his tax valuation. Providing more information to the tax auditor, as in regimes 2 and 3, allows for more elaborated audit strategies. More information has two effects on tax revenues and audit frequency that work in the same directions: (1) The tax audit probabilities increase with more information and, thus, increase audit frequency and increase tax revenue.
(2) Taxpayers with high tax valuation may deviate to reporting strategies with low tax valuation but lower audit risk. Then, tax revenue and audit frequency decrease. With respect to tax revenues the first effect dominates the second. Tax revenue increases with better information independent of the relative importance of financial statement valuation. The same argument holds for the tax audit frequency in the case of important financial statement valuation. Because taxpayer types b,t and b,t report b,t with the same probability in regimes 2 and 3a, the second effect is not strong enough to outweigh increased audit probability and thus audit frequency increases. In regime 3b, taxpayer types b,t and b,t lower the probability of report b,t compared to regime 2. However, audit frequency still increases because of the higher audit probability for an uncorrected report b,t in regime 3b .

Conclusion
Is it possible to simultaneously increase tax revenues and reduce tax audit frequency simply by providing the tax auditor with access to statutory audit adjustments? To examine this question, we integrate two novel features in a tax compliance game: In the first modification of the standard inspection game, the tax auditor can observe the financial statements that are already audited by a statutory auditor. The statutory auditor is a strategic player who acts conservatively, i.e., he or she only corrects overstatements. Thus firms can misreport both their book and their tax income. In the second modification, we extend this model by assuming that the tax auditor additionally receives a report from the statutory auditor that contains information on audit adjustments regarding the firms' book values. The report is an informative signal about the true tax values of a firm because a firm's incentive to conduct upward management of book values depends on the respective tax value.
If tax auditors have medium-powered incentives and firms place high weight on book income, we indeed find that providing the tax auditor with access to statutory audit adjustments reduces tax evasion and tax audit frequency and increases tax revenues.
In this case, granting access to statutory audit adjustments clearly improves the efficiency of the tax audit regime. Information sharing between statutory and tax auditors could therefore be an important but so far overlooked policy instrument to combat tax evasion and increase the efficiency of tax audits. In particular, in countries such as Canada or the United States, where private firms signal that they place high weight on book income by voluntarily opting for an audit of their financial statements, the obligation for auditors to share their information with tax auditors could improve tax audit efficiency. However, our model also highlights the limitations of this approach. We find that the effect on tax audit efficiency strongly depends on the tax auditors' incentives and the importance that firms place on book income.
If tax auditors have medium-powered incentives but firms' place a low weight on book income, then tax revenues may even decrease because tax audit frequency decreases while average tax evasion is not affected by the additional information provision. Moreover, we obtain no effect on efficiency for high-powered tax auditor incentives. Highly motivated tax auditors do not decrease their audit frequency due to the additional information, whereas firms' change their evasion behavior. Increasing the information level of the tax auditors induces some firms to engage in more tax evasion, while other firms reduce their evasion. However, in equilibrium, average tax evasion remains at the same level. Moreover, because highly motivated tax auditors do not differentiate their audit probability with respect to book-tax differences, we do not observe that firms conduct downward management of book income. Finally, for low-powered tax auditor incentives, the audit frequency increases with increasing ad-ditional information. This also raises tax revenues but the overall efficiency effect is ambiguous.
Before giving tax auditors extended access to statutory audit information, the incentives of tax auditors in the respective country should therefore be taken into account.
Moreover, countries should differentiate with respect to the type of firm, as the efficiencyincreasing effect requires firms to place high weight on book income.
The focus of this study is to examine the implications of additional information on tax compliance and tax audit effectiveness. However, as we have seen, the strategic interaction between statutory auditors and tax auditors also affects firms' financial reporting. Interestingly, the observability of the statutory audit report affects both downward and upward earnings management of book income. Future research could investigate these implications in greater detail and might also consider different incentives of the statutory auditor, as we limit our analysis to conservative statutory auditors.
Moreover, our analysis concerns firms in which managers place greater weight on tax savings than on the disclosure of high book income. Although we also study variations in the importance of book income, we explicitly exclude from our analysis those firms that prioritize financial reporting over tax outcomes. Thus our analysis does not fully cover public firms, which prior research finds place greater emphasis on financial reporting outcomes than private firms (Mills, 1998;Mills and Newberry, 2001;Beuselinck et al., 2015;Lynch et al., 2018). Nevertheless, we can still transfer some of our considerations to those firms that prioritize financial reporting over tax outcomes.
First, for tax auditors with high-powered incentives, the result that the additional information does not affect tax audit efficiency holds independent of the prioritization of book and tax outcome. Second, firms that prioritize book over tax income will not engage in downward book income management. Instead, we expect that some firms would overstate tax income to conceal upward book income management (as reported in Erickson et al. (2004) and modeled in Niggemann (2018)). This should expand the settings in which additional information increases tax audit efficiency. Third, similar to our above analysis, we expect that additional information should, in general, increase tax audit frequency and tax revenues if tax auditors have low-powered incentives. In sum, when applied under the right conditions, information sharing between statutory and tax auditor has the potential to significantly improve tax audit efficiency.

