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Competition in auditing : a spatial approach Chan, Derek Kwok-Wing 1995

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COMPETITION IN AUDITING: A SPATIAL APPROACHbyDEREK KWOK-WING CHANB. SSc., The Chinese University of Hong Kong, 1988M. A., The University of Western Ontario, 1990A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESTHE FACULTY OF COMMERCE AND BUSINESS ADMINISTRATIONWe accept this thesis as conformingto the required standardTHE UNIV SITY 0 RITISH COLUMBIAAugust 1995© Derek Kwok-Wing Chan, 1995In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.•_____(Signature)ACCOUNTINGDepartment of________________The University of British ColumbiaVancouver, CanadaDateAUGUcT 18, 1995DE-6 (2)88)ABSTRACTThis dissertation develops variants of the well-known Hotelling’s location model to examine the nature of competition in the audit market where audit firms make strategic specialization and pricing decisions.In a multi-period spatial oligopoly model of auditing competition, audit firms obtainmarket power through their service specialization with respect to client characteristics relevant to audit production. This market power allows audit firms to price discriminate amongclients. Competition among audit firms is localized: an audit firm optimally charges a client,to whom it has the lowest auditing cost to serve, the marginal auditing cost of the secondlowest-cost audit firm. These equilibrium audit firms’ pricing strategies result in an allocation of clients’ surplus and audit firms’ profits that lies in the core of the economy. Theexistence of a specialization-pricing equilibrium is also established. In equilibrium, given itsrivals’ specializations, each audit firm’s profit is maximized by choosing a specialization thatmaximizes the social welfare (the sum of clients’ surplus and audit firms’ profits). Moreover,audit firms never choose the same specialization in equilibrium. Instead, in order to earnrents as ‘local monopolists’, audit firms differentiate themselves from each other. This resultis consistent with a widely held notion that audit firms search for ‘niche’ markets, such asindustry specialization, to increase their profits.The dissertation then focuses on a two-period spatial duopoly model in which the marketpower created by audit firm specialization is now further fortified by the presence of auditors’learning and clients’ switching costs. In this case, audit firms optimally price discriminateamong clients by offering them ‘specialization-and-relationship-specific’ audit fee schedules.The practice of ‘low-balling’ is found to be a natural consequence of the competition amongaudit firms. However, low-balling occurs only in a certain market segment where auditfirms compete quite fiercely. The analysis also demonstrates how equilibrium audit fee11schedules, audit firms’ specializations and profits, clients’ surplus, and social welfare dependon the auditing costs, the learning rate, and the switching costs. Some interesting policyimplications are illustrated. Finally, the model is used to analyze the impact of banningaudit firms from the practice of low-balling. It is demonstrated that even though a policy ofbanming low-balling always reduces competition, it improves social efficiency in some cases.‘iiTABLE OF CONTENTSAbstractTable of Contents ivList of Figures viAcknowledgement viiChapter 1 Introduction 1Chapter 2 A Multi-Period Spatial Model of Auditing Competition 122.1 A Brief Literature Review of Spatial Competition 152.2 The Model 182.3 The Pricing Equilibrium 222.4 Specialization Equilibria 272.5 Concluding Remarks 30Chapter 3 A Two-Period Model of Auditing Competitionwith Learning and Switching Costs 323.1 The Model 373.2 Analysis of Pricing Subgames 413.2.1 The Second-Period Price Equilibrium 423.2.2 The First-Period Price Equilibrium 463.3 Equilibrium for the Full Game 503.4 Implications of Changes in Auditing Costs, Learning Rate, and Switching Costs 553.5 Concluding Remarks 62Chapter 4 ‘Low-balling’ and Efficiency 654.1 Low-balling 674.2 Welfare Implications of Low-Balling 714.2.1 Equilibrium Outcomes without Low-Balling 714.2.2 Welfare Comparison 73iv4.3 Concluding Remarks 74Chapter 5 Conclusion 76References 82Appendix A The Potential Benefits of External Auditing:An Information Economic Analysis 89A.l The Basic Borrower-Lender Model without Auditing 92A.l.l The Model 92A.l.2 Characterizing the Equilibrium 96A.2 The Setting with Auditing 100A.3 Concluding Remarks 112Appendix B Proofs of Propositions 113VLIST OF FIGURESChapter 3Figure 1 The Second-Period Equilibrium Audit Fee Schedule 63Figure 2 The First-Period Equilibrium Audit Fee Schedule 64viACKNOWLEDGEMENTI wish to express my most sincere appreciation to the members of my dissertation committee, Professor Dan Simunic and Professor Mukesh Eswaran, for their excellent assistance and,especially, to my dissertation supervisor, Professor Gerald Feitham, for his encouragementand professional guidance. I would also like to thank seminar participants at Duke, HongKong University of Science and Technology, Maryland, Pennsylvania State, UBC, Waterlooand Yale for helpful comments. Professor Jack Hughes at Duke University also deserves special thanks for providing early encouragement. Last, but not least, I wish to thank all thefaculty members and Ph.D. students at UBC for providing intellectual stimulation, constantencouragement and friendship. Financial support from the MacPhee Graduate Fellowshipand the University Graduate Fellowship is also gratefully acknowledged.viiChapter 1IntroductionPrior empirical studies have suggested that significant variation exists in accounting andauditing practices across industries.’ This is consistent with the conventional wisdom thataudit firms invest in specialized resources, such as SEC reporting expertise, taxation advice, computer audit and management consultation, to yield economies of scale and scopefor services rendered to particular market segments.2 This dissertation proposes that servicespecialization of audit firms is not only the result of an adaptation by audit firms to particulartechnological or institutional conditions but also reflects strategic positioning of audit firmsin the market. Through specialization, audit firms are able to create their OWII market nichesin which they possess some monopoly power and generate economic rents. The competitiveforces in the market then induce audit firms to achieve (constrained) efficient utilizationof specialized resources. Hence, this strategic scenario suggests the importance of servicespecialization considerations in modelling auditing competition. However, even though thenature of competition within the public accounting profession has received increasing attention from researchers and practitioners in recent years, there has been no formal model of1As stated in Danos and Eichenseher (1982): “The factors of production used in producing audit servicesare diverse in nature. Some can be used across all audit engagements, while others are unique to specificclient industries. Moreover, the importance of industry-specific factors tends to vary with the complexity ofaccounting and auditing rules unique to the client’s industry (p. 606).”2For example, Dopuch and Simunic (1980), Danos and Eichenseher (1982), Eichenseher (1985), Rhode etal. (1974) and Schiff and Fried (1976) find evidence of industry specialization by the then Big Eight auditfirms.audit firm specialization.3Extant research on auditing competition considers the audit market as an ex-ante perfectly competitive market and has primarily focused on the pricing behaviour of audit firms.Especially, the issue of ‘low-balling’, i.e., setting audit fees below total current costs on aninitial audit engagement, has been singled out as a very important research topic and received considerable attention. This attention not only comes from academics (e.g., Coateand Loeb (1994), DeAngelo (1981a, 1981b), Dye (1991.), Gigler and Penno (1995), Kanodiaand Mukherji (1.994), Magee and Tseng (1990), and Simunic (1980)), but also from the profession itself (e.g., The Cohen Commission Report (1978)). The interest in low-balling stemsfrom the hypothesized link to impaired auditor independence.Based on a multi-period contestable market model of auditing, DeAngelo (1981a) suggests that low-balling occurs if there are rents to be earned by audit firms. In her model,switching audit firms is costly to a client, which in turn, provides a quasi-rent to an incumbent audit firm on future audit engagements with the client. But since the market forauditing services is ex-ante perfectly competitive, competition among audit firms for theright to become the incumbent in the initial engagement drives the total quasi-rent for theaudit firm to zero, implying below-cost initial audit fees. Thus, DeAngelo concludes thatthe existence of client-specific quasi-rents to incumbent audit firms both lowers the amountof auditor independence and leads to low-balling in the initial period. Magee and Tseng(1990) basically concur with DeAngelo’s conclusioll after modelling the audit pricing gameby a dynamic programming approach.4 However, the effects of quasi-rents on the auditor’sindependence are less significant in the Magee and Tseng’s framework than in DeAngelo’s.DeAngelo asserts that the existence of quasi-rents is a necessary condition for reducing au3The study of auditing competition was stimulated first by the Metcalf Staff Report (U.S. Senate 1976)and later by the Cohen Commission Report (AICPA 1978). The former argues that there is insufficientcompetition, whereas the latter believes it to be excessive.4To be more correct, Magee and Tseng (1990) look at ‘price-cutting’ rather than low-balling, where theformer is defined by them as the difference between the second- and first-period audit fees. While low-ballingimplies price-cutting, the converse is not true.2ditor independence. However, she does not describe how and to what extent the quasi-rentswould affect the auditor’s independence. Magee and Tseng extend DeAngelo’s model in agame theoretic setting and obtain five necessary conditions under which a client-specific rentmay lead to a compromise of auditor independence. They further argue that those conditions are usually not satisfied. Among other things, Magee and Tseng point out that whenaudit firms possess all the bargaining power and there is no disagreement among audit firmson the proper interpretation of generally accepted accounting principles (GAAP), clientshave nothing to gain by threatening termination of incumbent audit firms and there is noimpairment of auditor independence. Thus, Magee and Tseng conclude that there is littlepressure on the auditor’s independence despite the existence of client-specific quasi-rents.Dye (1991) argues that DeAngelo (1981a) implies a division of bargaining power (in termsof audit fees determination) favouring the audit firm; and low-balling would not exist in theabsence of the audit firms’ bargaining power over their clients.5 To provide an explanationfor the existence of low-balling, Dye turns to a model with asymmetric information betweenthe audit firm, client and financial statement users. He shows that low-balling is induced bythe auditor selection mechanism designed by the client, who has all the bargaining power.Similar to Dye (1991), Kanodia and Mukherji (1994) also assume that the client has almostall the bargaining power. In addition, they assume that the incumbent audit firm has aninformational advantage on the auditing costs in subsequent periods following the initialengagement. Kanodia and Mukherji then use contract theory to derive an equilibrium inwhich low-balling occurs. Using a setting similar to Kanodia and Mukherji, but basing theanalysis on auction theory rather than contract theory, Coate and Loeb (1994) also findlow-balling occurs in equilibrium. Dye, Kanodia and Mukherji, and Coate and Loeb all findthat the motivation for low-balling is to reduce informational rents accrued to the incumbentaudit firms, and it does not induce the audit firm to compromise its audit decisions.5As argued by Goldman and Barley (1974), since the attestation service the audit firm provides is valuableto the client, it confers power to the audit firm. However, it is now a commonly held notion that the powerheld by audit firms is diminishing due to fierce competition in the audit market.3The conclusions drawn from the abovementioned theoretical analyses lead to a consensusthat there is no causal relation between low-balling and impaired auditor independence.This conclusion does not depend on an assumption that the audit firm has all the bargainingpower, as in DeAngelo (1.981a) and Magee and Tseng (1990); or that the client has all ofit, as in Coate and Loeb (1994), Dye (1991) and Kanodia and Mukherji (1994); or thatthey somewhat share it. Hence, it seems that the hypothesized link between low-ballingand impaired auditor independence may be unwarranted. This dissertation shifts the focusfrom issues of auditor independence to the economic relation among audit pricing policies(includes low-balling), audit firms’ specialization decisions, and social efficiency in the auditmarket.It is clear now that the extant analytical literature on audit pricing is built on the assumption that the audit market is ex-ante perfectly competitive. Given a perfectly competitiveaudit market, and if there is no causal relationship between audit pricing policy and auditorindependence as suggested by the extant theoretical literature, then an audit firm’s pricingbehaviour will only affect how the benefits from an audit are divided between the client andthe audit firm, with the social efficiency being held fixed. Thus, in order to address a meaningful social efficiency issue, one has to depart from the perfectly competitive paradigm. Forthis reason, this dissertation shifts the focus from ex-ante perfect competition to imperfectcompetition and emphasizes the strategic interactions among audit firms.The only published work in the auditing competition literature which also emphasizesmarket imperfections is a recent article by Gigler and Penno (1995). However, their treatment of market imperfection is primitive. Gigler and Penno assume that audit firms havesubstantial market power because they are randomly endowed with different auditing costs.In other words, the cost differences which are modelled by them as the primary source ofmarket power are not the result of audit firms’ equilibrium behaviour. Rather, audit firmsin their model are assumed to be ex-ante heterogeneous. On the contrary, this dissertation examines a setting where audit firms are ex-ante identical and strategically choose to4become ex-post heterogeneous in terms of their audit production costs by means of theirservice specialization. Thus, audit firm specialization is not a mute issue as in the existingmodels on auditing competition. By explicitly recognizing the strategic purpose of auditfirm specialization, the models in this dissertation capture the widely held notion that auditfirm specialization is the primary source of the market power and, hence, the economic rentsaccrue to the audit firms. As such, the models in this dissertation enrich the traditionalaudit pricing models by expanding the strategy spaces of the audit firm; audit firms strategically make both pricing and specialization decisions. Our emphasis on an imperfect auditmarket and the importance of audit firms’ specialization decisions should provide insightsthat augment the studies of DeAngelo (1981), Coate and Loeb (1994), Dye (1991), Giglerand Penno (1995), Magee and Tseng (1990) and Kanodia and Mukherji (1994).More specifically, this dissertation postulates a spatial perspective to examine the natureof competition in the audit services market where audit firms make strategic specializationand pricing decisions.6 The spatial perspective is borrowed from the spatial economicsliterature which represents a recent breakthrough in the development of a new industrialorganization theory. Recently, many economists have increasingly recognized that the perfectcompetition paradigm is inappropriate to the explanation of pricing behaviour in manyreal life markets characterized by a significant separation between buyers and sellers (seeGreenhut, Norman and Hung (1987) and Beath and Katsoulacos (1991)). Unlike the perfectcompetition paradigm, the spatial perspective recognizes the dispersed nature of many reallife markets, and more importantly, the market power conferred to the suppliers becauseof natural market separation created by space. Moreover, because market activities areperformed at dispersed points in space, each supplier finds only a few competitors in itsimmediate neighborhood. Accordingly, competition in space occurs “among the few” whichare deemed as close substitutes by the buyers, thus leading to the analysis of the problem6The use of spatial analysis in audit market research is first discussed by Simunic and Stein (1987).However, they do not provide a formal spatial model to examine audit firms’ choices of both audit feeschedules and service specialization.5as a game of strategy.The essence of the spatial perspective is that it not only provides natural market separation, but also provides a powerful analogy for some apparently nonspatial issues. Themajor concern of this dissertation can then be viewed as making this analogy explicit byapplying the spatial analysis to audit firms’ specialization and pricing problems. Applyingthe spatial approach, the models in this dissertation explicitly recognize the dispersed nature of the audit market, namely that it embodies a large number of audit purchasers withdifferent ‘characteristics’ relevant to audit production and relatively few audit suppliers whodiffer in their area of specialization with respect to client characteristics. In this framework,all audit clients are unique. They operate in different businesses, have different management organizations, employ different philosophies, are subject to different risks, and havedifferent information and control environments.7It follows that audit firms bidding on auditengagements have to customize their production of services to meet the unique characteristics of each client. It also creates an incentive for audit firms to specialize their services.Through specialization, an audit firm achieves a comparative cost advantage over its rivalsfor all clients whose characteristics are closer to its area of specialization. The interest ofthis dissertation is on the kinds of audit firms’ specializations that the market mechanismwill provide and the implications for welfare distribution of alternative audit market environments. Moreover, explicit account is taken of the ability that audit firms have to acquireand exercise market power and of the strategic interactions among audit firms.It is noteworthy that the spatial models developed in this dissertation are different fromthe ones that are generally used in economic literature to explain the phenomenon of production differentiation. While the standard product differentiation models assume that productdifferentiation involves making a particular firm’s product either really or apparently different from its rivals, the starting point in the models of this dissertation is that the differences7One thing in common is that all audit clients, regardless of their own characteristics, require externalaudit services to carry out a systematic program of financial audit. This creates the demand for externalauditing services.6in audit firms’ specializations are grounded on the underlying differences of clients’ characteristics. More specifically, it is assumed that corporate financial statement audits arehomogeneous across audit firms from the viewpoint of the users.8 The rationale underlyingthis assumption is that professional standards impart homogeneity across audit reports, ineffect causing audit services to be identical across audit firms.9 Therefore, as long as professional standards and qualifications are maintained, the users of financial statements haveno reason to distinguish among audit firms. Consequently, a client’s perception of auditservices is assumed to be independent of the identity of the audit firm.’° That is, eventhough auditing activities are performed at dispersed points in client-characteristics space,each audit firm provides the same service at a given point.The assumption that audit services are identical across audit firms does not remove allthe frictions in the audit market. The reason is that audit firms differ in general in their areaof service specialization with respect to client characteristics. Hence, there is heterogeneityon the cost side of audit services simply because audit firms are responsible for customizingtheir production of audit services to meet clients’ characteristics and professional standards.The analysis in this dissertation will take as given the nature of clients’ characteristics,In its final report, the Cohen Commission states its belief that there is little or no product differentiationin the audit profession: “Public accounting firms go to considerable lengths to develop superior services fortheir clients, but there is little effective product differentiation from the viewpoint of the present buyer ofthe service, that is, the management of the corporation.... A ‘clean opinion’ obtained from one reputablefirm is about as valuable to the competent, honest financial manager as one from another reputable firm(AICPA 1978, p. 111).”9Audit firms are constrained to provide a minimum level of audit quality to comply with generally acceptedauditing standards (GAAS).‘°In contrast to this view, the importance of product differentiation in explaining observed market sharesin the market for audit services is first asserted by Dopuch and Simunic (1980). Subsequent studies tieaudit (quality) differentiation to the pricing of audit services and the relationship between audit firm sizeand auditor independence (see DeAngelo (1981a, 1981b) and Dopuch and Simunic (1980, 1982)). There isconsiderable amount of subsequent evidence for this issue which is consistent with the ‘quality differentiation’ (see Francis (1984) and Palmrose (1986)). In fact, one might argue that both differences on clientcharacteristics and quality differentiation exist in the real-world audit market. However, for simplicity, thisdissertation only considers the former, which has a greater effect on audit service production, and simplyassumes that audit quality is not at issue. In this way, one may argue that the models in this dissertation aremore relevant to a regulation requiring specific audit procedures (e.g., confirmation of accounts receivable orobservation of inventory) than to regulations aimed at quality control (e.g., rules requiring proper supervision or peer review). Allowing audit quality to vary in the spatial framework considerably complicates theanalysis but is, of course, a promising topic for future research.7the specification of audit technology, and a suitable notion of what would constitute an equilibrium in the particular problem under consideration. In all cases, the latter will be definedin terms of audit prices and audit firms’ specializations. The analysis provides predictionsof the nature of audit firm specialization that would emerge in market equilibrium. Thisof course is a question of the ‘positive’ economics of auditing. However, this dissertationalso considers the ‘normative’ economics of auditing. In particular, audit firms are subjectto increasing degrees of regulatory controls. For example, controls or guidelines have beenset with respect to audit pricing policies and switch of audit suppliers.11 In all cases, thequestion that would be central in the mind of a regulator is the implications of these marketequilibria for market power and the welfare of the participants in the audit services market.An answer to this question involves comparing the market equilibrium with a relevant socialoptimum. If the market equilibrium outcome is incompatible with the social optimum andthere are ways to reduce the efficiency loss, then there might exist a role for governmentintervention. However, the extant regulatory control on the audit market is primarily rootedin economic principles derived from classical competitive economics, while regulation is applied almost by definition to imperfectly competitive markets.’2 Therefore, it is necessarythat the methodological foundation of the regulation be re-examined, and perhaps, muchof the regulation should be re-evaluated. Adopting the spatial approach, the analyses andresults in this dissertation are shown to carry interesting policy implications with respectto policies concerning audit industry regulations such as policies concerning switching ofaudit suppliers studied in the chapter 3, and audit firms’ practices of ‘low-balling’ studiedin chapter 4. As such, the spatial framework developed in this dissertation may shed lighton a number of important audit industry issues and might provide regulators with a more“For example, Accounting Series Release (ASR) No. 250 requires disclosure of “fee arrangements wherethe audit firm has agreed to a fee significantly less than a fee that would cover expected direct costs in orderto obtain the client”, whereas ASR No. 165 et al. require disclosure of both the resignation of the prioraudit firm and the engagement of the new audit firm. See also footnote 13 for details.‘2For example, changes in CPAs’ codes of ethic during the 1980’s were designed to stimulate the competitiveness of audit firms, suggesting that the audit market was less than perfectly competitive when theprocess started.8adequate foundation on which to base regulatory judgements.This dissertation is organized as follows. Formal models of auditing competition in aspatial context are presented in chapters 2, 3, and 4. Conclusions and suggestions for futureresearch are the subject of the final chapter. A simple model demonstrating the existenceof a demand for voluntary external auditing is provided in appendix A. All proofs of resultsare given in appendix B.This dissertation is built on the belief that understanding the specialization and pricingdecisions of audit firms is the cornerstone of modern audit market research. Thus, to expandthat understanding, this dissertation introduces the spatial framework into the auditingcompetition literature. In chapter 2 a multi-period spatial oligopoly model is introduced tostudy how audit firms make strategic specialization and pricing decisions. Audit firms aremodelled as Bertrand oligopolists who simultaneously choose specializations with respectto client characteristics and then compete in setting audit fees. It is shown that throughspecialization, each audit firm obtains some market power and is able to price discriminateacross clients by offering ‘specialization-specific’ audit fee schedules. We find that, given aspecialization configuration, each audit firm optimally charges the minimum of the marginalauditing costs of its rivals on services to clients whose characteristics are closer to its ownspecialization. Given these pricing strategies of the audit firms, the assignment of audit firmsto clients is simply a function of audit cost conditions; clients purchase audit services fromthe least-cost supplier. The resulting allocation of clients’ surplus and audit firms’ profitsis shown to be in the core of the economy. This means that, at the induced allocation,no group of clients can move to another audit firm for a mutually advantageous auditorclient re-alignment. Turning to the specialization decision, each audit firm optimally choosesa specialization so as to maximize its own expected profits, given the equilibrium auditpricing strategies and the specializations of its rivals. The existence of a specialization-priceequilibrium is established. Surprisingly, we find that in a subgame perfect Nash equilibriumchoice of audit firm specializations, given its rivals’ specializations, each audit firm specializes9so as to maximize the expected social welfare (the sum of the total profits to audit firmsand the aggregate surplus to clients), rather than maximize its own expected profit. It isalso demonstrated that, in order to avoid intense price competition, and thus, earn rentsas ‘local monopolists’, audit firms would like to differentiate themselves from each otherin equilibrium. This result is consistent with a widely held notion that audit firms searchfor ‘niche’ markets, such as industry specialization, to increases their profits. As such,the model provides a theoretical link between audit firm specializations and the observedmarket segmentation in which clients with similar characteristics buy from the same auditfirm, which has a cost efficiency advantage in serving them.Chapter 3 simplifies the model developed in chapter 2 by focusing on a two-period spatialduopoly auditing competition model. This simplification allows us to add more institutionaldetails into the model. More specifically, auditors’ learning and clients’ costs of switchingaudit firms are introduced to capture salient economic features of an audit market with‘relationship-specific economic interests’. As in the multi-period spatial oligopoly model, audit firms make strategic specialization and pricing decisions. Through specialization, an auditfirm achieves a comparative cost advantage over its rival for all clients whose characteristicsare closer to its area of specialization. This comparative cost advantage is further fortified bythe presence of learning and switching costs. Thus audit firms are able to price discriminateacross clients by offering ‘specialization-and-relationship-specific’ audit fee schedules. Theanalysis demonstrates that the practice of ‘low-balling’ is a natural consequence of the competition among audit firms. However, low-balling occurs in a certain market segment whereaudit firms compete quite fiercely. Furthermore, the analysis shows how equilibrium auditfee schedules, audit firms’ specialization decisions and profits, clients’ surplus, and socialwelfare depend on the auditing costs, the learning rate, and the switching costs. Some ofthe results of the analysis are shown to carry interesting policy implications. For example,the analysis enables us to understand why there may be a conflict between the regulations(e.g., Securities Act Release No. 34-9344, ASR No. 165, ASR No. 194 and ASR No. 247)10the audit firms in the audit industry would like to adopt and those the regulators and/orthe clients might want to impose.13 In this respect, our results suggest that if the objectiveof the regulators (particularly the SEC) is to maximize the expected social welfare, thenthe regulators should impose regulations that induce lower switching costs. This policy mayraise audit firms’ profits at the expense of clients’ aggregate surplus, but it improves overallefficiency.The issue of low-balling is further scrutinized in chapter 4. We first compare the primarysimilarities and differences between the