the effects of business risk on audit pricing

Review of Accounting Studies, 3, 365–385 (1998)c© 1998 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.

The Effects of Business Risk on Audit Pricing

JOHN MORGANWoodrow Wilson School, Princeton University

PHILLIP STOCKENWharton School, University of Pennsylvania, 2426 Steinberg Hall-Dietrich Hall, Philadelphia, PA 19104-6365

Abstract. This paper examines the pricing of business risk by homogeneous auditors in a two period model.Incumbent auditors learn the client’s business risk type during the course of the engagement. They subsequentlycompete in prices with prospective auditors. In such an environment, we show that equilibrium audit fees do notfully reflect the cost of business risk. Moreover, there exists differential auditor turnover between high and lowrisk firms; cross-subsidization of the audit fees of high risk firms by low risk firms; and low-balling by auditors.

Auditors have been inundated with shareholder lawsuits. Malpractice-litigation costs of BigSix accounting firms, after insurance recoveries, have substantially increased, and by 1993amounted to nearly twelve percent of these firms’ total accounting and auditing revenue(Lambert (1994)). Furthermore, claims against non-Big Six firms rose by two-thirds from1987 to 1991. In 1990, the seventh largest accounting firm in the United States, Laventholand Horwath, filed for bankruptcy. The failure of Laventhol and Horwath was mainlyattributed to incurred and anticipated litigation costs. The firm’s chief executive officercontended that litigation arose, not from inadequacies in its professional performance, butfrom the perception that the firm had a “deep pocket” (Arthur Andersen,et al. (1992, 3)).Similarly, O’Malley (1993, 84–85), chairman and senior partner of Price Waterhouse,claimed that “unwarranted litigation and forced settlements constitute the vast majorityof claims against accountants” and that shareholders demand compensation from auditorseven if the auditor is not responsible for shareholders’ losses.

Against this background of increasing potential liability exposure due to circumstancesthat are largely outside the auditor’s control, it is important to consider the effect of businessrisk on audit pricing. Business risk, which is the focus of this paper, is defined by the Amer-ican Institute of Certified Public Accountants (AICPA) (1992) as having two components:client’s business risk, the risk associated with the client’s continued survival and well-being;and auditor’s business risk, the risk of potential litigation costs and other expenditure fromassociation with a client irrespective of whether or not an audit failure is asserted. Thus,we interpret business risk as the risk to the auditor of a lawsuit that remains after takingall steps required under the Statements of Auditing Standards (SAS) while performing theaudit and issuing an audit report. It must be stressed that it is often impossible for theauditor to avoid being sued regardless of due diligence efforts. O’Malley (1993, 93) arguesthat “the auditors may be sued by anyone who suffered a financial loss even though thatperson may never have so much as glanced at an audit report. The auditors need not havedone anything to cause the loss.”1

366 JOHN MORGAN AND PHILLIP STOCKEN

An essential feature of the auditing environment is that auditors learn about the businessrisk associated with a client over the course of the engagement. There are a number of factorsthat affect a firm’s business risk that may be known to an incumbent auditor but not to aprospective auditor. Such factors include: management integrity; ambiguous accountingprinciples that apply to a firm’s transactions; asset valuations that necessitate substantialjudgment; and nature of the ownership of the company.2 Knowledge of these factors mayprovide an incumbent auditor with an informational advantage relative to his competitors.

There appears to be little consensus in the accounting literature as to the appropriate modelof auditor competition for client engagements. Areas of minimal agreement are that it ismost common for a client, rather than an auditor, to make the appointment decision aftersoliciting proposals from competing auditors (Glezen and Elser (1996) and SAS No. 84),3

and that auditors effectively compete in audit fees (rather than quantities or other strategicvariables). Eichenseher and Shields (1983) find that the audit fee is generally the mostimportant choice variable in a firm’s auditor selection problem. Similarly, Simunic (1980)and Rubin (1988) find evidence of price competition among auditors in the private andmunicipal sectors, respectively. Thus, we model competition among auditors as a pricinggame. That is, all competing auditors simultaneously submit fee offers for the engagement,and the client selects the lowest.

Utilizing this framework, we examine the role of information about client business riskon audit fee determination in a two period setting. In such an environment, we show thatequilibrium audit fees do not fully reflect the cost of business risk. Moreover, there existsdifferential auditor turnover between high and low risk firms; cross-subsidization of theaudit fees of high risk firms by low risk firms; and low-balling by auditors. Finally, weoffer empirical predictions regarding audit pricing based on changes in the business riskand litigation environments in recent years.

Our paper is related to Schatzberg and Sevcik (1994) as well as Kanodia and Mukherji(1994). Both of these papers examine equilibrium audit pricing under circumstances inwhich the incumbent auditor has an informational advantage over his rivals as to the costof performing the audit. In Schatzberg and Sevcik, prospective auditors compete withincumbents by competing in audit fees with no cost to switching auditors. In Kanodiaand Mukherji, there are switching costs and the client chooses an optimal mechanism indetermining the price of the engagement.

Our paper differs from Schatzberg and Sevcik in that the cost of performing an audit iscommonto both prospective and incumbent auditors, but known only to the incumbent au-ditor; whereas Schatzberg and Sevcik have audit costs that arespecificto each auditor-clientpair. As a result of this difference, the pricing of the prospective auditor in Schatzberg andSevcik reflects the expected costs of the engagement; whereas, due to winner’s curse adjust-ments, such a strategy is inconsistent with competitive equilibrium pricing in our model.