A.2 Objective function parameters
Taxpayer ω financial statement valuation γ tax valuation F B penalty for incorrect financial stat. valuation F T penalty for incorrect tax valuation Statutory auditor λ return from detecting overstatement C S auditing cost } θ S = C S λ −C S Tax auditor δ return from detecting understatement C T auditing cost

A.3 Probabilities in mixed-strategy equilibria
Taxpayer Statutory auditor x 1 Statutory auditor audits report b with high effort given tax report t x 2 Statutory auditor audits report b with high effort given tax report t

Tax auditor
α Tax auditor audits t given report b,t (reg. 2 and 3) β Tax auditor audits t given report b,t (reg. 2) β 1 Tax auditor audits t given uncorrected report b,t (reg. 3) β 2 Tax auditor audits t given corrected report b,t (reg. 3)

B Legal obligations: Overview
Country Obligation to file financial accounting statements together with the tax return Obligation to file the audit report together with the tax return Yes Yes 1 Non-audited financial statements do not need to be filed. 2 The audit report is a publicly available information. 3 Survey respondents state that in practice the audit report is often enclosed along with the financial statements or that the revenue agency may request it as part of a tax audit.

C Equilibrium analysis in regimes 2 and 3: the details
This appendix contains the detailed equilibrium analysis for both information regimes.

C.1 Equilibrium analysis in regime 2
We begin the analysis with the derivation of the taxpayer's payoffs for those strategies that are not dominated. These are given in Table 6.

C.1.1 Equilibrium for high-powered tax auditor incentives
We first derive the equilibrium for low θ T and low θ S and present the restrictions for θ T and θ S under which this equilibrium will be valid.
First, high powered-incentives imply positive audit probabilities for all conditions under which an audit can take place. Therefore we conjecture x 1 ∈ (0, 1), x 2 ∈ (0, 1) as well as α ∈ (0, 1) and β ∈ (0, 1). Under this conjecture reporting the truthful valuations will be part of the equilibrium for the taxpayer. Then, the concrete audit probabilities for both auditors can be derived from the following equilibrium conditions: (1) The statutory auditor will set x 1 and x 2 such that ... Type b,t taxpayers are indifferent between reports b,t and b,t. Equating the payoffs and rearranging yields x 1 = ω ω+F B .
... Type b,t taxpayers are indifferent between reports b,t and b,t. This yields x 2 = x 1 .
(2) The tax auditor will set α and β such that ... Type b,t is indifferent between reporting b,t and b,t. Equating the respective payoffs yields α = γ γ+F T .
... Type b,t is indifferent between b,t and b,t yielding β = α. Moreover, α = β = γ γ+F T also make type b,t indifferent between reporting b,t and b,t.
Similar considerations yield restrictions on θ T and θ S defining the valid range of the equilibrium: 1. The first boundary condition ensures positive probabilities: 2. A taxpayer of type b,t has the strongest incentives to deviate from truthful reporting. The report b,t will be part of the equilibrium as long as ν 1 +ν 2 +ν 3 < 1. Inserting and reformulating yields the implicit condition Note that both conditions provide a valid space for θ S and θ T for all x 1 , p ∈ (0, 1). Condition 17 can be rearranged to and defines an upper bound for p. The RHS of (18) converges to 1 with θ S , θ T → 0.