predictions on low-balling as well as ‘price cutting’ ofour model developed in chapter 3 and those of the existing literature. Then, our focus shiftsto examining the welfare implications of low-balling which have not been fully considered byacademics and regulators concerned with low-balling by audit firms. To this end, we comparethe equilibrium outcomes derived in chapter 3 with those derived in an otherwise-equivalenteconomy where low-balling is not allowed, i.e., everything is the same as in the model inchapter 3 except that audit firms are required to price at or above their auditing costs. Theresult of this analysis provides theoretical support for banning the practice of low-balling.Finally, chapter 5 offers some conclusions on the dissertation and points out some directions for future research.13The principal reporting requirements under ASR No. 165 et al. are disclosure of both the resignationof the prior audit firm and the engagement of the new audit firm, and the existence of any significantdisagreement with the prior audit firm within the two most recent fiscal years. The client must request theprior audit firm to respond to the filing, and its response is appended as an exhibit. In addition, financialstatement disclosure of the effect of the disagreement, if material, is required.11Chapter 2A Multi-Period Spatial Modelof Auditing CompetitionThe purpose of this chapter is to expand our understanding about the specializationand pricing decisions of audit firms. To this end, the nature of competition in the auditservices market is re-examined from a spatial perspective, which is discussed in chapter 1.In the spatial framework, the dispersed nature of the audit market is recognized: auditclients are unique and have different ‘characteristics’ relevant to audit production.1The factthat clients have different characteristics leads to the natural consequence that audit firmsbidding on audit engagements have to customize their production of services to meet theunique characteristics of each client.2 It also creates an incentive for audit firms to specializetheir services with respect to client characteristics. Through specialization, an audit firmachieves a comparative cost advantage over its rivals for all clients whose characteristics area-For example, a client firm in a regulated industry requires the use of specialized financial rules forfilings with a government agency; some accounting rules and applications are unique to a given industry;and client-specific, as well as industry-specific, knowledge is necessary to the audit supplier in identifyingpotential problem areas and communicating with client personnel.2As stated in Arens and Loebbecke (1984): “An extensive understanding of the client’s business andindustry and knowledge about the company’s operations are essential for doing an adequate audit (p. 200).”Understanding the client’s business at least includes an appreciation for its business and related inherentrisks, and its information system and control environments. In addition, O’Keefe, Simunic and Stein (1994)find evidence of significant influences of client size, complexity, and inherent risk on the production of auditingservices.12closer to its area of specialization.3 Together with the assumption that audit outputs, i.e.,audited financial statements, are homogeneous across audit firms from the viewpoint of theusers,4 the assignment of audit firms to clients is thus simply a function of audit firm costconditions. That is, clients purchase audit services from the least-cost supplier.5The assumption that audit services are identical across audit firms does not remove allthe frictions in the audit market. The reason is that audit firms differ in general in their areaof service specialization, and therefore, are heterogeneous in terms of their auditing costs to aparticular client. These cost differences then create market power for the audit firms. Thus,as a result of specialization, audit firms possess some monopoly power even though clientsperceive alternative audit firms as perfect substitutes. In particular, since clients cannotresell audit contracts, their perception of homogeneous audit services does not preclude theaudit firms from having the ability to engage in price discrimination.To illustrate the above argument, this chapter presents a dynamic oligopoly model ofspatial competition with price discrimination to analyze the nature of auditing competition.The model falls in the domain of a hybrid framework of Hotelling (1929) and Hoover (1937).In this framework, audit firms behave as Bertrand oligopolists and price discriminate acrossclients with respect to the client characteristics when providing audit services to spatiallydispersed clients.6 That is, audit firms quote different audit fee schedules for services to different clients according to their characteristics. Owing to the competition among audit firms,the natural inference is that the cost effectiveness of audit firms determines their ultimatemarket shares. It is demonstrated that given a configuration of audit firm specializations,3The incentive of an audit firm to specialize with respect of client characteristics should not be takenas a pure theoretical conjecture. As stated in Stevens (1991): “With the audit process widely viewed as acommodity services that is virtually identical from firm to firm, Andersen proposed the local setting as away to differentiate itself from the competition. While Peat and Coopers relied on the old saws about auditquality and proficiency, Andersen offered a tangible difference the client could relate to (p. 233).”4See chapter 1 and appendix A for a rationale for this assumption.5This is consistent with the argument of Johnson and Lys (1990) that audit firms achieve competitiveadvantage through specialization, and that clients purchase audit services from the least-cost supplier.6The spatial Bertrand model is desirable in the sense that it proposes differentiated specializations whilestill projecting substantial competitive fee impacts.13each audit firms serves a client, to whom it has a comparative cost advantage, at an audit fee equal to the auditing cost of the second lowest-cost audit firm to that client. Thisequilibrium audit firm’s pricing strategy is shown to be efficient as the induced allocationof clients’ surplus and audit firms’ profits is contained in the core of the economy. That is,at the induced allocation no group of clients can move to another audit firm for a mutuallyadvantageous auditor-client re-alignment. When making their specialization decisions, auditfirms respond to the pricing and specialization decisions of other rivals. The competitiveforces in the market then induce audit firms to achieve constrained efficient utilization ofspecialized resources. Under some innocuous assumptions commonly used in spatial models,it is demonstrated that a specialization-price equilibrium is obtained when each audit firmmaximizes expected social welfare given the specializations of its rivals.7 Moreover, it cannever be beneficial for an audit firm to choose a specialization arbitrarily close to any of itsrivals’. This result implies that audit firms tend to differentiate themselves from each otherby means of service specialization. As such, the model provides a theoretical link betweenaudit firm specializations and the observed market segmentation in which clients with similar characteristics buy from the same audit firm, which has a cost efficiency advantage inserving them.The rest of the chapter is organized as follows. As is obvious from the discussion thus far,the dissertation heavily draws on the field of spatial economics for input. A brief review of thisliterature appears in section 2.1. Section 2.2 presents a very general multi-period oligopolyspatial auditing competition model, provides a precise specification of the demand and supplysides of the audit market, and defines a specialization-price game for the audit firms andan appropriate solution concept. Section 2.3 derives the unique audit pricing equilibriumfor the model. Section 2.4 establishes the existence of a specialization-price equilibrium.In addition to existence, some interesting properties of the specialization equilibria are alsodemonstrated. Section 2.5 concludes the chapter.7The definition of social welfare is given in section 2.4.142.1 A Brief Literature Review of Spatial CompetitionBeginning with the work of Hotelling (1929), the study of spatial competition has provided important insights into markets for differentiated product. The distinctive feature ofthe Hotelling’s spatial model, compared to other models of product differentiation, is thatit allows an explicit representation of product choice by oligopolistic firms. Specifically, theform of differentiation introduced by Hotelling (1929) can be described as ‘horizontal’ in thesense that no product (location) is unanimously preferred by all consumers.8 In this context,products differ only because they are offered at different locations. In his model, Hoteffingconsiders two identical firms that produce a single homogeneous product with a constantproduction cost in a bounded linear market over which consumers with inelastic demand areuniformly distributed. The firms compete in location and price and the consumers purchasethe product from the cheapest source and pay a transport cost which is assumed linear withrespect to the distance between the locations of the consumer and the firm. For each pairof locations chosen by the firms, Hotelling calculates the equilibrium prices they would set.To study the location tendencies, Hotelling introduces these equilibrium prices back into thefirms’ profit functions. In this respect, Hotelling can be said to have studied a subgameperfect Nash equilibrium in a two-stage location-price game. Hotelling claims that a Nashequilibrium in locations for the two firm market exists and yields ‘back-to-back’ locations atthe center of the market.Some problems with Hotelling’s analysis later become apparent. D’Aspremont, Gabszewicz and Thisse (1979) find that no equilibrium (in pure strategies) in prices exists whenfirms are located too close to each other. But, if no price equilibrium exists for certainlocational choices, then there is no way for firms to estimate the profitability of those locations. That is, the disturbing result of D’Aspremont, Gabszewicz and Thisse is that locationtendencies cannot be derived because the outcome of the price game is not well-defined.8J contrast, two products are said to be ‘vertically’ differentiated if all consumers unanimously rank unitquantities of them. Thus, if they are sold at the same price, all consumers purchase the same product.15Many resolutions of the existence problem have been discussed in the literature. Basically,they can be divided into four areas: (1) changing the transport cost function, (2) allowingfor mixed strategies over prices, (3) focusing on vertical as opposed to horizontal locationproblems, and (4) allowing for discriminatory pricing.IJ’Aspremont, Gabszewicz and Thisse point out that the problem of nonexistence of anoncooperative equilibrium arises from the fact that, with linear transport cost, the firms’demand functions are discontinuous and their profit functions are discontinuous and non-concave. Consequently, the price-competition problem is not well-behaved. To obviate thisnonexistence problem they assume quadratic as opposed to linear transportation costs. Theequilibrium locations are at the two ends of the linear market. Similar results are derivedin Economides (1984). He shows that a pricing equilibrium exists and firms locate far apartwhen consumers have a maximal or reservation distance. Unfortunately, since the equilibrium depends heavily on the form of transportation cost function, few applications have beendeveloped.The fact that there is no equilibrium in pure strategies (over prices) does not preclude theexistence of a mixed strategy equilibrium (see Dasgupta and Maskin (1986)). Nevertheless, itis of interest that Hotelling’s model with linear transportation costs and bounded reservationprices possesses no equilibrium even in mixed strategies. In games where mixed strategyequilibria do exist (see Gal-Or (1982) and Osborne and Pitchik (1987)), their complexityeffectively rules out comparative static analysis.A completely different approach was taken by Shaked and Sutton (1982, 1983). Theyattain important positive results in price-location theory by examining vertical rather thanhorizontal product differentiation. In their model, firms compete over quality and price.Quality choice is a ‘vertical’ location problem because all consumers prefer higher quality to lower quality. By contrast, in ‘horizontal’ location problems, changing the productspecification is a move towards some consumers and away from others. Their results are16encouraging, but it has proved difficult to develop generalizations that include horizontalproduct differentiation.In the traditional Hotelling setting, Hotelling assumes there is no price discrimination.Consumers pay the costs of transporting the product from firm to home plus the mill priceset by the firm. An alternative approach to spatial competition, pioneered by Hoover (1937),relaxes this constraint and allows firms to price discriminatorily. This situation is plausible iffirms can identify consumer locations. It is of interest that even if consumer locations are notdirectly observable, price discrimination may still be possible if firms choose delivered priceschedules over a space which precludes consumer arbitrage. In such situations, unless thereexist regulations dictating otherwise, the firm has the potential for price discrimination. Inhis original work, Hoover analyzes spatial price discrimination for firms with fixed locations.He concludes that a firm serving a market point would have a local price constrained by themarginal cost of service of other firms. In situations where demand elasticity is not too high,this will result in delivered prices at market points equal to the marginal cost of the firm inthe market with the second lowest marginal cost. This research agenda has then been furtherdeveloped by Hurter and Lederer (1985), Lederer and Hurter (1986) and Hamilton, Thisseand Weskamp (1989). They show that a two-stage perfect Nash equilibrium of prices andlocations exists when firms are allowed to set discriminatory prices. Above all, the resultinggame typically involves, as strategic variables, price schedules specifying the delivery pricesat which each firm is willing to supply the consumer at each point of space. In other words,if firms are allowed to price discriminate in a spatial market, their decision variables areprice functions instead of price scalars. The models in this dissertation basically follow thisparticular line of research. However, the key difference in the models in this dissertationis that firms will compete for consumers over a time horizon rather than a single period.Such multi-period extensions give rise to the consideration of the firm’s opportunity to learnand the consumer’s switching costs, which are the subjects of examination in chapter 3.As one would expect, the resulting multi-period models are much richer than their single17period counterpart, since the firm’s strategy set is expanded. Moreover, the incorporationof learning-by-doing and switching costs into the spatial competition model is novel in theeconomic literature.2.2 The ModelConsider an economy that lasts for T periods.9 There is a continuum of client firms(henceforth called clients) distributed over a convex compact subset Z C J?N, where Z isthe domain of client’s ‘characteristics’ which are relevant to audit production.’° That is, thecharacteristics of each client are fully described by a vector z E Z. Moreover, the densityof clients at z e Z, h(z), is positive and continuous on Z. For expositional convenience, theclient(s) located at z will be referred to as ‘client z’.Each client would like to acquire one unit of audit service from an external audit firm inevery period, provided that the benefit is higher than the cost of the audit.’1 It is assumedthat auditing standards are maintained by each audit supplier, such that audit services arequalitatively homogeneous across audit firms from the viewpoint of the clients. Furthermore,a unit of audit service, regardless of the identity of the audit supplier, is assumed to givea client z a gross benefit of bZ per period, where bZ can be interpreted as the highest auditfee that client z would be willing to pay for a unit of audit service.12 It is assumed thatbZ is sufficiently large, so that each client would like to purchase the audit service in everyperiod. Other than the audit fee paid to his audit firm, there are no additional transactioncosts incurred by the client regarding the hiring or switching of audit firms.139T can be finite or infinite.‘°‘Characteristic space’ is a natural criterion for the separation of the market for audit services in thecontext of imperfect competition.11For a useful background on the various sources of demand for audit services, see Jensen and Meckling(1976), Ng (1978, 1979), Benston (1985), Watts and Zimmerman (1986), and Berry and Wallace (1986).A simple model to explain the existence of a voluntary demand for external auditing is also provided inappendix A.12There will be no qualitatively change of the analysis if b is also time-dependent. Here, the assumptionis made for notational convenience.‘3This assumption is relaxed in chapter 3.18The audit market consists of n independent (non-colluding) audit firms bidding on auditengagements. They are indexed by i C {i, 2, ..., ‘4 and may only differ in specializationof services with respect to client characteristics. At the beginning of the first period, auditfirms choose their area of specialization simultaneously; once chosen, the specializations arefixed forever. For simplicity, each audit firm is only allowed to choose a single type of servicespecialization.14 Then, for each period, they simultaneously quote audit fees to each client.It is assumed that no multi-period offers are permitted.’5Let 1 C Z denote the specialization of audit firm i in the economy. Audit firms areresponsible for customizing their production of audit services to meet clients’ characteristics.The audit technology available to each audit firm is the same. When the difference betweenaudit firm i’s specialization and client z’s characteristics is equal to liii — zil, the auditingcost per period is given by a function m(II1 — zil), where fl j is a norm defined onrn( I — zI I) is increasing and continuous in liii — zi I with m(O) = 0. Again, for simplicity,it is assumed that there is no learning-by-doing advantages by the incumbent audit firm, sothat rn(.) is independent of time.’7The assumptions of the audit production function imply that an audit firm’s auditingcost to a particular client can be reduced by simply choosing a specialization that is closer tothe characteristics of that client. That is, the benefit from cost reduction gives an incentive‘4Presumably, one can expect there is a fixed cost of specialization which is an irreversible investment(e.g., costs associated with the technology adopted and the human capital/expertise of professional staff),otherwise audit firms will simply specialize at all points at which there are clients and demand will beperfectly satisfied. More specifically, it is assumed that the fixed cost is low enough for the audit firm tomake nonnegative profit but high enough to prevent it from having more than one specialization. Similarly,the fixed cost is assumed to be high enough to prevent more than n audit firms to co-exist in the auditmarket. Put differently, the audit market defined by Z is assumed to be just large enough to allow exactly itaudit firms to make positive profits (net of fixed cost). However, it will be clear later in the analysis that thefixed cost does not play any important role in the analysis. Therefore, it is intentionally omitted to reducethe notational burden.15n other words, audit firms are not able to make binding long term commitments for future audit fees.In fact, such binding long term commitments are rare in practice, probably because of prohibitions on auditcontracts that are contingent on the content of audited reports.16Given arbitrary points z1, z2 and z3 C Z, a norm is a real-valued function which satisfies (i) ) )zj —z2 = 0if, and only if, z1 = z2; (ii) jJzi— z211 + liz2 — zall liz, — zsii; and (iii) liz, — z2lI = liz2 — z,lI 0.‘7Again, this assumption is relaxed in chapter 3.19for audit firms to specialize their services. By means of specialization, an audit firm becomesmore cost efficient, compared to its rivals, to serve clients whose characteristics are closerto its area of specialization. Consistent with this line of thinking, empirical researchers findthat audit firms specialize by industry (Dopuch and Simunic (1980), Danos and Eichenseher(1982), and Eichenseher (1985)). As one would expect, specialists in the client’s industryare likely to enjoy cost advantages over nonspecialists.Formally, the setting is a T-period, T + 1-stage complete information game. In the firststage (the specialization stage) which occurs at the beginning of the first period, the auditfirms simultaneously choose their specializations in Z. Then, each audit firm becomes awareof its rivals’ specializations. It implies that, after the audit firms choose their specializations,everyone knows who the most cost efficient audit firm is for a particular client. In the secondstage (the first period pricing stage), the audit firms simultaneously quote audit fees to eachclient.’8 The client at z acquires auditing services from the audit firm quoting the lowestaudit fee.19 When audit fees are equal, it is assumed the audit firm with the lowest auditingcost provides the auditing services to the client.20 This may be rationalized by noting thatthe most cost efficient audit firm can always offer a strictly better audit fee schedule to theclient. Furthermore, if two or more audit firms have equal lowest costs of auditing client zand quote equal lowest audit fees to him, the client chooses the one with the lowest index.Generally, the set of clients and audit firms for which m(I l — zi I) = m(I II — zi I) i i,is negligible.21 Then, it follows that the ‘tie-breaking’ rule used in this latter case is of noconsequence in the equilibrium analysis. The first-period pricing stage will then repeat fromperiods 2 to T. Common to both audit firms and clients is the assumption that there is aone-period time-independent discount factor 6 E (0, 1) for future revenues (benefits) and18The assumption on the sequence of audit firms’ decisions is motivated by the fact that the choice aboutspecialization occurs prior to the decisions on audit fees and specialization is an irreversible investment.19t is clear that the audit fee setting and competition stage is closely related to the model of pricecompetition studied by Bertrand (1883).20This assumption explicitly avoids defining an equilibrium in terms of an E-equilibrium where the costefficient audit firm slightly undercuts the other’s auditing cost.21See proposition 2.5.20costs.The equilibrium concept employed is Selten’s (1975) subgame perfect Nash equilibrium(SPNE) and attention is restricted to pure strategy equilibria only.22 A set of pure strategiesfor a game is an SPNE if it is a Nash equilibrium for the entire game and its relevant actionrules are a Nash equilibrium for every proper subgame. This is the appropriate equilibriumconcept for a complete information game. In this model, a strategy for audit firm i is anordered pair, (li, {F}’Z1), that consists of the audit firm’s specialization, l, and a sequence of time-dependent functions, {F}1mapping every possible observed combinationof (1k, 12, ..., 1) and audit fee history {(F1,., F2r, ..., F)}j into a sequence of period-taudit fee schedules quoted to client z, {f}zEz23 A strategy for client z is a sequence oftime-dependent functions {Q}..1 that maps every possible combination of (f1, f2, ..., f,)in period t into (1, 2, ..., n}, where {1, 2, ..., n} is the set of audit firms in the market.Hence, an SPNE strategy choice is an ordered pair, ({(1’, {F})}, {{Q*}Ll}zEz),such that (i) no player can improve his payoff by unilaterally deviating, (ii) {F2}constitutes SPNE choices of {f}zEz, i = 1, 2, ..., n and t = 1, 2, ..., T, for every possible priorchoice of (l1, 12, ..., I) and {(F1, F2, ..., and (iii) {Q*}zEz constitutes SPNEchoices of audit suppliers for every possible combination of (ft, f2, ..., f,), for each z E Z,and for all t = 1, 2, ..., T.The characterization of the SPNE proceeds in two steps. The first step is to characterize the SPNE for subgames starting from stages 2 to T + 1 (henceforth called the pricingsubgames) defined by every possible audit firm specialization choices of (1, 12, ..., i,j. Oncethis has been done, the SPNE for the specialization stage is readily solved.22The concept of subgame perfect Nash equilibrium captures the idea that, when audit firms choosetheir specialization, they all anticipate the consequences of their choice on future audit fee competition. Inparticular, they are aware that this competition will be more severe if their specializations are close to oneanother, rather than far apart.23Obviously, the audit fee history at the beginning of the first period is a null set.212.3 The Pricing EquilibriumConsider the pricing stage under a subgame defined by (1k, 12, ..., la). Given that (i)auditing cost to a particular client z in period t is time-independent and unaffected by thecosts to other clients, and (ii) clients are not able to resell audit contracts, audit fee schedulesquoted by an audit firm to a particular client across different periods or to different clients ina particular period are strategically independent. It follows that the equilibrium audit fee foreach cheilt should be the same for each period. Since audit fees are stable over time, no clientwill ever switch auditors in equilibrium.24 Thus, the pricing subgames are equivalent to aT-period repeated Bertrand pricing game at each point in client-characteristics space underasymmetric auditing cost conditions. The equilibrium of the pricing subgames can then becharacterised by a set of client-specific and time-independent Bertrand pricing equilibria,one for each client at z E Z repeated for T periods. Since all decision variables are time-independent, the subscript for time is suppressed from now on. In the sequel, the SPNE ofthe pricing subgames will be derived as in a one-shot pricing game.Notice that, as a result of specialization and the fact that clients cannot resell auditcontracts, audit firms possess some monopoly power and are able to price discriminate acrossclients as they take into consideration the heterogeneity (by characteristics) among clients.25In general, since the audit fee schedule if specifies the audit fee at which audit firm i iswilling to supply audit services to client z E Z, it must cover its auditing cost to that client.Moreover, in order to induce a client to accept an offer, the audit fee must not exceed themaximum the client would be willing to pay. Thus, formally, ft is in the set:{if: a nonnegative function defined on Z, measurable and such that,24Re-alignment of clients and audit firms are possible if the clients’ characteristics change over time.However, this is assumed away from our model. For empirical test on auditor-client re-alignments, seeJohnson and Lys (1990).25Contrary to general belief, spatial discriminatory pricing is ‘pro-competitive’ compared to uniform pricing. This is because spatial discriminatory pricing provides more flexibility to an audit firm to respond toits rival’s strategies. Since this flexibility is available to each audit firm in the market, audit firms end upgetting trapped into a Prisoner’s Dilemma-type situation and their profits will be driven down by intenseprice competition. See Thisse and Vives (1988) for details.22for all z E Z, bZ f1 m(IIIj — zII).}It is also worthwhile to mention that, in a complete information game, each audit firmknows the characteristics of the client, and knows the specialization of its rivals. Thus, eachaudit firm can calculate the audit fee offered by the other audit firms, and respond with anaudit fee that is attractive to the client but still leaves it with a monopolistic rent. As aresult, each client receives a set of audit fee schedules which depends on his characteristicsrelative to the audit firms’ specializations.Suppose client z receives and accepts an audit fee f from audit firm i, the one-periodsurplus for him is defined asSz(f)_=bz_f.The client accepts the offer which gives him a nonnegative and highest surplus. As statedbefore, it is assumed that when a client receives equal surplus from two or more audit firms,the client chooses to patronize the audit firm that has the lowest auditing cost to serve him.If client z rejects all offers, his surplus is zero. Recall that an audit is not mandated in ourmodel. Therefore, it is clear that SZ(ffl 0 since client z has the right to reject any offerthat gives him a negative surplus. Thus, given a configuration of specializations (Ii, L) anda set of audit fee schedules, (fr, fj), where —i {l, 2, ..., i — 1, i + 1,..., n}, quoted toclient z, the one-period profit that audit firm i earns from client z is26llz(1. 1 fZ fZ)_f ft—m(II1—zII)ifSz(f)>Sz(f7)forallji,‘ ‘ ‘ 1 0 if S(f7) SZ(f) for at least one j i.Hence, given any specialization configuration (1k, 12, ..., i,j, an SPNE audit fee schedulein pure strategies is an n-tuple (F1*, F, ..., F,) of audit fee schedules such thatrrZul. 1 ;Z* Z*’ -. Zf 1 . .