Accurate audits and honest audits

Abstract

A regulator hires an auditor and designs the audit to be performed on a firm. The anticipation of an accurate audit can induce a non-compliant firm to bribe the auditor. An inadequate salary from the regulator can induce the auditor to accept the bribe. Yet, we show that (1) as the budget available to the regulator increases, the optimal audit might become less accurate, and (2) as the regulator gets access to complex contractual forms to deal with the auditor, the odds of collusion can increase. Key to these results is the observation that a regulator might induce the firm to invest more in compliance by tolerating collusion rather than by preventing it.

1. INTRODUCTION

Audits are meant to spot transgressions of companies, tax payers and public officials. At times, the subject of an audit might prefer bribing the auditor rather than facing the risk of an unfavorable audit outcome. The recent Public Company Accounting Oversight Board (PCAOB)/KPMG scandal is an example. The Sarbanes–Oxley Act of 2002 gave the PCAOB the task of inspecting the operations of the major accounting firms operating in the United States, including KPMG. Between 2015 and 2017, executives at KPMG were illegally told by PCAOB employees which operations would be inspected; those operations were then reviewed to fix any deficiency before the next inspection.¹ In hindsight, we can say that the cost of engaging in bribery, for instance, the risk for PCAOB employees of losing their jobs, was too small compared to its benefit, that is, the gains for KPMG of tampering with the inspections. A higher salary at PCAOB and/or less threatening inspections for KPMG could have prevented the scandal.

In this article, we study the optimal design of audits when collusion between auditors and audited subjects is a concern. In the spirit of Laffont (2005) and Estache and Wren-Lewis (2009), we focus on situations in which the regulator’s resources are limited. These situations are common in high-income countries and are the norm in low-income countries, where limited fiscal efficiency often results in low salaries for public auditors.² We show that the optimal audit may be purposely inaccurate, so as to prevent collusion between the auditor and the audited subjects and we highlight non-trivial relationships between (1) the accuracy of the audit and the regulator’s budget as well as (2) the prevalence of collusion and the regulator’s access to “complex” contracting tools.

In our model, a regulator hires an auditor to inspect a firm’s conduct. If the audit deems the firm’s conduct harmful, the firm has to implement a costly remedy. The firm has two ways of avoiding the remedy: investing in making its conduct safe, and/or bribing the auditor to conceal any unfavorable evidence that the audit might eventually uncover. Accepting a bribe is a risk for the auditor: at times the regulator detects the bribe, and when this happens the auditor loses his salary.

The objective of the regulator is twofold: inducing the firm to invest ex-ante in making its conduct safe and screening the firm’s conduct ex-post to remedy any unsafe conduct. The regulator determines the accuracy of the audit, that is, the probability that the audit recommends a remedy, conditional on the firm’s conduct being harmful. Our main contribution amounts to showing a non-monotonic relation between the optimal accuracy of the audit and the budget of the regulator, proxied here by the (exogenously fixed) salary of the auditor. High and low salaries are optimally associated with accurate audits, while for intermediate salaries the optimal audit is imprecise. In other words, as the budget available to the regulator increases, the optimal audit might become less accurate.

The intuition is as follows. The salary determines the opportunity cost for the auditor of accepting a bribe. The accuracy of the audit determines the benefit for the firm of paying a bribe. The smaller the salary, the less accurate the audit has to be if the regulator wants to prevent collusion. We distinguish three scenarios corresponding, respectively, to high, intermediate, and low salary. For a high salary the regulator should request an accurate audit, as the size of the salary discourages collusion between firm and auditor. For an intermediate salary, the optimal audit is inaccurate, so as to ensure that the benefit of a bribe for the firm does not exceed the cost of a bribe for the auditor. In this case the audit is honest, if not fully accurate. For a low salary, it is optimal for the regulator to “give up” on collusion, that is, to require an accurate audit even though this audit will induce the firm to bribe the auditor.

Why would tolerating collusion ever be optimal? Consider the two objectives of the regulator: encouraging ex-ante investment and ensuring ex-post screening. Collusion disrupts screening. As the firm pays a bribe only in case of harmful conduct however, collusion need not disrupt investment. On the contrary, to deter collusion a regulator might have to limit the accuracy of the audit. An honest but inaccurate audit gives the firm little reason to invest, as harmful conduct might go unnoticed. A firm that resorts to bribery might in fact invest more than a firm that anticipates an honest, but loose, audit. The regulator thus faces a trade-off: by tolerating collusion the regulator can get stronger investment but poorer screening than by deterring collusion. We provide conditions under which the trade-off resolves in favor of tolerating collusion.

We move on to consider a regulator that chooses the auditor’s salary, as well as the accuracy of the audit. The case in which the same subject chooses both the auditor’s salary and the audit accuracy is arguably the most common in practice. The Securities and Exchange Commission, for instance, oversees at the same time standards, rules, and funding of the PCAOB.³ With an endogenous salary, the regulator’s budget is proxied by the salary’s shadow cost: a smaller budget is associated with a higher shadow cost. We show that the non-monotonic relation between the optimal accuracy and the budget of the regulator continues to hold. The optimal audit is accurate if the shadow cost is high or low, and it is inaccurate for intermediate shadow costs.

Next, we consider a regulator endowed with a richer set of contracts. We give the regulator the option to reward the auditor when the audit deems the firm’s conduct harmful. As it turns out, tolerating collusion becomes extremely convenient for the regulator. The intuition goes as follows. Suppose the regulator were to compensate the auditor only if the audit deems the firm’s conduct harmful. As the compensation gets larger, so does the amount needed to bribe the auditor. For a large (but not too large) bribe, the firm optimally chooses the second-best level of investment and then bribes the auditor whenever needed. The regulator can ensure this level of investment without even having to pay the auditor, as collusion ensures that the audit always deems the firm safe.

Tolerating collusion is so convenient with report-contingent compensation that the regulator never opts for an honest yet inaccurate audit. The optimal audit is accurate, and the non-monotonic relation between budget and audit accuracy we discussed above does not hold here. Interestingly, collusion is tolerated even for parameter values for which collusion is prevented in the baseline model. Hence our second result: as the set of contracts available to the regulator becomes larger, collusion can become more frequent.