Kanodia and Mukherji also have a cost of performing the audit engagement that is commonto both incumbent and prospective auditors, but differs in the form of the mechanism used bythe client in setting audit fees and the presence of a cost to switch auditors.4 In the absenceof costs associated with changing auditors, the optimal mechanism from the perspective ofthe client has the property that the client is able to obtain audit services at cost, and theclient changes auditors every period. Giving all of the bargaining power to the incumbent

THE EFFECTS OF BUSINESS RISK ON AUDIT PRICING 367

auditor reverses this result in that an optimal mechanism entails no auditor turnover andpricing at the highest possible cost.5 One may then view our assumption that audit pricingoccurs on the basis of fee competition as a middle ground, giving some bargaining powerto both the client and the incumbent auditor. Here, we show that the presence of frictionsis not required for differential rates of auditor turnover to occur.

Underlying this difference is the nature of the informational asymmetry generating therents (and hence low-balling) for the incumbent auditor. In Kanodia and Mukherji the centraltension occurs in the negotiations between the client and the incumbent auditor. Specifically,the client designs an incentive compatible contract to extract information about the true costof the audit; naturally, this leads to the incumbent auditor receiving information rents. Theprospective auditor plays the role of an outside option in the negotiations between the clientand the incumbent auditor. In contrast, we focus on the role of private information in thecompetitionbetweenauditors.

More broadly, our paper is related to Engelbrecht-Wiggans, Milgrom, and Weber(EWMW) (1983) who consider bidding strategies in first-price common value auctionswith a continuum of possible valuations. Notice that our second period game is similarin that it may entail a dominantly informed bidder competing against others who are lessinformed. However, unlike EWMW, our model of price competition has the feature thatwith positive probability, the incumbent auditor’s information may beidenticalto that of theprospective auditor; thus the incumbent auditor is onlypotentiallydominantly informed.Moreover, the dynamic nature of the model also differs from EWMW, which is concernedwith equilibria in static bidding games.

The rest of the paper proceeds as follows: Section 1 describes a model of audit competitionin the presence of business risk. The audit and litigation environment and the extensiveform of the game are formulated. In Section 2, the equilibrium of the game is identified.In Section 3, the pricing of business risk, auditor turnover, and low-balling are shown toarise endogenously in this setting. We conclude this section by examining the implicationsof changes in the stringency of auditing standards regarding business risk on equilibriumpricing behavior. Finally, Section 4 draws conclusions. All proofs are contained in theAppendix.

1. Model

Auditors engage in fee competition in an environment where it is mandatory for the firm’sfinancial statements to be examined by an auditor each period. Since Statements of Au-diting Standards place requirements on the nature of the auditor’s activities, and the au-ditor’s ultimate responsibility is the issuance of an audit opinion on the fair presentationof the financial statements, it is assumed that audit quality is homogeneous among audi-tors. Moreover, since auditors use remarkably similar audit technologies, it is supposedthat the cost of performing the audit is constant across the auditors that compete for theengagement.6 Auditors, however, may have differing beliefs about the business risk as-sociated with the (potential) client. Accordingly, strategic interaction between competingauditors is driven mainly by auditors’ beliefs about a firm’s business risk. Because thefocus of this paper is pricing of the risk of potential litigation cost irrespective of whether


or not an audit failure is alleged, it is assumed auditors’ potential liability arises when someadverse circumstance occurs, such as financial distress, management fraud, or some illegalact.

The audit pricing decision is examined within a two period game which has the followingextensive form. In the first period, there are two risk-neutral auditors and an audit clientor firm.7 Initially, both auditors are assumed to be symmetrically uninformed about theclient’s risk type. The competing auditors have common prior beliefs about firm typei ,wherei = (L)ow risk or (H)igh risk. The prior beliefs are Pr(i = H) = λ > 0 that thefirm is of the high business risk type and Pr(i = L) = 1− λ that the firm is of the lowbusiness risk type. The high type firm has a probability of litigation ofπ > 0 in eachperiod. The low type firm has a zero probability of litigation.8 The probability of litigationis common knowledge.

In both periods, auditors simultaneously choose fees, which are denoted with the vari-able R, for performance of the audit. This assumption is analytically equivalent to thefirm receiving the offers at different times, but not revealing the offers that are submitted.However, even if the client were to disclose the fee offers, competing auditors recognizethat the firm has a strategic incentive to under-report fee offers. In addition, the clientmay choose to re-bid the engagement or otherwise offer the incumbent auditor the right torespond to offers by prospective auditors. However, if, as seems likely, the client cannotcommit to truthfully disclose the current “best” offer, then the incumbent auditor will haveno incentive to revise his current offer. Thus, the game form remains simultaneous despitesequential firm disclosures or allowing the client to re-bid the engagement.9

The firm chooses the lower of these fee offers; however, if the fee offers are equal, thenthe firm randomizes between the auditors, giving equal weight to each.10 The appointedauditor is referred to as the incumbent auditor and the auditor who is not appointed as aprospective auditor. The incumbent auditor then performs the audit, and with commonlyknown probability,α, successfully determines the client’s business risk type. It is useful tothink of α as a parameter for the degree of informational advantage held by the incumbentauditor.

Finally, the incumbent auditor issues an audit report. Following this, nature determineswhether litigation occurs, and this event (or non-event) is publicly observed. If litigationoccurs, the incumbent auditor incurs an expected liability ofB, after which the game ends.11

If litigation does not occur, the game continues. The prospective auditor updates his priorbeliefs as to the firm’s type in the usual Bayesian fashion following this event (or non-event). The posterior beliefs formed as a result of this updating process are denoted byλ′,the revised probability that the firm is a high risk type.

If no litigation occurs, then in the second period auditors once again compete in price bysimultaneously offering fees for the performance of the audit. The simultaneous nature ofthe strategies chosen by the two firms implies that the fee offered by the incumbent auditoris not observable by the prospective auditor and vice versa. The firm then chooses theauditor with the lower fee; however, in this period, in the case of a tie the client retainsthe incumbent auditor.12 At the end of the second period, nature again determines whetherlitigation occurs, and the second period auditor incurs an expected liabilityB. A key featureof the second period competition is that the incumbent auditor may have private information


Figure 1. Time line of events.

about the firm’s risk type, whereas the prospective auditor only knows the firm’s risk typeprobabilistically.

