Legal Liabilities Audit Accuracy and the
Journal of Business Finance & Accounting, 29(3) & (4), April/May 2002, 0306-686X
Legal Liabilities, Audit Accuracy
and the Market for Audit Services
Sankar De and Pradyot K. Sen*
1. INTRODUCTION AND OVERVIEW
In recent years, liability lawsuits against auditors seem to have
reached an epic proportion. The liability claims against the big
six audit firms in the US is estimated to be about $30 billion,
which exceeds their total partner's capital by a sizable amount
(Balachandran, 1993). Several large jury decisions and out of
court settlements often get reported in the financial press. The
seriousness of the problem is appreciated by academics and
practitioners alike.1 Since the early 1990s, the practitioners felt
that the existing liability regime put an ever increasing burden
on the audit firms and their clients (Weinbach, 1993; Cook, 1993;
and Freedman, 1993). The litigation phenomenon is not limited
to the US alone. US audit firms routinely face lawsuits for work
done outside the US.2 Although the US remains the preferred
* The authors are respectively from the Center for Professional Development in Finance,
Berkeley, California and the Haas School of Business, University of California, Berkeley;
and the University of Cincinnati. They wish to thank Bala Balachandran, Robert Chatov,
Srikant Datar, Jere Francis, Angela Gore, Bob Hagermann, Jack Hughes, Bjorn Jorgensen,
Ken Klassen, Rick Lambert, Bob Magee, Nahum Melumad, V. G. Narayanan, Zoe-Vonna
Palmrose, Victor Pastena, Stefan Reichelstein, Kevin Sachs, Toshi Shibano, Brett
Trueman, the workshop participants at University of California at Berkeley, SUNYBuffalo, University of Iowa, Center for Applied Mathematics ± New Jersey Institute of
Technology, New York University, University of Wisconsin-Madison, European
Econometric Society Annual Meetings and European Economics Association Annual
Meetings, several anonymous referees and the editor of this journal for their comments
on different versions of this paper. Part of the work was done when both the authors
visited the Indian Institute of Management Calcutta, India. Typing assistance of Rona
Velte and Kathy McCord is greatly appreciated. (Paper received May 2000, revised and
accepted January 2001)
Address for correspondence: Pradyot K. Sen, Department of Accounting and Information
System, College of Business, 302 Lindner Hall, University of Cincinnati, Cincinnati OH
45211-0211, USA.
e-mail: [email protected]
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and 350 Main Street, Malden, MA 02148, USA.
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forum for litigation, audit litigation has begun to become
prevalent in the common law countries such as England,
Caribbean, Canada, Australia, New Zealand, and in many Asian
countries such as Hong Kong and Singapore. Even in countries
such as Japan and Germany, audit related litigation has become
more common.3 The Bank of Credit and Commerce
International (BCCI) litigation, perhaps involving the most
amount of money in any litigation, was filed in both Luxembourg
and the UK. It is believed that in countries such as New Zealand
and Australia, the rate of litigation, considering the size of the
economy, far exceeds professional malpractice litigation
anywhere in the world (Grobstein and Liggio, 1993). Given the
differences in the liability levels and regimes in different parts of
the world, it is important to understand how a rational auditor
would respond to changing liability levels and regimes and
modify her efforts level and fees and how such choices would
influence the supply of audit accuracy and the demand for audit
services. This paper is an attempt to study this problem through a
model, where in a market characterized by asymmetric
information between the firm-managers and the shareholders,
firms demand auditing to signal their private information and,
anticipating the demand and responding to various liability levels
and regimes, rational auditors of different operating efficiency
choose the level of care (effort).
