auditors’ liability, investments, and capital markets: a potential unintended consequence of the...

DOI: 10.1111/j.1475-679X.2012.00458.xJournal of Accounting Research

Vol. 00 No. 0 xxxx 2012Printed in U.S.A.

Auditors’ Liability, Investments,and Capital Markets: A PotentialUnintended Consequence of the

Sarbanes-Oxley Act

M I N G C H E R N G D E N G , ∗ N A H U M M E L U M A D , † A N DT O S H I S H I B A N O †

Received 2 October 2009; accepted 2 December 2011

ABSTRACT

To restore investors’ confidence in the reliability of corporate financial dis-closures, the Sarbanes-Oxley Act of 2002 mandated stricter regulations andarguably increased auditors’ liability. In this paper, we analyze the effects ofincreased auditor liability on the audit failure rate, the cost of capital, andthe level of new investment. We focus on a setting in which, with imperfectauditing, a firm has better information than investors about its prospects andseeks to raise capital for new investments in a lemons market. The equilib-rium analysis derives corporate reporting and investing choices by the firm,attestation opinions by the auditor, and valuation by rational investors. Threeempirically testable predictions emerge: although increasing auditor liabilitydecreases the audit failure rate and the cost of capital for new projects, it alsodecreases the level of new profitable investments.

∗University of Minnesota †Columbia Business School. Accepted by Richard Leftwich. Wethank the editor, an anonymous referee, David Abody, Tim Baldenius, Shira Cohen, FrankGigler, Jack Hughes, Chandra Kanodia, Volker Laux, Tong Lu, Paul Newman, Bugra Ozel,Haresh Sapra, Brett Trueman, Rick Young, and the participants at the Copenhagen 2008 Inter-disciplinary Accounting Conference, the 2008 Carnegie Mellon University Accounting TheoryConference, and UCLA for helpful comments and suggestions.

1

Copyright C©, University of Chicago on behalf of the Accounting Research Center, 2012

2 M. DENG, N. MELUMAD, AND T. SHIBANO

1. Introduction

A number of highly publicized controllership failures of public companiesand their auditors led to the Sarbanes-Oxley Act of 2002 (hereafter, SOX).SOX aims to improve the reliability and usefulness of corporate financialdisclosures. Section 404 of SOX requires auditors to attest not only to thefairness and representational faithfulness of financial statements, but alsoto the effectiveness of the internal controls system. In case of an audit fail-ure, the auditor may be subject to criminal prosecution, and face greaterlegal liability.1 Recently, in the aftermath of the Satyam controllership fail-ure in India, questions about the appropriateness of auditors’ legal respon-sibility and their liability exposure have surfaced once again in the businesspress as well as among regulators in India, the United States, Europe, andelsewhere.2

Capital market participants and regulators, however, have been con-cerned about raising legal liability for auditors. In the United States, theCommittee on Capital Markets Regulation recommends “Congress shouldexplore protecting audit firms against catastrophic loss through the provi-sion of caps or safe harbors.”3 Treasury Secretary Henry Paulson and SECChief Accountant Conrad Hewitt have both concurred with the views thatthe Big Four accounting firms are exposed to potentially large legal liabil-ity and that they should receive legal protection against damage claims.4,5,6

Meanwhile, Charlie McCreevy, the European Union internal market com-missioner, has already come out in favor of a fixed cap on the auditors’liability claims.7 The UK Department of Trade has proposed the Compa-nies (Audit, Investigations and Community Enterprise) Bill to cap auditors’liability when they audit publicly traded companies, arguing that the con-sequence of growing litigation risk may result in more economic damagethan it prevents. As part of the proposal, auditors would be able to ne-gotiate proportionate liability by contracting with their clients, subject toshareholders’ approval.8

1 We should note that SOX did not explicitly raise auditors’ liability. However, early evi-dence suggests audit liability is higher after SOX. For example, Rashkover and Winter [2005]argue that civil monetary penalties have increased because SOX empowers federal courts andthe SEC to impose equitable remedies for violations of federal securities laws. Ghosh andPawlewicz [2008] document that audit fees after SOX are higher partially because of higherlegal liability due to an audit failure. Also see the references cited therein.

2 http://www.atimes.com/atimes/South Asia/KA10Df02.html.3 http://www.capmktsreg.org/research.html.4 http://www.treas.gov/press/releases/hp174.htm.5 http://www.sec.gov/news/speech/2007/spch020907cwh.htm.6 Global Capital Markets and the Global Economy: A Vision from the CEOs of

the International Audit Networks. http://www.pwc.com/Extweb/pwcpublications.nsf/docid/A3263860A2AC0059802572220057EF35.

7 http://ec.europa.eu/commission barroso/mccreevy/allspeeches en.htm.8 For example, see David Reilly, The Wall Street Journal (Europe), April 13, 2004, “U.K. Body

Could Cap Liability of Auditors; If Britain Adopts Standard, Accounting Firms Hope Move

AUDITORS’ LIABILITY, INVESTMENTS, AND CAPITAL MARKETS 3

The literature has established that, in the presence of information asym-metry between firms and investors, efficient firms may forego profitableinvestment opportunities (see Myers and Majluf [1984]). To reduce thecost of information asymmetry, firms may convey the quality of investmentprojects by issuing signals such as financial reports to investors. Such dis-closures are most effective in reducing information asymmetry pertainingto hard information, that is, information that is verifiable once disclosed.9

This disclosed information, in turn, creates demand for auditor attestationregarding the reliability of financial reports. Imposing higher auditor lia-bility will not only change the auditors’ attestation strategy, but also affectinvestors’ ex post assessment of the financial report. As a result, the firms’ex ante optimal investment decisions are also affected.

To analyze consequences of increased auditor liability, we focus on a set-ting in which the auditor’s role is to alleviate the information asymmetrybetween investors and firms. Specifically, our model combines and extendsthe “lemons” investment model of Myers and Majluf [1984] and the imper-fect auditing model in Shibano [1990]. We introduce an auditor to Myersand Majluf [1984] and extend Shibano’s [1990] hidden information auditmodel by incorporating investors with rational expectations, thus endog-enizing the valuation of the firm. In the Myers and Majluf model, a firmhas an investment opportunity with a positive net present value. It mustissue equity to raise the investment funds needed; otherwise, the opportu-nity will be lost. The firm invests in the new project if its cost of capital islow enough. When a firm has private information regarding the value ofits existing assets and liabilities, investors’ assessment of firm value may belower than the firm’s assessment. In such a case, the firm’s cost of capitalincludes an asymmetric information premium that may be so high that thefirm underinvests; that is, it foregoes a profitable investment project.

Mandatory audits of corporate financial disclosures serve to reduce theinformational asymmetry, but potential audit failures may lessen such aneffect. The role of the auditor is to provide attestation to corporate fi-nancial disclosures about existing assets and liabilities.10 When the audi-tor’s attestation is perfect, the investors’ confidence in the reliability andaccuracy of information reported is assured, thereby reducing the informa-

Will Resonate in EU,” p. M1: “Even as the corporate-scandal body count rises in Europe, audi-tors, led by the Big Four accounting firms, are trying to limit the amount of money they can beforced to pay out when a company blows up on their watch. And they may be on the verge ofa significant victory. Capping auditor liability has been a long-held objective of the accountingindustry around the world, which has seen some success in the past in countries such as Ger-many. But the issue became a tough sell with legislators and the public in the wake of scandalsin which auditors were cast in a harsh light, from Enron in the U.S. to, most recently, Parmalatin Italy.”

9 See Grossman [1981], Dye [1985], Kwon and Jung [1988], and Gigler [1994].10 Other roles of auditing include monitoring managerial contracts (e.g., Watts and Zim-

merman [1983] and Antle and Demski [1991]), risk sharing with managers or investors (asin Antle [1982]), regulatory compliance, and informational needs of lenders (as in Melumadand Thoman [1990a,b]).


tion asymmetry and the lemons premium. However, the auditor’s attesta-tion is imperfect; the auditor must make a judgment call based on limitedinformation; therefore, the auditor’s opinion is subject to unintentional er-rors. The auditor, after observing imperfect audit evidence, may unknow-ingly attest to financial statements that under- or over-value existing equity.

An increase in the liability for audit failures may reduce the incentivefor new profitable investments. Because the auditor’s attestation is subjectto possible audit failures, an increase in the liability for such audit failuresinduces the auditor to become more conservative in his interpretation offinancial disclosures. As a result, the firm needs to choose its corporate dis-closure strategy and investment decision based on its anticipation of boththe investors’ valuation and a possible audit failure. When a firm has privateinformation about its existing equity, the chance of receiving understatedfinancial statements increases. As long as the new shareholders cannot in-fer the true value from the understated reports, the lemons problem willdepress the firm’s valuation, thereby increasing the firm’s cost of capitaland making new investments less likely. In sum, our results identify an eco-nomic consequence of SOX the literature has not previously asserted: theincrease in the auditor liability after SOX has the countervailing effects ofdecreasing both audit failures and new investments.

Our study suggests several implications of SOX for investors, auditors,and policy makers. In the presence of information asymmetry between thefirm and investors, an increase in auditor liability attributed to SOX resultsin greater investor protection and confidence and, as such, encourages in-vestment. However, if the information asymmetry regarding the value ofexisting equity is large, higher legal liability may exacerbate a lemons prob-lem and result in reduced levels of new investments. This countervailingeffect, nevertheless, can be mitigated by improving the informativeness offinancial disclosures, thereby reducing the cost of information asymmetry.

Our paper relates to several papers in the extant literature analyzing theimpact of auditor liability on a firm’s investment decisions (e.g., Antle andNalebuff [1991], Schwartz [1997], and Lu and Sapra [2009]). These pa-pers model a capital market valuation rule, an auditor’s attestation strategy,and a firm’s reporting strategy and its investment decision. In these papers,because no information asymmetry is present, the auditor’s role is not toverify a firm’s strategic report, but rather to identify the nature of the firm’soperation activities. Naturally, the equilibrium investment level depends ona firm’s audit fees/liability ratio. A low fee-to-liability ratio makes an auditormore conservative, which, in turn, reduces audit quality and therefore thedegree of investment inefficiency. Similarly, Laux and Newman [2010] an-alyze the effect of auditor liability on client acceptance. In their setting, thesource of potential over- or under investment is the auditor’s moral hazardproblem with respect to the client evaluation task.11

11 In Laux and Newman [2010], an auditor exerts effort to evaluate the type of a client;the auditor rejects the client if he discovers the client is high-risk. An increase in the auditor’s


In the presence of information asymmetry, the relation between auditconservatism and the firm’s reporting strategies is more involved. On theone hand, the firm may have various reporting strategies, depending on theauditor’s attestation strategy and the investor’s valuation. We narrow ninepossible reporting strategies down to two unique equilibriums (see table 2).On the other hand, whether the auditor’s attestation strategy can effectivelymitigate information asymmetry depends on the level of auditor liabilityand on the firm’s reporting strategies. An increase in the liability for auditfailures induces the auditor to become more conservative in his interpre-tation of financial disclosures and, in turn, to issue an unfavorable reportmore frequently. The firm’s reporting strategies and the level of auditor li-ability jointly determine the equilibrium strategies in two different regimes(detection and deterrence) and the extent of investment inefficiency in ourmodel (see table 1). Focusing on the auditor’s role of mitigating the lemonsproblem, our study contributes to Laux and Newman [2010] and Lu andSapra [2009] by showing that, in the presence of information asymmetry, amixed reporting strategy may emerge in equilibrium when the audit adjust-ment cost is nontrivial and the auditor cannot effectively identify the firm’strue type, which subsequently leads to investment inefficiency.