C.1.2 Equilibrium under medium tax auditor incentives
We conjecture x 1 ∈ (0, 1), x 2 = 0, α ∈ (0, 1) and β ∈ (0, 1) as the equilibrium strategies for the statutory and tax auditors. The probabilities can be determined from the following: (1) The statutory auditor will set x 1 such that type b,t is indifferent between reporting b,t and b,t. This yields x 1 = ω ω+F B .
(2) The tax auditor will set α and β such that ... Type b,t being indifferent between b,t and b,t yields α = γ γ+F T .
• Type b,t is indifferent between b,t and b,t. This yields β = (γ−ω) γ+F T . These probabilities imply that type b,t is also indifferent between reporting b,t and b,t.
Given the conjectures above, the taxpayer will use an equilibrium reporting profile with φ 1 , φ 2 , η, ν 3 ∈ (0, 1) and ν 1 = 1 − ν 3 ∈ (0, 1). Equating the payoffs for the statutory and tax auditors for a S = 1, 0 and a T = 1, 0 and using Bayes' rule gives three equations that can be used to determine the four equilibrium reporting probabilities.
• The tax auditor being indifferent given b,t and prob{t|b = b,t = t} = • The tax auditor being indifferent given b,t and prob{t|b = b,t = t} = The solution to this system of three equations is The equilibrium derived above will be valid if the following conditions are fulfilled: 1. Our conjecture x 2 = 0 must be fulfilled.
The lower and upper bounds for φ 2 will be determined as follows: (1) ν 3 ≤ 1 yields . Combining this with condition φ 2 ≥ 0 we obtain: (2) φ 2 has to be determined such that (23) holds. Inserting and rearranging leads to . The minimum of the two expressions above determines the upper bound φ 2 .

C.1.3 Equilibrium under low tax auditor incentives
Under weaker incentives, the tax auditor will not audit report b,t or b,t. We conjecture that α > 0 and β = 0 because the taxpayer has stronger incentives for the former report. We also guess x 1 > 0 and x 2 = 0. The probabilities can be determined as follows: (1) The audit probability x 1 will make type b,t indifferent between reports b,t and b,t. Therefore, x 1 = ω ω+F B . (2) The tax audit probability α is set such that b,t is indifferent between b,t and b,t. We obtain α = ω γ+F T . (3) Given these considerations, the taxpayer's equilibrium profile will consist of φ 1 > 0, φ 2 = 1 − φ 1 , η > 0 and ν 3 = 1. We can determine the two probabilities from two equilibrium conditions: • The statutory auditor is indifferent between a S = 1 and a S = 0. Using prob{b|b = b} = pη pη+1−p+pφ 1 ) yields • Tax auditor indifference and prob{t|b = b,t = t} = pφ 1 Solving these two equations with respect to η and φ 1 yields (4) The following equilibrium conditions have to hold: 1. Our conjecture β = 0 has to be valid. Bayesian updating leads to 2. η > 0 and φ 1 > 0 yields θ S ≤ 1 θ T (1−x 1 ) . 3. η ≤ 1 yields the stricter condition θ S ≤ p 1−p+θ T (1−px 1 ) and φ 1 ≤ 1 ⇔ θ T ≤ p 1−p+θ S (1−x 1 ) gives the upper bound for θ T . If θ T or θ S exceed these upper bounds, the tax or statutory auditors will no longer audit because their incentives are too low.