The time line in Figure 1 summarizes the sequence of events.In this model, we take the game form as given, rather than being chosen by the client

as in Kanodia and Mukherji; thus, the focus is on strategic interaction between competingauditors with the firm being a non-strategic player. This model also does not consider theeffects of the audit opinion on the pricing of business risk. Provided that the space of firmrisk types is richer than that of the possible opinions (i.e., going concern modificationsand the like) that the auditor may issue and that “opinion shopping” (see Dye (1991) andSchatzberg (1994)) is not a first order strategic concern, the opinion issued by the incumbentauditor will not be fully informative of the firm’s risk type. The issuance of an opinion onlymitigates the information asymmetry between the competing auditors without eliminatingit.

The respective auditor’s problem in the two period game may be formalized as follows.If the incumbent auditor learns the client’s business risk type in the first period, his problemis expressed as:

If the firm is a high type then,

maxRH

E(W2

H

) = (RH − πB)Pr(RH ≤ RP) . (1)

If the firm is a low type then,

maxRL

E(W2

L

) = RL Pr(RL ≤ RP) . (2)

If, on the other hand, the incumbent auditor does not learn the client’s business risk typein the first period, then his problem is expressed as:

maxRU

E(W2

U

) = (RU − λ′πB)

Pr(RU ≤ RP) , (3)

whereRi is the incumbent auditor’s second period fee offered to thei type firm,i ∈ {H, L}∪U ,

whereU denotes a firm of unknown type;


RP denotes the prospective auditor’s second period fee; andPr(Ri ≤ RP) is the probability that the incumbent auditor offers a lower fee than the

prospective auditor.The prospective auditor’s problem, given probabilityα that the incumbent auditor has

learned the client’s type, is represented as:

maxRP

E(W2

P

) = α(λ′ (RP − πB)Pr(RP < RH )+

(1− λ′) RP Pr(RP < RL)

)+ (1− α) (RP − λ′πB

)Pr(RP < RU ) , (4)

wherePr(RP < Ri ) is the probability that the prospective auditor submits a lower offer than the

incumbent auditor and thereby obtains appointment.The posterior beliefs are updated based on the publicly observable litigation state and are

determined by Bayes’ Rule as follows:

λ′ = (1− π) λ1− πλ . (5)

Thusλ′ is the probability that the firm is a high type given that it has survived to the secondperiod.

Finally, auditor 1’s (respectively 2’s) problem in the first period is formalized as:Given some equilibrium strategies in period 2, chooseR1 such that:

maxR1

E (W1) =(

Pr(R1 < R2)+ 1

2Pr(R1 = R2)

) (R1− λπB+ E1

(E(W2

I

)))+(

Pr(R2 < R1)+ 1

2Pr(R1 = R2)

)E1(E(W2

P

)), (6)

whereR1, R2 denote the audit fees offered by auditor 1 and auditor 2, respectively;E1(E(W2

P)) is the expected profit earned by the prospective auditor in the second periodevaluated using the auditor’s prior beliefs as they existed in the first period; and

E1(E(W2I )) is the expected profit earned by the incumbent auditor in the second period

evaluated using the auditor’s prior beliefs as they existed in the first period. Specifically,

E1(E(W2

I

)) ≡ α((1− λ) E

(W2

L

)+ λ (1− π) E(W2

H

))+ (1− α) (1− λπ) E

(W2

U

).

2. Analysis

Before characterizing the unique Perfect Bayesian Equilibrium of the game in Theorem 1,let FP(R) denote the cumulative distribution of audit fees,RP, charged by the prospectiveauditor.13Similarly, letFi (R) represent the cumulative distribution function for the informedincumbent auditor’s fee,Ri , conditional on the firm’s known business risk typei . Finally,let FU (R) represent the cumulative distribution function for the uninformed incumbentauditor’s fee,RU .


Theorem 1 The unique Perfect Bayesian Equilibrium14 for the two period game is givenby:

The client chooses the lowest bid in both periods.A prospective auditor randomizes his bid according to:

FP (R) =

0 where −∞ < R< λ′πB1− λ′πB

R where λ′πB ≤ R≤ λ′πB1−α+αλ′

1− λ′πBα(1−λ′)R−λ′πB where λ′πB

1−α+αλ′ ≤ R< πB1 otherwise

and earns zero expected profits.An informed incumbent auditor who knows that the firm is a low business risk type

randomizes his bid according to:

FL (R) =

0 where −∞ < R< λ′πBR−λ′πBα(1−λ′)R where λ′πB ≤ R≤ λ′πB

1−α+αλ′1 otherwise

and earns expected second period profits of

E(W2

L

) = λ′πB.

An informed incumbent auditor who knows that the firm is a high business risk type bids:

RH = πB

and earns expected second period profits (losses) of

E(W2

H

) = 0.

And an uninformed incumbent auditor randomizes his bid according to:

FU (R) =

0 where −∞ < R< λ′πB

1+α+αλ′R(1−α+αλ′)−λ′πB(1−α)(R−λ′πB) where λ′πB

1+α+αλ′ ≤ R≤ πB1 otherwise

and earns expected second period profits of

E(W2

U

) = λ′πBα(1− λ′) .


Finally, in the first period the competing auditors bid:

R1 = λπB− E1(E(W2

I

)).

The beliefs of all parties are specified by Bayes’ rule wherever possible. Off-equilibriumbeliefs of all players are that the strategies specified above will be played regardless ofhistory.