In the stylized market characterized by us, on the supply side of
the market (in Section 2), two different types of audit firms
(differing in their operating cost structures, which is well known
in the market) decide optimally their level of care (characterized
as high quality or low quality audit services), based on the
(anticipated) demand, the legal liabilities arising from audit
failures and other considerations such as potential loss of
business. We characterize a high quality service by lower type I
and type II errors.4 The identity of auditors and, therefore their
efficiency, as well as the audit findings are public information
whereas the (two) client firm types (referred to as `good' and
`bad' or equivalently, type 1 and type 2) are known only to the
firm-managers. While the marginal benefits to auditors are
represented by the audit fees offered for the services, the
marginal audit costs typically have three components; operating
cost structures which vary across the audit firms, potential costs of
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loss of business arising from a type 1 audit error, and potential
costs of legal liabilities arising from a type 2 audit error. In our
model, some audit firms are more efficient than others, due to
their operating cost structure, and this advantage in operating
cost structure of some firms is a common knowledge in the
market. However, all auditors are generally efficient in the sense
that they all generate favorable findings for a good client firm
with a higher probability than for a bad client firm. However, no
audit service achieves perfect audit accuracy (zero type I and type
II errors) because the costs involved may be prohibitive. In this
scenario, if more efficient auditors are subjected to a higher level
of liability and business loss arising from type I error, we
characterize an equilibrium where more efficient auditors also
provide a higher quality audit service and receive higher audit
fees.5
On the demand side of the market (in Section 3), the potential
client firms and outside investors who cannot directly observe
firm types, engage in a game of market valuation under
asymmetric information. Since direct communication of the
private information (unobserved types) to the outside investors is
prone to adverse selection problems, the client firms signal their
type through their choice of audit quality in order to generate
favorable market valuations. In equilibrium, given that the
available audit services are heterogeneous in terms of accuracy
and required fees,6 in an asymmetrically informed market, choice
of a particular type of audit service (more efficient or less
efficient) by a client firm together with the audit report on the
firm provide additional value relevant information. Because of
the way investors use audit information in their evaluation of a
firm, good client firms demand high quality auditing because
they have a high probability of being identified as good firms.
Bad client firms, on the other hand, would prefer a low
probability of correct identification and would like to choose a
low quality audit service. However, if the market sees through this
incentive and perceives that no good client firm is likely to opt
for a low quality audit service, they run the risk of being singled
out through their action. For the bad firms, it is essentially a
choice between two unfavorable options. The choice is decided
by the differential in fees for the two audit services. The (bad)
client firms, therefore, strategically consider the possible reaction
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of the market, the audit fees, and decide whether to demand
high quality or low quality audit services.7
In this model, we show (in Section 4) that under fairly general
conditions, depending on the level of the differential in fees for
the two audit methods, three Nash Sequential Equilibrium (NSE)
outcomes, which are robust to refinements8 are possible:
1. For cost differential below a certain level, a pooling
outcome in which all firms choose high quality auditing
(DP).
2. For the differential exceeding that level, a semi-separating
outcome in which all good firms opt for high quality
auditing while bad firms randomize between the two audit
methods (DSS).
3. If the differential is too high, a third possibility exists where
both types forego high quality auditing.
We focus on the second equilibrium where all good and either
some or all bad client firms demand high quality audit services in
equilibrium, unless the differential in fees for high and low
quality services is too much. On the other hand, client firms who
choose low quality audit services are always bad themselves. The
intuition behind the second equilibrium is that as more and
more bad firms choose a high quality audit, the expected value of
such a firm in the pool is diluted and, given sufficiently high fees,
eventually no higher than what could be obtained with the help
of a low quality audit. As a result, in the second equilibrium, on
the margin, the bad firms become indifferent to the choice of
audit method and consequently, randomize between the two
audit methods. Interestingly, the proportion of bad quality firms
which choose high quality auditing in this equilibrium is unique
given the parameters of the system and, further, higher for a low
differential between the respective costs of the two audit
methods.
So long as the market has a high proportion of good client
firms, our finding suggests perhaps an overwhelmingly high
demand for the high quality audits. If big audit firms offer high
quality audit services, our result that all good and all or some bad
client firms opt for high quality auditing in equilibrium in spite
of the higher fees offers an explanation why a high proportion of
all US firms use the services of the big six (big eight) audit firms
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for audit purposes. Table 1 shows that, during 1982±1987, 82% of
all audited firms in the US used big eight audit firms.9 In the
higher annual sales range of $250 million and more, this
proportion was 95%. In terms of value, the big eight audit firms
audited about 97% of the total sales of all US audited firms in this
period, though the existing empirical evidence suggests that the
bigger firms also charge higher fees. Though several reasons have
been suggested for this overwhelming preference for the services
of the big audit firms in spite of their higher fees, there is no
formal model of demand for audit services that explains this
phenomenon in the existing literature on auditing.10
Using this scenario, the study offers a number of insights into
the impact of legal liabilities on the demand for and supply of
audit services. First of all, it shows that if there are auditors with
(two) different skill levels and their marginal costs of providing
the same level of service systematically differ, and if the more
efficient firms are subjected to a `higher' level of liability (in a
sense to be specified later) for audit failure, audit services with
(two) different accuracy levels could be simultaneously available
in the market where the more efficient auditors exert higher
efforts leading to higher accuracy and command higher fees.