çz ;Z*\J , J_) — i—:, J;, J)forallfe2,i=1, 2,..., nandzEZ.26Note that audit firm i’s profit is not directly influenced by its rivals’ specialization choice. Instead,audit firm i’s direct concern is only the current fees offered by its rivals. However, those fees are in generaldependent on the rivals’ specializations.23Following a standard Bertrand argument in spatial models, in equilibrium the audit firmwith the lowest auditing cost to serve client z will exclusively audit him since it can profitablyundercut any audit fee set by a rival. Thus we have:Proposition 2.1. There exists a unique SPNE audit fee schedule which is given by— f min m(IIl — zil) ifm(II1 — zil) < m(1113 — zil) for allj i,— 1 m(fl1—z) otherwise,fori=1, 2,..., n.Proposition 2.1 is a very strong result and depends on only two innocuous assumptions:(i) audit firms are able to set discriminatory audit fees according to client characteristics,arid (ii) clients cannot resell audit contracts. Both of them are believed to be prevalent inthe audit markets.27 Other than these two assumptions, the existence of an SPNE audit feeschedule equilibrium (in pure strategies) is guaranteed for any configuration of specializations(1k, 12, ..., 1), audit technology m(.), and distribution of clients. Moreover, the generalstructure of the equilibrium audit fee schedule is robust to arbitrary client distributions,auditing cost functions, multidimensional client-characteristics space, and many audit firms.The SPNE audit fee schedule is such that the lowest-cost audit firm serves a client at anaudit fee equal to the auditing cost of the second lowest-cost audit firm to that client. Thisimplies that competition among audit firms becomes ‘localized’. An audit firm’s pricingstrategies will have a powerful impact on those rivals whose specializations are very similarto it, but will only have a weak impact on those rivals whose specializations are very differentfrom it.The SPNE pricing strategies characterized in proposition 2.1 provide some interestingempirical implications. Prior empirical studies on audit pricing have suggested that there isa direct relationship between client characteristics and the audit fee charged to that client27The client-specific nature of audit services guarantees the satisfaction of condition (ii), and makes itvirtually impossible for regulators to impose restrictions that would violate condition (i).24(e.g., Simunic (1980) and Palmrose (1986)). Our result suggests that this relationship isindirect. Instead, the equilibrium audit fee to a client is directly related to the cost of theclosest substitute in the audit market, i.e., the auditing cost of the second lowest-cost auditfirm to that client. Since the cost of the closest substitute depends on the difference betweenthe client’s characteristics and the specialization of the second lowest-cost audit firm, theclient’s characteristics only indirectly affect the audit fee charged to him by the lowest-costaudit firm through their influence on the auditing cost of the second lowest-cost audit firm.In other words, while client characteristics directly affect the supplier’s cost, the supplier’sfee charged to the client is only indirectly affected by the effect of client characteristics on theclosest competitor’s cost. Nevertheless, our result is not at odds with the empirical findingswhich document a positive relationship between client characteristics and audit fee. It isbecause the client’s characteristics may be a good proxy for the difference between the client’scharacteristics and the specialization of the second lowest-cost audit firm. However, ourresult implies that one has to be cautious when interpreting the empirical results regardingthe relationship between client characteristics and audit fee.The following caveat is in order before proceeding. The above analysis implicitly assumesthat audit firms make a take-it-or-leave-it offer to the client, while the latter is not allowedto respond with a counter offer (which would start a process of bargaining). This impliesthat the lowest-cost audit firm is assigned superior bargaining power relative to that of theclient in the auditor-client matching game. In spite of the lowest-cost audit firm’s superiorbargaining position, the availability of the other audit firms in the market provides the clientwith an option that defines his bargaining position (i.e., his reservation surplus) when thelowest-cost audit firm makes an offer to him. That is, the client uses an offer from thesecond lowest-cost audit firm to obtain a lower offer from the lowest-cost audit firm. In thisrespect, one might argue that a more direct approach would be to use a bargaining game tostudy the interaction among audit firms and clients. However, the difficulty of this approachis that the outcome is sensitive to the way the extensive form of the bargaining game is25defined (see Bester (1989)). Above all, even though our approach is somewhat arbitrary (thesame comment can equally apply to almost all the existing models on audit pricing), theequilibrium pricing strategies in our model can be shown to result in an allocation of clients’surplus and audit firms’ profits that lies in the core of the economy. It means that, at the finalallocation, no group of clients can move to another audit firm for a mutually advantageousauditor-client re-alignment. To see this, let Z Z U {1, 2, ..., n} be the set of participantsin the audit market (clients and n audit firms), S Z be an arbitrary coalition, and A(S)be the set of feasible allocations for a coalition S. An allocation 9 A(Z) is said to be inthe core if, and only, if there does not exist a coalition S c Z and an allocation 9 A(S)in which all members of S are better off. As such, if profits to audit firms and surplus toclients in the coalition can be increased, the current allocation is not in the core. Clearly,since clients cannot resell audit contracts and we do not allow clients to collude, there isno interaction among clients. Similarly, since audit firms are not allowed to collude, auditfirms can only gain by making deals with clients. Hence, a blocking coalition must containa nonempty subset of clients and at least one audit firm. To see that, given a configurationof specializations, the allocation induced by the equilibrium pricing strategies described inproposition 2.1 is contained in the core, let us recall how equilibrium audit pricing strategiesare determined. For each client z E Z, the equilibrium audit fee is constructed to maximizingprofit for the supplying audit firm (the lowest-cost audit firm) by extracting all the surplusin excess of the surplus the client could obtain from the second lowest-cost audit firm. Thusno other offer could make both the client z and the supplying audit firm better off, and noother offer for the client z could be profitably provided by the other audit firms. Therefore,given a configuration of specializations, the allocation of clients’ surplus and audit firms’profits induced by the SPNE audit fee schedule is in the core, and hence, is efficient.262.4 Specialization EquilibriaIn the specialization stage, audit firms choose specializations looking ahead to the pricingstage outcome derived in the previous section. Under the equilibrium audit fee schedule andthe assumed tie-breaking rule, the audit markets served by each of the audit firms can bedefined asZ1(12,1_i) = j(12,1_i) U L1),wherej(1j,1_j) {z Z m(II1—zIJ) <m(11I—z)for allj i},ô1(12,i) {z e Z m(II1 — zil) m(1113 — zil) for all j , andi+2,..., n}}.Z1 is the audit market exclusively served by audit firm i. It is assumed that U=1Z1 =such that the whole audit market will be covered. It is also easy to see that under the SPNEaudit fee schedule, the audit firms in ãj earn zero profit. Thus, audit firm i’s total expectedprofit (in present value) at the beginning of the first period can be written asll1(1,1,f* f) = 8(1ST)f [pin m(IIIj -zil) - m(III - zII)Jh(z) dz.A noncooperative SPNE choice of audit firm specializations in pure strategies is an ntuple (1, 1, ..., 1) of audit firm specializations such thatTT.(1* 1* 4Z* çZ*’ - (1 1* çZ* IZ*.LL:l1i, i_i Ji , J—i) — t’’t’ i_i, Ji , J —ii’for all 1, E Z, i = 1, 2, ..., n. That is, in the specialization stage, an SPNE of specializationsobtains when each audit firm chooses its specialization so as to maximize its total expectedprofit given its rivals’ specialization choices.Before proceeding to analyze an SPNE choice of audit firm specializations, a few preliminary results will prove helpful and add insight to the properties that equilibrium specializations of audit firms must obey. Let define C(11, 12, ..., 1) and S(11, 12, ..., I) be the total27expected costs for auditing services (in present value) and the aggregate expected surplus toclients (in present value) given a specialization configuration (ii, 12, ..., 1), respectively.C(11, ‘2, “•, i)= ST) j [mm m(II1 - zII)]h(z) dz,S(11, ‘2, )S(i ST) J [bZ — mm m(1113 — zII)]h(z) dz.Furthermore, denote by W(11, 12, ..., 1) the expected social welfare (in present value) whichis taken to be the sum of the total expected profit to audit firms and the aggregate expectedsurplus to clients, i.e.,28W(11, 12, ..., 1) = fl(i, 12, ..., i) + S(11, 12, ..., 1).It follows immediately that W(11, 12, ..., 1) can be rewritten asW(11, 12, •••, 1)=ST)f bzh(z) dz — C(11, 12, 1).The following proposition states the economic relation between the cost-minimizationand welfare-maximization audit firm specializations.Proposition 2.2. A specialization configuration (1k, 12, ..., 1) that maximizes theexpected social welfare also minimizes the total expected costs for auditing services.Because demand for audit service is perfectly inelastic, if social welfare is defined asW(i1, 12, ..., ia), then the maximization of social welfare with respect to specialization reduces to the minimization of total expected auditing costs.The next proposition states that a specialization configuration that maximizes the expected social welfare (or equivalently, minimizes the total expected costs for auditing services)always exists.28Since the model assumes that corporate financial statement audits are homogeneous across audit firmsfrom the viewpoint of the users, the welfare of the end users of audited financial statements should not beaffected by the result of the auditor-client matching.28Proposition 2.3. There always exists a specialization configuration that maximizesthe expected social welfare.Next, it is demonstrated that the existence of an SPNE choice of audit firm specializationsdepends on the existence of specializations that maximize the expected social welfare. Since,proposition 2.3 provides the existence of such specializations, then the existence of an SPNEchoice of audit firm specializations is always assured.Proposition 2.4. An SFNE choice of audit firm specializations, (I’, 1), exists andsatisfiesT’V(1, l) T’V(I,for all 1 e Z, i = , 2, ..., n.Proposition 2.4 states that the existence of an SPNE choice of specializations hingeson the existence of specializations such that each specialization chosen by an audit firmmaximizes the expected social welfare given its rivals’ specializations. Such specializationsexist by the result of proposition 2.3. Thus, under the equilibrium audit fee schedule derivedin the previous section and given the specializations of its rivals, each audit firm chooses aspecialization that maximizes the expected social welfare. However, it does not imply thatthe welfare-maximization specializations are the only equilibrium specializations. In fact, theset of welfare-maximization specializations is likely to be a proper subset of the equilibriumspecializations.29 This is because, even though 1’ must locally maximize W(I, l) given 1,it does not imply that (1, 1) globally maximizes W(1, 1_:).The next proposition states another important property of an SPNE choice of special291n other words, there may exist multiple equilibia. As such, the predictive ability of the model willbe reduced. In order to avoid multiple equilibia, more structure has to be given to the model such thatthe equilibrium is unique. The simplifications we made in chapter 3 ensure that the equilibrium is uniqueand, hence, there is no ambiguity regarding the predicted consequences on welfare of changes in the model’sparameters.29izations.Proposition 2.5. Audit firms will choose different specializations in equilibrium.The intuition behind proposition 2.5 is that if an audit firm chooses the same specialization with at least one rival, profits are driven to zero by intense price competition. Anticipating this outcome, audit firms will never choose the same specialization. Therefore,Bertrand competition drives audit firms to choose different specializations in order to earnpositive profits. In other words, in an SPNE of audit firm specializations, audit firms havea tendency to differentiate themselves in order to relax price competition.3°2.5 Concluding RemarksThis chapter re-examines the nature of competition in the audit market from a spatialperspective. Audit firms in the model make strategic specialization and pricing decisions.Through specialization, an audit firm achieves a cost advantage over its rival for all clientswhose characteristics are closer to its area of specialization. Thus, each audit firm obtainssome market power and is able to price discriminate across clients by offering ‘specializationspecific’ audit fee schedules.The analysis demonstrates that the unique SPNE choice of audit fee schedules requires301n the terminology of spatial economics, our result finds that the principle of minimum differentiationdoes not hold. On the contrary, using a modified model as in D’Aspremont, Gabszewicz and Thisse (1979) butassuming firms can collude on prices, Friedman and Thisse (1993) find that the unique equilibrium outcomeinvolves all firms choosing the same specialization right at the middle of the market. That is, Friedman andThisse restore the principle of minimum differentiation. The same result is obtained in Chan (1993) wherehe assumes inelastic demand and firms can set discriminatory prices. All in all, it is not surprising that thepresence of price collusion induces more supplier concentration because, contrary to Bertrand competition,price collusion does not lead to zero profits when firms choose the same specializations. What is perhaps moresurprising is that price collusion induces no differentiation at all. The reason for this seemingly surprisingresult is as follows. In the spatial framework, choosing the same specialization means that firms’ abilityto punish each other for defection is maximized once the equilibrium specializations are selected. In thiscase the non-cooperative equilibrium profits are zero. Thus once the same specialization has been chosen byall firms, the punishment for defecting is naturally the most severe punishment possible within the model.ilence, the collusion is sustainable.30each audit firm to charge the minimum of the marginal auditing costs of its rivals on servicesto clients whose characteristics are in the vicinity to its own specialization. Given thespecializations of audit firms, these pricing strategies induce an allocation of clients’ surplusand audit firms’ profits that is in the core of the economy. The existence of an SPNE choiceof specializations is also established. We find that, given the specializations of its rivals, anSPNE choice of specializations requires each audit firm to specialize such that the expectedsocial welfare is maximized. Moreover, audit firms will not choose the same specializationin equilibrium. Instead, in order to earn rents as ‘local monopolists’, audit firms will searchfor ‘niche’ markets such as industry specialization. Thus, the model provides a theoreticallink between audit firm specializations and the observed market segmentation.31Chapter 3A Two-Period Spatial Modelof Auditing Competitionwith Learning and Switching CostsIn the first year of an audit, audit firms incur substantial ‘start up’ costs when learningabout new clients’ operations and checking their financial statements.1 If clients terminatethe relationship with their incumbent audit firms and establish another with new audit firms,these start up costs must be incurred again. In addition, once an audit firm has performedan initial audit for a given client, it has acquired specialized knowledge of that client andcan therefore reduce its auditing costs when serving this particular client in future periods.Hence, the existence of ‘learning’, which includes both the start-up costs and learning-by-doing advantages, in the provision of audit services provides comparative cost advantages toan incumbent audit firm when recontracting occurs.21Arens and Loebbecke (1984) provide three reasons for the existence of significant start-up costs entailedin initial audit engagements: (1) it is necessary to verify the details making up those balance sheet accountsthat are of a permanent nature, such as fixed assets, patents, and retained earnings; (2) it is necessary toverify the beginning balances sheet accounts on an initial engagement; and (3) the audit firm is less familiarwith the client’s operations in an initial audit (p. 150-1).21n practice there is a cycle to recurring audits. The completion of one year’s audit naturally leads to andprovides inputs for the planning phase of the following year’s audit. The knowledge gained from previousaudits accumulates and contributes to cost advantages. For example, assessment of a client’s inherent risk isbased on the audit firm’s cumulative audit knowledge and its updated understanding of the client’s business,information, accounting and control systems. The nature and level of inherent risk directly influences the32On the other hand, termination of an audit firm can also impose costs on the client.In general, a client has to incur ‘switching costs’ if he employs an audit firm that he didnot hire in the previous period. The client’s switching costs arise from the need to solicitpresentations from a potential audit firm and, therefore, include the cost of adapting fromone audit firm to another.3 Needless to say, it does not pay for the client to build up a newrelationship with another audit firm if the benefit from switching to a new audit firm doesnot fully cover the switching costs. Thus, clients may display loyalty and continue to usetheir incumbent audit firms simply because of the existence of switching costs.The above discussion suggests that the presence of audit firms’ learning and clients’costs of switching audit firms creates ‘relationship-specific economic interests’ which providethe joint incentive to continue an auditor-client relationship once established.4 That is,both audit firm and client tend to lose in economic terms if an established relationship isterminated. This creates a ‘lock in’ effect and provides the incentive for audit firms toenlarge their market shares in competing for initial audit engagements. As a result, owing tothe competition among audit firms, the natural inference is that the existence of economicinterests of an established relationship induces ‘low-balling’, i.e., audit firms bid below totalauditing costs in their initial audit engagements. In this way, the phenomenon of vigorousprice-cutting on initial audit engagements can be viewed as a competitive weapon utilizedby audit firms seeking to achieve market dominance. Consequently, the competition amongaudit firms, both for the initial audit and at the time of recontracting, will govern the extentto which an incumbent audit firm can benefit from learning and switching costs. In fact, thecosts of changing audit firms in the future period partially induce clients to continue usingtype and extent of audit evidence required.3The client’s switching costs may include search costs of finding a new audit firm, the costs of theadditional time spent by management explaining the system to the new audit firm, and the costs of complyingwith regulation which mandates disclosure of the circumstances surrounding a change of audit firms, etc.. Inaddition, non-economic costs may also be incurred. For example, the client has to interact with an auditorwhose style and personality is quite different from his incumbent auditor.4A relationship-specific economic interest is an asset that is non-marketable or non-transferable in transactions involving different trading partners other than the old ones. In this way, the existence of a relationshipspecific economic interest creates a ‘lock in’ effect by making it costly to switch trading partners.33the audit firms they initially selected. As a result, clients display loyalty and audit firms havean incentive to raise the fees for their audit services in the future period. Thus, in a subgameperfect Nash equilibrium, the audit market is ‘less’ competitive with higher profits after theinitial period. However, the dependence of future profits on the number of ‘locked-in’ clientsalso leads to ‘more’ competitive behaviour in the initial period (before clients have attachedthemselves to audit firms) than if there were no relationship-specific economic interests. Thismeans that relationship-specific economic interests will change the structure of demand inthe first period as well as the future period. In the first period, audit firms are more willingto cut their fees. In other words, in the audit market with relationship-specific economicinterests, audit firms are willing to compete quite fiercely to build up a larger client base,i.e., they behave more competitively, in the first period. On the other hand, audit firmsalso have an incentive to exploit their previous clients, i.e., they behave less competitively,in the future period. Therefore, the effect of the presence of relationship-specific economicinterests on overall competition is potentially ambiguous. Thus, one of the goals of theanalysis in this chapter is to identify the conditions under which the overall competitionis ‘excessive’ or ‘insufficient’ (from a social welfare perspective) in an audit market withlearning and switching costs.5 Among other things, it is shown that while the relationshipbetween learning and the strength of overall competition is monotonic, the relationshipbetween switching costs and the strength of overall competition is not. Nevertheless, theanalysis establishes the conditions under which overall competition will be further fortifiedwhen switching costs increase.This chapter is related to the growing literature on audit pricing which is reviewed inchapter 1. Our work is distinguished from this area of research on the basis of its focus on animperfect competitive audit market. The only published work in the auditing competitionliterature which also emphasizes market imperfections is a recent article by Gigler and Penno(1995). While Gigler and Penno look at audit firms who have substantial market power5The Metcalf Staff Report (U.S. Senate 1976) argues that there is insufficient competition in the auditmarket, whereas the Cohen Commission Report (AICPA 1978) believes it to be excessive.34because of their stochastic endowment of different auditing costs, we examine a settingwhere audit firms are ex-ante identical and strategically choose to become differentiated (interms of their audit production costs) by means of service specialization. The focus of theanalyses is also very different. Gigler and Penno, following Magee and Tseng (1990), focuson the pricing contest between two audit firms for serving a single client. In contrast, ourmodel is a market setting where audit firms compete for an infinite number of clients byoffering client-specific audit fees. The spatial approach that we adopt explicitly recognizesthe dispersed nature of the audit market, namely that it embodies a large number of clientswith different characteristics relevant to audit production and relatively few audit supplierswho may differ in their area of service specialization with respect to client characteristics.It is a widely held notion that audit firm specialization is the primary source of the costdifferences among audit firms. The cost differences, in turn, are believed to be the sourceof market power and, hence, the economic rents which may accrue to the audit firms. Inthis respect, we provide the first formal spatial model to examine how audit firms acquiremarket power by means of service specialization and the effect of audit firms’ specializationson their audit pricing decisions.In this model, the audit finns’ learning is exogenous, but the audit firms’ specializationsas well as the clients’ switching costs are controlled (at least partially) by the audit firms.The endogeneity of the audit firms’ specializations and the clients’ switching costs is notfound in either the existing auditing competition models in the accounting literature orthe existing switching cost models in the economics literature.6 In this framework, it isdemonstrated that social welfare (the sum of audit firms’ profits and clients’ surplus) isinfluenced by the audit firms’ specialization decisions, which in turn are influenced by thelearning and switching costs. As such, it is possible to examine the welfare implications ofchanges in learning and switching costs. These welfare implications are absent in the extantaccounting literature. It is because the social welfare is fixed in a pure pricing game with6For discussion and models of switching costs see Kiemperer (1987) and the references cited there.35inelastic demand. Any changes in the learning and switching costs will simply lead to atransfer of economic interest from one party to the other, and will not change the sum.One of the conclusions in this chapter is that the audit market is less efficient in the presence of switching costs (i.e., social welfare is lower). This inefficiency is driven by the factthat, in the presence of relationship-specific economic interests created by switching costs,audit firms are able to relax price competition and achieve partial collusion by differentiating themselves through specialization of services. However, whether the switching costs,which are often cited as the source of audit firms’ economic rents, may actually increase theeconomic rents to the audit firms is not obvious. In fact, it may happen that increasingswitching costs decreases the audit firms’ profits, and increases the benefit to clients. Such acase is possible if an increase in switching costs induces more aggressive pricing and specialization decisions of audit firms in order to compete for clients in the initial period. Then,audit firms may be worse off if the increased competition drives their first-period fees so lowthat their final profits are reduced even if their future-period rents increased by the increasein the switching costs. Clients are better off in this case since they pay lower audit fees owingto a more competitive audit market. This result is consistent with the finding of Gigler andPenno, although our conclusion is based on an intuition that is different from theirs.The effect of the presence of learning on social welfare is equally subtle. On the onehand, the presence of learning has a direct effect of reducing total auditing costs, given auditfirm specialization choice. On the other hand, it also has an indirect effect of distortingthe audit firms’ specialization decisions. Therefore, the impact of learning on social welfareis potentially ambiguous. A similar argument applies to the effect of learning on the totalprofits to audit firms. An increase in learning allows each audit firm to reduce its totalauditing costs to its clients in the future period. This in turn provides each audit firm anincentive to enlarge its own market share by pricing more aggressively in the initial period.While it is individually rational for each audit firm to do so, all audit firms taken togetherare made worse off by the increased competition. Unlike the cases of social welfare and audit36firms’ profits, the effect of learning on clients’ surplus is clear. Clients are better off becausethey pay lower audit fees owing to increased competition as learning increases.The rest of the chapter is organized as follows. Section 3.1 presents the model, and definesa specialization-price game for the audit firms and an appropriate solution concept. Section3.2 derives a duopoly equilibrium by first analyzing how the second-period price equilibriumdepends on the first-period market shares. Knowledge of this dependence allows one to solvefor the first-period price equilibrium, and hence the first-period specialization equilibriumarid the outcome of the full game in section 3.3. Section 3.4 studies the implications ofchanges in auditing costs, learning rate, and switching costs. Section 3.5 concludes thechapter.3.1 The ModelThe model presented in the previous chapter is very general. In order to get somequalitatively stronger results, it is necessary to give the model more structure. To this end,the analysis in this chapter focuses on a simple two-period spatial duopoly model whereclient characteristics are distributed on a one-dimensional compact space.7 Formally, itis assumed that there is a common index function, G(z) : —* mapping a client’svector of characteristics z into a one-dimensional point z, where for sake of simplicity, z isfurther assumed to be uniformly distributed along the line segment [0, 1] with unit density.8On the other hand, in order to capture salient economic features of an audit market withauditor-client relationship-specific economic interests, audit firms’ learning and clients’ cost7The analysis becomes increasingly complex, if not unmanageable, as additional periods, audit firms ordimensions of client characteristics are added. Nevertheless, the basic economic forces that drive the resultsin a two-period duopoly model are believed to be present in a more general model.8While such a distribution is empirically rare, it provides a setting in which interesting parametric variations can be investigated. Specifically, the assumption of a uniform distribution has the advantage ofeliminating the effect of nonuniformity of distribution as a possible explanation of equilibrium specialization.A nonuniform distribution (e.g., unimodal or bimodal) may lead to agglomeration (i.e., audit firms choosethe same specialization) or differentiation (i.e., audit firms choose different specializations), and confoundsthe effect of competition, which is what this chapter attempts to analyze.37of switching audit firms are introduced.The audit market consists of two independent audit firms which bid on audit engagementsand may oniy differ in specialization of services with respect to client characteristics. Let 11and 12 denote the respective specializations of audit firm 1 and audit firm 2, where 0 1112 i. The auditing technology is the same for each audit firm. It costs m(l) = cjl—c > 0 to audit a client z E [0, 1] in the first period, where l — zI measures the absolutedifference between audit firm i’s specialization and client z’s characteristics.’0In the secondperiod, because of learning by the incumbent audit firm, the auditing cost of audit firm i toclient z is— I /3cl — zI if audit firm i audited client z in the first period,m.2(1) —I cl1 — zI otherwise,where 0 < /3 < 1 characterizes the degree of learning of the incumbent audit firm (i.e., thelower the beta, the higher the learning rate, which is equal to 1 — /3). The effect of /3 is togive the incumbent audit firms a comparative cost advantage over their rivals in the secondperiod.On the demand side, each client voluntarily acquires one unit of audit service from oneof the audit firms in every period. If an audit firm is hired, the audit services it provides willgive a client a gross benefit of b per period, where b > c(2 + /3)/2.” The cost of the auditto the client is an audit fee, f. In the second period, in addition to an audit fee paid to anaudit firm, a client has a switching cost k(12 — 1,), 0 < k < /3c/2, of hiring an audit firm thathe has not perviously hired. The switching costs of a client are assumed to be proportional9This assumption imposes a coordination device concerning the ranking of the audit firms’ specializationalong the line segment [0, 1]. This device can be interpreted as a collusive rule which restricts the auditfirms’ strategy spaces. In the absence of this restriction the two audit firms find themselves in a coordinationgame. This results in an infinity of mixed strategy equilibria. See Bester et al (1991) for