Our study is also related to the finance, economics, and accounting liter-ature studying the role of various mechanisms used to alleviate the lemonsproblem. These mechanisms include reputation, licensing, signaling by re-tained ownership, minimum quality standards, entry restriction, optimalcontracting, and choice of capital structure (see Akerlof [1970], Datar,Feltham, and Hughes [1991], Leland and Pyle [1977], Leland [1979], My-ers and Majluf [1984], Dybvig and Zender [1991]). We assume practicallimitations exist that prevent achieving the optimal contracts between thefirms and new investors described in Dybvig and Zender [1991].12 Specif-ically, because the audit technology is noisy, the firm may not always beable to signal its type via auditors’ attestation. This analytical approach is

liability provides stronger implicit insurance to investors. However, investors may not necessar-ily benefit from higher legal liability, because of the cost of litigation frictions. When the costof litigation frictions is high, higher auditor liability may give rise to a higher client rejectionrate and thereby reduce overall investment. In contrast, our paper focuses on the role of auditattestation service in a lemons setting and analyzes the link between audit liability and an au-ditor’s reporting conservatism. We study a setting involving information asymmetry betweena firm and its investors. An auditor certifies the firm’s report to investors. Under symmetricinformation, investors would invest in “ good type” but not in “bad type” firms. In the presenceof asymmetric information and no auditing, no investment would be made. Auditing enablesgood types to creditably signal their type and thereby attract investments. When an auditor’sliability increases, he behaves more conservatively and is more likely to reject the high-typefirm’s report, consequently reducing the firm’s incentive to invest in a capital market.

12 Dybvig and Zender explain how limitations induced by the inability to precommit, byrestrictions on indentured servitude, and by corporate governance rules can preclude a com-plete solution to the underinvestment problem through contracting. For instance, if the man-ager’s tenure is likely to end prior to the revelation of type, and if penalties on the managerbeyond his tenure are not feasible, optimal contracts are not implementable.


not foreign to the literature on how a middleman (an auditor in our con-text) helps guarantee and monitor product quality, thereby mitigating in-formation asymmetry between buyers and suppliers (see Biglaiser [1993],Biglaiser and Friedman [1994], Albano and Lizzeri [2001]). We contributeto this literature by showing that, although the auditor’s noisy attestationreduces the cost of asymmetric information, his conservative behavior maydistort the firm’s incentives and affect the investors’ valuation.

Interestingly, the extant literature in auditing suggests that increasing au-ditor liability may have positive effects on audit quality and investors’ con-fidence. Two notable studies—Schwartz [1997] and Newman, Patterson,and Smith [2005]—specifically look at the interactions among audit penal-ties, audit quality, and investor investments. Their results suggest that an in-crease in auditors’ liability may give rise to overinvestment. Schwartz [1997]argues that the auditor’s liability provides insurance for investors againstbad states of nature, which may lead to overinvestment. She also establishesthat a strict liability rule with a damage measure independent of the ac-tual investment can induce both the socially optimal investment and thesocially optimal audit effort. Newman, Patterson, and Smith [2005] showthat markets with greater auditor penalties for audit failure and greaterinsider penalties may lead to larger total investment levels, a higher pro-portion of the firms held by outsiders, and higher investment returns. Sim-ilarly, Laux and Newman [2010] analyze the effect of auditor liability onclient acceptance rather than capital investment.

Early theoretical studies in the accounting literature tend to focus onstudying the impact of audit quality on firm investments, showing that, be-cause higher audit quality provides better information to investors, the legalregime that induces the highest audit quality also generates the most effi-cient investment. For example, Dye [1993] suggests that equilibrium auditfees depend on both the informational value of the audit and on the audi-tor’s wealth. Melumad and Thoman [1990a,b] demonstrate that the pres-ence of a court system, increasing the exogenously specified damages, maylead to a Pareto improvement. Smith and Tidrick [1998] examine a modelof auditing and settlement under the U.S. and British systems and showthat more auditing can be a function of the unit cost of audit. Thoman andZhang [1999] point out that audit effort increases with the size of the dam-age award on audit effort, but may decrease with the rigor of the auditingstandards. Pae and Yoo [2001] indicate that an increase in auditors’ legalliability may result in underinvestment in the internal control system andoverinvestment in auditors’ effort.

In recent years, theoretical research on auditing has focused on char-acterizing the impact of specific institutional penalty alternatives on au-ditors’ effort, such as proportionate liability rules supported by the Com-panies Bill in the United Kingdom. Narayanan [1994] illustrates that, un-der the proportionate liability regime, the auditor pays only his share ofthe damages, so his litigation cost is more sensitive to his effort; hencehe has greater incentive to minimize his litigation cost by working harder.


Radhakrishnan [1999] finds that the presence of the recovery friction,whereby the lawyer retains a portion of any damages awarded as a con-tingent fee, leads to second-best efforts by the auditor and the manager.Hillegeist [1999] demonstrates that a change in a proportional auditor’s li-ability can decrease the audit failure rate, despite the concurrent decline inaudit quality. In contrast, Patterson and Wright [2003] consider the case inwhich the probability of detecting a misstatement depends on the auditor’seffort and on the evidence of fraud. They investigate whether the prescrip-tion of large marginal liability relief for audit quality can reduce audit riskrelative to joint and multiliability when the audit evidence of fraud is in-conclusive. Large marginal liability relief increases audit effort, reduces theauditor’s expected liability cost, and in turn provides less incentive for theauditee to reduce the fraud rate. In the meantime, the auditor’s evidenceevaluation becomes more liberal, which increases the auditor’s expected li-ability costs and provides more incentive for the auditee to reduce the fraudrate. In the authors’ equilibrium model of fraud, the net effect of these twoforces can motivate the auditee to commit fraud more often, and increasesaudit risk.

The analysis proceeds as follows: the model settings without and withauditing are described in sections 2.1 and 2.2, respectively. Section 3 estab-lishes the existence, uniqueness, and characterization of the equilibriumwith auditing. Section 4 develops the empirical predictions. We concludein section 5. We relegate proofs to appendix A.

2. The Model

2.1 THE SETTING WITH NO AUDITING: THE LEMONS INVESTMENT MODEL

In this section, we describe a variation of the Myers-Majluf investment“lemons” model in which a firm foregoes a positive net present valueproject due to asymmetric information. Then, in the following section, weintroduce a public auditor who, through attestation of the firm’s financialreport, serves to mitigate the lemons problem.

The risk-neutral firm is privately informed about the value of its existingequity (defined as assets minus liabilities). The value of existing equity isdenoted w t (where subscript t ∈ {h, l} denotes high or low values and p(h)denotes the common knowledge prior probability of the value being high).We refer to the firm with the high (low) value of existing equity as the high(low) type. (Because all parties are risk neutral, all results go through if thevalue of the existing equity is stochastic with mean w t .)

The firm also has a valuable new investment project that yields a positiveexpected net present value in excess of the initial investment, denoted E[x]> 0.13 Unlike Myers-Majluf, we assume common uncertainty exists regard-ing the present value of the new project. The firm must immediately issue

13 Myers and Majluf [1984] effectively made the same assumption by arguing that the firmwould have discarded an investment opportunity with negative present value before seekingnew investment from investors (p. 191).


equity to fund the project; otherwise, the opportunity is lost. (The resultsof this paper go through if the firm issues risky debt and is better informedabout its bankruptcy risk than are investors.) The assumption that invest-ment cannot be postponed captures the idea that a project loses value iffunding is delayed. The new issue amount, denoted I , represents the re-quired investment net of any funding the firm has available in the form ofcash, marketable securities, risk-free debt, and solicitations from existingshareholders. We assume existing assets are necessary for undertaking theproject, so the project cannot be spun off as a separate firm. We furtherassume the project is indivisible. We assume the firm invests whenever it is-sues new equity I . When we say the firm invests (does not invest), we meanthe firm both issues and invests (neither issues nor invests).

If the firm invests in the project, and investors know the firm’s type, thefull information market value of the type t firm is V t ≡ w t + E[x] + I . If it doesnot invest, the no-investment value of firm type t is the value of its existingequity w t . If both types of firms are assumed to invest and investors do notknow the true value of the firm’s existing equity, investors assess the firm’sno-information market value as

Vφ ≡ E [w | φ] + E [x] + I,

where E[w | φ] ≡ p(h)w h + (1 − p(h))w l and the φ denotes the fact thatinvestors have no information when forming expectations. Because we as-sume investors are risk-neutral in a competitive market and the risk-freerate of return is zero, investors provide I in exchange for the fraction I /V φ

of the firm. That is, investors expect a return of IVφ

· Vφ = I from an invest-ment of I . The existing shareholders retain the fraction (V φ − I )/V φ ofthe firm; their retained firm value as a function of the firm’s market valueis denoted

vt [Vφ] ≡ Vφ − IVφ

(wt + E [x] + I ) = wt + I{

E [x]I

− wt − E [w | φ]Vφ

}.

As in the Myers-Majluf model, we assume the firm invests if the retainedfirm value exceeds the no-investment firm value; that is, vt[V φ] > w t . Tohighlight the trade-off the firm faces in deciding whether to invest, one canrestate the retained firm value as follows:

REMARK 1. Retained Firm Value: The retained firm value of the type t firm is

vt [Vφ] = wt + I{

E [x]I

− wt − E [w | φ]Vφ

}.

The first term is the no-investment value of the firm, whereas the secondterm is the expected return of the new project in excess of the initial in-vestment. The first term in the brackets is the rate of return on the newinvestment, whereas the second term in the brackets is the cost of capitalnew investors require when they do not know the true value of the firm’s


existing equity.14 In other words, the second term is the lemons premiumabove the zero rate of return investors require. Note that, if investors knewthe firm’s type, the cost of capital is zero and all new projects would beundertaken.

In our model, a lemons problem occurs when the low type not the high type un-dertakes the project. Note the low type always invests because its cost of capitalis negative; that is, (w l − E[w | φ])/V φ < 0. To have a lemons problem,we assume the retained firm value for the high-type firm is less than theno-investment firm value:

wh > vh[Vφ]. (1)

In other words, the cost of capital for the high-type firm exceeds the newproject’s expected rate of return. That is,

wh − E [w | φ]Vφ

>E [x]

I. (2)

Because the firm with high value of existing equity does not undertake thevaluable project, the expected level of new investment is (1 − p(h))I andthe expected lost investment is p(h)I .