Second, the differential in fees for the different services and the
degree of their audit accuracy play a crucial role in determining
the demand of the client firms for such services and the extent of
revelation of their unobserved types in equilibrium. An increase
in legal liability induces an auditor to increase her efforts level
and improve her accuracy. However, we show (in Section 5) that,
for all client firms, good and bad, together, demand for a high
quality audit is inversely related to its fees. It is comforting to note
that the inverse relationship between audit fees and demand,
argued in the price theoretic analyses (e.g. Benston, 1985), holds
in a more complex game-theoretic setting. More interestingly, we
show that the demand for a high quality audit is also inversely
related to its accuracy as well (Proposition 3). The inverse
relationship of audit demand with audit accuracy goes somewhat
counter to the prevailing wisdom where a higher quality is
assumed to increase demand. The importance of this result is as
follows. An increase in legal liabilities (a) increases auditor's cost
and hence the fees; leading to increased marginal cost to client
firms, (b) increases audit accuracy or quality through increased
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Table 1
Description of Clients for Big Six and Non Big Six Audit Firms, 1982±1987
Number of Clients Total Sales Audited
($billions)
Year
NB6
b
1±25 mill.c
26±50 mill.
51±100 mill.
101±250 mill.
250 mill and over
B6
NB6
B6
NB6
B6
NB6
B6
NB6
B6
NB6
B6
NB6
6942
6716
6543
6488
6173
5727
1336
1457
1417
1452
1572
979
3664
3123
3525
3465
3170
3290
111
105
104
104
104
96
2943
2553
2691
2694
2566
1879
911
862
986
912
1097
815
833
815
770
772
766
684
119
156
135
137
171
102
765
776
791
764
699
638
135
121
112
116
115
88
855
839
794
779
765
788
88
94
104
107
109
39
1546
1433
1488
1479
1377
1468
73
78
78
71
71
75
6430
1369
3373
104
2554
930
773
137
739
115
689
90
1465
74
82
18
97
3
73
27
85
15
87
13
88
12
95
5
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Notes:
a
B6 represents big six
b
NB6 represents non big six.
c
This represents the range of company size audited in terms of million dollar (US).
Source: Computed using data from various issues of Who Audits America, The Data Financial Press.
DE AND SEN
1987
1986
1985
1984
1983
1982
Yearly
Average
Average
%
B6
a
Number of Companies Audited
THE MARKET FOR AUDIT SERVICES
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auditor's efforts; and therefore, provides incremental value to the
client firms. So far, the literature on audit markets seems to have
concluded that the joint effect of legal liabilities, as manifested
through these two factors, is at best, an empirical issue and may
even offset each other in determining the demand for audit
services (see for example, Lys, 1993).
Our result that an increase in both fees and accuracy reduces
the overall demand of high quality audits, raises serious questions
to the arguments used in the empirical auditing literature. Since
legal liabilities play a crucial role in providing incentives to
auditors, we examine (in Section 6) the role of the legal system,
through assessment of penalties consequent to audit failures, in
providing efficiency-improving incentives to the auditing
profession. We consider two regimes; (1) a `verifiable due care
regime' when it is possible for the litigation process to observe
the (type II) error level being allowed by the auditor and (2) a
`non-verifiable due care regime' where, although the (type II)
errors themselves could not be verified, the value of the firm,
were it to receive a `low' report, could be inferred from the
market value of other firms and similar other sources. We find
that though legal penalties for audit failures increases auditor's
effort, and therefore, accuracy, a penalty regime that relates the
penalty to the type II error (verifiable due care regime) induces
an effort level that is less than that induced by an exogenous
penalty regime (the benchmark case). The present legal practice
in the US of awarding damages based on the actual losses
suffered by the plaintiffs, is consistent with the non-verifiable due
care regime. According to our results, it induces an effort level
that is higher than that induced by the (benchmark) exogenous
penalty regime. We are also able to conclude that a proportional
liability regime induces a higher effort than a joint and several
liability regime where the level of awards may be independent of
the auditors care level or the effort. If the more established and
larger audit firms are the ones who are more wealthy and
therefore, believed to have `deep pockets', our results suggest
that the practice of `deep pocket' awards in the US, would
penalize more established and larger audit firms more severely
than the smaller firms for the same kind of audit failures. This
differential penalty may be undesirable from the point of view of
promoting efficiency in the smaller firms.