details.10A higher c implies that it is more difficult for an audit firm to ‘adjust’ its production process so asto service clients whose characteristics are different from its specialization. For example, suppose the linesegment [0, 1] represents the domain of the client industry. Then when c is high, ‘nonspecialists’ will find itmore difficult to efficiently service clients whose industries do not fall into their area of specialization.11Actually, b just needs to be sufficiently large, so that every client would like to purchase the audit services.Notice also that b can be client-specific and time-dependent without affecting the qualitative results derivedin this chapter.38to the difference between the specializations of the two audit firms. This assumption reflectsthe fact that the switching costs of a client arise from the need to adapt from his incumbentaudit firm to a replacement audit firm.’2The following assumption summarizes the specifications of the parameter values in thismodel.Assumption (Al). Let c> 0, b> c(2 + /3)/2, 0 < k </3c/2 and 0 <3 < 1.Notice that the ranges chosen for b and k are sufficient to ensure that in equilibrium, evenin the presence of relationship-specific economic interests in clients, the whole audit market iscovered and either audit firm can compete for the whole audit market in both periods. It alsoimplies that audit firms cannot charge the monopoly audit fee to their previous purchasersin the second period.Clients are assumed to have rational expectations in the sense that they foresee at anytime the equilibrium of the rest of the game and behave accordingly. The one-period surplusto client z at an audit fee f, if he hires audit firm i in period I is given byQZffZ’ — J 4Z—‘1Ji1) — — Jii’Z(fZ— f b — f if client z hired audit firm i in the first period,b— f2 — k(12 — l) otherwise.Each period, the client chooses which audit firm to hire given the audit fees offered to himand the transaction costs of switching audit firms.On the supply side, in order to induce a client to accept an offer, the audit fee mustnot exceed the maximum amount the client would be willing to pay. However, audit firmsi2j is assumed that clients face no adaption cost with the initial audit engagement. In fact, there is nochange in the analysis if such an adaption cost is independent of the audit firm’s specialization (it can bepart of b). The analysis would change if the adaption cost was specialization-specific (which is the casewith the switching cost). However, it can be shown that there would be no qualitative change in the resultsobtained in this chapter. Above all, incorporating a client’s adaption cost in the initial period does not giveany additional insights to the analysis. Hence, it is omitted for simplicity.39are not restricted to price at or above their marginal auditing costs in this model.’3 Infact, an audit firm might price below its marginal auditing cost if, say, audit firm i wereconfident that audit firm j would undercut its audit fee in the initial period and thus servethe client in question. Such behaviour is of no direct benefit to audit firm i, but serves toforce audit firm j to charge a lower audit fee. Therefore, this behaviour might be strategicallyimportant to audit firm i in attempting to discourage audit firm j from choosing a particularspecialization. Nevertheless, the range chosen for k ensures that, in equilibrium, audit firmsalways charge positive audit fees. Hence, in each period audit firms offer audit fees from thefeasible range [0, b] to each client. Suppose that the audit fees offered to a client z in periodt by audit firms i and j are f, and f, respectively. If the client purchases from audit firmi, then the one-period profit for audit firm i earned from this client is equal toi, f, f) = p — m(lj.Without loss of generality, it is assumed that there is no discounting to the second-periodrevenues (benefits) and costs.Formally, the setting is a two-period, three-stage complete information game with the firsttwo stages occurring in the first period and the third stage occurring in the second period.In the first stage of the game (the specialization stage), the audit firms simultaneouslychoose their specializations in the line segment [0, 1]. In the second stage (the first-periodpricing stage), the audit firms simultaneously quote audit fees to each client. Given the first-period audit fee schedules set by the two audit firms, clients make their audit firm choices.At the beginning of the second period, auditor-client relationships have been established.Learning is realised by the incumbent audit firms and transactions costs must be incurredif the clients choose to switch audit firms. As a result, in the third stage (the second-periodpricing stage), both audit firms simultaneously quote audit fees which reflect the effects oflearning and switching costs. Clients have rational expectations. At each point of time,13This restriction and its welfare implications are examined in chapter 4.40client z acquires auditing services from the audit firm giving him the highest, nonnegative(expected) total surplus.Once again, the equilibrium concept employed is Selten’s (1975) subgame perfect Nashequilibrium (SPNE) and attention is restricted to pure strategy equilibria only. For thefollowing analysis, a variable with an asterisk denotes an SPNE strategy for the subgame inquestion. Moreover, the following tie-breaking rules are adopted for simplifying the analysis:(i) a client will patronize the audit firm with the lower auditing cost if he is indifferent betweenthe audit firms’ offers in the first period; and (ii) a client will stay with his incumbent auditfirm if he is indifferent between his incumbent audit firm’s offer and that of the replacementaudit firm in the second period. A strategy for audit firm i is a triplet (12, F,,, Ff2), thatconsists of the audit firm’s specialization, l, a function Ff1, mapping every possible observedcombination of 11 and 12 into a sequence of first-period audit fee schedules, {f 1], anda function, F2, mapping every possible observed combination of (li, 12, F11, F21) into asequence of second-period audit fee schedules, {f}Zo, 1], where f,’ is the audit fee schedulequoted to client z by audit firm i in period t. A strategy for client z is a pair, (Q, Q),that consists of functions Q that map every strategy of f, and f into {1, 2} in period 1,where {1, 2) is the set of audit firms in the market.3.2 Analysis of Pricing SubgamesThis section finds and characterizes the SPNE for subgames starting from stages 2 to3. It is demonstrated that there exists a symmetric equilibrium in which all clients to theleft (right) of (l + 12)12 purchase from audit firm 1 (2) and clients do not switch auditfirms in equilibrium.’4 Then, the symmetric equilibrium will be proven to be the uniqueequilibrium for the game given the restriction on the audit firms’ strategies on specialization,i.e., 0 11 12 1. This conjecture can be rationalized by noting that audit firms have the‘4Looking for a symmetric equilibrium seems the natural procedure given the symmetric structure of thegame.41same audit cost functions before clients have established relationships with them in the firstperiod, and that audit firm 1 (2) has a cost advantage over the other audit firm with respectto all clients whose characteristics are on the left (right) of (l + 12)/2. This allows auditfirm 1 (2) to offer a strictly better audit fee schedule to all cliellts whose characteristics areon the left (right) of (1, + 12)/2. Unless audit firms choose the same specialization, the setof clients whose characteristics are ‘equidistant’ from both audit firms is of measure zero.As stated in the previous chapter, since audit firms possess some monopoly power bymeans of specialization and clients cannot resell audit contracts, the two audit firms areable to price discriminate across clients as they take into consideration the heterogeneity(by specialization and purchase history) among clients.’5 Thus, in each period, each clientreceives a pair of audit fee schedules, which depends on his characteristics and his relationshipto the two audit firms, and chooses which audit firm to hire. Given complete information,rational players in the game accurately anticipate all actions taken by all the other players.Under the configuration described above, the first-period audit fee schedules {f1,}E[o, 1]and {f1}zE[o, result in first-period profits H, and 1121 and market segments [0, (1, + 12)121and ((1, + 12)12, 11.16 It then follows that audit firms’ second-period choices of fee schedules{f2}o, 1) and {f2}E[o, 1] and their second-period profits 1112 and 1122 depend on thesemarket segments. As usual, this dependence must be examined first since to compute thefirst-period equilibrium one must know how future profits depend on first-period marketsegments.3.2.1 The Second-Period Price EquilibriumThis subsection analyzes the second period of an audit market with the presence oflearning and switching costs, given auditor-client relationships established in the first period.The audit firms’ optimal fee schedules are computed as functions of their first-period market15The purchase history of a client tells which audit firm audited the client in the previous period.16Wjthout loss of generality, the client at (l + 12)12 is assigned to audit firm 1 for technical convenience.42segments, which are characterized by their specializations in the first period. More formally,consider the second-period pricing stage under a subgame defined by (li, 12, F11, F21). Thetwo audit firms are able to set discriminatory audit fees according to a client’s purchasehistory and characteristics. Since the economy only lasts for two periods, the second-periodaudit fee schedule Th which specifies the audit fee at which audit firm i is willing to supplyaudit services to a client at z C [0, 1], must cover the unit marginal auditing cost to theclient. Again, in order to induce a client to accept an offer, the audit fee must not exceedthe maximum amount the client would be willing to pay. Therefore, in equilibrium, fA mustbe in the range [cIl — zi, b] for all z C [0, 1].As stated before, the cost advantage of audit firm 1 (2) over the other audit firm withrespect to all clients whose characteristics are on the left (right) of (l + 12)12 in the firstperiod ensures that, in equilibrium, all clients in the interval [0, (l + 12)/21 ((li + 12)/2, 1])hire audit firm 1 (2) in the first period. As such, clients will switch from the incumbentaudit firm to the replacement audit firm in the second period if, and only if, the differencein client surplus from the two audit fees is strictly greater than the client’s cost of switching.By assumption, an indifferent client stays with the incumbent audit firm. Hence, clientswho patronized audit firm 1 will stay with it in the second period if 112 f2 + k(l2 — li).Similarly, none of the clients who bought from audit firm 2 in the first period will purchasefrom audit firm 1 in the second period if ft2 112 + k(12 — li).Now, consider any client z in the interval [0, (l + 12)/21 who bought from audit firm 1in the first period. In the second period, to any client z in the interval [0, (l + 12)/2], themost favourable audit fee schedule that audit firm 2 can offer, subject to it at least breakingeven, is f = c(l2 — z). The surplus of client z if he accepts the offer is then given bys(f;;) = b — c(l2 — z) — k(l2 — l) > 0, (3.1)for all c> 0, 6> c(2 + j3)/2, 0 < k </3c/2, 0 < /3 < 1 and 0 11 12 1.Being an incumbent audit firm for any client z C [0, (l + 12)12] in the second period,43audit firm 1 must give client z a surplus no less than (3.1) to induce him to stay with it.Thus, the optimal audit fee schedule quoted by audit firm 1 to client z solves: (P3.1)z 11 7 CZ CZ*maxfr2 12k’1,2,J12,J22Q’Zf4’Z > Q’ZfçZ*S.. L72i.J12J— -‘2J22It is not difficult to show that (3.3) is binding and the unique maximum solution for (P3.1)isf=c(12—z)+k(1li).That is, in the second-period price equilibrium, the incumbent audit firm (i.e., audit firm1) matches the rival’s audit fee such that the difference in client surplus from the two auditfees exactly equals the client’s switching costs. In other words, audit firm 1 offers to a client[0, (l + 12)/2] an audit fee that makes him indifferent between staying and switching;and by assumption, the client stays. Hence, in the second period, audit firm 1 would haveits audit fee for client z E [0, (l +l2)/2] constrained by the marginal auditing cost of auditfirm 2 and the switching costs of the client. Accordingly, the profit that audit firm 1. earnsby offering this audit fee schedule to client z in the second period is1112(11,12, f1’, f;;) = c(l2 — z — /31i — zi) + k(l2 — l) > 0,for l1 12, 0< k < and 0< 0 < 1.Notice that the profit that audit firm 1 earns from client z in the second period iscomposed of two basic elements: (i) the difference between the auditing costs of servicingclient z by the incumbent audit firm and that of the replacement audit firm; and (ii) theswitching costs that the incumbent audit firm extracts from client z. Other things beingequal, an increase in the auditing costs (as c increases), the learning (as /3 decreases), orthe switching costs (as k increases) increases the second-period profit that audit firm 1 canearn from client z [0, (l1 + 12)/2]. The effects of the latter two factors on the secondperiod profit are very easy to understand. Only the effect of the first factor needs some44comment. Audit firms’ profits (both in the first-period and the second-period) rise with cbecause an increase in the cost parameter increases the barrier to competition and resultsin higher profit mark-ups earned by the audit firms.’7 The key here is that the auditproduction cost is proportional to c and the learning benefit for any given client is alsoproportional to c. The former would hold even if the learning benefit was additive insteadof multiplicative. Similarly, one can calculate the equilibrium pairs of audit fee schedules forall clients in the interval ((1, + 12)/2, 1]. A complete solution for the second-period pricingstage, (F12, F22, {Q}E[o, ‘i), is then obtained. In the second-period price equilibrium, theincumbent audit firm must price at the marginal auditing cost of its competitor plus theclient’s switching costs on services to clients whose characteristics are in the proximity of itsown specialization. Hence, the unique second-period SPNE audit fee schedules of audit firm1 and 2, respectively, are given by, for z E [0, 1],— I c(l2 —z) + k(12 — 11) if 0 z (l,+ 12)12,f12( 1, 2)— c(z— 1,) if (1, + 12)12 <z <1,— f c(12—z) if 0 z (11+12)/2,b 2 — c(z — 1,) + k(l2 — l) if (l + 12)12 <z 1.(Figure 1 about here)The second-period equilibrium audit fee schedule is illustrated in figure 1 where it isrepresented by the heavy line. The audit market is segmented at (l + 12)/2; audit firm 1(2) serves segment [0, (1, + 12)12] (((1, + 12)/2, 1]) at audit firm 2’s (l’s) marginal auditingcost plus the client’s switching costs. Over the interval [li, 121 the second-period equilibriumaudit fee to client z falls as the difference between client z’s characteristics and his incumbentaudit firm’s specialization rises since the incumbent audit firm has to meet the competition.18Notice that clients are loyal to their incumbent audit firms in the second period because of17t is interesting to point out that if the cost parameter c is firm-specific and audit firms can do somethingto affect their own cost, then while it may be individually rational for an audit firm to decrease its own cost,all audit firms taken together may be made worse off by the decreased profit mark-ups.18Notice that the interval [li, 12] is the market segment where services supplied by the two audit firms aredeemed as ‘close substitutes’ to the clients in that segment.45the presence of clients’ switching costs. This in turn weakens price competition between theaudit firms. Thus, given the audit firms’ specializations, switching costs make the outcomemore collusive in the second period.3.2.2 The First-Period Price EquilibriumNow, go back one stage and consider the subgame defined by (li, 12) in the first period.After the two audit firms have chosen their specialization simultaneously, each audit firm setsits fee schedules while taking into account not only the effect on its first-period profitability,but also the effect on its first-period market segment and hence its second-period profitability.Similar to the second-period pricing stage, in the first period each client receives a pairof audit fee schedules which depend on his characteristics relative to the two audit firms’specializations (although there is no client’s purchase history in the first period on whichto condition the audit fee). Every client has rational expectations and accepts the mostfavourable offer which gives him the highest, nonnegative total two-period (expected) surplus.Thus, given a pair of audit fee schedules in the first period, each client wants to maximizehis total (expected) surplus over the two periods. Again, since audit firm 1 (2) has a costadvantage over the other audit firm with respect to all clients whose characteristics are onthe left (right) of (l1 + 12) /2 in the first period, in equilibrium, all clients whose characteristicsare on the left (right) of (l + 12)/2 purchase from audit firm 1 (2).It is worthwhile mentioning that, because of the presence of learning and transactionscosts of switching audit firms in the second period, there exists a nonempty interval ofclients inside [li, (l + 12)12] ([(li + 12)12, 12]) who will stay with audit firm 2 (1) if theypurchased from it in. the first period. To see this, suppose there exists a client z who is in theinterval (li, (l + 12)/2) and bought from audit firm 2 in the first period. The client will staywith audit firm 2 (his incumbent audit firm) in the second period if f2 f2 + k(l2 — li).Since by assumption client z will stay with his incumbent audit firm if he is indifferent46between his incumbent audit firm’s offer and that of the rival audit firm in the secondperiod, then following the standard Bertrand argument, audit firm 2 will optimally offerf2 = c(z — l) + k(12 — 1) to client z. It is because the best audit fee schedule for client zthat audit firm 1 could offer without suffering a loss is c(z — li). Audit firm 2 offers clientz the same amount of surplus using the above audit fee. Any higher audit fee would meanaudit firm 2 loses client z whereas any lower audit fee would mean it gives up potentialprofits.Needless to say, it is oniy rational for audit firm 2 to make such an offer to client zif its second-period audit fee can cover its second-period auditing cost, i.e., /3c(12 — z)c(z—li)+k(12— i). Let z1 11+ (73C(l2_l1).19 Then f2 = c(z—li)+k(12l1)is the optimalsecond-period audit fee that audit firm 2 could offer to client z only if z e [z1, (li + 12)/21.On the other hand, if z E [0, z1), /3c(l2 — z) > c(z — l) + k(12 — li). Then, the bestsecond-period audit fee that audit firm 2 could offer to client z is its marginal auditing cost,i.e., f2 = /3c(12 — z). In this case, audit firm 2 earns zero profits from client z in the secondperiod.The next step is to determine the audit fee f1 that audit firm 2 would like to offer toclient z e [0, (l + 12)/2] in the first period. At first, notice that the second-period profitthat audit firm 2 can earn from any client z E [0, (l1 + 12)12] isw _f 0 if0z<z’,22—j c(z — l) + k(12 — l1) — /3c(l2 — z) if z1 z (l + 12)12.Consequently, in the first period, to any client z in the interval [0, (l + 12)/21, the mostfavourable audit fee schedule that audit firm 2 can offer, subject to it at least breaking even,isc(12—z) ifOz<z’,—c(12 — z) — [c(z— l) + k(12 — l) — c(l2 — z)] if z’ z (l + 12)12,19z is obtained by setting /3c(12 —z1) = c(z1 — l) + k(12 — li). It is easy to verify that z1 E (li, (1 + 12)12)for alic> 0,0< k</3c/2 and0</3< 1.47i.e., in the first period audit firm 2 is willing to offer an audit fee to client z e [0, (l + 12)/21which is so low that its second-period potential profit from client z is exactly cancelled.Clearly, f < c(l2 — z) for z E [z’, (11 + 12)12]. It means that in order to attract clientz E [z’, (l + l2)/2] to patronize it, audit firm 2 is willing to turn over its second-periodpotential profit to the client in the form of an initial discount.Given the above audit fee schedules, a rational client z E [0, (l + 12)/21 who purchasesfrom audit firm 2 in the first period will have a total two-period (expected) surplus ofS+S=2b—c(1+f3)(l—z)>0. (3.4)Hence, in the first period, looking ahead to the second-period equilibrium audit fees,audit firm 1 must give him a total two-period (expected) surplus no less than (3.4) to induceclient z E [0, (l + 12)/2] to patronize it. Thus, the optimal first-period audit fee schedulequoted by audit firm 1 to client z E [0, (1 + 12)12] solves: (P3.2)maxf1rI(l,fr;’)+Hf;,f;;) (3.5)s.t. S(f1)+S(f12) S+ S. (3.6)It is not difficult to show that (3.6) is binding and the unique maximum solution for (P3.2)is( (1 \ 11z*_J Cj2Z) liuZ<Z,— ‘I, /3c(l2 — z) — k(l2 — l) if z1 z (l + 12)12.Noticethatforallc>0, 0 <k<j3c/2and0</3< 1,f >OforallzE [0, (11+12)12].Accordingly, the profit that audit firm 1 earns by offering this audit fee schedule to clientz in the first period is—f c(l2—z—Ili—zI) if0z<z’,ll( 1, 2, c[(l2 — z) — (z—ii)] — k(l2 — l) if Z1 <Z < (l + 12)12.Similarly, one can calculate the equilibrium pairs of audit fee schedules for all clientsin the interval ((li + 12)/2, 1]. A complete solution for the first-period pricing stage,48(F11, F21, {QflZElo, ‘i), is obtained. The unique first-period SPNE audit fee schedules ofaudit firm 1 and 2, respectively, are given byc(l2—z) ifOz<z’,— /3c(12 — z) — k(l2 — l) if z1 z (li + 12)12,Jii1, 2)— c[(1+/3)(z—li)—(1—z)] k(ll if(li+l2)/2<zz,c(z—li) if z2 <z< 1,c(l2—z) ifOz<z1,lc[(l+/3)(l—z)—(z—li)]—k(li) ifz’z(li+l2)/2,J21 1, 2)— /3c(z — l1) — k(l2 — l1) if (l + 12)/2 < z z2,c(z—li) if z2 < z 1,where z2— (i+,3)C(l2 — l) is obtained by setting /3c(z2 — l) = c(l2 — z2) + k(l2 — li).20(Figure 2 about here)The first-period equilibrium audit fee schedule is illustrated in figure 2 where it is represented by the heavy line. Again, the audit market is segmented at (l + 12)/2 and over theinterval [li, 12] the first-period equilibrium audit fee to client z falls as the difference betweenclient z’s characteristics and the supplying audit firm’s specialization rises. Audit firm 1 (2)serves segment [0, z’) ((z2, 1}) at audit firm 2’s (l’s) auditing cost, which is greater than itsown auditing cost. However, over the interval [z1, z2] the first-period equilibrium audit fee toclient z is below the auditing cost of the supplying audit firm since the audit firm has to meetthe competition. That is, low-balling occurs only over the interval [z’, z2] where competitionbetween audit firms is quite keen.21 It is also easy to show that the interval [z1, z2] increasesas /3 decreases and/or k increases. The reason is that market share is more valuable in thesecond period in the presence of relationship-specific economic interests created by learningand switching costs. Thus, given specializations of audit firms, each audit firm competesmore aggressively than it otherwise would to capture that market share. As a result, bothaudit firm 1 and 2 choose lower first-period audit fees. Moreover, an increase in learningor switching costs has the effect that the competitor will anticipate higher future profits if20 is easy to verify that for all c> 0, 0 < k < /3c/2 and 0 </3 < 1, z2 E ((li + 12)12, 12) and f, f > 0for all z E [0, 1].21A detailed discussion of low-balling is deferred to chapter 4.49it manages to attract the client. Higher future profits imply that the competitor can standa lower audit fee today. Therefore, this drives the supplying audit firm’s fee downwards.As a result, audit firms’ first-period profits are lower if learning and/or switching costs increase. Recall that audit firms’ second-period profits increase with an increase of the learningand/or switching costs. Thus, the effects of learning and switching costs on audit firms’ totaltwo-period profits are potentially ambiguous. A clear conclusion cannot be reached withoutconsidering the corresponding effects on audit firms’ specialization decisions.3.3 Equilibrium for the Full GameIn the specialization stage, audit firms choose their specializations looking ahead to theoutcomes of the pricing stages derived in the previous section. Using the results of the pricingstages, audit firm l’s total two-period profit is given byll(l, 12) = 12, Ff1F;1) + 1112(11,12, Ff2F;2)J c(12 — z — l — zi) dz + J {c[/3(l2 — z) — (z — li)] — k(l2 — l)} dz02[c(12 — z— — zi) + k(12 — li)] dz,since q = 0 for all z E ((11 + 12) /2, 1] and I = 1, 2. By the same token, one can obtainaudit firm 2’s total two-period profit, which is symmetric to that of audit firm 1. In thespecialization stage, each audit firm will choose its specialization to maximize its total two-period profit given its rival’s specialization choice.Before proceeding to solve the SPNE choice of audit firm specializations, it is helpfulto define some useful terms. Similar to chapter 2, let define ll(l, 12), C(11,12), S(11, 12) andW(l, 12), respectively, as the total profit to both audit firms, the total costs for auditing services, the aggregate surplus to clients and the social welfare given an audit firm specializationpair (li, 12). Then, under the equilibrium audit fee schedules, they are given byll(l, 12) = 11(l, 12) + 112(11,12)50flC(l1, 12) I (1 + /)cIli — zi dz + I (1 + /3)c112 — zi dz,Jo JJ!IS(11,12) = I (b—fr) dz+ / (b—f2)dzJ0+ I (b—f7i)dz+ I (b—f;;)az,Jo J!i±!2.T47(l, 12) = H(11,12) + S(l1,12)= 2b—C(l1,1).It is easy to show that W(11,12) is continuous and strictly concave in (li, 12) (or, equivalently,C( l, 12) is continuous and strictly convex in (li, 12)). Then, it follows immediately fromthe results of propositions 2.1 and 2.2 that a specialization pair that maximizes the socialwelfare also minimizes the total costs for auditing services, and such a specialization pairalways exists. The next proposition states the unique welfare-maximization (or equivalently,cost-minimization) audit firm specialization pair.Proposition 3.1. Given (Al), the audit firm specialization pair which maximizes thesocial welfare is (lv, lv’) = (1/4, 3/4).Because of the assumption of perfectly inelastic demands for audit services and uniformdistribution of client characteristics, social welfare is maximized when the pair of audit firmspecializations is (1V, l”). In this case, the two audit firms cooperatively choose symmetricspecializations, each at a difference of one-fourth from the middle of the whole market segment, and split the market in half. The maximum difference between the characteristics ofany client and the specialization of the supplying audit firm is only one-fourth of the totalmarket space. As such, by choosing the specializations (lv, l’), the audit firms maximizethe social welfare. However, these welfare-maximization specializations are not likely to besustainable as the outcome of a noncooperative equilibrium.The next proposition shows the SPNE choice of audit firm specializations. The SPNE51choice of audit firm specializations is a pair (l, l) such that each audit firm’s profit ismaximized given its rival’s specialization. It is demonstrated that, in the presence of learningand switching costs, the noncooperative specialization choices that maximize audit firms’profits are in general different from the ones that maximize social welfare.Proposition 3.2. Given (Al), the unique symmetric SPNE choice of audit firmspecializations, (l, l), is given by1* — 1 1*‘1 —— (3 + /3)(c2 + /32c + 2ck — 2/3ck + 2k)— 2(4c + 4/3c2 + 6/32c+ 2/33c+ 7ck — 2f3ck — /32ck + 6k2 + 2/3k)It is clear that given the equilibrium audit fee schedules, audit firms never choose anidentical specialization in equilibrium. The reason is that when audit firms choose thesame specialization, profits are driven down by intense price competition.22 Anticipatingthis outcome, audit firms will never choose an identical specialization. Therefore, Bertrandcompetition drives audit firms to disperse in order to earn positive profits. In other words,in an SPNE specialization-price equilibrium, audit firms have a tendency to differentiatethemselves in order to relax price competition. To see this, notice that the equilibrium auditfirm specialization choices are a consequence of conflicts among three effects: the market-share effect, the strategic effect, and the cost effect. The market-share effect induces eachaudit firm to choose a specialization that is closer to the middle of the market so as toenlarge its market share. The strategic effect, on the other hand, gives an incentive to eachaudit firm to differentiate itself from the other so as to soften the price competition with itsrival.23 The cost effect induces each audit firm to specialize so as to minimize its total costs.As a result, audit firms choose distinct specializations in the interior of the market segment[0, 1].22J is easily shown that if /3 = 1, Bertrand competition will drive profits to both audit firms to zero whenaudit firms choose the same specialization.23The movement towards audit market agglomeration (differentiation) decreases (increases) an audit firm’sprofit mark-up but increases (decreases) its market share, given the specialization of its competitor.52Now, let us define the equilibrium specialization difference, l — l, as the degree ofcompetitiveness of the audit market. As a benchmark, the welfare-maximization equilibriumspecialization difference is equal to l’ — lv’. Given this, we have the following definition.Definition: From a social welfare perspective, competition in the audit market is‘excessive’ (‘insufficient’) if, and only if, l—l < (>) l’—l’, or equivalently, l > (<) ir’.24In the spatial framework, competition can be excessive if an increase in total auditingcosts (as a result of an undesirable pair of audit firm specializations) exceeds an increase inaggregate surplus to clients. It happens when the SPNE audit firm specializations, (l, 1),are closer together compared to the welfare-maximization ones, (lv’, 1V). On the otherhand, competition can be insufficient if an increase in total auditing costs comes alongwith a decrease in aggregate surplus to clients. It is the case when the SPNE audit firmspecializations are more distinct from each other compared to the welfare-maximization ones.The next corollary provides some preliminary insights about the roles of learning andswitching costs in determining the equilibrium audit firm specializations. A more detailedanalysis and discussion is deferred to the next section.Corollary 3.1. Given (Al), then (i) as /3 approaches one in the limit, competition inthe audit market is insufficient, i.e., 1im÷ l < l’; (ii) as k approaches zero in the limit,competition in the audit market is excessive, i.e., limjo l > lv’; (iii) as /3 approaches oneand k approaches zero in the limit, the SPNE audit firm specializations are the same as thewelfare-maximization specializations.Corollary 3.1 demonstrates that the presence of learning brings in the market-share effectand the presence of switching costs brings in the strategic effect. Hence, in the absenceof switching costs (learning), i.e., as k approaches zero (/3 approaches one) in the limit,24Since l = 1 — l and l’ = 1 — l’ in equilibrium.53the equilibrium audit firm specialization choices are a consequence of conflicts between themarket-share (strategic) effect and the cost effect. As a result, audit firms choose distinctspecializations inside (outside) the welfare-maximization specializations (lv, lv’). When /3is very close to one and k is very close to zero, the market-share effect and the strategic effectare negligible. Then the cost effect alone dictates the welfare-maximization specializations.Finally, given the equilibrium audit firms’ profit-maximization specializations (l, l),where l = 1 — l, the total two-period profit to both audit firms in the duopoly auditingmarket is given byH*(l*) ll(l,1 — l)= 2c(l /3)2+ 7/3c2 + /32c — — 4ck + 8/3ck + 4/3Ck — 6k2 —2/3k2+4l(c2 — 3/3c2 + /32c + /33c2 + 5ck — 6/3ck — 3/32ck + 6k2 + 2/3k)—2l2(5c + /3c2 + 7/32c+3flc2 + l2ck — 8/3ck — 4/32ck + 12k2+4/3k2)], (37)and the corresponding aggregate surplus to clients and social welfare, respectively, are givenbyS*(l) S(l,1 — l)= 2b— 4c(1 + /3)2[3c2 + 17/3c2 + 5/32c— /33c2 — 8ck + 16/3ck+8/32ck — 12k2 — 43k2 + 4l(c2 — 9/3c2 — /32c + /33c2 + lOck — 12/3ck—632 ck + 12k2 + 4/3k2) —4l2(3c — 5/3c2 + /32c + /33c2 + l2ck — 8/3ck—4/32ck+ 12k2 + 4/3k2)], (3.8)T’V(l,1 — l)= 2b—c(1 +/3)(1 —4l + 8l2)The presence of switching costs has no direct impact on social welfare. To see this,54observe that in equilibrium no switching costs are incurred. It is because incumbent auditfirms always set their second-period audit fees to prevent entry so that clients will not changeaudit firms in equilibrium. The presence of switching costs only influences the transfer ofeconomic interests from the clients to their incumbent audit firms. However, there is anindirect effect of switching costs on the social welfare through their influence on equilibriumspecializations of audit firms. The next section will discuss this further.3.4 Implications of Changes in Auditing Costs, Learning Rate, and SwitchingCostsThis section analyzes the implications of changes in auditing costs, learning rate, andswitching costs on the audit firms’ specializations and profits, the aggregate surplus to clients,and social welfare. Before going on, lemmas 3.1 and 3.2 below provide the indirect effects(i.e., the effects through the changes of the equilibrium audit firms’ specializations) of theauditing costs, the learning rate and the switching costs, respectively, on the total profit toboth audit firms, the aggregate surplus to clients and the social welfare. The correspondingdirect effects will be given in lemmas 3.3-3.5.Lemma 3.1. Given (Al), (i) ãl/öf3 < 0. Furthermore, (ii) 6l/Oc < (>) 0 and (iii)ôl/ôk> (<) 0 if,c2 — 3/32c + 4ck + 413ck + 2k > (.