We complement our results with a simple comparative statics exercise. We focus on the extent to which the auditor faces competition from other auditors, for example because different auditors are given overlapping jurisdictions. As our model has a single auditor, we mimic the effect of stronger competition by weakening the auditor’s bargaining power vis-à-vis the firm. We show that whatever weakens the relative bargaining strength of the auditor makes collusion less desirable for the regulator and calls for inaccurate, but honest, audits. In sum, our analysis highlights the interconnections of three policy levers: (1) the audit accuracy, (2) the auditor’s compensation, and (3) the extent to which the jurisdictions of auditors overlap.

In the next subsection, we discuss the related literature. In Section 2, we present the baseline model, which we analyze in Section 3. In Section 4, we modify the baseline model, first to make the salary endogenous, then to allow report-dependent compensation. In Section 5, we provide some concluding remarks. All proofs can be found in the  Appendix.

1.1 Related literature

Our work shares some features with the literature on collusion in principal-supervisor-agent models that stemmed from the seminal work of Tirole (1986).⁴ Within this literature, our work is most closely related to Asseyer (2020). Akin to us, Asseyer (2020) considers the problem of a regulator that designs the signal available to the auditor.⁵ Asseyer’s paper belongs more firmly than ours in this mechanism-design literature in which the principal offers an actual contract to the agent, whereas we focus on incomplete contracts. Another relevant difference is that the agent’s type is exogenous in Asseyer (2020). This assumption is typical in this literature, with some notable exceptions, such as Hiriart et al. (2010), Khalil et al. (2010), and Ortner and Chassang (2018). While we consider loose audits as a way to curb collusion, Ortner and Chassang (2018) focus on the provision of random, and privately known, compensation schemes to the auditor. Broadly speaking, while loose audits reduce the gains from collusion, random compensation makes it harder to share those gains. Our work is also closely related to Khalil et al. (2010) in that they show that collusion can provide the right incentives to invest ex ante as it acts as a penalty for bad quality. Tolerating collusion is desirable in Khalil et al. (2010) because the auditor can credibly threaten to frame the agent. This threat is absent in our model. In our setting, collusion is optimal as a result of a tension between ex-ante investment incentives and ex-post screening efficiency, whereas Khalil et al. (2010) only focus on moral hazard. How audit accuracy resolves this trade-off is a novel insight of our paper.⁶ Finally, Che et al. (2021) relate the choice of tolerating collusion with the accuracy of the audit. However, in their model, test accuracy is exogenous, and collusion is desirable as it allows the agent to correct a wrong signal.

Beyond the principal–supervisor–agent literature, Perez-Richet and Skreta (2022) and Pollrich and Wagner (2016) have shown that lowering the signal precision can reduce the applicant’s incentives to falsify a test. Perez-Richet and Skreta (2022) study the optimal test design by a regulator under the threat of direct falsification by an applicant. The relationship between the regulator and the applicant is not mediated by a (corruptible) auditor in their model. In Pollrich and Wagner (2016), collusion takes place between a certifier (which plays the role of the auditor in our model) and a sequence of applicants. In their model there is no benevolent regulator who designs the auditing policy and pays the certifier. Okat (2016) and Gonzalez Lira and Mobarak (2021) make the point that random tests might be more informative than regular ones as the latter make it easy for the agent to learn how to falsify a test. Gonzalez Lira and Mobarak (2021) also provide evidence that random audits of Chilean fish vendors are more effective in finding violations than regular ones. Our results provide a complementary rationale for this observation.⁷

Our regulator faces a simple information design problem, in the spirit of those described in the literature on Bayesian persuasion (Rayo and Segal 2010; Kamenica and Gentzkow 2011). Within this literature, we are mostly related to those models where the design of the test affects the distribution of the state being tested (Farragut and Rodina 2017; Saeedi and Shourideh 2020; Zapechelnyuk 2020; Bizzotto and Vigier 2024).

2 THE MODEL

We model a regulator (she) hiring an auditor (he) to evaluate a firm (it).

2.1 The setting

2.1.1 Firm’s conduct

We say that the firm (or, interchangeably, its conduct) is safe if $ω=0$ and harmful if $ω=1$⁠. A harmful firm causes a non-verifiable social loss of size $H$⁠.

2.1.2 Audit

Signal $s=1$ is hard evidence of harmful conduct, whereas $s=0$ amounts to a lack of such evidence. After observing the audit signal $s$⁠, the auditor publishes a report $r∈{0,1}$⁠. The auditor can conceal the evidence: upon observing $s=1$⁠, the auditor can report $r=1$ (i.e., reveal) or $r=0$ (i.e. conceal). The auditor cannot fabricate the evidence: upon observing $s=0$⁠, he must report $r=0$⁠.⁸ Following a report $r=1$⁠, the firm must implement a remedy. The remedy costs the firm all its profits, which amount to $k∈(0,1)$⁠, and yields a social benefit denoted $Δ$⁠. We assume that $H≥Δ>k$⁠.⁹

2.1.3 Collusion

Firm and auditor have the option to collude to tamper with the audit. After the firm observes its conduct $ω$⁠, but before signal $s$ realizes, with some probability $α$⁠, the auditor gets to demand a bribe amount $ba∈R$⁠; the firm can pay the requested amount or refuse to do so. With the remaining probability $1−α$⁠, the firm gets to offer a bribe amount $bf∈R$⁠, and the auditor can accept the amount or reject it. As discussed in the next subsection, the parameter $α$ effectively captures the auditor’s bargaining power vis-à-vis the auditor. If the bribe is paid, collusion occurs.¹⁰ Whenever the bribe is paid, the auditor publishes $r=0$⁠. If a bribe is not paid, i.e., if collusion does not occur, then the auditor publishes $r=s$⁠. Collusion is detected with some probability $ϵ∈(0,1)$⁠. If collusion is detected, the firm must implement the remedy, and the auditor forgoes his salary, $t∈R+$⁠.

2.1.4 Timeline

The timeline is as follows:

1. The regulator announces the audit accuracy $π$⁠;

2. The firm selects $p$ and observes its conduct $ω$⁠;

3. Nature determines who sets the bribe; the bribe is set and then accepted/paid or rejected;

4. Signal $s$ realizes, and report $r$ is generated. If collusion has occurred, it is detected with probability $ϵ$⁠. The players obtain their payoffs.