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The role of legal liabilities and operating cost structure has
been addressed by Nelson, Ronen and White (1988) in a supply
side model that uses similar characterization to our Section 2. As
Demski (1988) notes, there is no demand for auditing per se in
their model. More recently, Gigler and Penno (1995) have
addressed the problem of demand for auditing in a model where
different auditors have different cost structures. However, their
model is essentially different from ours in that the auditors in our
model have fixed types and are identified unambiguously with
their types. In Gigler and Penno, the auditors switch their types
randomly in different time periods in a Markovian manner, thus
making any type-identification impossible. More recently,
Hillegeist (1999) studied the role of different liability rules on
audit quality choice. Though Hillegeist uses a similar game
theoretic model and asks some of the same questions, the focus
of his study and ours is fundamentally different. In the Hillegeist
model, there is only one type of auditor, who makes a uniform
quality choice. Thus, quality differentiated audit does not exist in
the Hillegeist model and some of the issues we raise regarding
the differential impact of legal liabilities on different auditors
cannot be addressed through the Hillegeist model. We do not
explicitly model financial reporting and the litigation process
that Hillegeist does, and therefore, cannot address some of the
interesting reporting related issues addressed by Hillegeist. The
audit technology used in Hillegeist is also less general than ours
in that in the Hillegeist model, there is never an audit failure with
a high type firm, or when a low type reports low. Thus, the
auditor in the Hillegeist model never pays a cost for a type II
error. Since auditors in our model can commit type II errors,
they also suffer a cost associated with business loss when type II
errors occur, and must trade off that cost when deciding on the
audit quality. In the Hillegeist model, after the firm type is
revealed (exogenous) bankruptcy can occur to the owner.11 A
deep pocket award, thus transfers wealth from the owner to the
auditor. In our model, we do not explicitly model such a
bankruptcy and we focus on the inefficiency a deep pocket award
may cause by differentially affecting the different types of
auditors. In spite of these differences, Hillegeist arrives at
fundamentally the same conclusions as ours, in that an exposure
to a higher liability level leads to a higher audit quality.12
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However, since any penalty paid by the auditor can possibly
reduce the penalty to be paid by the insider, auditor penalty has
an insurance role in the Hillegeist model and adversely affects
the owner's disclosure behavior. We do not explicitly model any
penalty to the insiders. Taken together, our study and the
Hillegeist study provide some confirming and complementary
findings on the issue of audit quality choice and auditor liability.
Other studies that have addressed the issue of demand for
auditing, particularly in special contexts such as new issues
markets, include Titman and Trueman (1986), Datar, Feltham
and Hughes (1990), Bachar (1989) and Sarath and Wolfson
(1989). Though they contribute useful insights, the scope of these
studies is different from ours.13 In an insightful study, Melumad
and Thoman (MT) (1990) address some of the same issues examined in the present study. However, there are important
differences between the two models as well as the results derived
from them. In MT, the manager can disclose the true type apart
from selecting the auditor. In our model, adverse selection
problems rule out direct disclosure of type. Rather, the audit
findings together with the choice of the auditor constitute an
(imperfect) signal of firm type. Further in MT, the auditor makes
no error while auditing good client firms. Thus, a `low' report
unmistakably identifies a bad client firm, leading to a separating
result in their scenario.14 This is different from and less general
than our model where the audit is imperfect for both types. We
have shown that, if the probability of an error in auditing good
client-firms is non-zero, though small, a separating equilibrium
cannot exist. Most importantly, in either of the two viable
equilibrium outcomes with a high quality audit pool in our study
(named DP, for Differentiated Pooling and DSS, for Differentiated
Semi-Separating), the firms of different types in the same pool ex
ante expect different market outcomes which are imperfectly
related to their true firm types. As a result, for sufficiently high
audit costs, though the same for both types of client firms, high
quality auditing may be economical for good client firms but
uneconomical for bad client firms. This feature of our framework
enables us to eliminate a number of un-intuitive equilibrium
outcomes. Unlike our model, there is no intuitive correspondence
between audit costs across different equilibria in MT.