Examination of assumption (2) indicates that the underinvestment prob-lem occurs ceteris paribus when the expected returns E[x] are too low, therequired equity issue I is too large, the probability of the high type is toosmall, or the variance between high and low values, w h − w l , is too high.Furthermore, assumption (2) indicates that the cost of capital for the high-type firm is decreasing in investors’ perceptions of the firm’s existing equityvalue. If the high firm could improve investors’ perception of its existing eq-uity value sufficiently, the firm’s cost of capital could be sufficiently reducedsuch that the new project would be undertaken.

In the absence of a mechanism to penalize misstatements, both typeswould report high and the report would not be credible. Consequently,investors would assess the firm as having the no-information market valueand, given the assumption in (2), the underinvestment problem persists. Inthe next section, we examine the situations in which a mandatory audit isrequired, and how an audit may mitigate the underinvestment problem.

2.2 THE SETTING WITH AUDITING

In this section, we focus on attestation by an auditor as the mechanismby which the firm can credibly communicate its value. Previous analysesfocusing on auditors in financial markets have studied how the choice to

14 We define the cost of capital as the expected return on the firm’s market value. Thedenominator of the second term in the brackets is the firm market value when no informa-tion is available to investors. The numerator is the expected return, which is the differencebetween the actual firm value and the expected firm value. The definition is consistent withstandard asset pricing models in finance (e.g., Fama and Miller [1972]) and disclosure modelsin accounting (see Lambert, Leuz, and Verrecchia [2007]).


hire an auditor, the choice of auditor type, or the choice of auditor feesmay convey information to the market. In contrast, we focus on the infor-mation content of the auditor’s attestation of the firm’s financial report.To highlight the information content of attestation, we abstract from otherinformational roles auditors serve, by assuming a single-period setting inwhich auditing is mandatory, only one auditor type is present, and the au-dit fee is unobservable. In a single-period setting, repetition, learning, andreputation of the firm and auditor do not play a role. Because we assumeauditing is mandatory, the firm cannot signal its type through the decisionto hire an auditor (as in parts of Melumad and Thoman [1990a,b]). Be-cause only one type of auditor is present, no signaling is possible throughthe firm’s choice of auditor (as in Titman and Trueman [1986]). Further-more, because the firm does not have spare capital unless investors fundthe project, it cannot choose the level of investment to signal its type either.

In our model, the firm hires the auditor and pays him an audit fee F thatinvestors are unable to observe.15 We assume a competitive audit market,so the auditor receives a zero profit. Thus the audit fee F just equals theauditor’s costs of sampling CS and expected costs of audit errors (CI andCII ), which will be described in detail below.

The firm privately observes the value of its existing equity (i.e., its type)and privately submits an unaudited report of the value of its equity to theauditor for attestation. That is, the firm’s type is unobservable to the au-ditor and the investors, and the unaudited report is unobservable to theinvestors. We denote the two possible private reports as the unaudited highreport h or the unaudited low report l . We denote the firm’s reporting strat-egy as ρ(t | t), the probability of reporting t ∈ {h, l} when t ∈ {h, l} is thetrue type.16

The auditor tests the hypothesis that the firm’s unaudited report t isequal to its true type t. He collects audit evidence from the firm’s financialrecords incurring the cost of sampling Cs. The auditor’s audit evidence isan informative signal ω whose conditional distribution p(ω | t) is correlatedwith the true value of existing equity and independent of the new project’s

15 For ease of exposition, we assume an audit fee F is unobservable by investors. However,this assumption is not crucial. Though F differs across the different equilibria, observing F willnot impact our results because the amount of F is uninformative about the auditee’s type. Aswe illustrate later, F is endogenously determined ex ante for a given equilibrium. Moreover, wenote this assumption is in line with common IPO practice. When firms go public, disclosure ofIPO audit fees is only voluntary and many firms do not disclose them. Consequently, empiricalstudies of auditing in IPOs (e.g., Beatty [1993], Willenborg [1999], and Mayhew and Wilkins[2003]) use nonunderwriting expenses or accounting fees that issuing firms pay in the IPOas a proxy for the audit fees. According to the SEC guidelines, this amount may be estimatedand include all payments to the audit firms of issuing firms.

16 The assumption of binary firm types has greatly simplified the analysis. We expect thatour key results will continue to hold qualitatively when there is a continuum of firm types. Butthe calculations would be tedious and we would be unable to identify explicit expressions forthe parameter ranges under which the equilibrium arises.


future returns.17 The audit evidence is private information to the auditor;that is, it is unobservable to the firm and investors.

We consider the audit technology {p(ω | t), t ∈ {h, l}} consisting ofdiscrete conditional distributions, which is called a discrete audit tech-nology.18 The audit technology {p(ω | t), t ∈ {h, l}} captures all infor-mation about existing equity that is available solely to the auditor dueto his expertise in observing and interpreting financial records and valu-ing assets and liabilities. We assume the audit technology is imperfect inthat no audit evidence exists from which the auditor can infer the truetype with certainty. Further, we assume the audit technology is informa-tive in the sense of strict first-order stochastic dominance (FOSD); that is,∫ ω

ωp (ω | t = h)dω <

∫ ω

ωp (ω | t = l)dω.

Based on the audit evidence ω, the auditor either accepts (denoted a)or rejects (denoted r) the unaudited report t . The decision rule for accep-tance or rejection is denoted δ(d | t, ω); that is, the probability of makingdecision d ∈ {a, r} given the unaudited report is t and the audit evidence isω. Shibano [1990] shows that the ex post problem of accepting or rejectingthe firm’s unaudited report after observing the audit evidence is (almosteverywhere) equivalent to the ex ante problem (i.e., before observing theaudit evidence) of choosing the probability αt of rejecting a true unauditedreport, called the Type I error probability, and the probability βt of acceptinga false unaudited report, called the Type II error probability.19 Furthermore,Shibano [1990] shows βt is uniquely determined by αt according to a well-behaved functional relationship βt = Bt [αt ].

If the auditor rejects, we assume the firm undertakes the adjustments theauditor requires, revises its unaudited financial report to satisfy the auditor,and privately (i.e., unobservable to investors) incurs an audit adjustmentcost R ≥ 0.20 Audit adjustment costs include the costs of adjusting the firm’sfinancial records to conform with the auditor’s required adjustments, thecosts of additional evidence collection undertaken in response to evidence

17 Independence is a simplifying assumption that is not critical for our results.18 For tractability, we derive our predictions for discrete distributions. There is very little loss

in generality in confining attention to discrete audit technologies because, as Shibano [1990]shows, any audit technology can be approximated arbitrarily closely in an informational senseby a discrete audit technology.

19 Patterson [1993] extends the results of Shibano [1990] to include the auditor’s choiceof sample size. Alternatively, the auditor may exert costly effort to increase the precision ofthe audit technology. We assume that the auditor simply picks a fixed sample size that meetsminimum auditing standards, and consequently the only strategic decision is the choice ofaccept/reject regions. In other words, we focus on the auditor’s trade-off between Type I andType II errors given a fixed sample size (fixed audit evidence) and a fixed level of precision ofaudit technology.

20 In effect, we are assuming the auditor retains all the bargaining power over audit adjust-ments (see Antle and Nalebuff [1991]). None of our results are sensitive to this assumptionas long as the auditor has some bargaining power. Our analysis also assumes away the newinformation acquired by the additional evidence.


that is inconsistent with the firm’s unaudited report, and the costs of auditdelays.

The auditor releases a public report that may be either an audited highreport H or an audited low report L. In practice, both the initial unauditedfinancial report from the firm to the auditor as well as the adjustments theauditor requires prior to the public release of the audited financial state-ments are confidential. To capture the confidentiality of the auditor–firminteraction, we assume the audited high report H is released if the unau-dited high report was accepted or if the unaudited low report was rejected.Similarly, the audited low report L results from the unaudited low reportbeing accepted or from the unaudited high report being rejected.21

Investors observe only the audited report and update their prior expec-tation of the firm’s market value using rational expectations. That is, in-vestors revise beliefs using the equilibrium reporting strategy of the firm{ρ(h | h), ρ(h | l)} and the errors inherent in the audit evidence and theequilibrium audit decision rule {αh , αl}, and those beliefs are fulfilled inequilibrium. Investors’ valuation of the firm denoted VT , is conditional onthe audited report T , where T ∈ {H , L}, and on the firm’s equilibrium in-vestment decision; the investors’ valuation determines the firm’s cost ofcapital. Comparing the cost of capital with the project’s rate of return, thefirm decides whether to undertake the investment. We now characterizethe firm’s objective function. First, based on Remark 1 and the payoff spec-ification in section 2.2, we can show the following:

REMARK 2. The firm of type l chooses a reporting strategy ρ(h | l) to maximize itsobjective function:

Maxρ(h | l)

ρ(h | l)[(1 − Bh[αh])(vl [VL] − R − F ) + Bh[αh](vl [VH ] − F )]

+ (1 − ρ(h | l))(vl [VL] − F ), (3)

where vt [VT ] ≡ VT −IVT

(wt + E [x] + I ) is the type-t firm’s retained firm value, given

an audit report T ∈ {H , L} and before the payment of the audit fee F .

The first line of (3) is the expected utility to the low-type firm if it reportshigh. With probability (1 − Bh[αh]), the auditor correctly rejects the highreport so the low type receives vl [VL] and the firm pays an audit adjustmentcost R . With probability Bh[αh], the auditor commits a Type II error, falselyaccepting the high report, so the low type nets vl [VH ]. Because the hightype has no incentive to mimic the low type, only the low type is believedto report low, and the auditor always accepts a low report (i.e., αl = 0 and

21 We could accommodate the possibility of a publicly announced reject opinion in ourmodel by having a small probability that, when the auditor privately rejects, the negotiationsover audit adjustments break down and the rejection is announced publicly. The qualitativeresults in this paper would not change.


Bl [αl = 0] = 1).22 Thus, when the low-type firm reports a low report, it re-ceives vl [VL] without having to pay a rejection cost. Second, the firm of typeh chooses ρ(h | h) to maximize:

Maxρ(h | h)

ρ(h | h)[(1 − αh)(vh[VH ] − F ) + αh(wh − R − F )]

+ (1 − ρ(h | h))(wh − F ). (4)

We can interpret the maximization problem in (4) similarly. The expres-sion in the square brackets in (4) represents the expected utility to the highfirm if it reports truthfully. With probability αh, the auditor incorrectly re-jects the high report, and the firm pays the audit adjustment cost R .23 Withprobability (1 − αh), the auditor accepts the firm’s high report and the firmreceives vh[V H ].