C.2 Equilibrium analysis in the high-information regime
In Table 7 we present the taxpayer's payoffs for those strategies that are not dominated in information regime 3.
... Type b,t taxpayers are indifferent between reports b,t and b,t. This yields x 2 = x 1 .
(2) The tax auditor will set α and β 1 , β 2 such that ... Type b,t is indifferent between reporting b,t and b,t. Equating the respective payoffs yields α = γ γ+F T .
... Type b,t is also indifferent between b,t and b,t. This yields (3) The conjectures and considerations above have shown that the taxpayer will use an equilibrium strategy profile consisting of φ 1 > 0, η > 0, ν 1 > 0, ν 2 > 0 and ν 3 > 0. Using Bayes' rule, we obtain the following equations from the tax and the statutory auditors' indifference conditions: • The statutory auditor is indifferent given b,t. The updated probability prob{b|b = b,t = t} = • The statutory auditor being indifferent after report b,t and prob{b|b = b,t = t} = ν 1 (1−p) • Tax auditor indifference after b,t and probability update prob{t|b = b,t = t} = • The tax auditor is indifferent after observing b,t without corrections by the statutory auditor. The posterior probability prob{t|b = b,t = t, no correction} = (1−p)ν 3 (1−p)ν 3 +p(1−η) leads to • The tax auditor is indifferent between audit and not audit after report b,t is corrected from original report b to b. The posterior probability prob{t|b = b,t = t, correction} = (1−p)x 1 ν 2 (1−p)x 1 ν 2 +x 1 pη yields Equations (30) to (34) have the following solution: (4) The following conditions for θ T and θ S have to be met to ensure the existence of the equilibrium: 1. φ 1 ≤ 1 ⇔ θ T ≤ p 1−p . This also implies ν 1 ≥ 0.
3. The condition ν 1 + ν 2 + ν 3 ≤ 1 leads to the constraint C.2.2 Medium incentives for the tax auditor and unimportant financial statement valuation If the tax auditor's incentives become less high powered, θ T will increase beyond the upper bound outlined in section C.2.1. Then, the equilibrium depends on the relative magnitude of the preference parameters γ and ω. We first consider the case of γ > ω + F B . Since tax valuation is very important, a report of b,t will be less attractive for type b,t, and therefore, we conjecture that x 2 = 0 and x 1 , α, β 1 , β 2 ∈ (0, 1).
Given these conjectures, the equilibrium mixed strategy probabilities can be computed: (1) As above, the probability x 1 = ω ω+F B ensures that type b,t will be indifferent between reports b,t and b,t.
(2) The tax auditor's equilibrium audit probabilities will be set such that ... Type b,t is indifferent between reporting b,t and b,t. Equating the respective payoffs yields α = γ γ+F T .
... Type b,t is indifferent between reports b,t and b,t. This yields β 1 = (γ−ω) γ+F T . Note that α and β 1 imply that type b,t will also be indifferent between reports b,t and b,t.
The solution of system (37) is given by (4) The following conditions ensure that the strategies computed above constitute an equilibrium: 1. The conjecture x 2 = 0 must hold under the given equilibrium.