2.1.5 Payoffs

The strategy of the regulator selects an accuracy $π∈[ϵ,1]$⁠.¹¹ The firm’s strategy maps: (1) any $π$ into an investment $p$⁠, (2) any pair $(π,ω)$ into an amount $bf$⁠, and (3) any triple $(π,ω,ba)$ into a choice to accept or reject. The auditor’s strategy maps any $π$ into an amount $ba$ and any pair $(π,bf)$ into a choice to accept or reject. We focus on perfect Bayesian equilibria in pure strategies, or equilibria for short.

The continuation game associated with each level of accuracy is a proper (sub)game. We call the continuation equilibrium honest if collusion does not occur. If instead collusion occurs whenever $ω=1$⁠, then we talk of a collusive continuation equilibrium.

2.2 Discussion of the assumptions

2.2.1 Three Players

Our analysis focuses on a three-player setting. To fix ideas, in the context of the example discussed in Section 1, KPMG, the PCAOB, and the U.S. Security and Exchange Commission would be, respectively, the firm, the auditor, and the regulator.¹²

2.2.2. Firm’s investment

The firm’s investment $p$ captures the precautions taken to ensure that the firm’s operations are safe. These precautions could involve internal monitoring to avoid misconduct or mistakes that can jeopardize the firm’s financial security, or the workers’ safety. Else, these precautions could involve monitoring damage caused to third parties, such as environmental damage. In the context of our motivating example, the PCAOB provides a list of standards that accounting firms, including KPMG, have to adhere to:¹³ making sure all audits comply with all these standards is likely to be a costly burden, but would reduce the risk of incurring a sanction from the PCAOB.

2.2.3 Harmful conduct and remedy

Harmful conduct causes a welfare loss. This loss could capture the risk of environmental damage, workplace accidents, or financial turmoil. The loss can be mitigated by way of some costly remedy. Each dollar spent by a harmful firm on this remedy reduces the loss by a factor of $Δ/k$⁠. As $Δ/k>1$⁠, it is socially desirable that a harmful firm spends the entire profit $k$ to remedy the loss. Both the firm and the auditor are protected by limited liability: the firm cannot be punished beyond the loss of its profits, and the auditor cannot be punished beyond the loss of his wage.

2.2.4 Audit

We assume that the regulator can influence the accuracy of an audit. In practice, this can be achieved by setting more or less strict rules for what constitutes safe conduct. Alternatively, the parameter $π$ could capture the probability that an audit takes place, or the probability that the audit finds evidence of harmful conduct (conditional on the conduct being harmful). The case of PCAOB is exemplary. As explained on the PCAOB’s webpage “The inspection team selects the audits and the audit areas, including non-financial areas such as independence, that it will review”, and “A PCAOB inspection is not designed to review all aspects of a firm’s quality control system, to review all of the firm’s audits, or to identify every deficiency in the reviewed audits.” In other words, the PCAOB does not carry out a complete review, but rather a sample-based one. Mandating the size of the sample is a way to determine the probability that any wrongdoing is discovered.

2.2.5 Audit report

We assume that the auditor can report $r=0$ upon observing $s=1$ but cannot report $r=1$ upon observing $s=0$⁠. This asymmetry seems appropriate as a firm will likely challenge an unfavorable report if it believes that it has been treated unfairly. Indeed, in most contexts, an auditor would have to substantiate a report that a firm’s conduct is harmful with some material proof that could hold up in court, which dramatically limits the threat of framing.¹⁴ In Section 4.3 we briefly discuss how the predictions of our model would change if we were to assume that the signal $s$ is soft information so that the auditor can publish a report $r∈{0,1}$ upon observing any signal. Intuitively, this gives rise to the possibility that the auditor might extort money from a firm by threatening to publish $r=1$ upon observing $s=0$⁠.

We assume that the auditor reports $r=s$ when collusion does not occur. We could have written a richer model and given the auditor the possibility to report $r=0$ upon observing $s=1$ even in the absence of collusion. Yet, this would result in a multiplicity of equilibria, as the auditor would be indifferent among different reports. We find this multiplicity uninteresting and somewhat spurious, as letting the regulator discover concealed evidence even with a small probability (and punish the auditor for concealing) would be enough to ensure that, absent collusion, the auditor reports $r=s$⁠.

2.2.6 Collusive agreement

We assume that firm and auditor collude before observing the realization of signal $s$⁠. This describes, for instance, a firm cultivating long-term ties with an auditor, so as to be tipped off about an upcoming inspection, or to ensure that an eventual audit will not flag any issues. This is exactly what happened in the PCAOB/KPMG scandal discussed in Section 1. In a different context, Vannutelli (2022) empirically finds that a reform that severed the connection between mayors and their auditors has significantly improved the fiscal efficiency of Italian municipalities, implying that preexisting relationships are critical to implementing collusive outcomes.¹⁵ In line with the agency theory of corruption, the collusive agreement is enforceable. Several mechanisms can be invoked here, such as reciprocity, reputation, or reliable intermediaries (for a discussion, see Tirole 1992).

2.2.7 Bargaining protocol

For simplicity, in our model the firm and the auditor bargain non-cooperatively. The results of the analysis would be unaffected if the players bargained cooperatively according to the generalized Nash bargaining solution in which the auditor receives a share $α∈[0,1]$ of the expected surplus from collusion. We provide more details in the  Appendix. Here we just add two remarks. First, the standard equivalence between the two bargaining models usually requires symmetric information across the bargaining parties; in our model, information is asymmetric, yet the equivalence still holds because both forms of bargaining result in collusion only if the firm is of a specific type (i.e. it is harmful). Second, the generalized Nash bargaining solution is essentially mute on the way bargaining actually takes place in practice, and as such it applies to a large variety of contexts: the equivalence with the Nash Bargaining suggests that our model can also apply to a variety of forms of bargaining.¹⁶

2.2.8 Collusion detection

We assume that the regulator can detect collusion with some probability. This could be due to a leak that reveals that collusion took place, as occurred, for instance in the PCAOB/KPMG scandal discussed in Section 1.¹⁷ Else, $ϵ$ can be interpreted as an additional signal that might come from an investigation uncovering malfeasance, or from a second auditor disclosing contradicting evidence, as in Kofman and Lawarrée (1993) and Khalil and Lawarrée (2006).