The nature of the equilibrium strategies arises largely from the need for the prospectiveauditor to avoid the adverse selection problem inherent to his potentially disadvantagedposition. Specifically, the prospective auditor is faced with a “lemons” problem in decidingwhat fee to charge. In the event that he follows some pure strategy of pricing above theexpected costs of unknown types, but less than the expected cost of a high type firm, it isclear that he will be undercut in the case where the client is an unknown or low risk typeand left with only high risk firms. Likewise, the case where the prospective auditor pricesabove zero but below the expected costs of unknown types, leads to an adverse selectionproblem where prospective auditors end up solely with unprofitable high and unknowntypes. Thus, the prospective auditor mixes to avoid this undesirable outcome. While onemay be troubled that the only equilibrium to this game is in mixed strategies, the fact that thegame has an unique mixed strategy equilibrium is entirely consistent with the practitioners’complaint that they are unable to price business risk in engagements (see Lochner (1993),Bell, Landsman, and Shackelford (1995)). Moreover, it is useful to keep in mind that thesestrategies may be thought of as the limit of pure strategies in a sequence of “perturbedgames.” See Osborne and Rubinstein (1994, 41-43) for a more detailed discussion.

We illustrate the equilibrium strategies derived above with a simple graphical example.Suppose that incumbent auditors learn the client’s business risk type with certainty, i.e.,α = 1. Then, the second period pricing distributions may be illustrated as in Figure 2.

Since incumbents with low risk firms have the lowest costs, it follows that informedincumbents auditing these firms bid more aggressively to ensure their retention than doincumbents of high risk firms or prospective auditors, who face the possibility of takingon high risk engagements. This may be seen by the fact that the distribution of bids bylow incumbents lies above that of all other types of auditors. Clearly, prospective auditorsstill have strong incentives to try to “steal” the low risk firms away from incumbents, butcannot afford to be too aggressive in their bidding due to the lemons problem discussedabove. Thus, the equilibrium bidding strategies reflect a balance between aggressivenessin retaining (or acquiring) low risk firms and caution in avoiding high risk firms. This maybe seen by noticing that the distribution of prospective auditors lies between that of lowand high type incumbents. Finally, incumbents with high risk firms, faced with competitivepressure from prospective auditors, have little recourse but to bid on engagements at cost,knowing that they will often fail to retain the client.

3. Features of Equilibrium Audit Pricing

In this section, we consider the implications of equilibrium behavior on the pricing ofbusiness risk, auditor turnover, and “low-ball” pricing by auditors. Finally, we explore the


Figure 2. Pricing strategies of competing auditors when incumbent auditor learns the firm’s risk type (α = 1).

comparative static effects of changes in the stringency of accounting standards regardingbusiness risk on equilibrium behavior.

3.1. Pricing of Business Risk

A consequence of the equilibrium fee strategy profiles set out in Theorem 1 is that the auditfees offered to high risk firms for the performance of the second period audit,E (R|H), arehigher on average than those offered to low risk firms for the performance of the secondperiod audit,E (R|L). Nevertheless, the risk premium or difference between the feesoffered to the high and low risk firms is less than the difference between the expected costof servicing the high risk type and that of servicing the low risk type firm in the secondperiod. Thus, the model is consistent with Bell, Landsman, and Shackelford (1995) whoempirically find that although audit fees are higher for high risk clients, business risk maynot be fully priced. These observations are formalized in Proposition 1.

Proposition 1 Given the strategy profiles stated in Theorem 1, then in the second period

0< E (R|H)− E (R|L) < πB.


From Proposition 1 it follows that, in equilibrium, the expected litigation costs of highrisk firms are subsidized by the low risk firms. To be precise, since uninformed incumbentand prospective auditors are charging prices for the second period audit above the cost ofservicing low business risk firms, but below the cost of servicing high type firms, thenprofits from low type firms are, in effect, subsidizing the added cost of servicing high typefirms. Of course, informed auditors price high types appropriately, but still reap profits onlow types due to their informationally advantaged position in the market. This incompletepass-through of expected litigation costs to high risk firms is consistent with practitioners’claims that competitive pricing prevents them from fully adjusting audit fees to reflectbusiness risk (see Lochner (1993)).

Notice that the expected fee offered to the high risk firm,E(R|H), is less than the expectedlitigation cost,πB, associated with auditing that firm in the second period. Thus, in thesecond period, prospective auditors incur an expected loss on bids to high risk firms. It isimportant to emphasize that this aggressive bidding does not arise due to the prospectiveauditor anticipating future rents from securing the engagement. This is in contrast to themotives for aggressive auditor bidding for engagements offered in much of the existingliterature, and indeed in first period bidding in our model. The prospective auditor bidsaggressively for the engagement in the second period in an attempt to secure second periodrents from low risk clients. The empirical implication of this pricing strategy is that oneshould not necessarily expect to observe future period audit fees that reflect a recouping oflow-ball costs incurred in the first period.

3.2. Auditor Switching

An important feature of the equilibrium is that, with some probability, auditor turnoverarises as a consequence of the efficient pricing of audit services and is observed forbothtypes of firms in the absence of switching costs or other frictions. In particular, given thata firm has survived to the second period, low risk firms are less likely to change auditorsthan high risk firms. This relationship is stated in Proposition 2, where the probability of aswitch given ani type firm is defined as Pr(s|i ) wherei ∈ {H, L}; this result adjusts forthe fact that high risk firms may have dropped out at the end of the first period.

Proposition 2 Given the strategies stated in Theorem 1, then

Pr(s|H)− Pr(s|L) > 0.

The higher probability of auditor switching for high risk firms than for low risk firms,or the probability of auditor switching increasing with business risk, is consistent with theempirical observations of Schwartz and Menon (1985).15 They find that firms experiencingfinancial distress are more likely to switch auditors; moreover, the likelihood of switchingincreases as the firm approaches bankruptcy.

Our analysis suggests that the market may respond negatively to a firm’s decision to switchauditors in the absence of collusion between the firm and the new auditor, or disagreementover the audit opinion issued. Since a switching firm is more likely to be a high risk type,


this publicly observable event may be informative to investors about the firm’s type. Ifbeing a high risk firm harms the firm’s value, the market will respond negatively to anauditor switch. While we have not formally modeled this effect in the payoffs of the clientfirm, clearly, this anticipated downward change in the market value of the firm may act asa deterrent to switching auditors.