Two types of auditor liability exist. First, investors in a firm with an au-dited high report may discover the firm’s true type was low. If those in-vestors had paid more than the value of a low-type firm, they may sue theauditor for damages. In other words, the auditor’s failure to detect a mis-statement in the firm’s audited report, referred to as an audit failure, mayresult in a liability cost to that auditor. In this case, we assume the audi-tor is assessed a penalty, A, to be paid as a damage award to investors. Aswe illustrate later (see Remark 5), investors incorporate this damage awardinto their market value of the firm.24 Because we are interested in study-ing the effect of varying the size of the damage payment on the resultingequilibria, we assume A is exogenous; in other words, the damage award Amay not perfectly reflect the size of the actual damages incurred (see alsoMelumad and Thoman [1990a,b]).25 In addition, the auditor may incur a

22 The detailed analysis is presented in the proof of Proposition 1.23 Audit adjustment costs include the costs of adjusting the firm’s financial records to con-

form with the adjustments the auditor requires, the costs of additional evidence collectionundertaken in response to evidence inconsistent with the firm’s unaudited report, and thecosts of audit delay. Our analysis assumes away the effect of new information acquired by theadditional evidence and that audit adjustment costs are not a function of the firm’s report.Our results are not qualitatively sensitive to these simplifying assumptions.

24 We assume investors can sue the auditor but not the firm. We adopt this assumptionfor two reasons. First, this assumption simplifies the analysis, permitting us to avoid questionsof how to divide damage payments between a firm and its auditor. In addition, it makes thesignaling problem more significant because the court system cannot directly punish a firm forlying. However, because the audit fee is determined competitively, it will reflect any expecteddamage payment; thus, ex ante, the auditor’s expected utility does not depend on the waydamages are assigned. Further, we assume penalties are imposed on the auditor but not onthe firm. In general, both the auditor and firm are subject to penalties if the auditor does notdetect misstatements. However, in an audit failure, the firm is often bankrupt, whereas theauditor has “deep pockets.” Assuming penalties on the auditor but not the firm is an extremecase capturing this differential ability to pay the penalty. This assumption is not uncommon inthe literature (see, e.g., Dye [1993, 1995], Chan and Pae [1998], Schwartz [1997], and Lauxand Newman [2010]).

25 We assume the damage award, A, is less than the firm’s value differential w h − w l . Ifthe investor is fully insured (i.e., A = w h − w l ), the auditor’s report would be irrelevant.Irrespective of the audit’s report, the firm would be valued as high type and all types of firms


legal cost, L, imposed through the court, legal fees, and opportunity costsof time spent in court. Investors will not recover this legal cost L, and it isconsidered the cost of litigation frictions (see Laux and Newman [2010]).In this paper, we denote the expected auditor liability due to a Type II au-dit failure by the sum of the penalty A and the legal cost L, that is, CII =A + L.26

Second, if the auditor erroneously rejects a high type who has reportedhigh, the auditor incurs expected losses associated with the possibility ofbeing replaced, as well as reputational cost.27 We denote the expected lia-bility under this circumstance as CI and refer to it as the cost of a Type Ierror. Our analysis focuses on policies affecting Type II costs, and thus wetake Type I costs to be exogenously given.28

The auditor’s problem can be rewritten by incorporating the auditor’sliability into (3). As per Remark 2, the auditor examining the firm’s reporth chooses αh to maximize:

Maxαh

p (h)ρ(h | h)[(1 − αh)(F − CS) + αh(F − CS − CI )]

+ (1 − p (h))ρ(h | l)[(1 − Bh[αh])(F − CS) + Bh[αh](F − CS − CI I )].(5)

In expression (5), the auditor receives his fee net of sampling costs, F −Cs , in all cases.29 But if he rejects (accepts) the high report from the high(low) type, he pays an additional Type I error cost CI (Type II error cost

would invest. Moreover, litigation friction may be such that investors cannot claim all thedamage from the auditor.

26 We abstract from the complexity involved in endogenously determining CII . Instead, weexamine how new investment, audit failures, and cost of capital are affected by legislative,judicial, and regulatory policy changes that increase or decrease audit liability. We also assumethe auditor is assessed an exogenous penalty, A, to be paid to investors as a damage award.If the damage award A were endogenous, investors would conjecture the auditor’s reportingstrategies and incorporate this damage award into their valuation of the firm. Moreover, theauditor would also need to conjecture investors’ reaction given the audit report, which in turndetermines the firm’s valuation and the audit liability. Our equilibrium analysis herein wouldstill be valid under this setting, but it would be difficult to characterize explicitly the firm’svaluation and the auditor strategies.

27 See, e.g., DeAngelo [1981], Dye [1991], and Kanodia and Mukherji [1994].28 Audit standard setters claim Type I error probability is relatively insignificant because an

auditor would perform additional audit procedures until he confirms a truthful disclosure.AICPA [1986] states “This definition of audit risk does not include the risk that the auditormight erroneously conclude that the financial statements are materially misstated. In such asituation, he would ordinarily reconsider or extend his auditing procedures. . . . These stepswould ordinarily lead the auditor to the correct conclusion.” The purpose of this paper is toaddress the effect of higher legal liability on the probability of accepting a false unauditedreport (a Type II error). Thus we take the costs of a Type I error to be exogenously fixed (seealso Melumad and Thoman [1990a,b], Pae and Yoo [2001], and Laux and Newman [2010]).

29 If we were to relax the assumption of a perfectly competitive audit market and study anoligopoly audit market, the auditor would expect to earn a positive payoff and a higher auditfee compared with that under perfect competition. This assumption would subsequently in-crease an efficient firm’s cost to separate itself from an inefficient firm. As a result, an efficient


FIG. 1.—Timeline of events.

CII ). Again, because only the low type is assumed to report low, the auditoralways accepts a low report (i.e., αl = 0, and Bl [αl ] = 1).

The equilibrium is specified by the reporting and investment strategiesfor the firm, {ρ(t | t), {invest, not invest}}, the audit opinion strategy forthe auditor, {δ(d | t, ω)}, and the equilibrium market valuation by investors{VT , T ∈ {H , L}}. See figure 1 for a timeline of the setting with auditingand appendix B for a summary of the notation.

3. Equilibrium Analysis

3.1 BENCHMARK CASE: NO INFORMATION ASYMMETRY

In this section, we first consider a benchmark case in which neither thefirm nor investors observe the value of existing equity w t . The firm is re-quired to hire an auditor and issue an audited accounting report. At thebeginning of the period, the firm hires the auditor and pays him an auditfee F unobservable to investors.

Both firm types submit an unaudited high report (i.e., t = h andρ(h | h) = 1). As shown in (5), the auditor chooses the probability of re-jecting a true unaudited report α

hto maximize the expected payoff. In the

absence of information asymmetry, the optimal probability, denoted by αf

h,

is given by

B′h

[α

fh

] = − p (h)CI

(1 − p (h))CI I. (6)

The second-order condition implies that when the auditor liability CII in-creases, the auditor will reduce the probability of a Type II error even ifdoing so means increasing a Type I error.

firm would sometimes find financing the project to be too costly, because of a higher auditfee. Consequently, the loss of funding positive NPV projects may decrease a social warfare.


REMARK 3. When the auditor liability CII increases, the auditor increases theType-I error αh (∂α

fh/∂CI I > 0) and reduces the Type II error Bh[α f

h].

Conditional on the audited report T , where T ∈ {H , L}, investors accessthe firm’s market value as

VT ≡ E [w | T] + E [x] + I,

where E [w | T] ≡ Pr(wh | T)wh + (1 − Pr(wh | T)) is the expected equityvalue conditional on an audit report T . Investors provide I in exchangefor the fraction I/VT of the firm. The existing shareholders retain the frac-tion (VT − I )/VT of the firm; their retained firm value as a function of thefirm’s market value is denoted

vt [VT ] ≡ VT − IVT

(E [w | T] + E [x] + I )

= E [w | T] + E [x] > 0.

As in Myers-Majluf, we assume the firm invests in the project when theretained firm value exceeds the expected no-investment firm value; thatis, vt [VT ] > E [w | φ] where T ∈ {H , L}. Given a high audit report H , theretained firm value is strictly higher than E[w | φ], and thus the project isfunded. In contrast, when the auditor issues a low report L, the retainedfirm value is not necessarily higher than the expected no-investment firmvalue. Whether investors will fund the project depends on the magnitudeof E[x]. When E[x] is relatively small or when the variance between highand low firm value w ≡ w h − w l is large (so that E [w | L] is small), thefirm will not invest in the project.

As the auditor liability CII increases, the auditor becomes more conserva-tive by selecting a higher α

fh

and a lower Bh[α fh

]. Consequently, the auditorissues a low report more often to reduce the likelihood of incurring theoverstatement penalty (a Type II error). However, a countervailing indirecteffect occurs when the auditor report is conservative, a high audit reportbecomes more informative (d Pr(wh | H )/dα

fh

> 0),30 thereby giving rise toa higher Pr(wh | H ) and a higher retained firm value vt [VH ].

Suppose the new investment is funded only when the audit report is high.Denote the probability of new investment by G ≡ p (h)(1 − α

fh

) + (1 −p (h))Bh[α f

h], where α

fh

is characterized by (6). From Remark 3, as the audi-

30 The derivative

d Pr(wh | H )

dαf

h

=−p (h)(1 − p (h))

(Bh

[α

fh

] + (1 − α

fh

)B′

h

[α

fh

])[p (h)

(1 − α

fh

) + (1 − p (h))Bh[α

fh

]]2 > 0,

because Bh[α fh

] + (1 − αh)B′h[α f

h] < 0 from Proposition 0.


tor liability CII rises, the auditor increases the Type I error (∂αf

h/∂CI I > 0)

and the level of new investment decreases.31

If the firm’s existing equity value is not observable, investors and thefirm reach the same expected equity value E [w | H ]. The cost of capital in-vestors require is determined by (E [w | H ] − E [w | H ])/VH , which is zero.Because the required cost of capital is zero, all new projects given an auditreport H should be undertaken.

The results are summarized as follows.

REMARK 4. In the absence of information asymmetry, as auditor liability for au-dit failures increases, the retained firm value vt [VH ]increases and new investmentdecreases. But the level of auditor liability does not affect the cost of capital.

3.2 CHARACTERIZATION OF THE EQUILIBRIUM

In this section, we characterize the equilibrium when the firm privatelyobserves the value of existing equity and privately submits an unaudited re-port of the value of its equity to the auditor for attestation. The existenceof equilibria involves a complex relationship between the behavior of theauditor and the firm during the audit engagement and investors’ expecta-tions regarding firm value based on the information content of the auditedreport. The reporting strategy of the firm both depends on and affects in-vestors’ expectations. The auditor’s decision rule affects investors’ expecta-tions and simultaneously depends on and affects the firm’s behavior.