2.2.9 Institutional constraints

We assume that the auditor receives a salary independent of the report. This is reasonable in most contexts, as typically the legislative branch of the government allocates money to regulatory agencies, and the latter have limited latitude in setting their employees’ wages. For instance, the U.S. Congress sets the budget of federal agencies through the federal appropriations process. Similarly, in developing countries, public agencies’ budgets require government approval and public officials’ salaries follow civil service pay scales.¹⁸ We explore the role played by institutional constraints in Section 4.

2.3 First- and second-best benchmarks

3. ANALYSIS

The amount $b_$ measures the auditor’s opportunity cost of collusion.

Our first lemma establishes that the asymmetry of information between firm and auditor at the time of collusion does not prevent them from extracting all available surplus.

 

The regulator can induce an honest continuation equilibrium by selecting $π∈[ϵ,min{π°,1}]$⁠. If $π°<1$⁠, the regulator can induce a collusive continuation equilibrium by selecting $π∈(π°,1]$⁠. We now focus on honest and collusive continuation equilibria.

3.1 Optimal honest accuracy

We restrict here attention to accuracy levels for which an honest continuation equilibrium exists, and, for each level, we calculate the regulator’s expected payoff associated with this continuation equilibrium. We refer to the accuracy level yielding the highest expected payoff as the optimal honest accuracy.

In line with the insights of Becker and Stigler (1974) and the findings of Di Tella and Schargrodsky (2003), a higher probability of ex-post detection is associated with a more accurate audit (i.e., the optimal $π$ is a weakly increasing function of $ϵ$⁠).

3.2 Optimal collusive accuracy

We now do the same exercise for collusive continuation equilibria and look for the optimal collusive accuracy.²⁰

Also in this case, more accurate audits result in a larger investment in precautions. The logic goes as follows. A higher accuracy $π$ results in a larger $ba$⁠, while the amount $bf$ does not depend on the audit accuracy (see the proof of Lemma 1). A higher accuracy hence results in a larger marginal benefit of investment for the firm, as a harmful firm pays a bribe, while an honest one does not. The investment level for any $π∈(π°,1]$ is smaller than the first-best level, and is increasing in the audit’s accuracy. The optimal collusive accuracy is therefore $π=1$⁠.²¹

Summing up, we have so far established that:

• If $t≥t°$ (so that $π°≥1$⁠), then every accuracy level is associated with an honest continuation equilibrium and the optimal honest accuracy is $π=1$⁠;

• If $t<t°$ (so that $π°<1$⁠), then the optimal honest accuracy is $π=π°<1$ and the optimal collusive accuracy is $π=1$⁠.

3.3 The equilibrium

All is left to characterize the equilibrium of the game is to establish, in case $t<t°$⁠, whether the regulator is better off setting $π=π°$ and ensuring an honest (yet imprecise) audit or setting $π=1$ and tolerating collusion. The following proposition characterizes the essentially unique equilibrium of the game.²²

 

The optimal audit accuracy is a non-monotonic function of the salary amount $t$⁠. Figure 1 illustrates. The solid red curve describes the equilibrium audit accuracy as a function of $t$ (the dashed curves will be discussed below). If the budget is tight (⁠$t<t˜$⁠), collusion is tolerated, and the audit is accurate: $π=1$⁠. For larger budgets (⁠$t>t˜$⁠), collusion is deterred. For $t˜<t<t°$ deterring collusion calls for an inaccurate audit (⁠$π<1$⁠), while for $t≥t°$ an accurate audit is consistent with the objective of deterring collusion.

It is therefore optimal to tolerate collusion and set $π=1$ if the salary is small, while it is instead optimal to deter collusion by setting $π<1$ if the salary is large.

Finally, the blue densely-dashed curve denoted $F$ and the green loosely-dashed curve denoted $A$ in Figure 1 describes, respectively, the firm’s and the auditor’s expected payoffs in equilibrium. Interestingly, a salary increase can leave the auditor worse off.²⁵ The reason is that as the salary crosses $t˜$ from the left, the regulator switches from an accurate to a loose audit. A looser audit discourages collusion, thus depriving the auditor of the income associated with the bribe.

3.4 Policy implications

Our results imply that audits should be tailored to the budget available to the regulator. As stressed by Estache and Wren-Lewis (2009), the regulators’ limited fiscal efficiency and capacity in low-income countries make it impractical to apply the same regulatory policies as in high-income countries. Seen through the lens of our model, these institutional limitations translate into a smaller budget available to the regulator and, hence, a smaller salary for the auditor.

Our results also imply that the optimal audit accuracy depends on the relative bargaining power of the auditor in the bribe-setting process, as measured by $α$⁠. The greater the auditor’s bargaining power, the higher the expected bribe and, hence, the stronger the firm’s incentive to invest if collusion is anticipated. Thus, the higher $α$ the stronger the case for tolerating collusion. Accordingly, the optimal accuracy is non-decreasing in $α$ and the cutoff $t˜$ is an increasing function of $α$⁠. Figure 2 illustrates. The dashed red lines describe the utility of the regulator in the collusive candidate equilibrium, for different values of $α$⁠. The solid blue curve describes the utility in the candidate honest equilibrium (this utility does not depend on $α$⁠). When $α=0$⁠, the regulator is always better off without collusion. When the firm has all the bargaining power, collusion results in both poor ex-post screening and little ex-ante investment, as the firm anticipates that it will have to pay a small bribe to get a favorable report. For higher values of $α$⁠, the cutoff value of $t$ above which the regulator prefers to deter collusion increases. Moreover, an increase in the auditor’s bargaining power makes the regulator better off. Specifically, when the auditor has all the bargaining power, i.e., $α=1$⁠, tolerating collusion has the highest impact on the investment incentives, as the firm anticipates that a harmful conduct carries a high price in terms of the bribe that must be paid to the auditor.

The relation between the optimal level of accuracy and $α$ has policy implications, because a regulator can influence the relative bargaining power of auditors and firms. One way to do so is to determine how many auditors are entitled to assess the firm’s compliance.²⁶ Competition between auditors (or agencies) is likely to reduce their bargaining power vis-à-vis a firm. As extensively discussed by Shleifer and Vishny (1993) and Rose-Ackerman (2013), when the jurisdictions of different auditors overlap, the amount necessary to bribe them can be very small.