Regardless, it is interesting to note that the switching differential, Pr(s|H) − Pr(s|L),which determines the informativeness of switching on stock price is non-monotonic in thedegree of informational advantage(α)of the incumbent auditor. Differential switching ratesare largest for intermediate values ofα; hence, we would expect downward stock pressureto be most severe under these circumstances. To be precise, if it is difficult to learn a firm’sbusiness risk relative to the chances of a high risk firm, i.e.,α ≤ 1

1+λ′ , then increases ininformational advantage of the incumbent(α increasing) create more downward pressureon stock prices if a switch is observed. In contrast, in the case whereα > 1

1+λ′ , an increasein the informational advantage of the incumbent has the opposite effect: switches becomeless informative and one expects to observe less downward price pressure as a consequence.

3.3. Dynamic Pricing

We now turn to the first period to consider the impact of business risk on the initial pricingproblem faced by auditors. From Theorem 1, it follows that the incumbent auditor earnsexpected continuation payoffs ofE1(E(W2

I )) while the expected costs of auditing a firm inthe first period areλπB.

Low-balling is traditionally defined as setting the initial audit fee below first periodexpected cost (DeAngelo (1981a, b)). Since the first period bid isR1 = λπB−E1(E(W2

I )),then we have the following proposition:

Proposition 3 Auditors bid below expected costs in the amount E1(E(W2I )).

The fee offered in the first period is set recognizing that the incumbent auditor benefitsfrom knowing the firm’s type when bidding for appointment as the firm’s auditor in thesecond period. Since the competing auditors are symmetrically uninformed in the firstperiod, it follows from Bertrand competition that the two period expected profit must bezero.

Thus, the anticipation of expected profits in the second period induces auditors to offera fee lower than the first period expected audit liability in the hope of being appointedauditor. Appointment potentially allows the auditor to learn the client’s type and therebyoffer differential audit prices in the second period. In short, the auditor has an incentive tolow-ball. Since entry is free and firms are assumed to have no fixed costs, this condition isconsistent with the audit market being in long-run equilibrium.

One can show that the amount of low-balling is increasing in the amount of the liability,B, which arises in the event of litigation. That is, the heightened prospect of adverse courtoutcomes in the event of litigation actually increases low-balling by auditors. Thus, ourmodel has the testable implication that the recent changes in the litigation environmentshould lead to a decline in the amount of the low-ball relative to the amounts observed bySimon and Francis (1988) and Ettredge and Greenberg (1990). This prediction arises since


many of the earlier changes in the law that had facilitated legal actions against auditors, (seeKothari, et al. (1988); Lys and Watts (1994)), have since been reversed. First, the 1983 NewJersey court’s decision inH. Rosenblum v. Adler, which substantially increased auditor’sexpected liability by holding them to the foreseeable privity standard was overruled bythe New Jersey legislature in 1995.16 This state has adopted the less severe, strict privitystandard (AICPA (1995)). Second, in 1993, the U.S. Supreme Court held inReves et al. v.Ernst & Youngthat since auditors do not participate in the management of an enterprise,they are not subject to the Racketeer Influenced and Corrupt Organizations (RICO) Actof 1970.17 Finally, the recently passed Private Securities Litigation Reform Act of 1995provides protection to auditors by adopting modified proportionate liability for defendantswho do not knowingly engage in fraud and includes provisions to prevent abusive litigationpractices (AICPA (1996)).

3.4. Informational Advantage of Incumbency

Finally, we consider the impact of changing the probability that the incumbent firm learnsthe client’s business risk type; that is, the degree of informational advantage accruing toincumbent auditors. Since efforts to reduce audit risk may have the benefit of mitigatingbusiness risk, one might expect that incumbency informational advantage will increasewhen audit standards become more stringent. For instance, Brumfield, Elliott, and Jacobson(1983) note that factors that indicate inherent risk, to which auditing standards apply, arecorrelated with business risk.

We summarize the effects of increasing the informational advantage of incumbents in thefollowing proposition:

Proposition 4 As incumbent informational advantage increases:

a) business risk is more fully priced;

b) second period profits to incumbent auditors increase while profits to prospective auditorsremain at zero; and

c) first period low-ball bidding increases.

Intuitively, greater incumbent informational advantage leads prospective auditors to bemore cautious in their bidding strategies. Incumbents serving low risk firms as well asuninformed incumbents take advantage of this caution by increasing their average price.However, the overall price effects to low risk firms are mitigated by the fact they aremore likely to be served by informed incumbents, who offer generally lower prices thanprospective or uninformed auditors. Incumbents serving high risk firms continue to priceat their expected cost. Thus, prices to high risk firms increase to levels more reflective ofthe business risk of the engagement.

Improving the informational advantage of incumbents enables pricing that better avoidsthe downside risk of being uninformed and auditing a high risk client; as a consequence,expected second period profits increase. Working backwards, this then immediately im-plies that such an environment will see more extreme low-ball bidding. Thus, increased


stringency in audit standards increases the prevalence of low-ball bidding in the short run,while raising long-run average fees charged for audits (even in the absence of any changesin the underlying costs of performing the audit engagement). We therefore hypothesizethat the introduction in 1988 of the Auditing Standards Board expectation gap standards(SASs 53 through 61), that were intended to increase the auditor’s responsibility to detectand report errors, irregularities, and illegal acts and improve audit effectiveness (AICPA(1993)), would increase low-balling by auditors while ultimately raising the price of auditengagements.

4. Conclusion

Auditing arises in an environment characterized by incomplete information. Consequently,it seems natural that the information asymmetry concerning the business risk of an auditclient should affect the pricing of audit services. We find that incomplete pricing of businessrisk, differential auditor turnover, the established empirical regularity of pricing an initialaudit below the expected cost of performing that audit, or low-balling, arise in our model asa consequence of price competition among auditors when information asymmetries aboutbusiness risk are present. Our paper shows that transaction costs, technological differencesbetween auditors, and operating efficiencies which arise through repeated performance ofan audit arenot required to explain these phenomena.