The following proposition demonstrates that the equilibrium in ourmodel always exists, is unique, and can be characterized in closed form.The equilibrium takes one and only one of two forms. The detection equilib-rium involves the auditor effectively reducing the underinvestment prob-lem while detecting but not deterring overstatements by the low-type firm. Inthe deterrence equilibrium, the auditor decreases underinvestment while de-tecting and deterring overstatements. Because the auditor is effective in re-ducing the underinvestment problem in both the detection and deterrenceequilibria, we refer to settings in which one of these equilibria hold aseffective-auditing settings.32

PROPOSITION 1. Existence, Uniqueness, and Characterization ofEquilibria: The equilibrium in the market for lemons setting with

31 This inequality can be verified by inspecting the derivative:

∂G∂CI I

= −p (h)∂α

fh

∂CI I+ (1 − p (h))B′

h[α

fh

]< 0.

32 We ignore a possible form of “ineffective-auditing” equilibrium wherein the auditor isineffective at reducing the underinvestment problem. In this equilibrium, the revised cost ofcapital (assuming both types invest) is still too high to induce the high type to invest. There-fore, rational investors know that, if a firm with the audited high report invests, it must be alow type. In equilibrium, the cost of capital for a firm with the audited high report that investsis based on the expectation that the firm has the low-value equity. Thus the high type does notinvest and the low type does. That is, similar to the lemons setting without auditing, only thelow type invests; thus the auditor is ineffective at mitigating the underinvestment problem.


T A B L E 1Equilibrium Strategies

Effective-Auditing Settings

Detection DeterrenceStrategies equilibrium equilibrium

High-type firm’s unauditedreport

h h

Low-type firm’s unauditedreport

h h with prob. ρ(h | l)

Auditor’s decision rule givenunaudited high report

α∗h α∗∗

h

Auditor’s decision rule given αl = 0. Belief: αl = 0. Belief:unaudited low report prob.(l | l) = 1 prob.(l | l) = 1

Investment decision of highfirm w/audited high report

Invest Invest

Investment decision of highfirm w/audited low report

Do not invest Do not invest

Investment decision of lowfirm w/audited high report

Invest Invest

Investment decision of lowfirm w/audited low report

Invest Invest

T A B L E 2Classes of Equilibrium Reporting Strategies

Low type reportshigh

Low type randomizesbetween high and

low reports Low type reports low

High type reportshigh

Class 1 Class 2 Class 3: Does notexist by Lemma 1

High typerandomizesbetween highand low reports

Class 4:Informationallyequivalent toClass 6


Class 6: Does notexist by Lemma 2

High type reportslow




auditing always exists, is unique, and is either a detection equilibrium or a deterrenceequilibrium.

Proof . See appendix A for all proofs. �

In the remainder of this section, we describe the behavior predicted ineach form of equilibrium. More precise statements and mathematical ex-pressions are in appendix A. Table 1 summarizes the equilibrium strategiesof the auditor and the firm for each type of equilibrium.

3.2.1. Detection Equilibrium. The detection equilibrium has three distin-guishing features. First, both firm types submit unaudited high reports andthe auditor rejects the unaudited high report with equilibrium probability


α∗h

specified in (A5). Second, if the auditor receives the out-of-equilibriumunaudited low report, he believes the firm is low and always accepts. (Thesebeliefs are robust to standard equilibrium refinements.) Finally, the lowtype invests in response to both the audited high report and the auditedlow report.

Although investors revise their valuation of the firm upward and, conse-quently, the firm’s cost of capital downward under the detection equilib-rium, the revised cost of capital in the latter case is low enough to inducethe high type with the audited high report to invest. Therefore, under adetection equilibrium, investors can rationally expect the firm with the au-dited high report to be a high type with positive probability consistent withthe auditor’s equilibrium decision rule. In equilibrium, the high type withthe audited high report invests; the auditor’s report is effective in partiallyreducing the underinvestment problem. We call this equilibrium “detec-tion equilibrium” because the auditor is effective in partially “detecting”the low type who overstates, but is ineffective at “deterring” the low typefrom overstating.

3.2.2. Deterrence Equilibrium. The deterrence equilibrium differs fromthe detection equilibrium in two ways. First, the low type does not alwaysoverstate. The low type is “deterred” from always overstating by the ex-pected costs of having the unaudited high report rejected. This deterrenceeffect of the auditor involves a complex equilibrium interaction betweenthe firm’s reporting strategy, the auditor’s decision rule, and the market’svaluation process. Second, the auditor accepts or rejects the unaudited highreport with a different equilibrium probability from that under the detec-tion equilibrium.

Similar to the detection equilibrium, the deterrence equilibrium involvesthe high type always reporting high, the auditor always accepting the unau-dited low report, the low type always investing, and the high type investingonly if it receives an audited high report. Therefore, the auditor is effectivein partially mitigating the underinvestment problem.

In what follows, we focus on analyzing the economic consequences ofincreasing the auditor’s liability in the detection equilibrium. The resultsof the deterrence equilibrium are qualitatively similar and hence are notpresented.33

4. Predicted Effect of Increases in Auditor Liability

4.1 EQUILIBRIA IN WHICH LITIGATION OCCURS

The motivation of this study is to shed light on the equilibrium effects ofincreasing auditor liability. In general, audit legal liability depends on threekey factors, including the probability of misstatements in the firm’s report,

33 The detailed characterization of the deterrence equilibrium is available on the authors’website.


the probability that the audit would fail to detect the misstatement, andthe probability that the auditor would incur a legal liability due to an auditfailure (Choi et al. [2008]). In the effective-auditing settings, investors valuethe firm with the audited high report sufficiently high to induce the hightype with the audited high report to invest. After equity is issued and theinvestment undertaken, a positive probability exists that the firm with theaudited high report will be discovered to actually be a low type. Investorsreceive I

VHVH with probability p (h)(1 − αh) if the true type is high, and I

VHVl

with probability (1 − p(h)) ρ(h | l) Bh [αh] (where ρ(h | l) may equal one)if the true type is low. In the latter case, the investors suffer a capital lossof I − I

VHVl . Because investors sustained actual damages, they have grounds

for litigation.

4.2 INCREASES IN AUDITOR LIABILITY

The following three propositions involve predictions of an increase inauditor liability in the effective-auditing equilibria.

4.2.1. Prediction Regarding New Investment. The key contribution of thispaper is the analysis of how and why increasing auditor liability may de-crease new investment. As we discussed in the introduction, the possibil-ity that raising legal liability for auditors will have a negative impact onthe capital market concerns many market participants. Unsurprisingly, thefour biggest audit firms have been attempting to reduce the threat of litiga-tion. However, the arguments market participants typically make are qual-itative and speculative in nature. In fact, the extant theoretical literature(Schwartz [1997], Newman, Patterson, and Smith [2005]) suggests that anincrease in auditors’ liability after SOX may result in increased investment.Our analysis establishes an equilibrium explanation for when and why in-creasing auditors’ liability may decrease investment; it provides theoreticalarguments consistent with the concerns we describe in the introduction.Specifically, Proposition 2, below, establishes that, when an auditor’s liabil-ity for audit failure increases, the audited firms are less likely to undertakenew investment.

In this model, the probability that the firm undertakes new investment isthe sum of the probabilities that the high firm and the low firm invest con-ditional on the probability of the firm’s type. Because investors always setthe cost of capital low enough to induce the low type (with either auditedreport) to invest, the joint probability that the firm is low and undertakesinvestment is simply (1 − p(h)). Because the high type firm only invests if itreceives the audited high report, the probability of the high type investingp (h)(1 − αh) is the joint probability of the type being high and the auditorattesting to the audited high report when the firm type is high.

In the detection equilibrium, both firm types submit unaudited high re-ports and the probability of new investment in the detection is denotedby G∗ = p (h)(1 − α∗

h), where the equilibrium probability α∗

his specified

in (A5): p (h)CI + B′h[α f

h](1 − p (h))CI I = 0. Thus, as auditor liability in-

creases, the probability of new investment is lower for


∂G∗

∂CI I= −p (h)

∂α∗h

∂CI I< 0.

The following proposition captures the effect of increasing liability on theprobability of the firm undertaking new investment.

PROPOSITION 2. Effect of Increased Liability on New Investment: As auditorliability for audit failures increases, new investment decreases.

Understanding the trade-offs the auditor faces clarifies the intuition un-derlying Proposition 2. As we discussed in section 2.2, the auditor tradesoff expected losses from attesting to an understated versus an overstated re-port. Thus, increases in the cost of an overstatement error lead directly tothe auditor rejecting the unaudited high report more often to reduce thelikelihood of incurring the overstatement penalty. However, a countervail-ing indirect effect occurs. In the proof of Proposition 2, we established thatthe more likely the auditor is to reject an unaudited high report, the morelikely it is that a firm with an audited high report is indeed a high type.34

As investors’ valuation of the firm with the audited high report goes up,the likelihood of the low type to overstate rises, increasing the likelihoodof the auditor incurring overstatement losses. Thus, it is not immediatelyclear that the auditor will reject a high report more often as the liabilityincreases.

The key step in proving Proposition 2 is showing that, in the effective-auditing settings, the auditor rejects more often as liability increases. Thatis, the direct effect always dominates the indirect effect. As the auditor in-creases the probability of rejecting an unaudited high report, the probabil-ity of new investment declines. To see why, note that both the low and thehigh types are having their unaudited high report rejected with a higherprobability. Every time a high firm’s unaudited high report is rejected, itreceives an audited low report and finds its cost of capital is too high.

Looking beyond this stylized setting, our analysis suggests that, as an audi-tor’s liability increases, any firm with private information about its existingequity will find its financial statement more likely to be understated. As longas investors cannot infer the true value from the understated reports, thefirm’s valuation will be depressed, thereby increasing its cost of capital andmaking it less likely to pursue new, economically desirable investments.

4.2.2. Prediction Regarding Audit Failure Rate. The audit failure rate is thepercentage of audits involved in litigation associated with the failure todetect a materially misstated financial report that caused investors actualdamages. In this model, the audit failure rate is the equilibrium joint prob-ability, in effective-auditing settings, of the firm type being low, the low firmsubmitting an unaudited high report, and the auditor observing evidencethat leads the auditor to erroneously accept the unaudited high report.

34 See also Gigler et al. [2009].


In the detection equilibrium, the audit failure rate is (1 − p(h)) ρ(h | l)Bh[α∗

h], where (1 − p(h)) is the prior probability of the firm being low

and Bh [α∗h] is the equilibrium probability of accepting the unaudited high

report from a low firm. Note that, in the detection equilibrium, the proba-bility of the low firm submitting an overstated report is one.

The following proposition answers the question of how increasing audi-tor liability affects the audit failure rate.

PROPOSITION 3. Effect of Increased Liability on Audit Failures: As auditorliability for audit failures increases, the audit failure rate decreases.