The prediction that increasing the auditors’ bargaining power vis-à-vis the audited firms should encourage investment is in line with the evidence in Vannutelli (2022) and Burgess et al. (2012). Vannutelli (2022) considers a reform that changed the way auditors are assigned to review the financial statements of Italian municipalities. Whereas mayors used to appoint their auditors, the reform introduced a random mechanism to assign auditors to municipalities, thus, arguably, increasing the auditors’ bargaining power vis-à-vis the mayors. This change is credited as one of the main channels through which the financial performance of municipalities has thereafter improved. Burgess et al. (2012) find that, in the context of the Indonesian logging industry, the higher the number of political jurisdictions from which a firm can (illegally) obtain logging permits, the more extensive the deforestation and the lower the price of timber.²⁷

4. EXTENSIONS

In this section, we let the regulator set the auditor salary together with the audit accuracy. In Suection 4.1, the salary cannot depend on the audit report. This extension allows us to show that the non-monotonic relation between budget and accuracy remains qualitatively unchanged whether the salary is set exogenously or chosen by the regulator. In Section 4.2 instead, the regulator can reward the auditor whenever the audit deems the firm harmful. In this case, we show the relation between budget and accuracy is monotonic: a “richer” regulator opts for a more accurate audit. In Section 4.3, we briefly discuss the case in which the auditor can fabricate evidence of harmful conduct.

4.1 Endogenous salary

Lemma 1 continues to hold. The rest of the analysis is similar to the one of the baseline model, if somewhat more convoluted. We tackle the intermediate steps in the  appendix and record all the key equilibrium properties in the following proposition.³⁰

 

The optimal audit accuracy is a non-monotonic function of $λ$⁠. If the budget is small (i.e. $λ$ is large), the regulator pays a small $t$⁠, demands an accurate audit and tolerates collusion; for intermediate values of $λ$⁠, the regulator makes sure that audits are sufficiently inaccurate/infrequent so as to deter collusion. For small $λ$⁠, the salary $t$ is sufficiently large to deter collusion regardless of the audit accuracy.

Summing up, the non-monotonic relation between budget and accuracy remains qualitatively unchanged whether the salary is set exogenously or chosen by the regulator.³¹

4.2 Report-dependent compensation

We now give the regulator the opportunity to pay the auditor a base salary together with a bonus for recommending a remedy. Let $tr$ denote the auditor’s compensation following report $r$⁠. A policy is now a triple $(π,t0,t1)$⁠, such that $t0≥0$ and $t1≥t0$⁠.³²

The next lemma establishes that also in this model the bargaining process is efficient.³³

 

• for policies that satisfy $π<(t0+k)ϵ/(k−t1+t0)$, the unique continuation equilibrium is honest;

• for policies that satisfy $π>(t0+k)ϵ/(k−t1+t0)$, the unique continuation equilibrium is collusive.

As intuition would suggest, the optimal honest policy amounts to paying the auditor only if the audit does recommend a remedy. As the shadow cost of funds becomes larger, the audit becomes less accurate and the compensation for recommending a remedy becomes smaller.

We then show that the optimal collusive policy is $(1,0,(1−ϵ)k)$⁠. The intuition here is as follows. In any collusive continuation equilibrium the auditor never recommends a remedy. As the regulator does not ever pay $t1$⁠, it is optimal to set this compensation as large as possible, so as to drive the bribe demanded by the auditor up. As we discussed already, the prospect of having to pay a large bribe when $ω=1$ encourages investment. In fact, in the collusive continuation equilibrium associated with the policy $(1,0,(1−ϵ)k)$⁠, the firm chooses the second-best level of investment.

In equilibrium, the regulator either announces the optimal honest policy, or the optimal collusive one. When the two policies coincide, as is the case for $λ<λ_†$⁠, the regulator’s favorite continuation equilibrium is played. The following proposition summarizes our results.

 

• if $λ>(Δ−k)/k$, then collusion occurs;

• if $λ<(Δ−k)/k$, then collusion does not occur.

Proposition 3 establishes that, even when a report-dependent compensation is feasible, the regulator opts for tolerating collusion when the shadow cost of funds is high. When instead the cost is relatively small, and/or the benefit of screening, as measured by $Δ/k$ is large, collusion is deterred. Similarities with the previous versions of the model end there. First, Proposition 3 establishes that the regulator never opts for less than the highest degree of accuracy. Secondly, the relative bargaining power of the auditor has no bearing on the optimal policy. As we show in the  appendix, the optimal policy leaves no surplus for the colluding parties. Without any surplus to share, the relative bargaining power is irrelevant.

Report-dependent compensation gives the regulator enough latitude so as to make it suboptimal to ever limit the audit accuracy. We have shown, for instance, that the regulator can induce the second-best level of investment at no cost, if she is willing to tolerate collusion. In fact, perhaps counterintuitively, there exist parameter combinations for which collusion occurs in equilibrium only if the compensation can depend on the report. In particular, if $λ∈((Δ−k)/k,λ¯⋆)$⁠, collusion occurs if the compensation of the auditor can depend on the report (Proposition 3), yet collusion is deterred if the auditor’s compensation cannot dependent on the report (Proposition 2).³⁴

4.3 Extortion

We now consider, informally, what would happen if the auditor could publish any report following any signal realization, that is, if the auditor could report $r=1$ even upon observing $s=0$⁠. At the same time, we allow the auditor to extort money from the firm: whenever the auditor can request a bribe, he might now (1) threaten to report $r=1$ irrespective of the signal generated by the audit if no sum is paid (we call this framing), (2) commit to reporting $r=s$ in exchange for a sum of money (we call this extortion), and commit, for a possibly different sum of money, to reporting $r=0$ regardless of the signal generated by the audit (we call this collusion). Importantly, we believe it is reasonable to assume that if the auditor fabricates evidence, that is, he reports $r=1$ upon observing $s=0$⁠, this can be discovered by the regulator with some probability, and accordingly punished by withdrawing the auditor’s salary.

Allowing for extortion and framing in the baseline version of the model would be inconsequential. Intuitively, a harmful firm cannot gain from giving in to extortion, whereas the safe firm would not believe the auditor’s threat to misreport the signal observed. The threat is not credible because, faced with a firm’s refusal to pay the sum requested, the auditor would prefer reporting truthfully rather than fabricating evidence and, potentially, losing his salary.