Our analysis has a number of testable empirical relations. We hypothesize that althoughaudit fees increase with business risk, high business risk firms on average are not fullycharged with the expected cost of litigation arising from the audit service. Thus, we wouldexpect that cross-subsidization between the high and low business risk firms would beobserved. Auditor turnover should be observed for both low and high business risk typefirms with high risk firms being more likely to change auditors than low risk firms. Also,low-balling and audit fees are shown to be positively correlated with litigation liabilityexposure. Finally, we posit that increases in the stringency of auditing standards willinduce a decline in the partial pricing of business risk and an increase in low-balling.

Appendix

Proof of Theorem 1: We begin by considering the strategies in the second period. Since it isnot difficult to show that an equilibrium in pure strategies for the actions in the second perioddoes not exist, we will consider the equilibrium in non-degenerate behavioral strategies only.

First, consider the expected payoff of the prospective auditor given the strategy adoptedby the incumbent auditor. Given beliefsα by the prospective auditor, his expected payoffis

E(W2

P

) = α(λ′(Rp − πB

) (1− FH

(Rp))+ (1− λ′) Rp (1− FL (RP))

)+ (1− α) (RP − λ′πB

)(1− FU (RP)) .


If RP < πB, then FH (RP) = 0. Consider the equilibrium payoffs over the non-degenerate support ofFL (RP). Notice that on this supportFU

(Rp) = 0. Thus

E(W2

P (RP)) = α

(λ′(Rp − πB

)+ (1− λ′) Rp

(1− RP − λ′πB

α (1− λ′) RP

))+ (1− α) (RP − λ′πB

)= 0

after some simplification.Similarly, equilibrium payoffs over the non-degenerate support ofFU (RP) where

FL(Rp) = 1 are

E(W2

P (RP)) = αλ′

(Rp − πB

)+ (1− α) (RP − λ′πB)(1− FU (RP))

= αλ′(Rp − πB

)+ (1− α) (RP − λ′πB

) (1− R

(1− α + αλ′)− λ′πB

(1− α) (R− λ′πB)

)= 0.

Consider deviations outside the non-degenerate strategy support by the prospective audi-tor:

SupposeRP < λ′πB; thenE(W2P(RP)) = Rp−λ′πB < 0.Suppose now thatRP > πB;

thenE(W2P(RP)) = 0.

Therefore the prospective auditor has no incentive to deviate outside the support specifiedin Theorem 1.

Second, consider the expected payoff of the informed incumbent auditor given the strategyof the prospective auditor. Suppose that the client is a low type firm. The expected payoffto the incumbent auditor forRL ∈ [λ′πB, λ′πB

1−α+αλ′ ] is

E(W2

L (RL)) = RL (1− FP (RL))

= RL

(λ′πB

RL

)= λ′πB.

Consider deviations by the informed incumbent outside the non-degenerate support. Sup-pose RL < λ′πB; then E(W2

L(RL)) = RL < λ′πB. Next, consider the case whereπB ≥ RL >

λ′πB1−α+αλ′ ; thenE(W2

L(RL)) = RL(λ′πBα(1−λ′)

RL−λ′πB ) < λ′πB sinceRL >λ′πB

1−α+αλ′ .Finally, whereRL > πB then profits are zero. Thus, the incumbent auditor remains indif-ferent between payoffs generated by fees on the supportRL ∈ [λ′πB, λ′πB

1−α+αλ′ ] which arestrictly preferred to payoffs outside this support.

Now, suppose that the client is a high type firm. The expected payoff to the incumbentauditor is

E(W2

H (RH )) = (RH − πB) (1− FP (RH )) .


Since the incumbent auditor setsRH = πB with probability one,FP(πB) = 1, and theexpected payoffE(W2

H (RH )) = 0. Notice that all deviations yield payoffs that are eitherlower (for the case whereRH < πB) or the same (for the case whereRH > πB) as thoseobtained in equilibrium.

Third, consider the expected payoff of the uninformed incumbent auditor given the strat-egy of the prospective auditor. The uninformed incumbent’s equilibrium payoff for allRU ∈ [ λ′πB

1+α+αλ′ , πB] is

E(W2

U (RU )) = (

RU − λ′πB)(1− FP (RU ))

= λ′πBα(1− λ′) .

Analogous to the previous cases, it is routine to verify that deviations outsideRU ∈[ λ′πB

1+α+αλ′ , πB] are not strictly profitable.Finally, first period pricing follows from Bertrand arguments.It remains to show uniqueness of the equilibrium payoffs. We begin by considering the

second period auction. By Bertrand arguments, equilibrium prices must be less than orequal toπB.

Define the non-degenerate supports of the bidding strategies of each of the bidder typesasSH ,SL ,SP,SU . Using standard arguments (see Engelbrecht-Wiggans, Milgrom, andWeber (1983) and Amann and Leininger (1996)), the supports of any set of equilibriumbidding strategies are:

1. Closed and atomless except possibly at upper endpoints;

2. SH ∪ SL ∪ SU is connected;

3. (SH ∪ SL ∪ SU )∖SP = ∅.

Claim In any equilibrium E(W2P) = 0.

Proof: Since theprospectiveauditor loses theauctionwithcertaintybypricingat sup(SP),then expected profits from the strategyRP = sup(SP) are zero. SinceSP is a closed set,sup(SP) ∈ SP. By definition of equilibrium, then for allRP ∈ SP, expected profits arezero. 2

Claim In any equilibriuminf (SP) = λ′πB.