The intuition for Proposition 3 is as follows: as auditor liability for au-dit failures increases, overstatement errors become more costly to the au-ditor. In equilibrium, the auditor reacts by rejecting the unaudited highreport more often. The audit technology is informative, so the auditor isincrementally more likely to reject a low type than a high type. A decreaseoccurs in the equilibrium probability that the firm with an audited high re-port is actually a low type. Thus, in the detection equilibrium, the auditor’sincreasing “conservativeness” leads directly to a lower probability of auditfailure litigation.

Propositions 2 and 3 therefore suggest the following remark:

REMARK 5. Tradeoff Between New Investment and Audit Liability: As audi-tor liability increases (decreases), both new investment and audit failures decrease(increase).

4.2.3. Prediction Regarding Costs of Capital. As we discussed in section 2.1,the firm’s cost of capital is a function of the investors’ valuation of the firm.In this section, we first discuss how investors value the firm and set its cost ofcapital, and then we develop predictions regarding how changes in auditorliability affect the cost of capital.

In making their valuation decisions, investors have rational expectationsin that they take into account both the equilibrium reporting and issu-ing/investing behavior of the firm and the equilibrium audit opinion. Thefollowing remark captures how investors incorporate their expectations re-garding the equilibrium interaction between the auditor and firm into theirvaluation of the firm.

REMARK 6. Equilibrium Market Valuation: (a) In both the detection and thedeterrence equilibria, the market value of the firm with the audited low report is

VL = Vl ≡ wl + E [x] + I .

(b) In the detection equilibrium, the market value of the firm with the audited highreport is

V ∗H

[α∗

h

] = γ[α∗

h

]wh + (

1 − γ[α∗

h

])(wl + A) + E [x] + I,


where

γ[α∗

h

] =p (h)

(1 − α∗

h

)p (h)

(1 − α∗

h

) + (1 − p (h))(Bh

[α∗

h

]) .

The intuition behind Remark 6(a) is that investors know that, in equilib-rium, any rational valuation of the firm with the audited low report wouldnot be high enough to induce the high type to invest. Therefore, any firmwith the audited low report that invests in equilibrium must be the low type.

In contrast, in the detection equilibrium, investors’ valuation of the firmwith the audited high report is high enough to induce the high type withthe audited high report to invest. Therefore, it is rational for investors toassign an expected value to the firm with the audited high report that isconsistent with the firm’s equilibrium reporting/investing behavior and theauditor’s equilibrium decision rule.

Therefore, in Remark 6(b), the market price V ∗H

[α∗h] in the detection

equilibrium of the firm with the audited high report is a Bayesian weightedaverage of the full information high and low values, where the weight γ [α∗

h]

on the high type (weight 1 − γ [α∗h] on the low type) is the conditional prob-

ability, given the audited high report, of the type being high (low), the high(low) type reporting high, and the auditor not rejecting the high reportfrom the high (low) type.

Now we can identify the cost of capital for the firm with the auditedreport. Recall that in equation (2), the high firm’s cost of capital in theno-auditing world is wt −E [w |φ]

Vφ. The lack of information about existing eq-

uity is denoted φ, signifying the empty set. The analogous expression inthe effective-auditing settings is (wt − E [w | T])/VT , where the audited re-port T could be high H or low L and where in equilibrium, E [w | L] = wl

and E [w | H ] = γ wh + (1 − γ )(wl + A), where γ equals γ [α∗h] defined in

Remark 6(b). Proposition 4 summarizes predictions regarding the cost ofcapital:

PROPOSITION 4. Effect of Increased Liability on Cost of Capital: Increasingauditor liability for audit failures

1. does not affect the cost of capital for a firm with an audited low report.2. decreases the cost of capital for a firm with an audited high report.

The effect of auditor liability changes for the audited low report isstraightforward. Because the only type that invests with the audited low re-port is the low type, the cost of capital to the firm with that report is wt −wl

Vl,

t = h, l . This cost is invariant to the auditor decision rule and thus alsoinvariant to any changes in the auditor’s liability.

In contrast, the cost of capital the firm with the audited high report facesdoes depend on the auditor’s decision rule in equilibrium. In effective-auditing settings, the expression for cost of capital is

wt − E [w | H ]VH

= wt − γ wh − (1 − γ )(wl + A)γ wh + (1 − γ )(wl + A) + E [x] + I

,


for t = h, l , where γ equals γ [α∗h] as defined in Remark 6(b). The impact of

higher auditor liability on the high type firm’s cost of capital is ambiguous.On the one hand, it makes the auditor more conservative and more likely toreject a high report increasing the high-type firm’s cost of capital. On theother hand, the market assigns a higher value to an audited high reportbecause investors know the report is more likely to have been issued by thehigh-type firm, thereby reducing its cost of capital. Thus a change in auditorliability influences both the equilibrium audit decision rule and the firm’sreporting, and subsequently affects the cost of capital.

The surprising feature of Proposition 4.2 is that, by construction, the ex-pected returns to investing in the firm with the high report is constant andequal to the investors’ required rate of return. However, each type of firmthat has the audited high report faces a lower cost of capital as auditor lia-bility increases. This seeming paradox is a variation of Simpson’s ReversalParadox studied in Sunder [1983]. In Simpson’s Paradox, a weighted aver-age consists of weighting on two factors. The paradox occurs when the fac-tors decrease but their weighted average increases. The increased weightedaverage results from putting sufficiently higher weight on the larger of two(now reduced) factors.

In our model, the expected return to investors from investing in the firmwith the audited high report is constant because we assume the investorsare competitive and require a zero rate of return. At the same time, thecost of capital for each type of firm with the audited high report is de-creasing, because the weight on the high-type firm with the high report hasincreased.

The distinctive feature of the Simpson Paradox in our model is that it isendogenously generated by the auditor’s optimal use of audit evidence inresponse to an increase in auditor liability. As liability increases, the audi-tor rejects the high report with greater frequency, but he does not do this“blindly” with respect to the audit evidence. If an auditor simply rejects ahigh report more often unconditional on the audit evidence, the conditionalprobability of the firm being high given that it has received the auditedhigh report stays constant. Therefore, the cost of capital for the firm withthe high report would stay constant.

The auditor uses the information in the audit evidence to reject the highreport more often when the audit evidence indicates the unaudited highreport is most likely coming from the low type. Consequently, the incre-mental probability of rejecting a high report from the low type is higherthan the incremental probability of rejecting a high report from the hightype. As a result, the conditional probability of the firm type being highgiven the audited high report increases. In this model, Simpson’s Paradoxarises as an endogenous response to higher liability and the optimal use ofaudit evidence. The apparent paradox is resolved by the observation thatincreasing auditor liability leads the auditor to reject both types more often,but he optimally uses his information to reject the high type incrementallyless often than the low type. Thus, the expected return to investing in the


firm with the audited high report is constant, but the cost of capital for eachfirm type with the audited high report decreases. We highlight this resultin Corollary 1:

COROLLARY 1. Effect of Increased Liability on Audit Conservatism: As auditorliability for audit failures increases, the auditor becomes more conservative.

Proposition 4.2, together with Corollary 1, confirms the intuition that, inresponse to increased auditor liability, new investment falls not because thecost of capital increases, but because the auditor becomes more conserva-tive and therefore more likely to certify understated audited reports for thehigh-type firm.

5. Concluding Comments

The economic trade-offs our model documents may be of interest to cap-ital market participants and regulators. On the one hand, imposing higherlegal liability may prevent the occurrence of audit failures, a desired conse-quence of the Sarbanes-Oxley Act. On the other hand, an increase in legalliability may curtail new investment, because of the information asymmetrybetween the firm and new investors. Our study identifies and characterizesthis economic trade-off, but is silent on how to resolve it. Future researchmay consider positing a social welfare function that trades off the disutilityof audit failures versus the utility of new investment, thereby determiningthe socially optimal level of liability.

Our study can be extended in several ways. First, the consequences ofSOX are much broader than the impact of high auditor liability we explorehere. For example, Section 2.1 of SOX establishes auditor independencestandards and limits the extent of non-audit services. This limitation couldbe a mixed blessing from investors’ perspective. Non-audit and audit ser-vices may be synergetic, and thus improve profitability and increase thevalue of audit attestation. Yet non-audit services may also reduce the au-ditor’s independence and possibly adversely affect the value of attestation.How limiting the extent of non-audit services would affect the auditor’s lia-bility and investors’ incentive to invest in capital markets is not immediatelyclear (see Lu and Sapra [2009]).

A second extension would be to allow for heterogeneous audit quality.One could measure audit quality by the extent to which an audit report in-creases the market value of a firm (see Dye [1993]). On the one hand, thepost-audit market value increases in the probability of rejecting an unau-dited high report, which consequently discourages the low type from over-stating its report. On the other hand, higher rejection probability leads tohigher market value for a firm with an audited high report; therefore, a lowtype would possibly be more willing to submit an unaudited high report.Thus, the net effect of higher audit quality, measured by the probabilityof rejecting an unaudited high report, is not immediately clear, as it is afunction of the relative magnitude of two opposite effects. We leave thesepotential extensions for further research.


APPENDIX A

Proofs

The results that follow depend on the general characterization of therelationship between error probabilities presented in Proposition 0 below:

PROPOSITION 0 (Shibano [1990]). The choice among decision rules δ(d|t, ω)is payoff-equivalent to the choice among error probabilities (αt , βt ), where t is thenull hypothesis.

Proof . A nondegenerate audit technology is defined as a pair of distri-butions {p(ω | t), t ∈ {h, l}}, such that p(ω | h) �= p(ω | l)on a set of ω ofpositive measure. An audit technology {p(ω | t), t ∈ {h, l}} has constantsupport if the set of ω that have positive probability when the type is h isidentical to the set of h that have positive probability when the type is l .That is, {z | p(ω | h) > 0} = {z | p(ω | l) > 0}.

For any nondegenerate audit technology with a constant support,

(i) The auditor can confine his choice of tests to the class of tests ofthe null t versus the alternative t ′ represented as an error functionBt [αt ] mapping [0,1] to [0,1], where Bt [αt ] is convex and strictlydecreasing on [0,1] and differentiable for almost every αt ∈ [0, 1].Bt [αt ] is the test with the lowest Type II error probability for testswith Type I error probability αt .

(ii) A well-defined inverse function B−1t [αt ] mapping [0,1] to [0,1]

exists. B−1t [βt ] is convex and strictly decreasing on [0,1] and dif-

ferentiable for almost every βt ∈ [0, 1]. B−1t [βt ] is the lowest Type

I error probability for tests with Type II error probability βt .(iii) Bt [0] = 1 and Bt [1] = 0. Furthermore, B−1

t [0] = 1 and B−1t [1] =

0.(iv) −B′

t [αt1] ≥ −B′t [αt2] if and only if αt1 ≤ αt2.

(v) 1 − αt− Bt [αt ] >0, ∀αt ∈ (0, 1). When this condition holds, we saythe audit technology discriminates between types.