<)0. (3.10)Part (i) of lemma 3.1 shows that a decrease in /.3 fortifies the market-share effect sothat audit firms choose specializations that are closer to the middle of the market. Thereason for this is that the monopolistic rents increase with higher learning (i.e., lower /3)if audit firms can enlarge their market shares in the initial period. This can be done bychoosing a specialization that is closer to the middle of the market. As such, it makes an55audit firm become more cost efficient to serve most of the clients in the market, and hence,increases its market share. Part (ii) and (iii) of lemma 3.1 show that, under some conditions(say, inequality (3.10) is positive), while an increase in c induces audit firms to be moredifferentiated, an increase in k induces them to choose specializations that are closer to themiddle of the market. These imply the audit market may becomes more or less competitiveas the cost parameter or switching costs increase.Lemma 3.2. Given (Al) and holding c, /3 and k constant. If l and 1 = 1 — l shiftexogenously, then we have (i) 5 <0 and (ii) > 0. Furthermore, (iii) < (>)0 if,2c — 2/3c + 5ck — 6/3ck — 3/32ck + 6k2 + 2/3k> (<)0. (3.11)Lemma 3.2 states the effects of an exogenous shift of l and l = 1 — l on the total profitto both audit firms, the aggregate surplus to clients and the social welfare. Here, we hold c, /3and k constant, and do not ask why l and l = 1 — l shift. Part (i) and (ii) of lemma 3.2 arevery intuitive. They say that if the equilibrium audit firms’ specializations are closer to eachother, intense price competition drives down the total profit to both audit firms but raisesthe aggregate surplus to clients. However, an increase in price competition may hurt thesociety as a whole as stated in part (iii) of lemma 3.2. In such a case, i.e., if inequality (3.11)is positive, social welfare decreases as the equilibrium audit firms’ specializations are closerto each other and, therefore deviate farther from the welfare-maximization specializations.As a consequence, the total auditing cost in the industry increases. In this sense, excessivecompetition is detrimental to the society as a whole.Depending on the sign of inequalities (3.10) and (3.11) there are two possible sets ofcomparative statics to be considered: (i) when both inequalities (3.10) and (3.11) are positive;and when (ii) both inequalities (3.10) and (3.11) are negative.25 Since the derivation of25J is easily verified that the other two cases (i.e., (iii) when inequality (3.10) is positive but inequality(3.11) is negative; and (iv) when inequality (3.10) is negative but inequality (3.11) is positive) do not existgiven the set of restrictions imposed on the parameters in the model.56comparative statics in both cases is similar, only the case in which both inequalities (3.10)and (3.11) are positive is examined. The other case can be similarly derived and is left tothe interested reader.It is easy to ascertain that solving the positive inequalities (3.10) and (3.11) simultaneously implies 0 < /3 < 0.885618 and mm {0.171573c, /3c/2} < k < /3c/2. For expositionalconvenience, the audit market in which the above conditions are fulfilled is referred to as ahigh learning (i.e., /3 is small), high switching cost audit market.Assumption (A2). Let c > 0, b > c(2 + /3)/2, mm {0.171573c, /3c/2} < k < ,6c/2and0 </3<0.885618.The following two propositions summarize the results of the above analyses.Proposition 3.3. Given (A2), competition in the audit market is excessive.In a high learning, high switching cost audit market, market share is very valuable in thesecond period. This provides an incentive for each audit firm to compete more aggressivelyto capture that market share. As a result, the SPNE choice of audit firm specializationsare closer together than the welfare-maximization ones. This implies that competition isexcessive in a high learning, high switching cost audit market.Proposition 3.4. Given (A2), in the SPNE, the audit market is more competitive(i.e., audit firm specializations are closer together) as (i) the auditing cost parameter, c,decreases; (ii) the learning parameter, /3, decreases; or (iii) the switching cost parameter, k,increases.Recall that proposition 3.3 states that in a high learning, high switching cost auditmarket, audit firms compete more aggressively and choose to specialize inside the welfare57maximization specializations (lv’, 1V). This implies that the total auditing costs given theSPNE audit firm specializations are higher as compared to that of the welfare-maximizationones. Hence, an increase in c fortifies the cost effect and induces audit firms to choosespecializations that are more distinct from each other but closer to the welfare-maximizationspecializations as stated in part (i) of proposition 3.4. The intuition underlying part (ii) ofproposition 3.4 is that a decrease in /3 fortifies the market-share effect since the monopolisticrents increase with higher learning if audit firms can enlarge their market shares in the initialperiod. Hence, a decrease in /3 enhances audit firms’ aggressive behaviour towards marketshare. The intuition behind part (iii) of proposition 3.4 is similar to that of part (ii). Ina high learning, high switching cost audit market, an increase in k increases the incentiveof the audit firms to enlarge their market shares. As a consequence, audit firms choosespecializations that are closer to the middle of the market and compete more aggressively.The following lemmas provide the direct effects (i.e., when specializations of audit firmsare fixed) of the auditing costs, the learning rate and the switching costs, respectively, onthe total profit to both audit firms, the aggregate surplus to clients and the social welfare.The results of these lemmas will be useful in the proofs of the remaining propositions.Lemma 3.3. Given (A2) and holding l and 1 = 1 — l fixed, then (i) 011*/ac> 0,(ii) OS*/Oc < 0; arid (iii) 8W*/Oc < 0.Lemma 3.4. Given (42) and holding l and l = 1 — l fixed, then (i) OS*/a/3 <0and (ii) aW*/0/3 < 0. However, (iii) 811*/8/3 is indeterminate.Lemma 3.5. Given (A2) and holding l and l = 1 — l fixed, then (i) 811*/Ok > 0,(ii) OS*/Ok < 0; and (iii) OW*/Ok = 0.Most of the results of lemmas 3.3-3.5 are very intuitive and will be discussed in thecontext of the remaining propositions. Only part (iii) of lemma 3.4 seems surprising and58needs immediate discussion. To see why the direct effect of learning on the total profit toboth audit firms is ambiguous, recall that the total two-period profit that audit firm 1 earnsfrom supplying services to a client z E [0, 1/2], given the equilibrium audit fee schedulesand audit firms’ specializations, isW* 1* 4. — f c[2(1 — l — z) — (1 + /3)Il — zi] + k(l — 2l) if 0 z <z1,()+ 12(1)_l(1+/3)c(1_2z) ifz’zl/2,since l = 1 — l in equilibrium. Observe that while a decrease in /3 increases the two-period profit that audit firm 1 earns from client z E [0, z1), it decreases those from clientE [z1, 1/2]. Moreover, the number of clients in the market segment [z1, 1/2] increases as/3 decreases.26 Therefore, the direct effect of 3 on the total profit to audit firm 1 dependson the equilibrium specializations of the two audit firms (which, together with /3, determinez1). As audit firms are cx ante identical, a similar argument applies to audit firm 2. Hence,in a high learning, high switching cost audit market, the direct effect of /3 on the total profitto both audit firms is ambiguous.27Now, the implications of changes in auditing costs, learning rate and switching costs onthe audit firms’ specializations and profits, the aggregate surplus to clients and the socialwelfare are readily presented in the following propositions.Proposition 3.5. Given (As), in the SPNE, an increase in the auditing cost parameter,c, increases the total profit to both audit firms, decreases the aggregate surplus to clients, buthas an ambiguous effect on the social welfare.Proposition 3.5 provides some comparative statics properties of an increase in the auditingcosts. Increasing c directly increases audit firms’ profit mark-ups when specializations ofaudit firms are fixed and, hence, it is not surprising that the total profit to both audit firmsincreases as c increases. This increase is further accentuated by the fact that increasing c25Recall that z1 l + — 2l). Therefore, decreasing / also decreases z1.27llowever, given specific values of parameters, the direct effect of 3 on the total profit to both audit firmsis easily determined.59results in a decrease in competition between audit firms as stated in proposition 3.4. As aresult, increasing c relaxes competition and allows audit firms to build up higher profits. Itis also clear that increasing c directly decreases the net value of an audit to clients whenspecializations of audit firms are fixed and, hence, the aggregate surplus to clients decreasesas c increases. This decrease is further aggravated by the fact that audit fees are also higherif audit firms choose more distinct specializations as a response of an increase in c. As aconsequence, clients are worse off when there is an increase in auditing costs. However, theeffect of an increase in c on the social welfare is ambiguous. On the one hand, increasing cdirectly decreases the total surplus available to be shared by the clients and the audit firmswhen specializations of audit firms are fixed. However, the full effect of an increase in c onthe social welfare is confounded by the fact that an increase in c improves also social welfaresince it drives the two audit firms’ specializations more distinct from each other but closerto the welfare-maximization specializations. In fact, social welfare may be higher if auditfirms are more efficiently specialized as c increases.Proposition 3.6. Given (A2,), an increase in the learning parameter, /3, decreasesthe aggregate surplus to clients, but has an ambiguous effect on the total profit to both auditfirms and the social welfare.To understand the intuition behind proposition 3.6, notice that while an increase inlearning allows a given audit firm to reduce its total auditing costs to its clients in thesecond period, which in turn induces it to enlarge its market share in the first period, itsimultaneously induces its competitor to do the same thing. Hence, in equilibrium, auditfirms compete more fiercely and choose specializations that are closer to the middle of themarket. It is then not surprising that clients are better off as they pay lower audit fees as/3 decreases. Whether audit firms are better off after an increase in the learning depends onwhether the change has improved or reduced their profitability under the more competitive60environment. Similarly, even though decreasing /3 directly increases social welfare, whenspecializations of audit firms are fixed, it also drives audit firms’ specializations farther awayfrom the welfare-maximization specializations. Hence, the full effect of a decrease in /3 onsocial welfare is ambiguous.Proposition 3.7. Given (A2), an increase in the switching cost parameter, k, decreasessocial welfare, but has an ambiguous effect on the total profit to both audit firms and theaggregate surplus to clients.As stated before, an increase in k has no direct impact on social welfare since clients donot switch audit firms in equilibrium. It only affects the amount being transferred from theclients to their incumbent audit firms. However, an increase in k induces the two audit firmsto choose specializations that are closer to the middle of the market in equilibrium, whichin turn increases the total auditing costs in the audit market. As a result, social welfaredecreases. However, increasing k has an ambiguous effect on the total profit to both auditfirms and aggregate surplus to clients. On the one hand, it is not surprising that increasing kdirectly increases the total profit to both audit firms and decreases the aggregate surplus toclients, when specializations of audit firms are fixed. This simply reflects the fact that morerents are being extracted from the clients by their incumbent audit firms. However, thiseffect on the total profit to both audit firms or the aggregate surplus to clients is confoundedby the fact that increasing k also results in an increase of competition between the two auditfirms. Thus, audit firms may be worse off if the increased competition drives their audit feesso low that their ultimate profits will be reduced even if k is increased. By the same token,clients may be better off as a result of an increase in k if they would have to pay lower auditfees owing to a more competitive audit market. This result is consistent with that of Giglerand Penno (1995), even though the economic mechanism is quite different.613.5 Concluding RemarksThis chapter develops a simple two-period spatial duopoly model to analyze the effectsof audit firms’ learning and clients’ costs of switching audit firms on auditing competition.In the model, audit firms make strategic specialization and pricing decisions. Through specialization, an audit firm achieves a comparative cost advantage over its rivals for all clientswhose characteristics are closer to its area of specialization. This comparative cost advantageis further fortified by the presence of learning and switching costs. As a result, each auditfirm obtains some market power and is able to price discriminate across clients by offering‘specialization-and-relationship-specific’ audit fee schedules. The analysis demonstrates howequilibrium audit fee schedules and audit firm’s specialization depend on the auditing cost,the learning rate, and the switching costs. In this respect, the analysis may shed light on thepotential conflict between the regulations that affect switching costs (e.g., Securities Release#34-9344, ASR-165, ASR-194 and ASR-247) the audit firms in the audit industry would liketo adopt and those the regulators and/or the clients might want to impose. In particular,the analysis demonstrates that the economic forces considered in the model are such thatlower switching costs can result in efficiency gains. Therefore, if the objective of the regulators (particularly the SEC) is to maximize social welfare (or equivalently, minimize totalauditing costs), then they might want to consider regulations that induce lower switchingcosts. These regulations may raise audit firms’ profits at the expense of clients’ aggregatesurplus, but they improve overall efficiency.62c12+k(13.—1)Figure 1c(1—li)+k(la—li)c(1-4)c(1—l2)8c(1—l2)The second-period equilibrium audit fee scheduleAuditfee ($)Audit finn l’sauditing costEquilibrium.fee schedule63The first-period equilibrium audit fee schedule____Audit firm l’s profit[1 Audit firm l’s lossc(1-l2)flc(1 — 12)Auditfee ($)cl2cl2Audit firm l’sauditing cost— ii)Equilibriumfee schedulec(1 — I)—11) k(l2cl1fldl—k(12—ll)Figure 20 11 Z’ Z 12264Chapter 4‘Low-balling’ and EfficiencyThe practice of ‘low-balling’ has been cited by both the Securities and Exchange Commission (SEC) and the Commission on Auditors’ Responsibilities (Cohen Commission) as afactor which impairs auditor independence.’ Specifically, low-balling could impair auditorindependence since it could provide clients with a credible threat of terminating incumbentaudit firms should they refuse accounting concessions.2 However, based on a multi-periodcontestable market model of auditing, DeAngelo (1981a) argues that low-balling does not itself impair auditor independence.3Instead, she claims that it is the existence of ‘relationship-specific economic interests’ (or ‘quasi-rents’ as defined by DeAngelo (1981a)) between clientsand their incumbent audit firms and the competition among audit firms that may lower the‘optimal’ amount of auditor independence and lead to low-balling. Thus, she concludes thatthere is no causal relationship between low-balling and impaired auditor independence. This1A typical definition for ‘low-balling’ is provided by DeAngelo (1981a), in which DeAngelo defines low-balling as setting audit fees below total current costs on an initial audit engagement. Auditor independence,on the other hand, can be measured in different ways. For example, DeAngelo (1981a) measures the level ofauditor independence as the conditional probability that an audit firm will truthfully report a misstatement.On the other hand, Magee and Tseng (1990) take auditor independence to mean an auditor’s decisions areconsistent with his or her beliefs about a reporting issue.2For example, the Cohen Commission Report (1978) contends that accepting an audit engagement withthe expectation of offsetting early losses with future fees gives the auditor an interest in the financial successof the client and might influence the auditor’s independence in carrying out the examination. The SEC is sowary of this pricing practice that it requires disclosure of any audit fee that is significantly less than whatwould cover expected direct costs.3Since initial fee reductions are sunk in future periods, they have no effect on auditor independence.65point is further explored by Magee and Tseng (1990) who provide five necessary conditionsunder which a relationship-specific economic interest that leads to low-balling may also leadto a compromise of auditor independence.4However, they argue that those conditions areusually not fulfilled. While the issue of auditor independence is admittedly interesting andimportant, this chapter focuses on the economic relation between low-balling and efficiencyin the audit market.5 That is, the purpose of this chapter is to examine the impact of banning audit firms from the practice of low-balling on social welfare, an important issue thathas not been fully considered by the academic researchers or the regulators concerned withlow-balling by audit firms.The conclusions of this chapter depends on a comparison of the equilibrium outcomesderived in chapter 3 with those derived in an otherwise-equivalent economy where low-ballingis not allowed, i.e., everything is the same as in the basic model in chapter 3 except thataudit firms are required to price at or above their marginal auditing costs. It is demonstratedthat although a policy of banning low-balling always reduces competition, it improves socialefficiency in some cases. The key factor that drives this result is the adverse effect of competition on the total auditing costs in the audit market. In a world without regulations onaudit pricing policy, audit firms cannot credibly commit to refuse to price below its marginalauditing cost in competing for initial audit engagements. As a result, audit firms seekingmarket dominance would like to strategically utilize their service specialization to advancetheir competitive position in the audit market. As mentioned in the earlier chapters, throughspecialization with respect to client characteristics, an audit firm can achieve a cost advan4These conditions are: (1) auditors must disagree among themselves on a client’s reporting issue; (2) atthe time of initial engagement, auditors do not know their own positions on the reporting issue; (3) when thereporting issue arises, the client must not know the incumbent auditor’s position on the reporting issue; (4)the reporting issue must affect the client for more than one reporting period; and (5) the client must benefitfrom the preferred reporting strategy even after an auditor switch (Magee and Tseng (1990), p. 317).5Among other things, Magee and Tseng (1990) point out that when audit firms possess all the bargainingpower and there is no disagreement among audit firms on the proper interpretation of generally acceptedaccounting principles (GAAP), clients have nothing to gain by threatening termination of incumbent auditfirms and there is no impairment of auditor independence. Given that the assumptions adopted in our modelare consistent with those of Magee and Tseng, one might argue that the same conclusion could be reachedhere.66tage over its rivals for all clients whose characteristics are closer to its area of specialization.More specifically, given the specialization of its rival, an audit firm has an incentive to choosea specialization that is closer to the characteristics of the average client. This action willincrease the audit firm’s profit by enhancing the audit firm’s competitive power. However,while it is individually rational for each audit firm to choose a specialization that is closer tothe characteristics of the average client, all audit firms in the audit industry taken togetherare made worse off by the increased competition. In fact, competition is excessive even froma social welfare perspective if the equilibrium specializations of audit firms are so ‘close’that the total auditing costs in the audit market become higher. Therefore, social welfareincreases as the policy of banning low-balling allows audit firms to partially collude theiraudit pricing policies and induces them to specialize in a more efficient way.The rest of the chapter is organized as follows. Section 4.1 compares our predictions onlow-balling with those of the existing literature. Section 4.2 analyzes the impact of banningaudit firms from the practice of low-balling. Section 4.3 concludes the chapter.4.1 Low-ballingThe existence of low-balling in the market for auditing services is suggested by the extanttheoretical literature on audit pricing which is reviewed in chapter 1. This section comparesthe primary similarities and differences between the predictions on low-balling of our modeland those of the existing literature. However, since not all the predictions on low-ballingprovided from the existing literature are readily comparable to ours, in the sequel, we onlyconfine our comparison with the predictions provided from DeAngelo (1981a), Kanodia andMukherji (1994) and Magee and Tseng (1990).From the analysis in the chapter 3, it is easy to see that the unique first- and secondperiod SPNE market audit fee schedules offered by the supplying audit firm to each client67z e [0, 1] are given by6c(l—l—z) if0z<z1,‘‘r 1 — /3c(l—l—z)—k(l—2l) ifz1 z <1/2,‘ —— /3c(z — l) — k(1 — 2l) if 1/2 < z z2,c(z—l) if z2 < z 1,1* 1 1* — f c(1 — l — z) + k(1 — 2l) if 0 z 1/2,f2 (, — )— 1 c(z—l)+k(1—2l) if 1/2<z< 1,where z’ l + (3C(1 — 2l) and z2 1 — l — ((1 — 2l).Following DeAngelo (1981a), the low-ball magnitude is defined as the difference betweenthe first-period total auditing cost and the audit fee. Thus the low-ball magnitude in ourmodel is given bylow-ball max {O, mm {cIl — zi, cli — l — zl} — f*} for all z E [0, 1]o if0z<z1,— c[z—l—fl(1—l—z)]+k(1—2l) ifz’ z1/2,— c[l—l—z—/3(z—l)]+k(1—2l) ifl/2 <zz,o ifz<z1.In DeAngelo’s (1981a) model, the predicted low-ball magnitude is the sum of the incumbent audit firm’s learning and the client’s switching costs. On the other hand, the low-ballmagnitude found in Kanodia and Mukherji (1994) is strictly less than that amount. Ourresult predicts that low-balling only occurs in the market segment [z’, z2] where competitionis fierce. Moreover, observe that for any client z E [z1, 1/2] (a similar argument applies toany client z E (1/2, z2]), the low-ball magnitude is given byc[z — l — /3(1 — l — z)] + k(1 — 2l) c(i — /3)(z — l) + k(1 — 2l),where the equality holds only when z = 1/2. Thus, the predicted low-ball magnitude in ourmodel is less than the sum of the incumbent audit firm’s learning and switching costs almosteverywhere. This result is consistent with that of Kanodia and Mukherji, even though ouranalysis is based on a different economic mechanism.6Recall that 1 = 1 — l in the unique symmetric SPNE choice of specializations.68The extant literature also uses ‘price-cut’ to measure the magnitude of the initial feereductions. Following Magee and Tseng (1990), price-cut is defined as the difference betweenthe second- and first-period audit fees.7 Thus the price-cut magnitude in our model is givenbyprice-cut E max {0, f* — f1z*}k(l—2l) if0z<z1,— c(l—/3)(1—l—z)+2k(l—2l) ifz’ zl/2,— c(1—/3)(z—l)+2k(1—2l) ifl/2 < zk(1—2l) ifz<z<1.Magee and Tseng show that the first-period price-cut should at most equal to the client’sswitching costs (i.e., price-cut = k(1 — 2l) in our model). They argue that “the first-periodprice-cut observed by Simon and Francis (1988) should be correlated with the client’s costsof switching to new auditor, not with the auditor’s learning cost (p. 320).” Thus, if clientswitching costs are less than the first-period price-cut, additional explanations for price-cutting are required. Our result shows that Magee and Tseng’ suggestion is valid oniy inthe market segments where low-balling does not occur, i.e., if z E [0, z’) U (z2, 1]. Whenlow-balling occurs, i.e., if z E [z’, z2], the first-period price-cut is always higher than theclient’s switching costs. The reason for this observation is as follows. For any client z inthe market segments [0, z’) and (z2, 1], the second lowest-cost audit firm realizes that evenif it were the incumbent audit firm for client z, it cannot offer an audit fee to him that isas attractive as that of the lowest-cost audit firm in the second period. In other words, thesecond lowest-cost audit firm knows for sure that it will lose client z to the lowest-cost auditfirm in the second period. Thus, it has no incentive to offer an audit fee that is lower than itscurrent auditing cost to client z in the first period. Anticipating this behaviour of the secondlowest-cost audit firm, the lowest-cost audit firm will not charge less than the auditing costof the second lowest-cost audit firm to client z in the first period. As such, the price-cut7The term ‘pricing-cutting’ is empirically motivated. Empirical researchers define it as the differentbetween the first-year audit fee and either prior auditor’s fee or an estimated fee based on a cross-sectionalmodel. See Francis and Simon (1987) for details.69to client z in the first period just reflects the maximum amount of economic rent that thelowest-cost audit firm can extract from him in the second period owing to the existence ofthe client’s switching costs, i.e., price-cut = k(1 — 2l). On the other hand, for any clientz in the interval [z1, z2], the existence of learning and switching costs in the second-periodwill increase the second-period profit of the second lowest-cost audit firm if it can attract theclient to patronize it in the first-period. Thus the second lowest-cost audit firm, anticipatingan economic rent earned as an incumbent in the second period, is willing to cut its fee in thefirst period to the extent that its second-period economic rent is exactly turned over to theclient. This in turn drives the first-period audit fee of the lowest-cost audit firm downwardsin order to meet the competition and results in a price-cut that is greater than the client’sswitching costs.Notice that our result has a clear empirical implication for price-cutting. It suggests thatempirical researchers should always observe price-cutting. However, empirical evidence onprice-cutting is mixed. On the one hand, Baber, Brooks and Ricks (1987), Ettredge andGreenberg (1990), Francis and Simon (1987) and Simon and Francis (1988), find evidence ofprice-cutting in first-year audits. On the other hand, Francis (1984), Palmrose (1986) andSimunic (1980) find no significant evidence of price-cutting. We do not have an explanationfor this mixed empirical evidence, and neither does the existing literature. Rather, we admitthat our understanding of the dynamics of audit pricing is far from complete. We wouldexpect that more important questions can be addressed by studying a T-period model withlearning and switching costs, where T> 2. For an example, one may want to know whetherprice-cutting will persist beyond the initial period. Empirical evidence on this issue is alsoinconclusive. Simon and Francis (1988) find that there is price-cutting on initial auditengagements and the lower audit fee persists into the second and third years following anauditor change. On the other hand, Baber, Brooks, and Ricks (1987) provide evidence ofprice-cutting on initial audit engagements in the public sector. Their results do not indicatethat price-cutting persists beyond the initial