Proof: inf (SP) = sup(SP) can only happen in a pure strategy equilibrium, which weruled out earlier. Thus, inf(SP) < sup(SP) . But since by playingRp = inf (SP) theprospective auditor wins with certainty, then inf(SP) = λ′πB for zero expected profits tohold. 2

Claim In any equilibrium,sup(SH ) = inf (SH ) = sup(SP) .

Proof: Clearly, in no equilibrium would informed incumbents win with positive proba-bility for any RH < πB. Since sup(SP) ≤ πB, then if inf(SH ) < sup(SP), we have a


contradiction. Likewise, for sup(SH ) < sup(SP) . Combining these contradictory condi-tions with(SH ∪ SL ∪ SU )

∖SP = ∅ yields the result.

Claim Given some interval[a,b] ⊆ [λ′πB, πB] and an endpoint condition FP (a) = k,then there exists a unique bidding function FP (R) played by the prospective auditor whichkeeps a low (respectively uninformed incumbent) type indifferent over[a,b] .Moreover, thesolution to FP (·) differs depending on the type of player being made indifferent.

Proof: Consider the case of a low type. For low types to remain indifferent over[a,b]requires for allRL ∈ [a,b]

RL (1− FP (RL)) ≡ z

wherez is some constant. Differentiating with respect toRL ,(1− FP (RL)− RL

d

d RL

(Fp (RL)

)) = 0 (7)

which is a first order linear differential equation. Combined with an endpoint condition andthe fact that the range ofFP (·) is convex, we know thatFP (·) is unique. An analogousargument holds for uninformed types(

1− FP (RU )−(RU − λ′πB

) d

d RL

(Fp (RU )

)) = 0 (8)

and it is clear that for a fixed endpoint, the solutions to equations (7) and (8) are different.The proof for a prospective type is identical. 2

Claim There exists no interval[a,b], a < b, such thatSL ∩ SU = [a,b] . Moreover, forany interval[a,b] over which uninformed types are indifferent, aÂL b. Likewise, on anyinterval [a,b] over which low types are indifferent bÂU a, where the relationÂi denotesstrict preference by type i∈ {L ,U } .Proof: The first part of the claim follows directly from the uniqueness ofFP. The secondpart follows from uniqueness combined with the absence of profitable deviations for eachof the types given in the existence proof above. 2

It then follows that inf(SP) = inf(SL) = λ′πB. And, by the definition of a cumula-tive density function,FP(λ

′πB) = FL(λ′πB) = 0. Using uniqueness of the solution to

the differential equation generating indifference, we then reproduce the strategies given inTheorem 1. The uniqueness of first period strategies follows from Bertrand pricing argu-ments.

Proof of Proposition 1: Recall that

E (R|L) = αE(R|L̄)+ (1− α) E (R|U )

E (R|H) = αE(R|H̄)+ (1− α) E (R|U )


whereL̄ (respectivelyH̄) denotes that the incumbent auditor has successfully learned thefirm’s type, whileU again denotes the type of firm being unknown to the incumbent.

Differencing,

E (R|H)− E (R|L) = α (E (R|H̄)− E(R|L̄)) .

DefineG(R|i ) to be the distribution of the minimum order statistic of the pair(R̃P, R̃i ),i ∈ {H, L}, and letg(R|i ) be the associated density. Observe that wheni = H , thenfor all R > πB, g(R|H) = 0; henceE(R|H̄) ≤ πB. Similarly, observe that for allR < λ′πB, g(R|L) = 0;henceE(R|L̄) ≥ λ′πB. Combining these inequalities yieldsE(R|H̄)− E(R|L̄) < πB.

Next, observe that theFH (R) first order stochastically dominatesFP (R) which firstorder stochastically dominatesFL (R) . Consequently,G(R|H) first order stochasticallydominatesG(R|L); henceE(R|H̄)− E(R|L̄) > 0.

Proof of Proposition 2: Note that Pr(s|L) = α Pr(RP < RL) + (1− α)Pr(RP < RU ).SinceFP(R), FL(R), andFU (R) are mixed strategies,Rp, RL , and RU are independentrandom variables. Therefore, Pr(RP < RL) may be expressed as a convolution ofFP(R)andFL(R). Also, definef j (R) = ∂

∂R Fj (R) for j ∈ {H, L ,U, P}. Thus,

Pr(s|L) = α Pr(RP < RL)+ (1− α)Pr(RP < RU )

= α

∫ ∞−∞

FP (R) fL (R)d R+ (1− α)∫ ∞−∞

FP (R) fU (R)d R.

Then using Theorem 1,

Pr(s|L) = α

∫ λ′πB1−α+αλ′

λ′πB

(1− λ

′πB

R

)(λ′πB

α (1− λ′) R2

)d R

+ (1− α)∫ ∞−∞

FP (R) fU (R)d R

= 1

2

(1− λ′) (α)2+ (1− α) ∫ ∞

−∞FP (R) fU (R)d R.

Now consider turnover among high types. By similar reasoning:

Pr(s|H) = α Pr(RP < RH )+ (1− α)Pr(RP < RU )

= α

∫ ∞−∞

FP (R) fH (R)d R+ (1− α)∫ ∞−∞

FP (R) fU (R)d R.

Then using Theorem 1,

Pr(s|H) = α (1− λ′α)+ (1− α) ∫ ∞−∞

FP (R) fU (R)d R.

Differencing

Pr(s|H)− Pr(s|L) = α(1− λ′α)− 1

2

(1− λ′) (α)2


= 1

2α(2− αλ′ − α)

> 0.

Proof of Proposition 3: The proof follows directly from Theorem 1.