(vi) B′t [αt ](1− αt )+ Bt [αt ] < 0,∀αt ∈ (0, 1).

(vii) Any function g(·) that maps [0,1] into [0,1] is everywhere con-vex and strictly decreasing, and almost everywhere differentiableis an “admissible” audit technology. That is, a pair of distributions{p(ω | t), p(ω | t ′)} exists such that g(·) represents the class of testsof the null t versus the alternative t ′ under those distributions.

(viii) An audit technology {p (ω | t), p (ω | t ′)} is a discrete audit technol-ogy if both p (ω | t) and p (ω | t ′) are discrete probability distribu-tions. For any arbitrary audit technology {p(ω | t), p(ω | t ′)} with er-ror function Bt [ · ], a discrete audit technology {p (ω | t), p (ω | t ′)}exists with error function Bt [ · ] such that Bt [ · ] is arbitrarilyclose to Bt [ · ], that is, for any ε > 0, and for any αt ∈ (0, 1)∣∣Bt [αt ] − Bt [αt ]

∣∣ <ε. Because {p (ω | t), p (ω | t ′)} is a discrete au-dit technology, Bt [ · ] is piece-wise linear, Bt [αt ] is differentiablealmost everywhere, and B′′

t [·] = 0 almost everywhere. �


Proof of Remark 2.Step 1: Consider the case in which the auditor examines the unaudited

high report. Let U (d | t, h) denote the auditor’s utility from decision d giventrue type t and unaudited report h. The ex post problem (i.e., after theauditor observes h) facing the auditor examining report h , for each ω ∈, is

Max{δ(d | h,ω),d=a,r}

∑t=l,h

∑d=a,r

p (t)ρ(h | t)p (ω | t)

p (h | ω)δ(d | h, ω)U (d | t, h).

Step 2: Factoring out 1/p (h | ω) because it does not affect the maximiza-tion, the auditor’s problem becomes


∑t=l,h

∑d=a,r

p (t)ρ(h | t)p (ω | t)δ(d | h, ω)U (d | t, h).

Step 3. The ex post problem in step 2 can be restated as the ex anteproblem (i.e., at the point in time before he observes ω) below:


∫�

∑t=l,h

∑d=a,r

p (t)ρ(h | t)p (ω | t)δ(d | h, ω)U (d | t, h)dω.

The solution of the ex ante problem is (almost everywhere) identical tothat of the ex post problem because the ex ante problem involves point-wise maximization for each ω.

Step 4: By algebraic rearrangement, an equivalent statement is


p (h)ρ(h | h)

{U (a | h, h)

∫�

p (ω | h)δ(a | h, ω)∂ω

+ U (r | h, h)∫�

p (ω | h)δ(r | h, ω)dω

}

+ (1 − p (h))ρ(h | l)

{U (r | l, h)

∫�

p (ω | l)δ(r | h, ω)∂ω

+ U (a | l, h)∫�

p (ω | h)δ(a | h, ω)dω

}.

Step 5: By definition of error probability {αh, βh}, the auditor’s maximiza-tion reduces

Max{αh ,βh}

p (h)ρ(h | h)[U (a | h, h)[1 − αh] + U (r | h, h)αh

]+ (1 − p (h))ρ(h | l)

[U (r | h, h)[1 − βh] + U (a | h, h)βh

]Step 6: By Proposition 0, the auditor’s problem can be equivalently stated

as a choice of the optimal αh only:

Maxαh

p (h)ρ(h | h)[U (a | h, h)[1 − αh] + U (r | h, h)αh

]+ (1 − p (h))ρ(h | l)

[U (r | h, h)[1 − Bh[αh]] + U (a | h, h)Bh[αh]

].


The simplification of the decision rule given the unaudited low report issimilar. �

Proof of Remark 3. The auditor’s optimal probability αf

his determined by

B′h

[α

fh

] = − p (h)CI

(1 − p (h))CI I.

Taking a partial derivative, we obtain ∂αf

h/∂CI I = −B′

h[α fh

]/(CI I B′′h [α f

h]) >

0. That is, the probability αh increases in CII , which subsequently decreasesthe Type II rate Bh[α f

h]. In addition, because Bh[·] is convex and strictly

decreasing on [0, 1], the second-order condition −(1 − p (h))CI I B′′h [α f

h] <

0 is satisfied by Proposition 0 (see Shibano [1990]). �

Proof of Proposition 1. The analysis in this section proceeds with the fol-lowing four steps:

Step 1: The analysis proceeds using backward induction: Assuming a de-terrence or a detection equilibrium exists, we then state the player’s objec-tive functions.

Step 2: Then we show the only reporting strategies that appear in equi-librium of the audit engagement involve both types reporting high or thehigh type reporting high and the low type randomizing between high andlow reports. See Lemmata 1–3.

Step 3: In Lemma 4, we identify auditor decision rules that satisfy oneor more conditions for the existence of equilibrium, and we show eachdecision rule exists for any given parameters. Then, in Remark 7, we statethe necessary and sufficient conditions for equilibrium in terms of theseauditor decision rules.

Step 4: In the supplementary material, we prove equilibrium exists, isunique, and is either a detection or deterrence equilibrium.35 Moreover,we provide a numerical example of the existence of each of the two equilib-rium forms to show the feasibility of satisfying the conditions we identified.

Step 1: The player’s objective functions. Recall that in the deterrenceequilibrium, the high-type firm reports high, the low type randomizes be-tween the high and low reports, and both types invest. The detection equi-librium is similar except that the low type always reports high. Because weassume only the low type reports low, the auditor always accepts a low re-port (that is, αl = 0 and Bl [·] = 1). Based on Observations 1 and 2 and thepayoff specifications in section 2.2, we characterize the objective functionsin the detection and deterrence equilibrium as follows: The firm of type lchooses ρ(h | l) to maximize:

ρ(h | l)[(1 − Bh[αh])(vl [VL] − R − F ) + Bh[αh](vl [VH ] − F )]

+ (1 − ρ(h | l))(vl [VL] − F ). (A1)

35 The supplementary material is available from the authors on request.


The firm of type h chooses ρ(h | h) to maximize:

ρ(h | h)[(1 − αh)(vh[VH ] − F ) + αh(wh − R − F )] + (1 − ρ(h | h))(wh − F ).

(A2)

The auditor examining report h chooses αh to maximize:

p (h)ρ(h | h)[(1 − αh)(F − CS) + αh(F − CS − CI )]

+ (1 − p (h))ρ(h | l)[(1 − Bh[αh])(F − CS) + Bh[αh](F − CS − CI I )].(A3)

The first line of (A1) is the expected utility to the low-type firm if it reportshigh. With probability (1 − Bh[αh]), the high report is rejected so the lowtype receives vl [VL] and pays an audit adjustment cost R . With probabilityBh[αh], the high report is accepted so the low type nets vl [VH ]. If the lowtype reports low, the auditor finds it optimal to accept that report as truth-ful. Thus the low type receives vl [VL] without having to pay a rejection cost.Expression (A2) is interpreted similarly. In expression (A3), the auditor re-ceives his fee net of sampling costs, F − Cs, in all cases. But if he rejects(accepts) the high report from the high (low) type, he pays an additionalType I error cost CI (Type II error cost CII ). Because both the detectionand deterrence equilibria involve only the low type possibly reporting low,the auditor examining the unaudited low report always accepts it.

Step 2: The possible equilibrium reporting strategies areClasses 4, 7, 8, and 9 are informationally equivalent to Classes 6, 3, 2, and

1, respectively, because, for instance, the report l is not the literal statement“ low,” but rather it is a report consistent in equilibrium with the statement“I am a low type.” For instance, when the high-type firm always reports l andthe low-type firm randomizes between h and l , the auditor must in equilib-rium interpret l to mean “ I am a high type.” Thus, for example, equilibriain Class 6, wherein the high-type firm randomizes between reports and thelow-type firm always reports low, is informationally equivalent to equilibriain Class 4, wherein the high-type firm randomizes between reports and thelow-type firm always reports high.

We show the equilibria in Classes 3, 7, 6, and 4 do not exist. Then we showthe equilibria in Class 5 do not exist generically. Then, in the followingsteps, we show the equilibrium with reporting strategies in Classes 1 or 2exists and is unique.

LEMMA 1. Fully separating reporting (in Classes 3 and 7) does not occur inequilibrium.

Proof . Assume the high type always reports high and the low type alwaysreport low. Then the auditor always accepts both the high report and lowreport because he suffers losses from incorrectly rejecting either report.Assume investors set firm values by assuming each report is correct. But nowthe low type defects to reporting high because his payoff from reportinghigh is vl [VH ] − F , whereas his payoff from reporting low is vl [VL] − F . �


LEMMA 2. Equilibria in Class 6 and 4 do not exist.

Proof . Assume the high type randomizes between reports and the lowtype always reports low. Then the auditor always accepts the high report(i.e., αh =1) because he suffers losses from incorrectly rejecting a report.By assumption, the auditor must choose an equilibrium αl that makes thehigh type indifferent between reports. Assume investors set firm values be-lieving the high report is correct and the low report could come from a highor low type. Then the high type’s payoff from reporting high is vh[VH ] − F ,whereas his payoff from reporting low is (1 − Bl [αl ]) (vh[VH ] − F − R)+Bl [αl ](wh − F ). Clearly, a dominant strategy is for the high type to reporthigh regardless of the αl the auditor selects, which contradicts the require-ment that the auditor must choose an αl that makes the high type indiffer-ent between reporting high or low. �

LEMMA 3. Equilibria in Class 5 do not exist generically.

Proof . Assume both the high type and the low type randomize betweenreports. Then the auditor must choose an equilibrium αh and αl that makeseach type indifferent between reports. Assume investors set firm values as-suming either report could come from either type. Then the high type’spayoff from reporting high must be equal to his payoff from reporting low.That is,

(1 − αh)(vh[VH ] − F ) + αh(wh − R − F ) = (1 − Bl [αl ])(vh[VH ] − F − R)

+ Bl [αl ](wh − F ).

Recall that

vh[VH ] = VH − IVH

(wh + E [x] + I ).

Rearranging yields

VH = (αh − Bl [αl ])(wh + E [x] + I )IBl [αl ](R − E [x] − I ) + αh(R + E [x] + I ) − R

≡ f (αh, αl ).

Similarly, the low type’s payoff from reporting low must be equal to hispayoff from reporting high:

(1 − αl )(wl − F ) + αl (vl [VH ] − F − R) = (1 − Bh[αh])(wl − F − R)

+ Bh[αh](vl [VH ] − F ).

Recall that

vl [VH ] = VH − IVH

(wl + E [x] + I + A),

where A is a damage reward from the auditor when the auditor’s report isH but the firm’s type is l . Note that if A = w h − w l ; then investors can fullyrecover the damage from an audit failure and vl [VH ] = vh[VH ]. We assumeA is strictly less than w h − w l . The low type’s indifference condition yields


the following restriction:

VH = (Bh[αh] − αl )(wl + E [x] + I + A)IBh[αh](R + E [x] + I + A) + αl (R − E [x] − I − A) − R

≡ g(αh, αl ).