engagement year. We propose to study this70issue on our future work.4.2 The Welfare Implications of Low-BallingWe now shift our focus to the welfare implications of low-balling. To this end, we firstderive the equilibrium outcomes in an economy where low-balling is not allowed, i.e., auditfirms are required to price at or above their marginal auditing costs. Then, the equilibriumoutcomes derived will be compared with those derived in chapter 3, where no restrictionsare imposed on audit pricing policy.4.2.1 Equilibrium Outcomes without Low-BallingSuppose that audit firms are required to price at or above their marginal auditing costs.Through competition and specification of the lowest bound on the audit price, it is straightforward to derive the equilibrium audit fee schedules as follows:f1(l, 12) =f1(l, 12) = cmax {Ili — zI, Il2 — zI},f2(l1,12) =f2(l1,12) = cmax {Iii — zI, 112 — zI} + k(l2 — li).That is, in equilibrium an audit firm will charge a client whose characteristics are in theproximity of its own specialization at the marginal auditing cost of its competitor in the firstperiod, and at the marginal auditing cost of its competitor plus the client’s switching costsin the second period.Given these audit fee schedules and the specialization of its rival, audit firm 1 chooses itsspecialization to maximize its total two-period profit, i.e., audit firm 1 solves the followingprofit-maximization problem given l2: (P4.1)max j2c(1—z—Ili—zI) dz+j2[c(l—z—Ili—zI)+k(lli ] dz,since q = 0 for all z E ((li + 12)12, 1] and t = 1, 2. By the same token, one can define71audit firm 2’s profit-maximization problem.The next proposition shows the unique SPNE choice of audit firm specializations whenaudit firms are not allowed to low-ball.Proposition 4.1. Given (Al), the unique symmetric SPNE choice of audit firmspecializations under the duopoly auditing structure without low-balling, (lv, lv), is given byc(3+/3)1 — 2— 2(5c + 3/3c + 2k)Corollary 4.1. Suppose (Al) holds and audit firms are not allowed to low-ball. Then(i) as /3 approaches one in the limit, competition in the audit market is insufficient, i.e.,lim1l < ir’; (ii) as k approaches zero in the limit, competition in the audit market isexcessive, i.e., limk.o l > lv’; (iii) as 3 approaches one and k approaches zero in the limit,the SPNE audit firm specializations are the same as the welfare-maximization specializations.The properties of the equilibrium audit firms’ specializations under the duopoly auditingstructure without low-balling are the same as those with low-balling. The intuition describedin proposition 3.2 and corollary 3.1 applies here.Accordingly, given the SPNE choice of audit firms specializations under the duopolystructure without low-balling, (lv, lv), where l’f = 1 — l, the total two-period profit toboth audit firms is given by11N(lN)ll(lr, 1 — lfl5c — /3c + 4k — 4l(c_:c+ 2k) — 8cl2(1 + /9)(4.1)and the corresponding aggregate surplus to clients and social welfare, respectively, are givenbyS(l, 1 — l) = 2b— 3c +2k —4l(c + k)(4.2)W”(l) W(l, 1 — l) = 2b— c(1 + 3)(1 —4l + 8l2)(4.3)724.2.2 Welfare ComparisonNow, the equilibrium outcomes derived in the previous section can be compared with theones derived in the duopoly auditing structure without restrictions on audit pricing policy.The major results of this chapter are presented as follows:Proposition 4.2. Suppose (Al) holds and audit firms are not allowed to low-ball.Then, compared to the outcome under the duopoly auditing structure with low-balling, auditfirms’ specializations are more distinct from each other (i.e., l < li), total profit to bothaudit firms is higher, and aggregate surplus to clients is lower. The effect on social welfareis ambiguous.It is easy to ascertain that the audit fees paid by clients are higher in both the firstand second periods and the total profit to both audit firms is higher in an audit marketwithout low-balling. The reason is that, without regulations on audit pricing policy, auditfirms cannot credibly commit, in the first-period pricing stage of the game, to refuse to pricebelow marginal auditing cost. Regulations on audit pricing policy provide a substitute forthis precommitment. As a consequence, the banning of low-balling effectively relaxes pricecompetition and allows the two audit firms to achieve a partial collusion in the audit market.However, it is of interest that the society as a whole may be either better off or worse offwhen low-balling is not allowed. To see this, subtracting (3.9) by (4.3), it is easy to showthatw*_wN=c(l+)(l_l)[l_2(l+lflj.Since l > l by the first part of proposition 4.2, thenWN > W* if, and only if, l + l> .This implies that there are two possible cases in which social welfare is higher without lowballing: (i) l > l> lv’; or (ii) l > l’ > l, with l — > l’ — l. In both cases, l > lv’,73i.e., competition in the audit market is excessive without any regulations on audit pricingpolicy. Then, the above analysis and tedious calculation lead to the following proposition.Proposition 4.3.. Suppose audit firms are not allowed to low-ball. Then, comparedto the outcome under the duopoly auditing structure with low-balling, social welfare is higherif, and only if, (i) /3 0.5, or (ii) 0.5 </3 < 0.778225 and c(l — /3)/2 < k </3c/2.Proposition 4.3 states that under some parameter values, social welfare increases as aresult of the policy of banning low-balling. This result is driven by the fact that the policyof banning low-balling effectively allows audit firms to partially collude their pricing policies. In particular, audit firms are less concerned about their competitive advantage in themarket segment [li, 121 where competition is the most fierce. Consequently, audit firms areinduced to choose specializations that are more distinct from each other but are closer tothe welfare-maximization specializations. Thus, increased efficiency justifies decreased competition. Nevertheless, in any case, total profit to both audit firms increases at the expenseof the aggregate surplus to clients (actually, all clients are worse off as audit fees increase).4.3. Concluding RemarksThis chapter compares the predictions on low-balling and price-cutting of our model withthose of the existing literature. Our work is distinguished from the existing literature on thebasis of its focus on an imperfect spatial audit market. While our analysis agrees with theexisting literature that the practice of low-balling is a natural consequence of the competitionamong audit firms, we are able to identify that low-balling occurs only in a certain marketsegment where audit firms compete quite fiercely. On the other hand, if price-cut is definedas the difference between the second- and first-period audit fees, our analysis suggests that itshould be observed as long as there are economic rents accruing to the incumbent audit firms.This chapter also examines the welfare consequences of banning the practice of low-balling.74Our analysis suggests that while a policy of banning low-balling always reduces competition,it improves social efficiency in some cases. Thus, the analysis in this chapter provides alegitimate reason for a regulator to consider the banning the practice of low-balling.75Chapter 5ConclusionThis dissertation develops variants of the well-known Hotelling’s location model to examine the nature of competition in the audit market where audit firms make strategic specialization and pricing decisions. In this chapter, the major conclusions of this dissertationare enumerated in their order of presentation. Suggestions for future research are also provided. As in most formal models, our models are also a stylized representation of real worldphenomena, employing a simple and restrictive framework to obtain tractability. The appropriateness of our assumptions hinges on the empirical validity of our predictions. Perhaps,future research will shed light on this issue.Chapter 2 presents a multi-period oligopoly spatial model of auditing competition thatcaptures all salient economic features of an audit market that involves a large number ofaudit clients with different characteristics relevant to audit production and relatively fewaudit suppliers who differ in their area of specialization with respect to client characteristics.In the model, audit firms strategically choose their area of service specialization and competein audit fees. It is demonstrated that, through specialization, an audit firm achieves a costadvantage over its rival for all clients whose characteristics are closer to its area of specialization. Thus, each audit firm obtains some market power through specialization and is ableto price discriminate across clients by offering ‘specialization-specific’ audit fee schedules.76As a result, given a configuration of audit firm specializations, the unique subgame perfectNash equilibrium audit fee schedule is such that each audit firm charges the minimum of themarginal auditing costs of its rivals on services to clients in the vicinity of its specialization.This structure of the equilibrium audit price schedule is robust to arbitrary client distributions, audit cost functions, multidimensional client-characteristics space, and n > 2 auditfirms. Given the equilibrium audit fee schedules, clients purchase audit services from theleast-cost supplier. This implies that the cost effectiveness of audit firms determines theirultimate market shares. The unique pricing equilibrium is also shown to induce an allocationof clients’ surplus and audit firms’ profits that lies in the core of the economy. That is, atthe final allocation, no group of clients can move to another audit firm for a mutually advantageous auditor-client re-alignment. When making their specialization decisions, audit firmsanticipate the pricing and specialization decisions of their rivals. The competitive forcesin the market induce audit firms to achieve constrained efficient utilization of specializedresources. We establish the existence of a subgame perfect Nash equilibrium choice of auditfirm specializations and find that such a specialization equilibrium is such that each auditfirm chooses a specialization that maximizes the expected social welfare, given the uniquesubgame perfect Nash equilibrium audit fee schedule and its rivals’ specializations. We alsodemonstrate that audit firms will not choose the same specialization in equilibrium. This isbecause the audit firms’ profits are driven down to zero by intense price competition if theychoose the same specialization. Instead, in order to earn rents as ‘local monopolists’, auditfirms search for ‘niche’ markets such as industry specialization. Thus, the model providesa theoretical link between audit firm specializations and the observed market segmentationin which clients with similar characteristics buy from the same audit firm which has costefficiency advantage in serving them.To obtain stronger results, chapter 3 focuses on a simple two-period spatial duopoly modelof auditing competition in which there is (i) learning by the incumbent audit firms, and (ii)clients incur transactions costs if they switch audit firms. As in chapter 2, audit firms achieve77a comparative cost advantage through specialization. However, in the chapter 3 model, thiscomparative cost advantage is further fortified by the presence of learning and switching costswhich create client-specific economic interests. Thus, audit firms optimally price discriminateamong clients by offering them ‘specialization-and-relationship-specific’ audit fee schedules.The analysis demonstrates that the practice of low-balling in initial audit engagements isa natural consequence of the competition among audit firms which are seeking to achievemarket dominance. However, low-balling occurs only in a strict subset of the market in whichaudit firms compete quite fiercely. We also examine how equilibrium audit fee schedules,audit firms’ specializations’ and profits, clients’ surplus, and social welfare depend on theauditing costs, the learning rate, and the switching costs. Some of our results are shown tocarry interesting policy implications. For example, the analysis enables us to understand whythere may be a conflict between the regulations (e.g., Securities Release #34-9344, ASR-165,ASR-194 and ASR-247) the audit firms in the audit industry would like to adopt and thosethe regulators and/or the clients might want to impose. In this respect, our results suggestthat if the objective of the regulators (particularly the SEC) is to maximize social welfare (orequivalently, minimize total auditing costs), then the regulators should impose regulationsthat induce lower switching costs. This policy may raise audit firms’ profits at the expenseof clients’ aggregate surplus, but it improves overall efficiency.The issue of low-balling has received considerable attention. In the past, the interestin low-balling has stemmed from the hypothesized link to a loss of auditor independence.While the issue of auditor independence is admittedly interesting and important, our focusin this dissertation is on the economic relation between low-balling and audit market efficiency, which has not been fully considered by the academics and regulators concerned withlow-balling by audit firms. Applying the model developed in chapter 3, chapter 4 examines the welfare consequences of banning audit firms from the practice of low-balling. It isdemonstrated that if low-balling is not allowed, while audit firms are better off and clientsare worse off, the effect on social efficiency is ambiguous. Our contribution lies in the fact78that we are able to identify those conditions under which a policy of banning low-balling canimprove social efficiency. Thus, the analysis in this chapter provides a legitimate reason fora regulator to consider the banning the practice of low-balling.By and large, this dissertation shows that inefficiency in the audit market arises from thefact that the market equilibrium audit firm specializations are in general different from thecorresponding welfare-maximization ones in an unregulated audit market. This suggests arationale for regulating the audit market. However, it is natural for the regulatory decisionon the audit market (particularly on audit pricing policy) to be made without consideringhow the anticipation of that decision might have affected audit firm specialization (which isalready in place at the time the decision is taken). The analysis in this dissertation suggeststhat this ‘case by case’ approach may lead regulators to ignore important externalities associated with their decisions. Instead, it is emphasized that regulations on the audit marketshould be formulated with an understanding of their likely effect on audit firm specialization.In conclusion, this dissertation represents an attempt to apply spatial economic analysisto understand the nature of competition in the audit market where audit firms make strategicspecialization and pricing decisions. On the contrary, the extant theoretical research onauditing competition considers the audit market as a concentrated and ex-ante perfectlycompetitive market, and has primarily focused on the pricing behaviour of audit firms butignored the issue of audit firm specialization. However, the analysis of spatial auditingcompetition in this dissertation is undoubtedly far from complete. The current models arerestrictive in that they do not permit an explicit examination of other important issues suchas auditor independence and the roles of different audit quality and asymmetric informationregarding auditing costs in audit markets. Without an explicit consideration of the effectof auditor independence on the welfare of the financial statement users (other than theaudit clients), it would be reckless to make any strong statements about the effect of aregulatory decision.1 In this respect, the purpose of this dissertation is less ambitious. It1n the dissertation, we simply assume that the regulators’ objective function is to minimize total auditing79provides structured models that highlight some of the basic trade-offs between the welfareof audit firms and clients that arise when regulations are changed. A formal explorationthat considers auditor independence and the welfare of the end users of audited financialstatements is left for future work.Another obvious deficiency in the models is the assumption that clients perceive auditquality as constant across audit firms notwithstanding suggestions in the literature thatclients correlate audit quality with ‘brand name’ of the audit firm (see DeAngelo (1981b)).Allowing audit quality to vary in the spatial framework would considerably complicate theanalysis, but is of course a promising topic for future research.Finally, the results of our models predict clients never switch audit firms in equilibrium.This is because incumbent audit firms always set their second-period audit fees to prevententry so that clients will not have any incentive to change audit firms in equilibrium.2 Ourresult is consistent with those of DeAngelo (1981) and Magee and Tseng (1990), wheredeterministic auditing costs and no disagreement among audit firms on the client’s reportingissue are assumed. On the other hand, clients do change their audit suppliers when thereare auditor-client disagreements, as in Dye (1991) and Teoh (1992); when cost-minimizationauditor-client matches change over time, as in Gigler and Penno (1995); or when auditorswitches are the clients’ rational response to limit the value of incumbency of the audit firmsowing to their superior knowledge of the auditing costs at the time of the switch, as in Coateand Loeb (1994) and Kanodia and Mukherji (1994). Other than Gigler and Penno (1995),where auditor switching is an artifact of their assumption that efficient auditor-client matcheschange over time, the above approaches emphasize that some sort of information asymmetryis necessary for auditor switching to be sustainable as an equilibrium phenomenon. Forour future work, we propose to incorporate information asymmetry regarding auditing costscosts or, equivalently, maximize social welfare (which, by definition, equals to the sum of the total profit toboth audit firms and the aggregate surplus to clients) without providing any economic justification for whythey would behave this way.2See Saloner (1987) and Milgrom and Roberts (1982) for theoretical discussions of predatory pricing.80into our models. This extension will check the robustness of the results obtained in thisdissertation.81References[1] Akerlof, G.A., 1970, The Market for Lemons: Quality Uncertainty and the MarketMechanism, Quarterly Journal of Economics, Vol. 84, 488—500.[2] American Institute of Certified Public Accountants, 1978, Commission on Auditors’Responsibilities (Cohen Commission), Report, Conclusions and Recommendations, NewYork.[3] Antic, R., 1982, The Auditor as an Economic Agent, Journal of Accounting Research,503—527.[4] Arens, A.A. and J.K. 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Abdel-Khalik and I.Solomon, eds., Sixth Symposium on Auditing Research (University of Illinois), 264—272.[30] Friedman, J.W. and J.-F. Thisse, 1993, Partial Collusion Fosters Minimum ProductDifferentiation, Rand Journal of Economics, 631—645.[31] Gale, D. and M. Heliwig, 1985, Incentive Compatible Debt Contracts: The One-PeriodProblem, Review of Economic Studies, 52, 647—663.[32] Gal-Or, E., 1982, Hotelling’s Spatial Competition as a Model of Sales, Economics Letters9, 1—6.84[33] Gigler, F. and M. Penno, 1995, Imperfect Competition in Audit Markets and Its Effecton the Demand for Audit-Related Services, The Accounting Review, 317—336.[34] Goldman, A. and B. Barley, 1974, The Auditor-Firm Conflict of Interests: Its Implications for Independence, The Accounting Review, 707—718.[35] Green, R.C., 1984, Investment Incentives, Debt, and Warrants, Journal of FinancialEconomics, Vol. 13, pp. 115—136.[36] Greenhut, M. L., G. Norman and C.S. Hung, 1987, The Economics of Imperfect Competition: A Spatial Approach, Cambridge University Press, Cambridge.[37] Hamilton, J., J.-F. Thisse and A. Weskamp, 1987, Spatial Discrimination: Bertrand vs.Cournot in a Model of Location Choice, Regional Science and Urban Economics 19,87—102.[38] Hoover, E., 1936, Spatial Price Discrimination, Review of Economic Studies 4, 182—191.[39] Hotellirig, H., 1929, Stability in Competition, Economic Journal 39, 41—57.[40] Hurter, A. and P. Lederer, 1985, Spatial Duopoly with Discriminatory Pricing, RegionalScience and Urban Economics 15, 541—553.[41] Jensen, M.C. and W.H. Meckling, 1976, Theory of the Firm: Managerial Behavior,Agency Costs and Ownership Structure, Journal of Financial Economics, 305—360.[42] Johnson, W.B. and T. Lys, 1990, The Market for Audit Services: Evidence from Voluntary Auditor Changes, Journal of Accounting and Economics 12, 281—308.[43] Kanodia, C. and A. Mukherji, 1994, Audit Pricing, Lowballing, and Auditor Turnover:A Dynamic Analysis, The Accounting Review, 593—616.[44] Kiemperer, P., 1987, The Competitiveness of Markets with Switching Costs, Rand Journal of Economics 18, 138—150.85[45] Lederer, P. and A. Hurter, 1986, Competition of Firms: Discriminatory Pricing andLocation, Econometrica 54, 623—640.[46] Magee, R. and M. Tseng, 1990, Audit Pricing and Independence, The Accounting Review, 315—336.[47] Melumad, N.D. and L. Thoman, 1990, On Auditors and the Courts in an AdverseSelection Setting, Journal of Accounting Research, 28, 1, 77—120.[48] Milgrom, P. and J. Roberts, 1982, Limit Pricing and Entry Under Incomplete Information: An Equilibrium Analysis, Econometrica, 443-460.[49] Ng, D.S., 1978, An Information Economic Analysis of Financial Reporting and ExternalAuditing, The Accounting Review, 910—920.[50] , 1979, Supply and Demand for Auditing Services and the Nature of the Regulatiolls in Auditing. In S. Davidson, ed., The Accounting Establishment in Perspective:Proceedings of the Arthur Young Professors Roundtable 1978.[51] Osborne, M. and C. Pitchik, 1978, Equilibrium in Hotelling’s Model of Spatial Competition, Econometrica 55, 911—922.[52] Palmrose, Z., 1986, Audit Fees and Auditor Size: Further Evidence, Journal of Accounting Research, 97—110.[53] , 1987, Litigation and Independent Auditors: The Role of Business Failures andManagement Fraud, Auditing: A Journal of Practice and Theory, 90—103.[54] Rhode, J.G., G.M. Whitsell and R.L. Kelsey, 1974, An Analysis of Client-IndustryConcentration for Large Public Accounting Firms, The Accounting Review, 772—787.[55] Saloner, G., 1987, Predation, Mergers, and Incomplete Information, Rand Journal ofEconomics, 215—236.86[56] Schiff, A. and H.D. Fried, 1976, Large Companies and the Big Eight: An Overview,Abacus 12, 116—124.[57] Schultz, J., Jr. and K. Pany, 1980, The Independent Auditor’s Civil Liability: AnOverview, The Accounting Review, 319—326.[58] Securities and Exchange Commission, 1971, Securities Exchange Act of 193, ReleaseNo. 94.4.[59] , 1975, Accounting Series Release No. 165.[60] , 1976, Accounting Series Release No. 194.[61] , 1978, Accounting Series Release No. 247.[62] Selten, R., 1975, Reexamination of the Perfectness Concept for Equilibrium Points inExtensive Games, International Journal of Game Theory 4, 25—55.[63] Shaked, A. and J. Sutton, 1982, Relaxing Price Competition Through Product Differentiation, Review of Economic Studies 49, 3—14.[64] , 1983, Natural Oligopolies, Econometrica 51, 1469—1487.[65] Shibano, T. 1993, Overguarding and the Guardians: Excessive Auditor Liability CanCause Suboptimal Investment, Working Paper, University of California, Berkeley.[66] Simon, D. and J. Francis, 1988, The Effects of Auditor Change on Audit Fees: Tests ofPrice Cutting and Price Recovery, The Accounting Review, 255—269.[67] Simunic, D., 1980, The Pricing of Audit Services: Theory and Evidence, Journal ofAccounting Research, 117—161.[68] and M. Stein, 1987, Product Differentiation in Auditing: Auditor Choice inthe Market for Unseasoned New Issues, Research Monograph No. 13, The CanadianCertified General Accountants Research Foundation.87[69] Stevens, M., 1991, The Big Six, Simon & Schuster, New York.[70] Teoh, S., 1992, Auditor Independence, Dismissal Threats, and the Market Reaction toAuditor Switches, Journal of Accounting Research, 1—26.[71] Thisse, J.-F. and X. Vives, 1988, On the Strategic Choice of Spatial Price Policy, American Economic Review, 122—137.[72] Titman, S. and B. Trueman, 1986, Information Quality and Valuation of New Issues,Journal of Accounting and Economics, 159—172.[73] U.S. Senate, 1976, Subcommittee on Reports, Accounting, and Management of theCommittee on Government Operations (Metcalf Staff Report), The Accounting Establishment: A Staff Study, Washington, D.C.: Government Printing Office.[74] Wallace, W.A., 1980, The Economic Role of the Audit in Free and Regulated Markets,Touch Ross.[75] Watts, R.L. and J.L. Zimmerman, 1986, Positive Accounting Theory, Englewood Cliffs,NJ: Prentice-Hall, Inc.88APPENDIX AThe Potential Benefits of External Auditing:An Information Economic AnalysisThe primary purpose of this appendix is to understand how voluntary auditing servicescan increase audit purchasers’ welfare.’ Prior research has used agency theory and information economics to suggest explanations for the production of audited information.2 All inall, the existing models of auditing assume that there is a welfare loss caused by informationasymmetry between an insider (firm in our model), who has private information, and anoutsider (lender in our model), who has not. This information asymmetry in the marketgives rise to a demand for auditing as a means of information transfer. Our model goesone step backwards and assumes that the firm does not have superior information over thelender. However, the firm can choose to become privately informed if it wishes. We arguethat, given the auditor’s liability system, the economic value of an audit lies in the auditor’s credibility in providing unbiased information compared to other means of providing thesame information. That is, while in the existing models of auditing that include asymmetric1A non-mandated auditing framework is assumed because we want to show a demand for auditing notdriven by exogenous regulation. Rather, we would like to treat the mandated auditing framework as aspecial case of the non-mandated framework. Moreover, Wallace (1980) points out that claims that auditingis prevalent due to regulation are inconsistent with the existence of audits prior to regulation.2For example, see Baiman, Evans and Noel (1987), Datar, Feitham and Hughes, Evans (1980), Melumadand Thoman (1990), and Shibano (1993), to name just a few.89information, the role of auditing is to attest and to verify the firm’s private information,we assume that the auditor has a stronger role of providing additional information that haseconomic value.3In our model, the owner of a firm (the audit client) has monopoly access to differentmutually exclusive investment projects which are classified by risk and return. The firmhas no resources and must turn to a competitive debt market for funding. We assume thatthe current financial condition of the firm not only affects the probability of project success,but also influences the firm’s attitude towards project risk. More specifically, we assumethat while the high risk project is optimal for a firm with a good financial condition, itis not optimal for a firm with a bad financial condition. However, the firm does not haveprivate information about its own financial condition. Moreover, we assume that withoutany information, the firm may shift to the high risk project once a loan contract is signed.Thus, a rational lender will make his loan decision based on his assessment of the firm’scurrent financial condition and his knowledge about the firm’s opportunistic behaviour. Weshow that given the uncertainty about the current condition of the firm and the fact thatthe firm may have an incentive to substitute a riskier project, the lender’s assessment of thefirm’s ability to repay may be so low that the firm will underinvest. That is, the projectchoice as well as the investment amount will be suboptimal. As such, the firm may attemptto discover its current financial condition through private information production. However,this means of private information production may lack credibility to the lender. Thus, thefirm may have an incentive to hire an external independent auditor to attest to the accuracyof its financial statements, such that the more appropriate project will be undertaken andmore favourable loan terms will be accepted by the lender. In executing her attest duties,the auditor acquires information about the firm’s true financial condition. Based on theinformation, the auditor either agrees to attest to the firm’s proposed report and issues anclean opinion, or disagrees with it and issues a qualified opinion. The audit is thus envisioned3Titman and Trueman (1986) make the same assumption.90as a means for independently producing additional information upon which loan contractbetween the firm and the lender is based.4Like the other players in the game, the auditor is also modelled as a rational economicagent who is subject to moral hazard.5 The moral hazard stems from the fact that neither thefirm nor the lender can costlessly observe the auditor’s action after the auditor is appointed.As a result, the auditor has an incentive to shirk if the audit is costly to perform. Sincecontingent fees for auditors are illegal, the court system is used to discipline the auditor.We assume that if the auditor is sued, then she is held liable if she fails to exercise duediligence or care as an auditor. However, what is due care is not always obvious, i.e., thereis a vague negligence standard. Generally Accepted Auditing Standards (GAAS) providesome useful guidance. Currently in the U.S. legal system, auditors usually defend againstcharges of having a negligent audit by demonstrating that they have complied with GAAS.However, courts do not always go by GAAS. As stated in Palmrose (1987): “... adherenceto GAAP/GAAS does not provide absolute assurance that would dismiss auditors fromany liability for material omissions or misstatement (p.91).” To illustrate this argument,we assume that the probability of conviction when an audit is conducted in accordancewith GAAS is greater than zero. That is, we maintain the assumption that the courts arefallible. On the other hand, to maintain that adherence to GAAS is a reasonable defensefor the auditor, we assume that the probability of conviction when an audit is conductedin accordance with GAAS is smaller than that when it is not. Moreover, we posit that theavoidance of litigation arising from substandard audits is the primary force motivating anauditor to adhere to auditing standards. There is no incentive for the auditor to exceed theprescribed standards as long as they are met. Nevertheless, the actual quality of an audit4Although auditing does not provide information about the firm’s project choice, it affects the firm’sproject choice because additional information about the firm’s current financial condition