C.2.3 Low incentives for the tax auditor and unimportant financial statement valuation
We now consider the case θ T > 1 and γ > ω + F B . We conjecture α, β 1 and x 1 ∈ (0, 1) and that x 2 = β 2 = 0. We furthermore suppose that type b,t never reports truthfully and never reports b,t.
(2) In equilibrium, α and β 1 will be set such that (1) type b,t is indifferent between reports b,t and b,t and (2) type b,t is indifferent between reporting b,t and b,t. We obtain the two equations and the solution (3) We show that type b,t indeed prefers report b,t over b,t with α, β 1 and x 1 as given above: Inserting and rearranging (use ω( yields the condition γ > ω + F B which is fulfilled by the assumption above. (4) The taxpayer's reporting probabilities will be determined from and amount to (5) The equilibrium will be valid if the following conditions hold: • The tax auditor will not audit in case of a report b,t (where b comes from a correction by the statutory auditor) if δ prob{t|b = b,t = t, correction} −C T < 0.
• Reformulating ν 3 ≤ 1 yields θ S ≥ θ T −1 θ T (θ T +2x 1 −1−px 1 (1+θ T )) C.2.4 Medium and low incentives for the tax auditor and important financial statement valuation Important financial statement valuation for the taxpayer is represented by condition γ < ω + F B . Our equilibrium conjecture for the two auditors is as in section C.2.3, which means α, β 1 , x 1 ∈ (0, 1) and x 2 = β 2 = 0. The taxpayer's preferences induce an important difference: If the financial statement valuation is important for the taxpayer, he will have incentives to avoid a strict statutory audit. Therefore, due to x 2 = 0 type b,t will report b,t instead of b,t. We determine the audit and reporting probabilities as follows: (1) The statutory auditor's high effort probability x 1 is determined as above (see section C.2.1).
(2) The tax audit probability α will make type b,t indifferent between reporting b,t and b,t. We obtain α = γ γ+F T . The second probability β 1 is set such that type b,t will be indifferent between b,t and b,t. This implies β 1 = (γ−ω) γ+F T . Note that α and β 1 also induce indifference between b,t, b,t and b,t.
(3) We show that type b,t indeed prefers report b,t over b,t under the audit probabilities given above: Inserting yields This is exactly the condition given above.
(4) The conjectures above yield φ 1 , φ 2 , η, ν 3 , ν 1 = 1 − ν 3 > 0. From statutory and tax auditor indifference together with Bayes' rule we obtain the following three equations to compute the concrete probabilities: The solution to this system is (5) The following conditions ensure that the audit and reporting probabilities derived above are an equilibrium:

D.1 Proof of proposition 1
The tax auditor's equilibrium audit probability can be derived from the indifference of a type-t taxpayer between truthful reporting and report t. Reformulating yields With prob{t = t|t = t} = θ T 1+θ T a type-t taxpayer's reporting strategy θ T = C T δ −C T is obtained by rearranging

D.2 Proof of lemma 1
The first three statements follow directly from the preferences of the tax and statutory auditors. A taxpayer of type b,t will never report b,t because this report is dominated by the (truthful) report b,t. By the same argument, a taxpayer of type b,t will never report b,t. The report b,t is also dominated by truth reporting, because ω < γ.

D.3 Proof of lemma 2
Part 1: In case of the reduced-information setting (regime R1), we obtain We now look at regimes 2 and 3. In a mixed-strategy equilibrium the taxpayer's equilibrium reporting strategies have to ensure that the tax auditor is indifferent between auditing and not auditing a certain report. The indifference conditions for the tax auditor in the intermediate-information regime (regime 2) yield the following two equations: Indifference at report t, b (α R2 > 0) ⇔ δ prob{t|b = b,t = t} −C T = 0.
Part 2: Suppose, for example, β R2 2 = 0. In this case, the corresponding equation changes to δ prob{t|b = b,t = t} −C T < 0. Summing both sides of the equation corresponding to α R2 > 0 and the inequality yields TE < 1 2 θ T . The same argument holds if α R1 , α R2 or one the three audit probabilities in regime 3 are equal to zero.

D.4 Proof of proposition 2
The equilibrium analysis (see Appendix C) reveals that for θ S and θ T close to zero the equilibrium in each regime is characterized by 1. Regime 2: α R2 = β R2 = γ γ+F T .
Appendix C provides the reporting probabilities in the two regimes:

D.7 Proof of proposition 5
The claims LTR R1 > LTR R2 and TA R1 < TA R2 are obvious because the lost tax revenue measure LTR can never exceed 1 2 and the audit frequency TA will never be lower than zero. However, these are the values for LTR and TA in the reduced information regime for θ T > 1.