Proof of Proposition 4: a) UsingE(R|H) − E(R|L) = α(E(R|H) − E(R|L)) fromthe proof of Proposition 1, substituting forE(R|H) and E(R|L) using the equilibriumstrategies set out in Theorem 1, and then taking the partial derivative ofE(R|H)− E(R|L)with respect toα yields:

∂

∂α(E (R|H)− E (R|L))

= ∂

∂α

(α(E(R|H)− E

(R|L)))

= ∂

∂α

(α

(πB−

∫ λ′πB1−α+αλ′

λ′πB

(α(1− λ′) R− α (1− λ′) (R− λ′πB

)(α)2 (1− λ′)2 R2

)Rd R

))

= ∂

∂α

(α

(πB− λ′πB

ln(1− α + αλ′)α (−1+ λ′)

))

= πB

1− α (1− λ′)(1− λ′) (1− α) > 0.

Since E (R|H) − E (R|L) ∈ (0, πB), the above sign implies less partial pricing ofbusiness risk.

b)

∂

∂α

(E(W2

I

)) = ∂

∂α

(α(1− λ′) λ′πB+ (1− α) λ′παB

(1− λ′))

= 2(1− λ′) λ′πB (1− α) > 0.

c)

∂

∂α

(E1(E(W2

I

))) = 2(1− λ) λ′πB (1− α) > 0.

Acknowledgments

We are grateful to Mary Barth, Mike Baye, Kalyan Chatterjee, Mark Dirsmith, John Felling-ham, Rick Lambert, Jane Mutchler, Massoud Yahyazadeh, the participants at the Universityof Michigan Accounting Workshop, Stanford University Accounting Workshop, the PennState Summer Economics Workshop, the Penn State Accounting Colloquium, and the 1996American Accounting Association annual meetings. We would especially like to thankMary Beth Stocken for helpful discussion. The second author gratefully acknowledges thefinancial support of the Deloitte and Touche Foundation.


Notes

1. For instance, KPMG Peat Marwick was sued for allegedly performing an inadequate review of the accountsof a bank which was subsequently placed into receivership following substantial losses. The jury foundin Peat Marwick’s favor, but Peat Marwick nevertheless incurred legal fees of $7 million defending it-self. The audit fee was only $15,000 (Berton (1995)). This anecdote is consistent with Dye (1995) whonotes that auditors have been sued irrespective of whether the audit complied with Statement of Auditing(SAS).

2. For additional factors see Brumfield, Elliott and Jacobson (1983), Schuetze (1993), and Pratt and Stice (1994).3. Recently, Statement on Auditing Standards No. 84 superceded No. 7 to reflect the extant proposal environment.4. Notice, however, that our qualitative results are unchanged by the inclusion of costs associated with changing

auditors.5. To see this, consider a mechanism where the prospective auditor makes a public bid, followed by the incumbent

auditor. The client accepts the auditor with the lower bid (and retains the incumbent in the case of a tie.) Forany bid less than the highest possible cost, the prospective auditor will only win clients whose costs are higherthan its bid. Thus, prospective auditors will only bid the highest possible cost, and incumbent auditors willmatch this bid and retain all clients. Clearly, this is in the class of optimal mechanisms from the perspectiveof the incumbent auditor.

6. This assumption is consistent with the findings of the AICPA Cohen Commission Report (1978), which notethat, “[p]ublic accounting firms go to considerable lengths to develop superior services for their clients, butthere is little effective product differentiation from the viewpoint of the present buyer of the service, that is,management of the corporation” (Commission on Auditors Responsibilities (1978, 111)). On the other hand,some evidence suggests that the cost of performing an audit for a given client may differ across auditors.Nevertheless, in our model, we abstract away from the effects on auditors’ behavior of possibly differentialaudit costs of performing a SAS-compliant audit.

7. The assumption of two competing auditors is made to facilitate explication and is without loss of generality.As we show later, the analysis extends to the n-auditor case.

8. Because of risk neutrality and since the qualitative results of the equilibrium will only depend on the differencebetween the probabilities of litigation rather than on levels, this assumption is without loss of generality.

9. Changing the assumption of simultaneous moves to one of sequential moves can significantly change theinference problem of the prospective auditor. For instance, see note 5.

10. Long-term contracts are not considered. In practice they are seldom observed because they may undermineauditor independence as set out in Rule 101 of the AICPA Code of Professional Conduct. In addition, underour assumptions, there are no gains to the firm from acting “strategically” and utilizing a strategy differentfrom merely choosing the lowest price. Thus, the firm is employing a dominant strategy in simply choosingthe lowest price offered.

11. The expected valueB is commonly known ex ante by all the auditors. The assumption of risk neutralityenables us to limit attention to the expected value of the litigation lottery rather than considering the specificlottery outcomes in determining the auditor’s optimization problem.

12. This tie-breaking rule, which is consistent with Magee and Tseng (1990), is chosen simply because it seemsa natural description of the firm’s auditor retention decision; however, it does not affect the equilibrium. Inparticular, an alternative tie-breaking rule, such as giving equal weight to each auditor would yield the sameequilibrium strategies.

13. While the equilibrium was derived assuming there are two competing auditors, the qualitative results arelargely unchanged by assuming that there are n competing prospective auditors. Since the prospective auditorsare symmetrically informed in the second period, the symmetric strategy profile,G(R), for each of then− 1prospective auditors is determined as follows:

FP (R) = Pr(min(R2, R3, . . . , Rn) < R)

= 1− Pr(R2, R3, . . . , Rn > R)

= 1− (1− G (R))n−1 .

Thus,G (R) = 1− (1− FP (R))1

n−1 .

Note that while equilibrium payoffs are still uniquely determined, there exist a continuum of non-symmetric


equilibrium strategies supporting these payoffs.14. In this model, any Perfect Bayesian Equilibrium is a Sequential Equilibrium. See Fudenberg and Tirole (1991).15. Of course, alternative explanations rationalizing this phenomenon are also possible. For instance, opinion

shopping by high risk firms might likewise account for such a switching differential (Dye (1991) and Schatzberg(1994)).

16. 444 A. 2d 66 (N. J. 1982), 461 A. 2d 138 (N. J. 1983).17. 113 S. Ct. 1163, March 3, 1993.

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