might provideindirect information about the firm’s project choice.5As pointed out by Antle (1982), if one seeks to understand the behaviour of the firm and the lender bymodelling them as expected utility maximizers, the same treatment has to be made to the auditor since theauditor’s incentives are also endogenous.91is not publicly observable when it is conducted, or when the audited report is issued. Atbest, the actual quality of an audit can only be perceived by the firm and the lender. Moreformally, we model the perceived audit quality as linked to the auditor’s attachable wealth,and assume that there always exist some auditors who are wealthy enough so that they willalways comply with the prescribed auditing standards when they are hired. Hence, the courtsystem and the auditing standards affect the interaction between the auditor and the firm.The interaction, in turn, determines the information content of the firm’s audited report.The firm’s ability to repay is then determined by both the lender’s rational expectationsabout the information content of the audited report and the firm’s investment decision.Since the debt market is competitive, the lender earns zero expected profits in equilibriumand, therefore, is indifferent about the existence of an audit. The firm, on the other hand,is better off with an audit since it improves capital allocation and facilitates investmentdecision. We demonstrate that if the marginal benefit to the firm of an increase in thequality of auditing standards is higher than the marginal cost, the gross benefit of an auditincreases as the quality of auditing standards increases.The organization of the rest of this appendix is as follows. Section A.1 describes andanalyzes a simple borrower-lender model without auditing. Section A.2 introduces an auditorwho is hired to investigate the borrower’s financial condition and report the findings. SectionA.3 briefly concludes the appendix.A.1 The Basic Borrower-Lender Model without AuditingA.1.1 The ModelConsider a universal risk-neutral economy in which the owner of a firm has monopolyaccess to M mutually exclusive investment projects, indexed by x X. The firm has noresources and must solicit at most a single loan from a competitive debt market in order to92fund any project. Without loss of generality, the net riskiess interest rate in the economy isset to be zero. There are N M types of observationally identical firms, indexed by y e Y.One may interpret Y as a possible set of the firm’s current financial condition which reflectsthe past performance of the firm and affects the probability of project success. Thus a lendermakes a loan decision based on his assessment of the firm’s current financial condition. Thebetter the firm’s current financial condition, the lower the default risk is. Because of this,the firm may want to submit a financial report regarding its current financial condition to apotential lender. However, even though the distribution of firm types is common knowledgeto the firm and the lender, the firm does not have superior information about its own type.6Moreover, it is assumed that there is no mechanism to penalize misstatements, such thatthe firm would always overstate its financial condition in order to get a better loan term. Inthis setting, a rational lender would then assess the firm’s financial condition as if there is nouseful information provided from the financial report.7 Later in this appendix, an auditorwho has access to an audit technology that can reveal the firm’s type will be introduced.In such a case, the firm must get the certification from an external auditor to show that itsfinancial report is a fair representation.The sequence of events is as follows. There are three dates, 0, 1 and 2. At date 0, the firmrequests and obtains a single loan from a lender in a competitive debt market. That lenderresponds by accepting or rejecting the loan request. If the offer is rejected, the firm thengoes to a new lender with another loan proposal, and so on, until an offer is finally accepted.After an agreement has been reached, the firm is prohibited from seeking additional loans.8At date 1, the firm privately chooses its optimal investment project given the loan amountobtained at date o. For an investment amount I, project x yields at date 2 a random terminal6This assumption avoids an obvious potential adverse selection problem. Dye (1993, 1995) employs thesame assumption.71t implies that the financial report would merely reflect a priori information in this case.8For example, a debt covenant that restricts future debt issues is included in the loan contract.9For simplicity, the agency problem between the owners and the managers of the firm is assumed away.93cash flow of R(I) > 0 if successful and zero if not.’° The support of the terminal cash flowis assumed to be independent of the firm’s type. This ensures that firm types cannot be expost distinguished by merely observing the firm’s performance. Moreover, the return functionR(I) satisfies the following usual regularity conditions: it is strictly increasing and strictlyconcave with R(0) = 0, limi. R(I) < cc and lim10÷OR/t9I = cc for all x C X.”It is assumed that the realized project cash flows, while observable, are not contractible.’2Because of this, we limit the extent to which the loan contract can be based on the observationof the project’s realized cash flow, and simply assume that a standard loan contract is optimalin this setting.’3 Specifically, a standard loan contract offered by the firm, (r, I), is a pairthat specifies a loan amount, I 0, provided by the lender if he accepts the contract, and anominal repayment amount, r I, that the firm must repay to the lender after the project’scash flow is realized (if it has sufficient funds).For simplicity, we focus on the case where there are only two types of firms and twomutually exclusive investment projects. The firm is either ‘good’ (U) or ‘bad’ (B), indexedby y C {G, B}. The common prior probability that the firm is a good type is 4’ C (0, 1).The two investment projects are classified by risk and return and are indexed by x C {L, H}.We refer to project L as the low risk project and project H as the high risk project. For an‘0The assumption that there is a single state in which the lender is paid is made for the sake of simplicity.Relaxing this assumption will not affect the results qualitatively.are the necessary and sufficient conditions for the existence of an interior solution.12This assumption rules out forcing contracts based on punishing the firm if the final project payoffindicates it chose the ‘wrong’ project. A justification for this assumption is that a single outcome is usedhere for simplicity. More generally, one could have multiple outcomes with constant support (i.e., investmentamount, I, and project choice, z, only shift the probability of project success). When the project is successful,the first realized outcome is sufficiently large to repay the debt when the loan contract terminates. However, asignificant time delay may exist between the loan repayment and the realizations of the remaining outcomes.Clearly, this justification is far from pleasing, but we believe this simplification does not destroy the essenceof the results.‘3A formal way to establish the optimality of the standard loan contract is to assume that the realizedproject cash flow can be observed by the lender only by spending some costs in monitoring at date 2. Thefirm then must pay the lender at date 2 whenever it is solvent, or otherwise the lender monitors the firmand seizes the entire proceeds of the project. Then, given the structure of our model and assuming thatthe lender can commit to follow his state-contingent monitoring strategy, it is well known that the optimalincentive compatibility contract must be a standard loan contract. See, for example, Diamond (984) andGale and Hellwig (1985) for details.94investment amount I, the payoff of the high risk project is RH(I) with probability PG (PB)and 0 with probability 1— PG (1 — PB) if this project is managed by a good (bad) type firm.On the other hand, the low risk project yields a payoff of RL(I) with probability ilL and 0with probability 1— ilL, regardless of the type of firm.’4 We assume that RL(I) = ciRH(I)for all I > 0, where 0 < PB < crpL < PG < 1. The restriction on the scaling factor, o,ensures that project H (L) generates a higher expected return than project L (H) does ifit is managed by a good (bad) type firm. This assumption captures the notion that highrisk is usually associated with a greater number of opportunities whose exploitation wouldplace a good type firm at an advantage relative to a bad type firm. We also assume thatall projects will generate a positive return irrespective of the firm’s type, i.e., ilLR(I) > Iand pR(I) > I for all x E {L, H}, y E {G, B} and I> 0. Let PH 4PG + (1 — )PBbe the firm’s expected solvency probability given its project choice is H. We assume thatPL > PH, which implies that the low risk project generates a higher expected return thanthe high risk project does if the firm’s type is unknown.’5 A sufficient condition for thiscondition to hold is that is sufficiently small. It is easy to see that this in turn requires<PL—PB(A.1)PG — PBThe equilibrium concept that we employ is Selten’s (1975) subgame perfect Nash Equilibrium (SPNE) and our attention is restricted to pure strategies only. Formally, a subgameperfect Nash Equilibrium (hereafter called equilibrium) requires that, at each node of thegame, the equilibrium strategy of each player maximizes his expected terminal payoff giventhe strategies of the other players. A strategy for the firm is a combination of the loanamount and the project choice. A strategy for the lender is a loan contract that he is willingto accept from the firm. Hence, an equilibrium is an allocation, [(r, I), xl, which is a pair14We could let firm type influence the success probability for the low risk project as well as the high riskproject without changing the qualitative nature of the results, provided that a good type firm has a higherprobability of success in the high risk project than in the low risk project.15This condition gives rise to the classical risk incentive problem in which a bad type firm has an incentiveto choose the high risk project even though the low risk project will yield a high expected return if the firm’sproject choice is unobservable to the lender. See, for example, Green (1984).95consisting of a loan contract, (r, I), and a project, x, chosen by the firm.A.1.2 Characterizing the EquilibriumThe equilibrium under the setting without auditing is characterized by solving the following principal-agent problem: (PA. 1)maxi->o, r>I, ‘,e p[R(I) — r] (A.2)s.t. j53,*r — I 0, (A.3)x” e arg max p[R(I) — r]. (A.4)zE{L, H}In words, an optimal loan contract is the one in which the expected terminal payoff of thefirm, (A.2), is maximized subject to some constraints.’6 Constraint (A.3) guarantees thatthe lender at least breaks even and constraint (A.4) is the firm’s incentive-compatibilityconstraint.To provide a benchmark, let us first derive the solution under the case where the firm’sproject choice is publicly observable and contractible. In this case, the equilibrium is characterized by solving (PA.l) without imposing the incentive compatibility constraint (A.4). Substituting the participation constraint (A.3) into the objective function (A.2), the principal-agent problem becomesmax pR(I)—I.aE{L, H}, 1>0By the assumption on RH, the first-order condition is both necessary and sufficient fora global maximum. If the high risk project is optimal, then the corresponding optimalinvestment level, ‘H, is characterized by the following equation:ÔRH(IH) — 1.(A 5)81 PH16Since the firm’s type is unknown to both the firm and the lender, all types of firms will offer the sameloan contract to the lender.96Similarly, if the low risk project is optimal, the corresponding optimal investment level, IL,is characterized by the following equation:8RH(IL) =(A 6)81 0PLClearly,8RH(IH)/9I> 8R(Ii,)/ãI since o > j3H/jiL. Then, ‘L > ‘H follows immediatelyfrom the fact that RH(I) is strictly concave. Hence, the investment amount is higher if thelow risk project is optimal. Indeed, the iow risk project is optimal since4TPLRH(IL)— ‘L IYpLRH(IH) — ‘H > PHRH(IH) — III,where the first inequality follows from the fact that ‘L is the maximum solution of €Jj3LRH(I) —I, and the second inequality follows from the assumption that o > liH/PL. Therefore, sincethe low risk project generates a higher expected return than the high risk project does, theequilibrium in the benchmark case is the allocation [(IL/lit, IL), L], where IL solves (A.6).Define U as the sum of the firm’s expected terminal payoff and the lenders’ expectedprofits. Since the lenders earn zero expected profits, the firm’s expected terminal payoff isthe same as U. In the benchmark case, UL is then given byUL = UPLRH(IL) — IL > 0. (A.7)Now we go back to the original setting in which the firm’s project choice is not publiclyobservable. The solution in the benchmark case may no longer be incentive compatible (i.e.,risk incentive problem exists). It is because the firm would choose the high risk project, H,when it receives the loan contract (IL/PL, IL), i.e., the incentive compatibility constraint(A.4) is violated. To ensure that the loan contract (IL/13L, IL) is not incentive compatible,we require- ‘L -PL JRH(IL) — <PH RH(IL) —PL PLwhich can be simplified as(A8)‘\ PL1PLRH(ILY97It is easy to see that, because ojiLRH(IL) > Ii,, (A.8) implies o < 1. Since it is not aninteresting issue if the risk incentive problem does not exist, we assume (A.8) from now on.17We show in the next proposition that underinvestment is needed to resolve the risk incentiveproblem.Proposition A.1. If there exists an I < ‘L satisfyingO’pLRH(I) — I PHRH(IH)— ‘H, (A.9)then the equilibrium under the setting with no auditing is the allocation [(l*/p_L, 1*), L],where 1* solves(JL —pH)RH(1) = (‘_ (A.1O)\ PL/Otherwise, the equilibrium is the allocation [(IH/PH, IH), H], where ‘H solves (A.5).Proposition A. 1 states that, with the risk incentive problem, underinvestment is requiredto motivate the firm to undertake the low risk project. The underlying intuition for whyunderinvestment can resolve the risk incentive problem is as follows. Notice that since(1 — )R(I) is strictly increasing in the investment I, the high risk project can be madeless attractive in the solvency state by reducing the loan amount. In addition, because theprobability of solvency is also lower for the high risk project, the low risk project may becomeoptimal if the loan amount is sufficiently small, i.e., 1* < ‘L. However, if condition (A.9)does not hold, there is no feasible loan contract that can induce the firm to choose the lowrisk project. In this case, undertaking the high risk project is the only credible outcome andthe firm receives J3HRH(IH) — ‘H.In order to study the equilibrium outcome in which the low risk loan contract specifiedin proposition A.l is indeed optimal, we assume condition (A.9) from now on. The firm’s17Note that condition (A.8) is not inconsistent with the earlier assumption that ‘7PL > PH since (1 —L ILPL )PLRH(IL)98expected terminal payoff, U, is then given byU = JPLRH(I ) — (A.11)The following corollary provides some interesting comparative static properties of theequilibrium outcome.Corollary A.1: Suppose the loan contract (I*/p_L, 1*) is optimal, then the equilibrium loan amount and the firm’s expected terminal payoff increase with an increase in theexpected solvency probability of the low risk project, PL, or a decrease in the expected solvencyprobability of the high risk project, jH.The intuition for corollary A.1 is that as the expected solvency probability of the low riskproject increases or that of the high risk project decreases, the low risk project becomes moreattractive compared to the high risk project. Hence, the risk incentive problem is reducedand less underinvestment is required to resolve the problem. Since investment is productive,larger loan size implies larger expected terminal payoff for the firm.The following numerical example illustrates the equilibrium outcome.Numerical Example: Consider an example where RH(I) = i and RL(I) = 0.8/i.Suppose that 4’ = 0.1, pG 0.7, PB = 0.405, and PL = 0.6, then = 0.1 x 0.7 + 0.9 x0.405 = 0.4345, PH/ilL = 0.72417, and ‘H = 0.047198, ‘L = 0.0576 solve (A.5) and (A.6)respectively. The corresponding expected terminal payoffs of the firm are UH 0.047198 x0.21725 — 0.047198 = 0.047198 and UL = 0.8 x 0.6 x 0.24 — 0.0576 = 0.0576 respectively.Observe that IL/PLRH(I’3)= 0.0576 ÷ (0.6 x 0.24) = 0.4, condition (A.8) is satisfied as0.8 <0.72417+0.27583 xO.4 = 0.8345. Thus the benchmark loan contract (IL/PL, IL) cannotbe attained. Then we find that 1* = 0.02721 <IH which solves (A.10). The correspondingexpected terminal payoff of the firm is U = 0.8 x 0.6 x 0.16496 — 0.02721 0.051968> UH.Therefore, the loan contract (l*/PL, 1*) is indeed optimal.99A.2 The Setting with AuditingNow we are ready to examine whether the firm can be made better off by the availabilityof external auditing services. Auditing services include: (1) performing an audit to thefinancial report; (2) expressing an opinion on the fairness of the report; and (3) supplyingan audited report to the client. Since audits are voluntary in this model, the firm hiresan auditor only if the firm’s expected terminal payoff increases by an amount greater thanthe cost of the audit. Our focus in this appendix is to understand the potential benefits ofexternal auditing. Thus the discussion of the audit cost is intentionally suppressed.It is worth mentioning that given the firm’s limited liability to pay, the function ofauditors in this model is not only to mitigate the inefficiency caused by uncertainty regardingfirm type, but also to provide ‘insurance’ to lenders. The first function arises from the factthat the audit provides information about the likelihood that the firm will be successful,thereby economizing on investment in a bad type firm. The second function, on the otherhand, comes from the fact that the loan contract offered by the firm with a clean auditedreport includes a claim on the auditor’s wealth if the firm subsequently fails and the auditis found to have been conducted negligently. The firm can pay damages up to its return ofinvestment, but even this amount is contingent on solvency. Auditors, on the other hand,are assumed to have ‘deep pockets’ sufficient to take care of the damages. However, theauditor is not hired to share risk since all players in the model are risk neutral. Instead, thepurpose of the auditor’s liability is to discipline the auditor’s behaviour.Formally, there are four stages in our incomplete information sequential game with auditing: the auditing stage, the borrowing stage, the investment stage and the litigation stage.The sequence of events proceeds as follows:At the beginning of the period:• Nature determines the firm’s type.100• The Auditing Stage:— The firm chooses whether to hire an auditor to audit its financial report.— Suppose the auditor is hired. Given the auditor’s liability rule and the prevailingauditing standards, the auditor completes the investigation, and issues an auditedreport.• The Borrowing Stage:— Based on the audited report, the firm proposes a loan contract to a lender whodecides whether to approve the loan contract or not. If approved, the firm borrows.• The Investment Stage:— The firm chooses one of the two investment projects.At the end of the period:• Either the firm realizes revenue R(I) and pays the lender or the firm goes bankrupt.• The Litigation Stage:— The lender sues the auditor if the firm received a clean audited report but wentbankrupt.— The court conducts its investigation and assigns damage awards.We now fully describe our game with auditing.In the auditing stage, the firm chooses whether to hire an external auditor to investigateits financial condition.’8 The payment to the auditor is the fee f. Conforming to rules18The firm does not need to hire an external auditor if it just wants to learn about its own type; presumablyit is cheaper to ask its own accounting manager to do the job. Besides the agency problem between theowners and the accounting manager, the firm would also like to avoid the adverse consequence owing to thewell-known signaling problem as the one described in Akerlof (1970) by making its financial report credible101governing auditor independence that prohibit contingent fees, the auditor’s fee cannot dependon her report. If the users of financial statements suspect a misstatement, the only remedyis to file a lawsuit.19 Although the auditor’s fee is usually privately known by the firm andthe auditor, the firm’s action of hiring or not hiring an auditor is public information. Forsimplicity, we violate our previous assumption a bit by assuming that the firm has sufficientresources to pay the required audit fee even though it does not have its own resources tofinance the project.2°The auditor is a utility-maximizing, risk-neutral agent who strategically decides her fee,f, and quality of audit service, q.2’ Without loss of generality, q is normalized such thato <q < 1 and represents the probability that the auditor successfully detects a discrepancybetween the firm’s claim and its true type. Each auditor is assumed to have the sameaudit technology described as follows. First, recall that since there is no penalty for thefirm’s misstatement of its type, the firm will always report itself as a good type. However,the firm’s claim is now subject to verification by the auditor. We assume that if the firmhas reported a false type, the auditor’s finding will be identical to the firm’s message withprobability (1 — q) and will contradict the message with probability q; but if the firm makesa correct claim, there is no error to discover, and the auditor’s finding will conform to thefirm’s message.22 To further simplify the analysis, we assume that the auditor will honestlyreport her finding in the auditor’s opinion. If her finding agrees with the firm’s reportedand transparent to lenders. One way of doing this is to issue an audited financial report. In this way, publiclyobservable audited financial statements and the fact that auditors are legally liable for material misstatementof accounting information prevent the less reliable and transparent private information production by thefirm. Moreover, the public good nature of auditing also prevents the less efficient private search for firm-specific information.19Note that a fixed audit fee plus a contingent liability is equivalent to a contingent fee scheme.20Alternatively, one might assume that the firm has no resources at all so that it must borrow the requiredaudit fee in addition to the necessary funds for the investment project. Our assumption that the firm hasmoney to pay the required audit fee is made to simplify the analysis. Relaxing this assumption will notaffect the results qualitatively.21While admittedly unrealistic, the assumption that the auditor is risk neutral may be less so for auditorsin large audit firms that are diversified across a large number of clients.22That is, we limit the auditor to making type II errors only. This assumption is consistent with thereal-world auditing procedure whereby the firm’s financial statements are modified only when the auditordiscovers a discrepancy between these numbers and her findings.102state, the opinion is said to be clean. Otherwise, it is a qualified opinioll. Although theauditor’s opinion is publicly observable, the quality of an audit is not.After the audit is completed, the firm submits it to a potential lender with its loanapplication. More specifically, in the borrowing stage, the firm with an audited reportrequests a loan by offering a loan contract (r, I) to a potential lender. We assume thatif an auditor is hired, then the audited report must be included with the contract offer nomatter what it contains. The firm knows the content of the audited report before it choosesthe contract to be offered. The lender, when he decides whether to accept a proposed loancontract from the firm, must infer the information content of the audited report. Clearly, thelatter is implied by the auditor’s quality choice. Even though the auditor’s quality choice isnot publicly observable, the lender can make such an inference because he knows the elementsof the auditor’s decision problem. The lender then uses the error inherent in the auditor’soptimal audit quality choice to revise his beliefs about the distribution of firm types uponobserving the firm’s audited financial report. In equilibrium, the lender’s conjecture mustbe fulfilled.Suppose the auditor’s individually rational audit quality choice is q. Then the posteriorprobability of good and bad type firms will equal qS+ (1— — q) and (1— qS)q, respectively.By our audit technology assumption, if the audited report is a qualified report, the firm is abad type for sure. That is, no mistake will be made by the auditor. Furthermore, supposethe following conditions hold:+—- IL, (A.12)PL PL PLRH(IL)where IL solves (A.6); and there exists an I < IL satisfyingOPLRH(I) — I PBRH(IH)— ‘H, (A.13)where ‘H solves (A.5) with PH replaced by PB 23 Then the optimal loan contract for thebad type firm will be the low risk loan contract, (lB/pB, IB), where ‘B solves (A.l0) with23The interpretation of (A.12) and (A.13) are similar to those of (A.8) and (A.9).103PH replaced by PB. Notice that because there is no uncertainty regarding firm types oncethe firm is identified as a bad type firm, the optimal low risk loan contract for the bad typefirm is independent of the audit quality q. On the other hand, suppose that the auditorissues a clean audited report to the firm. Then using Bayes’ rule, the posterior conditionalprobability that the firm is a good type is given byE@, 1).It is not difficult to verify thatthb(q) q(l—q) 0Oqandã2&(q) — 2qf(l —08q2 [+(1-)(1-q)]3>That is, upon observing a clean audited report, the posterior conditional probability that thefirm is a good type firm is strictly increasing and strictly convex in audit quality. Therefore,the uncertainty regarding firm type can be controlled by increasing audit quality.The firm stays in the borrowing stage until it obtains a loan, then it enters into theinvestment stage by choosing its optimal project given its loan amount. Neither the auditornor the lender can observe the firm’s project choice.We now know that if the firm gets a qualified opinion, the optimal loan contract will bethe low risk loan contract, (‘B/Pt, IB). Accordingly, the ex-post terminal payoff (excludingaudit fee) for the bad type firm is given by JPLRL(IB) — ‘B. In the sequel, we will focus onthe derivation of the optimal loan contract for the firm if it obtains a clean audited reportfrom the auditor.Suppose that the firm with a clean audited report obtains a loan contract (r, I) andundertakes project x. With audit quality q and cormnon posterior assessment of good typefirm ‘(q), the firm’s conditional probability of solvency if the low risk project is undertaken isgiven by &(q)jiL + [1— b(q)]L = Pt, which is the same as if there is no additional information104provided by the audited report. On the other hand, if the high risk project is undertaken,the firm’s conditional probability of solvency is given byTrH(q) b(q)pc + [1 — ?,b(q))pBçbpG+(l —b)(1—q)pB -= +(l-)(1-q)E(PH,PG).It is easy to see thatthrH(q) thb(q)aq=(PG—PB)‘9q>0.The firm’s conditional probability of solvency is strictly increasing in audit quality. In thiscase, the firm’s ex-post expected profit (excluding audit fee) is lrH(q)[RH(I) — r]. Noticethat given a clean audited report is issued, audit quality will have an effect on the firm’sconditional probability of solvency only when the high risk project is undertaken.For the loan contract (r, I), the lender’s expected loan repayment received from the firmwith a clean audited report is given by 7rH(q)r. However, in making a loan decision, thelender will also take into account the potential damage award receivable from the auditor.The availability and amount of the damage awards hinges on the auditor’s liability rule inplace.It is noteworthy that the auditor is a rational economic agent. If there is no liability,then given that contingent audit fees are illegal and audit quality is costly to the auditor,the auditor would provide the least amount of quality once she is hired. As such, the auditorwill issue a clean audited report regardless of the true type of the firm. Of course, all users ofthe audited report will recognize the auditor’s self-interested behavior and react accordingly.Thus the audited report will have no information content in the sense that the posteriorassessment of a good type firm will be the same as the prior. That is, we are back to thesetting where there is no auditor. Thus, we introduce a court system in which users of anaudited report can sue the auditor if they believe that there is a material misstatement madein the audited report. The threat of third-party suits produces equilibria in which the auditreport is informative.105Formally, in the litigation stage, the lender decides, based on the auditor’s report andthe solvency position of the firm, whether to sue the auditor.24 For simplicity, we imposethe following limitation on the types of suits that can be tried. The court only hears casesin which a party can claim to have been damaged by the auditor’s report.25 Hence, a lendercan sue only if the firm was bankrupt after the auditor issued a clean audited report.The auditor’s liability rule adopted by the court is assumed to be a vague negligencestandard under the joint and several liability regime, which seems to be the most descriptivescenario of the current situation. Under the vague negligence standard, the auditor is heldliable if she fails to exercise due care in her duties as an auditor. However, what is due care isnot clearly defined. Normally, GAAS provide some useful guidance, but courts do not alwaysgo by GAAS. Thus, there always exists some uncertainty in determining whether due care ismet. More specifically, let q* be the audit quality that is defined by GAAS.26 In a lawsuit,the court looks for discrepancies between the auditor’s report and the firm’s true type; wedefine such a discrepancy as an audit failure. We assume that, if an audit failure exists,based on the evidence provided by the auditor, the court discovers it with probability i/K > 0if the auditor chooses audit quality q q* and is sued. On the other hand, if the auditorchooses audit quality q < q* and is sued, the court discovers an audit failure with probability1,0> i/K. Thus, our model captures the notion that adherence to GAAS can be a reasonabledefense, and yet assuming that i/K is greater than zero, we maintain that the courts are notinfallible. As such, a reasonably diligent auditor who has conducted an audit in accordancewith GAAS still faces some probability of having an error in the report and not being ableto provide sufficient evidence of due care to clear herself. Moreover, we assume in our model24We assume that the lender can sue the auditor but not the firm. It is because the auditor, who has‘deep pockets’, is the only potential defendant that is still solvent in the model. In the other words, thisassumption is adopted to permit us to avoid questions of whether the firm can be sued when it is bankruptand how damage payments are to be divided between a firm and its auditor.25This is consistent with Schultz and Pany (1980) whereby they present an overview of the public accountant’s civil liability and address four elements of proof which are central to the determination of thisliability, namely, (i) the plaintiff must incur financial damages, (ii) there must be a material omission ormisstatement in the financial statements, (iii) there must be reliance on the statements by the plaintiff, and(iv) the accountant’s conduct must be deficient in some respect for determination of liability.26The standards are usually set by the accounting profession and are taken to be exogenously determined.106that the prospect of facing litigation arising from substandard audits is the primary forcethat motivates an auditor to adhere to auditing standards. As such, once the prescribedstandards are met, the only cost to the auditor is her cost of quality. Hence, the auditorwill never exceed the quality level defined by GAAS. It is also worthwhile mentioning thatthe liability losses fall when the auditor complies with the prescribed auditing standards fortwo reasons: there is less chance of an audit failure and the auditor is less likely to be foundnegligent even if an audit failure occurs.If the audit report is accurate (i.e., when there are no discrepancies between the auditor’sreport and the firm’s true type), there is no audit failure to uncover, and the court’s rulingagrees with the report. On the other hand, if the court discovers an audit failure and theauditor is judged negligent, the lender will get full compensation. Let D(I) 0 be thedamage award paying to the lender. The damage award in principle should be directlylinked to the actual loss incurred by the lender, which in turn is a function of the actual loansize I. Since I is endogenously determined, the penalty award, D(I), is also endogenouslydetermined.27 For simplicity, we assume that D(I) = 1.28 When the firm goes bankrupt inan audit failure, the auditor, who is held jointly and severally liable, must also pay damageson behalf of the bankrupt firm, even if the auditor is assessed as being responsible for onlya small fraction of the total liability.Suits also involve legal costs which have to be paid by both the auditor and the lender.Under the American Rule of litigation cost allocation, each side will bear its own cost.29 Thewinning party cannot recover its litigation cost from the losing party. Let the legal cost for271n general, the goal of the penalty has two parts. It aims at fairly compensating the victims andadequately deterring the auditor from any ‘wrong-doing’. The endogenity of the penalty award in our modelrenders any declared liability standard fair, and accordingly, narrows our focus to the deterrence aspect.25The qualitative results do not change as long as the damage award D(I) is assumed to be an increasingfunction of the actual loan size I. We argue that such a direct link between auditor’s liability and theloss incurred by the lender should bring out a more effective audit incentive structure because the auditor’sliability will be more sensitive to her audit quality in this case.29Note that many statutes on the state and federal levels in the United States also provide for the shiftingof fees under particular circumstances. Some of these rules resemble the English system whereby the loseris typically forced to bear the winner’s legal expenses. The English Rule of litigation cost allocation is notconsidered in this study.107the auditor be , e (0, I).