While a comprehensive analysis of the optimal audit accuracy and auditor compensation under the threat of framing is beyond the scope here, three remarks are worth making. First, tolerating extortion is likely to be suboptimal, as this would undermine the firm’s incentive to invest in compliance. Intuitively, the firm would anticipate that, irrespective of its conduct, it would have to pay a bribe to the auditor, or else he would report that the firm is harmful.³⁵ Second, even though deterring extortion would affect the design of the optimal honest and collusive policies, the possibility of extortion need not make contingent compensation useless, that is, the optimal policy need not be such that $t1=t0$⁠: as discussed above, the weaker condition $t0≥(1−η)t1$ is sufficient to deter extortion. Third, tolerating collusion no longer costlessly induces the second-best investment level: on the equilibrium path, the auditor will receive a positive payment equal to $t0$⁠.

5. CONCLUDING REMARKS

Accurate audits come with costs and benefits. The anticipation of an accurate audit might induce the audited subject to invest in compliance but might also encourage collusion with the auditor. We showed that a regulator might prefer tolerating collusion rather than compromising the standard of the audits.

It is worth stressing here the role of some modeling assumptions. Firstly, as in the PCAOB/KPMG scandal (see Section 1), we focus on a firm and an auditor that collude before knowing whether the audit will deem a costly remedy necessary or not. If collusion were to occur exclusively after the audit reveals a violation, then the regulator would have no strategic reason to reduce the audit accuracy. Second, our firm knows whether its conduct is harmful at the time of colluding. Our results would be qualitatively identical if the firm were to observe a noisy signal of its own conduct. Lastly, we have kept fixed the size of the remedy requested from a firm in case of harmful conduct. This assumption seems justified as the size of the remedy is often set by an entity other than the regulator that determines the audit accuracy (e.g. the courts of law), or because, for technological reasons, the remedy cannot be scaled up or down (e.g. a polluting firm could be asked to install an air filter or clean up a polluted lake). We believe it would be interesting, but beyond the scope of this paper, to explore the case of a regulator that can set both the accuracy of the audit and the size of the remedy. We just notice here that penalty $k$ and audit accuracy $π$ are quite different policy tools. The key difference is that, while both tools affect the benefit of collusion for a harmful firm (measured by $kπ$⁠), only the penalty affects the harmful firm’s cost of collusion (measured by $kϵ$⁠).

Our results inform the design of auditing policies. Optimal auditing policies in turn are critical to improve enforcement, as stressed by a growing body of empirical evidence (see, for instance, Gonzalez Lira and Mobarak 2021). In particular, we have shown how the optimal audit accuracy is a non-monotonic function of the size of the budget available to the regulator. As regulatory capacity, fiscal efficiency, and other institutional limitations (e.g. the availability of more flexible contractual arrangements with the auditor) are different between high and low-income countries, it follows that one-size-fits-all solutions should be avoided, as argued by Laffont (2005) and Estache and Wren-Lewis (2009).

Footnotes

Acknowledgements

We thank the editor Daniel Barron, two anonymous referees, Andreas Asseyer, Muxin Li, Caio Lorecchio, Ester Manna, Adrien Vigier, Martin Watzinger, and audiences at the Online Oligo Workshop 2021, at the 48th EARIE Annual Conference (Bergen, Norway), at the 2023 BSE Summer Forum workshop on Public Economics (Barcelona, Spain), and at the XXXVIII Jornadas de Economía Industrial (Seville, Spain) for useful comments. Donald Oswald helped with the editing. Alessandro De Chiara acknowledges the financial support of Ministerio de Ciencia, Innovación y Universidades through grants PID2020-114040RB-I00 and PID2021-128237OB-I00, and the Government of Catalonia through grant 2021SGR00678. Any remaining errors are ours.

Funding

Funding support for this article was provided by the Ministerio de Ciencia, Innovación y Universidades (PID2020-114040RB-I00), Ministerio de Ciencia, Innovación y Universidades (PID2021-128237OB-I00), Government of Catalonia (2021SGR00678).

Appendix

COOPERATIVE BRIBE-SETTING

That is, the bribe allows the auditor to capture a share $α$ of the gains from collusion.

Suppose $ω=0$⁠. Then, colluding never generates a surplus since $(1−ϵ)(t+k)<t+k$ where the right-hand side is the sum of the disagreement points in this scenario. As only one type of firm may be willing to bargain, asymmetric information does not undermine the success of collusion.

PROOFS FOR SECTION 3

   

Consider the continuation equilibria associated with some accuracy level $π>π°$⁠, so that $b¯>b_$⁠. An harmful firm is indifferent between paying and refusing to pay when the auditor requests a bribe $ba=b¯$⁠. Standard arguments ensure that a harmful firm pays (The argument is simple: it is not possible to construct an equilibrium in which a harmful firm refuses to pay such bribe, as it this case there exists no candidate for the optimal bribe amount.). As long as the auditor attaches positive probability to $ω=1$ then, the auditor demands $ba=b¯$⁠. Lemma A1 hence ensures that the auditor demands bribe $ba=b¯$⁠, and the firm pays if and only if $ω=1$⁠. Standard arguments ensure that the auditor accepts when the firm offers a bribe $bf=b_$⁠. The firm then offers bribe $bf=b_$ if $ω=1$⁠, and offers some bribe $bf<b_$ if $ω=0$⁠. We conclude that all continuation equilibria associated with accuracy level $π>π°$ are collusive and essentially identical to each other (i.e., all continuation equilibria are associated with the same probability distribution over outcomes).

Consider the continuation equilibria associated with some accuracy level $π<π°$⁠, so that $b¯<b_$⁠. The firm, regardless of the realization of $ω$⁠, offers some bribe $ba<b_$⁠, which the auditor rejects. The auditor in turn demands some bribe $bf>b¯$⁠, which the firm refuses to pay, regardless of the realization of $ω$⁠. We conclude that all continuation equilibria associated with accuracy level $π<π°$ are honest and essentially identical to each other.