A third restriction on VH is that investors use Bayes’s rule to set the firm’smarket value:

VH

= whp (h)[ρ(h | h)(1 − αh) + ρ(l | h)(1 − Bl [αl ])] + wl (1 − p (h))[ρ(h | l)Bh[αh] + ρ(l | l)αl ]

[ρ(h | h)(1 − αh) + ρ(l | h)(1 − Bl [αl ])] + [ρ(h | l)Bh[αh] + ρ(l | l)αl ]

+ E [x] + I

= h(αh , αl ). (A4)

Each of the three equations above, f (αh, αl ), g(αh, αl ), and h(αh, αl ), arefunctions of the exogenous variables and the auditor’s two decision vari-ables h(αh, αl ). Fixing the exogenous variables, the system of three equa-tions in two unknowns is “overdetermined.” That is, except for “knife-edge”conditions on the exogenous variables, the system of equations does nothave a solution. A more precise discussion of a nongeneric equilibrium in-volves the following: let the exogenous parameters of a given setting be apoint {w h, w l , E[x], I , R , CI , CII , CS} in the space R8. Then an equilibriumis nongeneric if the set of exogenous parameters for which an equilibriumexists is a set of measure zero in the set of parameters.

Together Lemmata 1–3 show the only reporting strategies that appear inequilibrium involve both types reporting high or the high type reportinghigh and the low type randomizing between high and low reports. �

Step 3:

LEMMA 4. If the equilibrium involves both types submitting unaudited high re-ports, the following auditor decision rules exist:

1. α∗h

∈ [0, 1] is the auditor’s optimal decision rule.2. αLR∗

h∈ [0, 1] at which the low type is indifferent between unaudited reports

and below which the low type prefers to report high.3. αH R∗

h< 1 at which the high type is indifferent between unaudited reports and

below which the high type prefers to report high.4. αH I ∗

hat which the high type with the audited high report is indifferent about

investing and above which the high type prefers to invest.

Proof .

1. From the first-order conditions of the auditor’s objective function in(A3) when ρ(h | l) = 1, the decision rule that maximizes the auditor’spayoff is the α∗

hthat solves

−B′h[αh] = p (h)CI

(1 − p (h))CI I. (A5)


The second-order condition (1 − p (h))CI I B′′h[αh] < 0 is satisfied by

Proposition 0(i). So α∗h

always exists in [0, 1].2. The low type is indifferent among reports in a detection equilibrium if

Bh

[αLR∗

h

] = Rvl

[VH

[α∗

h, l

]] − (wl + E [x] + A) + R. (A6)

Substituting into (A6) the expression for vl [VH [α∗h, l]] and the expres-

sions for VH [α∗h] from Remark 6(b), where ρ(h | l) = 1, results in a

quadratic equation in βh . Additional calculations show the positive so-lution to the quadratic equation that specifies βh as a function, de-noted k[·], of αh and other exogenous variables:

Bh = k[αh] = 12

− (wh + E [x]+ I )R + I (wh − wl − A)2(wl + E [x] + I + A)R

(1−αh)p (h)

1−p (h)

+{[

(wh + E [x] + I )R + I (wh − wl )2(wl + E [x] + I + A)R

(1 − αh)p (h)

1 − p (h)

]2

+ wh + E [x] + Iwl + E [x] + I + A

(1 − αh)p (h)

1 − p (h)

}1/2

.

k[·] is the locus of auditor tests that makes the low-type firm indif-ferent between reporting high and low, taking into account the in-vestors’ endogenously derived market value in the detection equilib-rium. Tedious calculations show k[·] is always strictly greater than zero,strictly less than one, and strictly increasing in α. Because Bh[·] is al-ways strictly between zero and one and is strictly decreasing in α, anαLR∗

hexists that solves Bh[αh] = k[αh]. For any αh <( > ) αLR∗

h, the low-

type firm strictly prefers to report high (low).3. From equation (A2), the high type always reports high if

α∗h ≤

vl[VH

[α∗

h

]] − wh

vl[VH

[α∗

h

]] − wh − R≡ αH R∗

h . (A7)

By construction, 0 < αH R∗h

< 1. In a detection equilibrium in whichα∗

h∈ [0, αH R∗

h], the high type would always report high, whereas if α∗

h∈

(αH R∗h

, 1], the high type would defect to reporting low.4. The high type with the audited high report invests in a detection equi-

librium if and only if

vh[VH

] =VH

[α∗

h

] − I

VH [α∗h]

(wh + E [x] + I ) ≥ wh . (A8)

Rearranging yields

VH

[α∗

h

] =whp (h)

(1 − α∗

h

) + wl (1 − p (h))(Bh

[α∗

h

])p (h)

(1 − α∗

h

) + (1 − p (h))(Bh

[α∗

h

]) + E [x]

+ I ≥ (wh + E [x] + I )IE [x] + I

, (A9)


or (Vh − VhI

E [x] + I

)p (h)(

VhIE [x] + I

− Vl

)(1 − p (h))

(1 − α∗

h

) ≥ Bh[αh], (A10)

where V t = w t + E[x] + I , t ∈ {h, l}. The coefficient on (1 − α∗h) is

positive because the numerator is positive by inspection and the de-nominator is also positive because

VhIE [x] + I

− Vl = whIE [x] + I

+ I − Vl

=(

whIE [x] + I

− wl − E [x])

>

(whI

E [x] + I− E [w] − E [x]

)> 0.

The coefficient on (1 − α∗h) is less than one because

(Vh− Vh IE [x]+I )p (h)

( Vh IE [x]+I −Vl )(1−p (h))

< 1

if and only if (Vh − VhIE [x]+I )p (h) < ( VhI

E [x]+I − Vl )(1 − p (h)), if and only ifVφ <

VhIE [x]+I , which follows from assumption (2).

A unique αH I ∗h

∈ (0, 1] exists that solves(Vh− Vh I

E [x]+I )p (h)

( Vh IE [x]+I −Vl )(1−p (h))

(1 − αh) =Bh[αh]. For all α∗

h∈ [αH I ∗

h, 1], the high type with the audited high report

would be willing to invest in a detection equilibrium. �

REMARK 7. The necessary and sufficient conditions for a detection equilibriumto exist are

(P1) The auditor’s decision rule given the unaudited high report maximizes hispayoff. That is, the auditor’s equilibrium decision rule is α∗

hand it exists

in the interval [0,1].(P2) The auditor’s decision rule given the unaudited low report maximizes his

payoff. Because the auditor believes an out-of-equilibrium unaudited lowreport comes from the low type, the auditor finds it optimal to always acceptthat report; i.e., α∗

l= 0.

(P3) The low type always reports high; i.e., α∗h

< αLR∗h

.

(P4) The high type always reports high; i.e., α∗h

< αH R∗h

.

(P5) The high type with the audited high report invests; i.e., αH R∗h

< α∗h.

(P6) The low type invests. This condition is always satisfied.

Step 4: In the supplementary material, we prove equilibrium exists, isunique, and is either a detection or deterrence equilibrium. Moreover, weprovide a numerical example of the existence of each of the two equilib-rium forms to show the feasibility of satisfying the conditions we identified.


Proof of Proposition 2.Step 1: We first prove

∂α∗h

∂CI I> 0. Implicit differentiation yields

∂α∗h

∂CI I= p (h)CI

(1 − p (h))C2I I B′′

h [α fh

]> 0. (A11)

�Step 2: The probability of new investment in the detection equilibrium

is denoted G∗ = p (h)(1 − α∗h). We can show that by Step 1,

∂G∗

∂CI I= −p (h)

∂α∗h

∂CI I< 0. (A12)

Proof of Proposition 3. Let the audit failure rate in the detection equilib-rium be denoted AF R∗ = (1 − p (h))Bh[α∗

h]. Then

∂AF R∗

∂CI I= (1 − p (h))B′

h[α∗h]

∂α∗h

∂CI I< 0, (A13)

by Proposition 0(i) and Proposition 2. �Proof of Proposition 4. Denote the cost of capital to the high type

with the high report in the detection equilibrium as kh(H ∗) = (wh −E [w | H ∗])/VH ∗ . Then

∂kh(H ∗)∂CI I

=p (h)(1 − p (h))(wh − wl − A)Vh

{Bh

[α∗

h

] + (1 − α∗h)}

VH ∗

∂α∗h

∂CI I.

(A14)

By Proposition 0(vi) and Proposition 2, the sign of the derivative is negativein the detection equilibria. Similarly, kl (H ∗) < 0. �

APPENDIX B

Notation

Types, Unaudited Reports, and Audited Reportsw t Value of the existing equity in firm type tE[x] Expected net present value of the investment

projectI Investment required to undertake the projectp(h) Probability of high-type firmt ∈ {h, l} Firm type, where w h > w l

t ∈ {h, l} Unaudited report from firm to auditorT ∈ H , L Audited report from auditor to investorsThe Firm’s Market ValueV φ = p(h)w h + (1 − p(h))w l + E[x] + I Firm’s market value conditional on no informationVT [αh, ρ(h|l)], T = H , L Firm’s market value conditional on audit report TShare of Firm Retained by Existing ShareholdersVφ −I

VφShare of firm retained by existing shareholders

given no information and both types investVT − I

VTShare of firm retained by existing shareholders

given audit report T and both types invest


Retained Firm Valuevt [Vφ] = Vφ−I

Vφ(wt + E [x] + I ) Retained firm value of firm type t given no information and

both types investvt [VT ] = V

T−I

VT

(wt + E [x] + I ) Retained firm value of firm type t given audit report T andboth types invest

Firm’s Strategiesρ(t |t) Probability of reporting t given type tAuditor’s PayoffsF Audit feeCS Cost of sampling for auditorCII Cost of a Type II error, including a damage award A and a

legal cost LCI Cost of a Type I errorAudit Technologyω ∈ � Audit evidence{p(ω | t}, t ∈ {h, l}} Audit technologyAuditor’s Strategiesδ(d | t, ω) Probability of making decision d ∈ {a, r} given the null is t

and the audit evidence is ω

αh Probability, under decision rule δ, of falsely rejecting the nullof t =h when the true value is t = h

βh Probability, under decision rule δ, of falsely accepting the nullof t =h when the true value is t = l

α∗∗h Type I error prob. where low type is indifferent between

reports in the deterrence equilibriumα∗

h Type I error prob. that is optimal for the auditor if both typesreport high

αH R∗h Type I error prob. below which the high type is willing to

report high in the detection equilibriumαLR∗

h Type I error prob. below which the low type is willing toreport high in the detection equilibrium

αH I∗h Type I error prob. above which the high type invests in the

detection equilibrium

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