° On the other hand, we are not going to model an explicit legalcost for the lender but simply assume that the lender always sues the auditor whenever thefirm goes bankrupt.3’ This assumption seems close to the current litigation environmentwherein trial lawyers always seem to be available to work on a contingency basis. Therefore,the lender has little to lose to sue the auditor. Of course, the lawyers must expect to recovertheir costs. We assume that trial lawyers are risk-neutral and incur a cost of t for litigating.If the lender wins the suit, the lawyers keep a portion of the damages collected. Let thefraction of the damages paid to trial lawyers as contingent fees be 1 — a, a E (0, 1). Weassume that the lawyers’ legal costs are always smaller than the expected contingent feessuch that they are always willing to take the case. Hence, in this framework, the lender’sstrategy about whether to sue the auditor is taken as fixed. As such, we can analyze theauditor’s optimal response to a potential legal liability.32Notice that the effective auditor’s liability for damage, L(I, w), also hinges on the auditor’s attachable wealth, w, i.e., L(I, w) = mm {I, w}. For simplicity, we assume thatthe liability rule/standard combination {[L(.), i,*}, q*} is typical as defined by Dye (1993),such that there always exists some auditors with attachable wealth, w, who would adopt astandard quality audit if she is hired, i.e., q = q*• Moreover, we assume that the firm wouldonly hire an auditor who is very likely to comply with the prescribed auditing standardsq* Since big audit firms are more likely to comply with the audit standards than smallaudit firms because, other things being equal, larger audit firms have more wealth to lose,we assume that the firm will go to a big audit firm for audit services.:34301f c I, the auditor is better off settling the suit out-of-court. The possibility of settlement is notconsidered in this study.31This is similar to the limited litigation case considered by Melumad and Thoman (1990).32J is worthwhile mentioning that an ‘effective audit’ equilibrium can exist only if the auditor is motivatedto be diligent because of her fear of being sued, and the lender is motivated to sue the auditor when the firmgoes bankrupt because of the potential to recoup his losses and legal costs. If the lender is not motivatedto sue, the auditor is not motivated to be diligent. In our model, the lender is motivated to sue the auditorbecause adherence to GAAS does not provide absolute assurance that would dismiss the auditor from anyliability and the lender has little to lose to sue the auditor. This is the key to why our approach works.33The auditor’s attachable wealth is a net amount after considering the audit fee, the cost of performingthe audit and the legal costs.34This consistent with the common perception that audit quality is positively correlated with audit firm108To further simplify the analysis, we assume that, given audit standard q* and a cleanaudited report, the optimal project for the firm (and also the lender) is the high risk project.That is, we require that, for all I> 0,{7r(q*)p+ [1 — > TJJLRH(I),which is equivalent to(aL—PB)—(PG—PB)e(0, 1). (AJ4)(1— cf.)(opL —PB)In the event that the audit is determined to have been conducted negligently, the lender’sexpected damage recoverable from the auditor is given by1(q*, I) = [1 — (q*)1(1 — pB)v*cr1— (1— qS)(1 _qj(1 _pB)v*cj—>0.Observe that= (1— 4)(l — q*)(1 _pB)v*a>0for all q* e (0, 1) and= q(1 q)(l _pB)v*aI <0[qS+(l_q)(1_q*)]2for all I> 0. That is, the lender’s expected damage recoverable from the auditor is strictlyincreasing in the firm’s investment level but strictly decreasing in prescribed audit quality.The optimal high risk loan contract for the ‘good type’ firm is then characterized bysolving the following principal-agent problem:35 (PA.3)max1>0,r>Ir(q*)[R(I)— r] (A.15)s.t. 7r(q*)r +f(q*, I) — I 0. (A.16)size (DeAngelo (1981b)).35The term ‘good type’ is in quotes because the firm may not actually be good, but rather deemed as goodunder the given audit technology.109Substituting the participation constraint (A.16) into the objective function (A.15), theprincipal-agent problem (PA.3) becomeslyax lrH(q )RH(I) + I) — I.By the assumption on RH, the first-order condition is both necessary and sufficient for aglobal maximum. The optimal investment level, ‘H, is then characterized by the followingequation:ÔRH(IH) — 1_fL81 — lr(q*)— 4) + (1 — çS)(1 q*)[1 — (1 — pB)v*a]A 17—Hence, the optimal high risk loan contract for the ‘good type’ firm is given by1 / I * 1- 1 *\1Hq ,, — Hq ,‘Hq )) I *\* ,1Hq)lrH(q)where I(q*) solves (A.17). The expected terminal payoff of the firm (excluding audit fee)under the setting with auditing is then given byU(q*) = [4) + (1 — q)(1 — q*)][7r(q*)R(I(q*)) + i(q*)) — i(q*)]— 4))q*[R(I)— ‘B]. (A.18)The gross benefit of an audit with audit quality q* can then be established by comparing(A.18) with (A.11). As uncertainty regarding the firm’s type is reduced by the audit, wewould expect such service to benefit the firm. Moreover, as the prescribed audit quality q*increases, one in general would expect the gross benefit of the audit increases since capital resources will be more efficiently allocated. The next proposition states the necessaryand sufficient condition under which the gross benefit of the audit is strictly positive andincreasing in the quality prescribed by the prevailing auditing standards.Proposition A.2. Given (A.14), the gross benefit of an audit, b(q*) U(q*) — (J, j8strictly positive. Furthermore, b(q*) is increasing in the prescribed audit quality, q*, if, and110only if,36(1 — p)v*.yj(q*) < [rjijRj(I)— 1B]— [p(I(q*)) — J(q*)] (A.19)Proposition A.2 demonstrates that if condition (A.19) holds, the gross benefit of anaudit, b(q*), which is strictly positive given (A.14), is strictly increasing in the prescribedquality, q*• Condition (A.19) has an intuitive economic interpretation. The right-hand-sideof (A.19) is the marginal benefit to the firm of an increase in the prescribed audit quality.The marginal benefit stems from the result of a better investment decision of the bad typefirm. On the other hand, the left-hand-side of (A.19) is the marginal cost to the firm of anincrease in the prescribed audit quality. The marginal cost arises from the fact that, if thebad type firm is identified, the lender will lose the expected damage award from the auditor.Since the debt market is perfectly competitive, this implies that the lender will require ahigher loan repayment from the firm. Thus, condition (A.19) simply says that the marginalbenefit to the firm of an increase in the prescribed audit quality is higher than the marginalcost. Of course, given that audits are voluntary in this model, the firm is willing to hire anauditor if, and, only if, the gross benefit of an audit is greater than the required audit fee,i.e., b(q*) f.Before ending this appendix, we use our previous numerical example to demonstrate thevalue of an audit with 95% audit assurance, i.e., if a clean audited report is issued, theposterior conditional probability that the firm is a good type is equal to 0.95.Numerical Example Continued: Suppose that the standard audit quality is q* =0.99415. Then i&(q) = 0.1 ÷ [0.1 + (1 — 0.1) x (1 — 0.99415)] = 0.95. That is, an auditconducted under the prevailing auditing standards provides 95% audit assurance. Supposefurther that ,/‘ = 0.1 and a = 0.5. It is easy to calculate that 7r(q*) = 0.95 x 0.7 +36Using (A.13), we have trj5LRH(IB)— ‘B PBRH(IH) — lii > p(I(q*)) — I(q*), where the secondinequality follows from the fact that ‘H maximizes PBRH(I) — I. Then the right-hand-side of (A.19) isstrictly positive.1110.05 x 0.405 = 0.68525, ‘B = 0.05325, I(q*) = 0.11774, f(q*,I(q*)) = 0.0001752 andU(q*) = [0.1 + (1 — 0.1)(1 — 0.99415)][0.68525 x 0.34313 — (0.11774 — 0.0001752)] + (1 —0.1) x0.99415 x [0.8 x 0.6 x 0.23076 — 0.05325) = 0.063836. Hence, the value of a standard auditservice is given by b(q*) U(q*) — U = 0.063836 — 0.051968 = 0.011868.A.3 Concluding RemarksThis appendix presents a model in which audit services purchased by the firm provideinformation to both the firm and the potential lender about the firm’s current financialcondition. The firm’s current financial condition affects the firm’s investment decision andrisk incentive, which in turn determine the firm’s ability to repay and the lender’s willingnessto contract. We do not assume that the audit is mandated; rather, the firm has an incentiveto hire an external auditor to attest its financial statements. We show that without theinformation provided from an audit, the firm will underirivest and the socially desirableproject will be foregone. Reducing such an inefficiency crucially depends on the optimaluse of the information on hand in the initial situation and additional information whichcan be used to further reduce the residual inefficiency. We then show that the presence ofan auditor mitigates the inefficiency caused by imperfect information and the resulting riskincentive problem between the lender and the firm. Thus, our model provides a theoreticallink between auditing and the efficiency of the capital market.112APPENDIX BProofs of PropositionsProof of Proposition 2.1.Suppose audit firm i’s auditing cost to serve client z, m(II1—zII), is lower than m(1113—zII)for all j i. Then audit firm i is said to have a cost advantage to serve client z. Let f bethe lowest profit-maximizing audit fee quoted to client z when audit firm i cannot chargemore than min m( 1113 — z II). Then, it is clear that f must take its greatest value, i.e.,if = mm m(II1 — zil).Thus, for a given client z, Bertrand competition drives audit fees down to the level of thesecond lowest-cost to that client, which then allows the lowest-cost audit firm to serve theclient and charge an audit fee of that amount. If the second lowest-cost audit firm chargedover its cost, the lowest-cost audit firm would charge at this amount and get the client. Thiswould induce the second lowest-cost audit firm to cut its audit fee.On the other hand, suppose audit firm i has no cost advantage to serve client z, i.e.,m(II1 — zil) > min, m(Il1, — zil). Then for any audit fee f,Z > m(II1, — zil), audit firm ihas no demand since it will always be undercut by at least one of its rivals. Similarly, auditfirm i makes no profit when m(III — zil) = min m(lI1j — zil). In either case, audit firm iearns zero profits and pricing at m(I,— zil) is optimal. U113Proof of Proposition 2.2.Since the first term in the expression of W(11, 12, ..., I) is independent of (1, 12, ..., i,j,then for any (1, l, ..., 1) that maximizes T47(1, 12, ..., 1), it also minimizes C(11, 12, ..., 1).UProof of Proposition 2.3.Since by assumption m( 1112 — z) ) is continuous, W(11, 12, ..., i,) is also continuousbecause minimization and integration preserve continuity. By the Weierstrass theorem,W(11, 12, ..., i,) has a maximum on the compact set Z. UProof of Proposition 2.4.Foralll2EZ,i=1, 2,..., n,f* fZ*) _ 6(1 - ST) J [mm m(11l - zil) - m(II1 — zJI)]h(z) dz1—6= 8(1 - ST) [J [mm m(IIl - zII)]h(z) dz - f min m(lI1 - zII)]h(z) dz]1—6 z 3 Z= 6(1 — 6T)j [mm m(IIi — zII)]h(z) dz — C(1,i_)1—6 z 3= 8(1 — 6T)j [mm m(llij — zil) — bZ]h(z) dz + W(I, ii).1—6 z 3If (1’, i) are equilibrium specializations, then ll(l’, l, f, fz*) ll(i, *., f*, fz),which is equivalent to6(1 — ST) J [mm m(111 — zIl) — bz]h(z) dz + W(1,i)1—6 z 226(1 — 6T) f [mm m(II1 — zil) — bZjh(z) dz + W(I, l).1—6 zThe condition specified in the proposition follows immediately. Finally, the existence of aspecialization configuration which satisfies that condition has been proved by proposition2.3. This completes the proof. U114Proof of Proposition 2.5.The proof is by induction. First, suppose that it is an equilibrium for all audit firms tochoose the same specialization, say 1. Then the unique audit fee equilibrium schedule isgiven by the Bertrand solution, i.e., f* = f2z* = fz* m(II1 — zil). Consequently, thecorresponding profits are necessarily zero. On the other hand, consider any configurationwith one audit firm, say audit firm i, who deviates from the proposed specialization equilibrium by choosing a distinct specialization, 1; 10. The market region exclusively served bythat audit firm is denoted by Z1 which is nonempty. Then, the equilibrium audit fee chargedby audit firm in its own market region Z1 will be m( I I 1 — z II) > m( Iil — z ), where theinequality follows from the definition of Z1. Thus, choosing a specialization away from lmust increase the profit of audit firm . Therefore, the proposed ‘n-firms pooling equilibrium’will be broken. The same logic can then be apply to break any proposed ‘n — 1-firms poolingequilibrium’, with n 3. This completes the proof. DProof of Proposition 3.1.Assume an interior solution exists. Then, since W(11,12) is strictly concave in (li, 12) (or,equivalently, C(11,12) is strictly convex in (li, 12)), the first-order conditions are necessaryand sufficient for a global maximum (minimum). Solving the first-order conditions yieldsthe result. DProof of Proposition 3.2.Observe thatpc—k11 fl1+(1+)C(12_41)H(l, 12)= / c(12 — l) dz + / c(li + 12 2z) dzJo J11+ f 2 — z) — (z — li)] — k(12 — l)} dzpc—k j(1+)c ‘2’1fll+1 {c[1—z—/3(li—z)]+k(1i }01152{c[1—z—/3(z—li) +k(1li } dz.11Then, by Leibnitz’s rule,8111(11,12)= 2(1 + /3)2c[(3 + /3)(l1 +l2)(c + /32c + 2ck — 2/3ck + 2k)—2l1(4c2+ 4/3c2 + 6/32c+ 2/33c+ 7ck — 2f3ck — /32ck + 6k2 + 2/3k2)],and8211(l, 12) — —(5c2 + 7/3c2 + 9/32c+ 3,8c2 + 8ck + 6k2 + 2/3k)8l — 2(1+/3)c <By the same token, one can get 8112(11, 12)1812, 82112(11, 12)1812 and easily verify that82112(11,12)812 <2Hence, the first-order conditions are necessary and sufficient for a global maximum. For thesymmetric specialization equilibrium, l = 1— l. Thus, solving the equation 8111 ( l, l ) /8l =0 yields l as reported in the proposition.Finally, it remains to show the global stability of the given symmetric specializationequilibrium. Uniqueness then follows immediately. This can be done by proving that bothaudit firms’ reaction functions are upward sloping and audit firm l’s reaction function iseverywhere steeper than that of audit firm 2 on the 11-12 plane, so that they intersect atmost once. DProof of Corollary 3.1.(i) It is straightforward to show that* w (2k—c)kli—il4(4c+ck+2k)<0,for alic> 0 and 0< k< c/2.116(ii) Similarly, one can geturn l — — 1+5/3+2/32kO 1 1— 4(2 +2/3+3/32 + /33)> 0,for all 0 < 6 < 1.(iii) Fillally, it is easy to show thatui1,kO14 0Proof of Lemma 3.1.Denote V = 4c2 + 4/3c2 + 632c+ 2/33c + 7ck — 2/3ck — /92ck + 6k2 + 2/3k > 0.(i) Partially differentiating l with respect to /3 yields— c(1 +[8c3 + 27ck + /3c2k+ 32ck + 20/3ck + 4/32ck2V+10k3 + 2/3k3 + (4/3c3 + 9/3c2k)(1— /3) + 3/3c2k(1 — /32)]<0,forallc>0,0<k</3c/2and0</3<1.(ii) Partially differentiating i with respect to c yields— k(3+fl)(1--/3)— 2D2(2— 3/32c+ 4ck + 4/3ck + 2k2)< (>) 0,if c2 — 3/32c+ 4ck + 4/’3ck + 2k> (<) 0.(iii) Similarly, partially differentiating i with respect to k yields— c(3 + /3)(1 + /3)2(c2— 3/32c+ 4ck + 4/3ck + 2k2)8k — 2D> (<) 0,ifc2—3/3c+4ck+4/3ck+2k>(<) 0. 0117Proof of Lemma 3.2.Denote V = 4c2 + 4/3c2 + 6/32c+ 2/33c+ 7ck — 2/3ck — 132ck + 6k2 + 2/3k > 0.(i) Partially differentiating 11* with respect to l yields= c(1 : /3)2[2— 3/3c2 + /32c + /33c2 + 5ck — 6/3ck — 3/32ck + 6k2 +2/3k—l(5c2 + /3c2 + 732c+ 3/3c2+ l2ck — 8/3ck — 4/32ck + 12k + 4/3k2)]_ [(7 — /34)c2 + 1O/3c2 + 14/32c+ 2/33c+ l2ck + 4/3ck + 12/3ck+4/33ck+ 8k2]<0,forallc>’0,0<k</3c/2and0</3<1.(ii) Partially differentiating S* with respect to l yields= c(1 + /3)2[9/3c2— c2 + /32c — /33c2 — lOck + 12/3ck + 6/32ck — 12k2 — 43k2+2l(3c — 5/3c2 + /32c + /33c2+ l2ck — 8/3ck — 4/32ck + 6k2 + 4/3k2)]= [(5 — 4/3)c2 + lO/3c2 + 16/32c+2/33c+ 7ck + (5c — 8k)/3k+ 7/33ck + 2k(1 — /32)]>0,forallc>0,0<k</3c/2and0<i3<1.(iii) Similarly, partially differentiating W with respect to l yieldsaw*= c(1+/3)(1—4l)c(1+ (2c2 — 2/3c + 5ck — 6/3ck — 3/32ck + 6k2 + 2/3k2)< (>) 0,if 2c — 2/3c + Sck — 6/3ck — 3/32ck + 6k2 + 2/3kc> (<) 0. D118Proof of Proposition 3.3.Denote V = 4c2 + 4/3c2 + 6/32c+ 2/33c+ 7ck — 2/3ck — 132ck + 6k2 + 2/3k > 0.The proposition then follows from the fact that1 — 2c — 2/3c + 5ck — 6/3ck — 3/32ck + 6k2 + 2/3k1_i— 4V>0,which is implied by (A2). DProof of Proposition 3.4.The proposition follows directly from applying the results of lemma 3.1 and imposing(A2).Proof of Lemma 3.3.Denote V = 4c2 + 4/3c2 + 6/32c+ 2/33c+ 7ck — 2/3ck — /32ck + 6k2 + 2/3k > 0.(i) Partially differentiating 11* with respect to c yields= 2c(1+ /3)2[c2 + 7/3c2 + /32c — /33c2 + 6k2 +2/3k + 4l(c2 — 3/3c2 + /32c+/33c2— 6k2 — 2/3k) —2l(5c + /3c2 + 7/32c+ 3/3c2 — 12k — 4/3k2)]=_ [(35 — 3/36)c4+ (99 — 37/36)/3c4+203/3c4+219/33c4+ 165/34c+53/3c4+16(7 — /34)c3k+ 160/3c3k + 224/3c3k+32/3ck+ 2(91 — 3/34)c2k+6(33 — /34)j c2k+33632ck+12833ck+ 144ck3 + 96/3ck3+ 160/32c1c3+96/33ck+ 16/34ck3+48k + 16/3k4]>0,for all c> 0, mm {0.171573c, /3c/2} <k < /3c/2 and 0 < /3 <0.885618.(ii) Partially differentiating S* with respect to c yields= 4c2(1+ /3)2[3C2 + 17/3c2 + 5/32c — /33c2 + 12k + 4/3k2 + 4l(c2 — 9/3c2119—/32c+ /33c2 — 12k2 — 4/3k2)—4l2(3c — 5/3c2 + /32c + /33c2 — 12k2 — 4/3k2)j=-_[— /37)c4 + 121/3c4+ 249/3c+ 285/33c4+223/34c+91j35c4+ l1/36c4+2(75 — 11/34)c3k+ 4(49 — 3/34)/3ck+ 2(129 — /34)f32c3k+72/33ck+255ck+ 251/3ck+35432ck+210/33ck+473c2k+3/35c2k+ 216ck3+144/3ck3+96/32ck3+48/33ck+ 8/34ck3+ 84k4 + 76/3k4 + 28/3k4+ 4/33k4]<0,for all c> 0, mm {0.171573c, /3c/2} < k < /3c/2 and 0 < /3 < 0.885618.(iii) Similarly, partially differentiating W with respect to c yields___— (1+/3)(1_4l+8l2)=_ [ioc4 + 22/3c4 + 46/32c+ 66/33c4+58/34c+38/35c4++2/37c4+ 2(19 — 3/34)ck+ 12(3 — 134)/ c3k+ 2(17 — /34)/32c3k + 40/33ck+73c2k+53/3c2k+18/32ck+82/33ck+ (53c — 8k)/34ck2+9/35c2k+8(9 — 8/32)ck3+ 48(1 — /32)/ ck3+ 36k4 + 60/3k4 + 28/3k4+ 4/33k4J<0,for all c> 0, mm fO.171573c, /3c/2} < k < 13c/2 and 0 < /3 < 0.885618. 0Proof of Lemma 3.4.Denote V = 4c2 + 4/3c2 + 6/32c+2/33c + 7ck — 2/3ck — /32ck + 6k2 + 2/3k > 0.(i) Partially differentiating S* with respect to /3 yields— (/3c — c — 2k)(2l — 1)2(llc + 4/3c + /32c + 10k + 2/3k)0/3 — 4c(1+/3)— c(1 + /3)(/3c — c — 2k)(c + /3c + k)2(llc + 4/3c + /32c + 10k + 2/3k)4V2<0,for all c> 0, b> (2 + /3)/2, mm {0.171573c, 3c/2} < k < /3c/2 and 0 </3 <0.885618.120(ii) Partially differentiating W* with respect to /3 yields____— c(1—4l+8l2)=_[ioc4 + 12/3c4 + 34/32c+32/3c4+ 26/34c+ 12/35c4++2(19 — /3)ck + 2(18 — 5/32)/c3k+ 2(2 — /32)/33ck+ (73 — 20/3)ck+38/32ck+ 4(l1/3c — lOk)/32c + (9/3c — 8k)/33ck2+ 24(3 — /3)ck + 36k4+24/3k4+ 4/92k4]<0,for all c> 0, mm {0.171573c, /3c/2} < k < /3c/2 and 0 < /3 < 0.885618.(iii) Similarly, partially differentiating 11* with respect to 3 yieldsc(1 : /3)3[(_411*)(5c2— 5/3c2 — 3/32c — /33c2 + l6ck + 10k2 + 2/3k)+2l2(9c — 13/3c2 — 9/32c — 3/3c2+ 32ck + 20k + 4/3k2)j(c4 + 14/3c — 25/3c4— 52/33c4— 45/3c — 18/35c4— 3/96c4+ 16c3k+128/3c3k+48/32c3k+22ck+ 216/3ck+88/32ck— 16/33c2k—6/94c2k+ 144/3ck3+ 96/32ck3+16/33ck — 16/34).To see the ambiguity of Oll*/a/3, notice that for all c> 0, mm {0.171573c, /3c/2} < k </3c/2and 0 </3 < 0.885618,awlim = 0.015625c> 0,3—O, k—Oôll*urn — = —0.05101415c < 0. 0/3—O.885618, k—O.171573c 9/3Proof of Lemma 3.5.Denote V = 4c2 + 4/3c2 + 6/32c+ 2/33c+ 7ck — 2/3ck — /32ck + 6k2 + 2/3k > 0.121(i) Partially differentiating 11* with respect to k yieldsjj*= [C_2/3C_/32C+3k+/3k_l(3C_2/3C_/32C+6k+2/3k)]= c2(1 + 13)2(c+ /3c + k)+ (3/3c - 2k)(1 - j32) + (9/3c - 8k)/3]>0,for all c> 0, mm {0.171573c, /3c/2} < k < /3c/2 and 0 < /3 < 0.8856:18.(ii) Partially differentiating S* with respect to k yields= 2(1—2l)[c—2/3c—/3c+3k+/3k—l(3c—2/3c---/+6 +2flJ )]c2(1 + /3)2(c+ /3c + k) [ + (3/3c - 2k)(1 - /32) + (9/3c - 8k)/3}<0,for all c> 0, mm {0.171573c, /3c/2} < k < /3c/2 and 0 < /3 < 0.885618.(iii) Similarly, partially differentiating W’ with respect to k yieldsDProof of Proposition 3.5.Differentiating 11* with respect to c yieldsdll* f* i* ãll*= —x—+—dc ãl ãc Oc>0,since by lemmas 3.1, 3.2 and 3.3, Ol/üc < 0, < 0 and 911*/öc> 0, respectively.Differentiating S* with respect to c yieldsdS* 9g* ãl ãS=dc ôl 9c ãc<0,122since by lemmas 3.1, 3.2 and 3.3, ôl/8c < 0, > 0 and ÔS*/ãc < 0, respectively.Differentiating W* with respect to c yieldsdW* — OW* Ol OWdc —X8+Since by lemmas 3.1, 3.2 and 3.3, 8l/ãc < 0, < 0 and aW*/oc < 0, respectively, thenit follows that the sign of dW*/dc is indeterminate. DProof of Proposition 3.6.Differentiating S* with respect to /3 yieldsdS* — 9S* ôl ôS—<0,since by lemmas 3.1, 3.2 and 3.4, 8l/8/3 < 0, > 0 and ÔS*/8/3 <0, respectively.Differentiating 11* with respect to 3 yieldsdll* — 011* Ol Orl—Since by lemmas 3.1, 3.2 and 3.4, Ol/O/3 < 0, < 0 and 011*/a/3 is indeterminate,respectively, then it follows that the sign of dll*/d/3 is also indeterminate.Differentiating W* with respect to /3 yieldsdW* — oW ai ow”d/3 — Ol 8/3’Since by lemmas 3.1, 3.2 and 3.4, 8l/O/3 < 0, <0 and OW*/8/3 < 0, respectively, thenit follows that the sign of dW*/d/3 is indeterminate. 0Proof of Proposition 3.7.Differentiating W* with respect to k yieldsdW* — OW”‘ Ol OW”dk — 8l Ok<0,123since by lemmas 3.1, 3.2 and 3.5, Ol/8k> 0, < 0 and oW*/ok = 0, respectively.Differentiating 11* with respect to k yieldsdiP J* 8i 9]J*= -x.+--.Since by lemmas 3.1, 3.2 and 3.5, 8l/8k > 0, < 0 and 811*/ok > 0, respectively, thenit follows that the sign of dH*/dk is indeterminate.Differentiating S* with respect to k yieldsds* as* 8l oS=Since by lemmas 3.1, 3.2 and 3.5, Ol/ak> 0, 5 > 0 and OS*/Ok < 0, respectively, then itfollows that the sign of dS*/dk is indeterminate. 0Proof of Proposition 4.1.Analogous to the proof of proposition 3.2.Proof of Corollary 4.1.Analogous to the proof of corollary 3.1. 0Proof of Proposition 4.2.Denote V = 4c2 + 4,6c2 + 6/32c+ 283c+ 7ck — 2flck — /32ck + 6k2 + 2,6k > 0. Define—L\IIN 11(l,l) — 11(l,l), IXSN S(l,l) — S(l,l) and LWNW(l, 1) — W(l, l’). Then1N — (3+/3)(c+13c+k)(c—/3c+2k)2>01— 2(5c + 3/3c + 2k)D11N =— 8(5c + 3/3c + 2k)V[1589c7 + 4507f3c7+ 8440f32c7+ 11128/33c7+1052634c7+78424,6c+4024/36c7+ 1048fl7c+61,68c7+ 13/39c7+ 91.20c6k+ 18320,6c6k124+24128/9c6k+21536/33c6k+1024/34c6k+3792/35c6k+2112/36ck+768/3ck+96/38c6k+ 4(6199 — 80/36)c5k2+ 4(9576 — 1l/36)/3c5k2+39160/32c5k+31428/3c5k2+12076/34c5k+1032/35ck2+ 39368c4k+47600/3ck3+33464/32ck+22560/33c4k+9016/34ck3+1520/3c4k3+72/36c4k3+ 38952c5k2+ 39176/3ck4+17584/32c3k4+7088/33ck4+2152/34c3k+264/35c3k4+23584ck5+20928/3ck5+6784/32ck5+1088/33c2k5+96/34c2k5+ 7968ck6+630418ck + 1696j32ck+160/33ck6+ 1152k7 + 768/3k + 128/32k7]<0,= 8(5c + 3j3c + 2k)V[(301 — 3/38)c6+ 720/3c6 + 1296/3c6+ 1456/33c6++832/35c6+30436c+32/37c6+ 1586ck+ 2402/3c5k+3114/32c5k+1930/33c5k+254/34ck+206/35ck+198/36c5k+38/37c5k+ 332(12 — /35)c4k2+4(1049 — 16/95)/3c4k2+4784/32ck+3112/3c4k2+192/94ck2+5752c3k+4032/3ck+3240/3ck+2424/33ck+736/34ck+72/35c3k+5048c2k4+ 2624/3ck4+736,32ck4+352/9c2k4+72/34ck+ 16(159 — /33)ck5+1328/3ck5+ 112/32ck5+ 576k6 +384/3k6+ 64/32k6]>0,forallc>0,0<k</3c/2and0</3<1.Similarly, it is easy to show thatWN — c2(1 +/3)(3 +/3)(c+ /3c+ k)(c— /3c+ 2k) (7 2_ 2 2_ 22 2— 2(5c + 3/3c + 2k)V2J—2/33c2— /34c2 + l4ck — 18/3ck — 22/3ck — 6/33ck + 16k2 + 8/3k2).To see the ambiguous effect of low-balling on the social welfare, notice that for all c > 0,1250<k<Bc/2and0</3<l,— 2ck(4c — 3k)(2c + k)— (4c+k)c+ck 2k)>0,F zXW— (3 — 1)(1 + 3)2(3 + 3)(7 +5/3+3/32+133)0 D— 8(5+3/3)2(2+2/3+3/32+133)2 <Proof of Proposition A.1.Suppose that the low risk project is optimal (i.e. = L). Substituting the lender’sbreak-even constraint (A.3) into the objective function (A.2) and the incentive compatibilityconstraint (A.4) yields the Lagrangian for (PA.1):5PLRH(I) — 1+ C[(oiiL —PH)RH(I)— (i —where 0 is the Lagrange multiplier for (A.4). By the assumption on RH, the first-orderconditions are necessary and sufficient for a global maximum. The first-order condition withrespect to I yields— 8RH(I*)OPL 81 —1— (_ 8R(I) — — (o 8RH(I) — 1PH 81 PL’ PL aiIt must be true that > 0. If not, = 0 would imply that- oRH(l*) - aRH(I*) ãRH(IL)PL ai— 1 °PL in — = 0,where the first equality follows from (A.6). Note that this in turn requires I = IL. Butthen it is not difficult to verify that (A.4) does not hold given (A.8). Thus, ( > 0 and (A.4)is binding. Solving (A.3) and (A.4) yields (A.10).Now, we show that 1 < ii,. Suppose not, 1* > IL would imply from the above analysisthat the numerator of the expression of C is negative as RH(I) is strictly concave. On theother hand, using (A.6) the denominator of the expression of C can be written as- ORH(l*) ÔRH(IL) - 8RH(l*) ÔRH(IL)PH — — °PL — in126- IORH(I*) ORH(l*) -- 9RH(IL) ORH(I*)= PH L 81 + (PL — PH) 81 — 81>0,where the last inequality follows from u < 1, 1* > IL and RH(I) is strictly concave. But thenis negative, which is a contradiction. Moreover, we have shown from the above analysisthat 1 ‘L Hence, we can conclude that 1* <IL.Now, suppose that the high risk project is optimal (i.e. &‘ = H), then there is noincentive problem. Hence, the optimal loan contract is where IH solves (A.5).Which project is indeed optimal then depends on condition (A.9) DProof of Corollary A.1:Total differentiating (A.11) with respect to U* and 1 and rearranging terms yieldsdU* - ORH(l*) - IORH(l*) ORH(IL)’\01 _l7PLL 01 — 01where the second equality follows from (A.6) and the inequality follows from the fact that1* <IL and RH(I) is strictly concave. This implies that the firm’s expected terminal payoffU and the investment amount 1* change in the same direction in equilibrium. Hence, if wecan show that OI*/OPL > 0 and9I*/0j5< 0, we are done.Differentiating (A.l0) with respect to j1L yields81* — JPLRH(1*) —— PL{(PH8RH(1j— PL) —(opL&R5I*)—> {vHRH(IH)_IH1+(1)1*— pL{(pH— ) — (aPLag*) —where the first inequality follows from (A.9). Clearly, the numerator of the expression ofOI*/OPL is positive. By the expression of and the proof in proposition A.1, the denominatoris also positive. Hence, OI*/OPL is positive.127Similarly, differentiating (A.l0) with respect to JH yieldsal — RH(I*) PPL— —— aRH(I*)_______(PH — — (opLaR$T*) — 1)—— PL(l — 0)RH(l*)<0,— ORH(1*)) (L8RH—(13L—13H){(PH — - — l)}where the second equality follows from (A.10). 0Proof of Proposition A.2.Using the result of corollary A.1, we know that rPLRH(IB)— ‘B > cTPLRH(l*) —1 sincePB <PH. Then, we haveU(q*) —> [qS + (1 — 4’)(l—+ f(q*, I(q*)) — J(q*)j_[0PLRH(1*) — l*]}>0,where the second inequality follows from the fact that, given (A. 14), the high risk project isoptimal and the optimal investment amount is I(q*).Furthermore, we have_____*1 ÔRHx — (1 —8U(q*)= [PG + (1 — q)(l — q )PBj 81 8q*8q*8’H 82H__—+ (1- )(1 - q )]- 8q*- 81 8q*+(1 — q)[i(q*) — f(q* I(q*))] + (1 — cb)[uLRjj(IB)— ‘B]= (1 — c){[oiiLRH(IB)— IB]— pR(I(q*))}+(1 - )[J(q*) - (q* i(q*))]++ (1— )(1 - q )j 8q*= (1— c){[JPLRH(IB) — IB]_pR(I(q*))+ [1— (1 _p)z,*a]I(q*)},where the second equality follows from (A. 17) thatÔRH_____ _________ai01 — .(q*) —1286t0jooidsjduioosiquotsodoidUtpjpdsuotipuocijou1Anbos!1:pNM‘(b)HI[4(&d———((b)H1)Hj&cf<—(&1)H1dDJ!‘Arnop0<be/(b)ng‘Ujj.(b)HI{4(&d_i)——=(b—t)(cb—t)+cb—(b)HI{[+(b——)]/1(d_i)—(b——i)+}(cb—I)—(b—T)(cb—T)+cb(b—T)(—T)+—(b)Hro(&d———(b)HI{[1(&d———t)(cb—‘)+çb}(çt—i)—--)++[((b)HJ’b)Hty-(b)HJ](-uioa;sA&oJo;Arnbp”qpu

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