Consider now the accuracy level $π°$⁠. It is easy to verify that for this accuracy level (a) an honest continuation equilibrium exists and (b) the honest continuation equilibrium ensures a strictly higher payoff to the regulator than any other continuation equilibrium (call this Remark 1). A standard equilibrium-selection argument ensures that if in equilibrium the regulator announces accuracy level $π°$⁠, then collusion does not occur. The equilibrium-selection argument goes as follows. We already established that, for any $π∈[ϵ,π°)$⁠, the (essentially) unique continuation equilibrium is honest. The regulator’s payoff associated with the honest continuation equilibrium is a continuous function of $π$ for $π∈[ϵ,π°]$⁠; we refer to this observation as Remark 2. Suppose that in a hypothetical equilibrium in which the regulator announces accuracy level $π°$ collusion occurs with some probability. Combining Remark 1 and Remark 2 it is easy to establish that the regulator gains from deviating to announce some $π$ smaller than, but sufficiently close to, $π°$⁠. The contradiction proves that if in equilibrium the regulator announces accuracy level $π°$⁠, then collusion does not occur. ▪

 

Consider now $t<t°$⁠. Combining Lemma 1 and arguments in the text establishes that in equilibrium either the regulator announces $π=π°$ (and collusion does not occur), or else she announces $π=1$ (and collusion occurs whenever $ω=1$⁠): the regulator selects among these two accuracy levels the one that ensures her the highest payoff.

Let $g(t)≡vN(t)−vC(t)$⁠. We prove a sequence of claims.

 

The proof is straightforward.

 

Let $f(x)≡k−(1−x·k)(H−(Δ−k)ϵ)−x2k2/2$⁠, so that $g(0)=f(ϵ)−f(α+(1−α)ϵ)$⁠. Note that $f$ is strictly concave and differentiable, and $f′(α+(1−α)ϵ)≥f′(ϵ)$⁠, hence $f(ϵ)−f(α+(1−α)ϵ)≤0$⁠.

 

Combining Claims 1-3 establishes that a cutoff $t˜∈[0,t°)$ exists such that, in equilibrium: (1) collusion is prevented if $t∈(t˜,t°)$ and (2) collusion occurs whenever the firm is harmful if $t<t˜$⁠. This observation proves the first two bullet points of the proposition.

All is left to show is that the above-mentioned cutoff is a strictly increasing function of $α$⁠.

  

Claim 4 ensures that $α=0⇒t˜=0$⁠. We know that $vN(t)<vC(t)⇔t<t˜$⁠, while $vN(t)=vC(t)⇔t=t˜$⁠; combining Claims 4 and 5 thus establishes that $t˜$ is strictly increasing in $α$⁠. ▪

PROOFS FOR SECTION 4.1

We consider here the model in which the salary is endogenous. As in Section 3, we define $t°≡(1−ϵ)k/ϵ$⁠. We now express $π°$ as a function of $t$⁠, so $π°(t)≡(t+k)ϵ/k$⁠. Lemmata 1 and A1 continue to hold.

Optimal honest policy

We restrict here attention to policies for which an honest continuation equilibrium exists. Lemma 1 ensures that the honest continuation equilibrium is unique. For each policy, we calculate the regulator’s expected payoff associated with the honest continuation equilibrium. We refer to the policy yielding the highest expected payoff as the optimal honest policy.

  

Optimal collusive policy

We restrict here attention to policies for which a collusive continuation equilibrium exists. For every policy, we calculate the regulator’s expected payoff associated with the collusive continuation equilibrium. We refer to the policy yielding the highest expected payoff as the optimal collusive policy.

 

The proof of the following corollary is straightforward.

   

The claim follows.

Claim 1 establishes that, for $λ≤λ_≡λ_N⋆$⁠, in equilibrium the regulator announces a policy such that $π=1$⁠, and collusion does not occur. The third bullet point of the proposition follows.

   

• $g⋆(λ)>0$  for $λ∈(max{λ_C⋆,λ_N⋆},λ˜)$;

• $g⋆(λ˜)=0$;

• $g⋆(λ)<0$  for $λ∈(λ˜,λ¯C⋆)$⁠.

Combining Lemmata A2 and A3 establishes that $g⋆$ is a continuous function over $R+$⁠. Combining Claims 1 and 2 ensures that $g⋆(λ)>0$ for $λ=max{λ_C⋆,λ_N⋆}$⁠. To verify the claim it is thus sufficient to show that $g⋆$ is a strictly concave function over the interval $(max{λ_C,λ_N⋆},λ¯C⋆)$⁠.

As $Δ/k>1>(1+(1−α)2)/2$⁠, the last highlighted inequality holds, and hence $g⋆$ is strictly concave over the interval $λ∈(max{λ_C⋆,λ_N⋆},λ¯C⋆)$⁠.

  

The claim follows.

Combining Claims 3–6 establishes that a cutoff $λ¯∈(λ_N⋆,λ¯N⋆)$ exists such that $g⋆(λ)<0$ if $λ>λ¯$ and $g⋆(λ)>0$ if $λ<λ¯$⁠. The first two bullet points of the proposition follow. All is left to prove is that $λ¯$ is a decreasing function of $α$⁠. The argument is identical to the argument used in in the proof of Proposition 1 to prove that the cutoff $t˜$ is strictly decreasing in $α$⁠. ▪

Proofs for Section 4.2

In this  appendix, whenever continuation equilibria are essentially identical to each other, we say that the continuation equilibrium is unique.

Lemma A1 continues to hold.

 

• (a) a safe firm pays any bribe $ba<−k·ϵ$  and refuses to pay any bribe $ba>−k·ϵ$;

• (b) a harmful firm pays any bribe $ba<b¯$  and refuses to pay any bribe $ba>b¯$;

• (c) the auditor accepts any bribe $bf>b_†$  and rejects any bribe $bf<ϵ·t0$;

• (d) if $ω=0$, then collusion does not occur.

  

• (e) the auditor accepts a bribe $bf=b_†$;

• (f) a harmful firm pays a bribe $ba=b¯$⁠.

• (g) the auditor rejects any bribe $bf<b_†$;

  

i.e., such that $b¯<b_†$⁠. The auditor demands some bribe $ba>b¯$⁠. To see that this is the case, note that (1) the firm refuses to pay any bribe $ba>b¯$ (Lemma A4, Properties (a) and (b)), (2) any bribe $ba≤b¯$ is accepted with some probability (Lemma A1, Lemma A4, Property (a), Lemma A5, Property (f)) and (3) as $b¯<b_†$⁠, then a bribe $ba$ that the firm refuses to pay strictly dominates a bribe $ba$ that the firm pays for some realization of $ω$⁠. A similar argument establishes that the harmful firm, in turn, offers some bribe $bf<b_†$ that the auditor rejects. We have just established that for $b¯<b_†$ in every continuation equilibrium a harmful firm does not pay a bribe. As a safe firm does not pay a bribe in any continuation equilibrium (Lemma A4, Property (d)), the unique continuation equilibrium is honest.

That is, $b¯>b_†$⁠. The auditor demands bribe $ba=b¯$⁠. To see that this is the case, note that (1) a harmful firm pays bribe $ba=b¯$ but refuses to pay any bribe $ba>b¯$ (Lemma A4, Property (b) and Lemma A5, Property (g)), and (2) a safe firm refuses to pay bribe $ba=b¯$ (Lemma A4, Property (b)). As $b¯>b_†$⁠, then $ba=b¯$ is the best choice among the bribes that a safe firm refuses to pay. The auditor hence demands $ba=b¯$ (Lemma A4, Property (e)). The harmful firm in turn offers bribe $bf=b_†$⁠. To see that this is the case, note that the auditor accepts bribe $bf=b_†$ but rejects any bribe $bf<b_†$ (Lemma A5, Properties (f) and (h)). As $b¯>b_†$⁠, the harmful firm is better off paying bribe $bf=b_†$ rather than offering a bribe that the auditor rejects. We just established that, for $b¯>b_†$ in every continuation equilibrium a harmful firm pays a bribe. As a safe firm does not pay a bribe in any continuation equilibrium (Lemma A4, Prop. (d)), the unique continuation equilibrium is collusive. ▪

  

In the first two cases, arguments based on the continuity of the regulator’s payoffs in the policy akin to the argument presented in the Proof of Lemma 1 ensure that if in equilibrium the regulator announces this policy, then the continuation equilibrium is the one associated with the highest expected regulator’s payoff. Suppose the third case holds, that is, all continuation equilibria associated with this policy ensure the same expected regulator’s payoff, and suppose that in equilibrium the regulator announces this policy. Then the policy must satisfy $t0=0$⁠, or else the regulator could deviate by marginally reducing $t0$⁠. By doing so the regulator would ensure an honest continuation equilibrium (Lemma 2), and it is easy to verify that the deviation is profitable. By a similar argument the policy must satisfy $π=1$ (or else the regulator could profitably deviate by marginally increasing $π$ and ensuring a collusive continuation equilibrium). We conclude that the policy must be such that $(π,t0,t1)=(1,0,(1−ϵ)k)$⁠. It is easy to verify that for this policy all continuation equilibria associated ensure the same expected regulator’s payoff if and only if $λ=(Δ−k)/k$⁠. ▪

 

The following properties can be easily verified:

1. $u†(·)$ is strictly concave if $λ<λ†$⁠, and it is convex if $λ≥λ†$⁠;

2. $du†dπ=0⇔π=π†(λ)$⁠;

3. $u†(ϵ)>u†(1)⇔λ>λ⁁$⁠, and $u†(ϵ)<u†(1)⇔λ<λ⁁$⁠;

4. $H≥H⁁⇔λ⁁≥λ†$⁠;

5. $H≥H⁁⇔dπ†dλ≥0$⁠;

6. $H≥H⁁⇔λ¯†≥λ†$⁠;

7. $π†(0)>1$⁠.

Let $H≥1+ϵ(2Δ−k)2$⁠. Then $λ†≤λ⁁$ (Property (3)). If $λ<λ†$⁠, then $u†$ is strictly concave (Property (1)) and $πN′(λ)>1$ (combine Property (4) and Property (6)). The optimal honest policy must thus be such that $π=1$⁠. If instead $λ≥λ†$⁠, then $u†$ is convex (Property 1). By Property 2, we can then distinguish two cases. If $λ∈[λ†,λ⁁)$⁠, the optimal honest policy is such that $π=1$⁠, while if $λ>λ⁁$ the optimal honest policy is such that $π=ϵ$⁠.

If instead $λ≥λ†$⁠, then $u†$ is convex (Property (1)). As $λ>λ⁁$ (Property (3)), the optimal honest policy satisfies $π=ϵ$ (Property (2)). The first part of the lemma follows. The proof of the last part of the lemma is straightforward. ▪

We now consider the optimal collusive policy (see the definition in the  Appendix of Section 4.1).

 

For policy $(π,t0,t1)=(1,0,(1−ϵ)k)$⁠, a collusive continuation equilibrium exists (see Lemma A6), and $pC†(1,0,(1−ϵ)k)=k$⁠.

The regulator’s utility depends on (1) the probability of a remedy conditional on $ω=1$⁠, (2) the size of the firm’s investment, and (3) the compensation paid to the auditor. To verify that policy $(π,t0,t1)=(1,0,(1−ϵ)k)$ is an optimal collusive policy it is enough to note that: (1) the probability of a remedy conditional on $ω=1$ is the same in any collusive continuation equilibrium, (2) $pC†(1,0,(1−ϵ))=k$ (recall that $p=k$ is the second-best level of investment), and (3) the policy ensures that the regulator does not pay any compensation to the auditor.

As $π≤1$⁠, we conclude that $π=1$⁠. The proof of the last part of the lemma is straightforward. ▪

The proof of the next corollary is straightforward.

  

Note first that $vN†(λ)$ is a strictly decreasing function as long as $λ<λ¯†$⁠, while $VC†$ does not depend on $λ$⁠. In light of this observation, to prove the proposition it is sufficient to show that the following two properties hold:

• $Δ−kk<λ¯†$⁠;

• $W(Δ−kk)=VC†⇔λ=Δ−kk$⁠.

  In order to prove Property (I), we show that a stronger property holds:

• $Δ−kk<λ_†$⁠;

  By definition of $λ_†$⁠, Property (III) holds if and only if the following two properties hold:

• if $H>1+ϵ(2Δ−k)2$⁠, then $Δ−kk<1k(1−k)(k·H+(Δ−k)−(1+ϵ)(2Δ−k)k2)$⁠;

• if $H<1+ϵ(2Δ−k)2$⁠, then $Δ−kk<k(2Δ−k)−k·H−Δ+kk(2k−1−k·ϵ)$⁠.

As the last inequality holds, so does Property (IV).

We thus conclude that Property (V) holds. This in turn proves that Property (III) holds, which in turn implies that Property (I) holds.

References