In Section 7, we offer some concluding remarks.
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2. THE NATURE AND SUPPLY OF AUDIT SERVICES
In this section we sketch a model of auditor's maximization
problem that is consistent with the observed reality. It is well
recognized that audit services are heterogeneous commodities15
and that some audit firms, perhaps the larger ones, provide
higher quality audit services than others, achieving lower error
rates (higher accuracy).16 Further, as Feltham, Hughes and
Simunic observe:
. . . the literature on audit quality . . . has only demonstrated the existence of
two or perhaps three distinct auditor quality levels (1992, p. 377).
Thus, we model the difference in audit quality by assuming the
existence of two separate types of audit firms, who differ in their
operating efficiencies. We show that the economic incentives of
the audit firms leads to supply of two levels of audit qualities
corresponding to their respective efficiency levels. Thus, an audit
firm does not merely assert that it is offering a `high' or a `low'
quality service. Given their efficiency types, their incentives are
well understood and the quality of the services they supply are
rationally anticipated by the market participants. For the
signaling equilibrium analyzing the demand side, we focus on
the fact that two audit services of different qualities, which are
priced differentially, exist simultaneously in the market. Both
audit services are informative, one more so than the other in a
sense to be made precise below. Neither of them, however, is
perfectly informative.17
(i) The Audit Firm's Role
The audit firm in our scenario provides a third-party evaluation
of the financial position of the client firms.18 There are n firms
which are potential audit clients. The insiders of each firm know
its true financial position which, however, is not directly
observable to the outside market. The financial position for any
client firm, denoted by a, can be either type 1 (good) or type 2
(bad). The types t "f1; 2g T denotes the two types such that
type 1 is the better of two types. Under full information, the
market would value a1 and a2 as V a1 and V a2 19 such that
V a1 > V a2 . Without full information, the market must use
the audit report for firm valuation.
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Audit firms often perform more than their duties prescribed
under the Generally Accepted Auditing Standards (GAAS).
Cushing and Loebbecke (1986) observe that:
many improvements in auditing standards and techniques are reflected in
the pronouncements of the AICPA . . . some time after such improvements
are made in the practices of individual firms.20
Senior members of the big six firms have often asserted that their
firms observe a higher standard than that which is required by
law.21 Since the audit firms need only to conform to the
Generally Accepted Auditing Standards (GAAS) for their audit
methods, regulation in the audit market cannot be the source of
such product differentiation. To wit, consider the following
quote from Elliot (1993):
There is clear undeniable evidence of demand for audit services before
(security) laws were passed, and there's enormous demand today for audits
that aren't required by law. My own firm (KPMG Peat Marwick), for example,
does more voluntary audits than statutory required audits ± by an order of
magnitude.
Providing a higher quality audit must therefore be in the best
interests of the concerned audit firms. Our auditors are strategic
in that they anticipate demand from the particular firm types and
provide an optimal level of effort consistent with their best
interests.22 However, they are independent in that they report
their findings truthfully.23 Audit quality (defined more precisely
later) increases with effort.
(ii) Auditors' Cost Functions
Let A and B represent two different audit services or methods
m; m "fA; Bg, where A is the higher quality method. The
suppliers of high quality audits are more cost-efficient than the
suppliers of low quality audits in that for a given level of effort,
higher quality auditors produce lower error rates. Since any audit
firm in our scenario supplies only one type of service, we shall use
A or B to indicate, interchangeably, both an audit firm and the
corresponding service. The operating costs are driven by the
efforts (e) spent by the auditors, which control the type I and type
II errors. We assume that the operating cost, C e, of providing a
given level of audit effort, denoted by e, is higher for the low
quality auditor (B) for all e, and let the operating cost differential
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increase with effort. The quality differential is characterized by
the difference function e CB e ÿ CA e such that 0 e > 0
and 0 0. Define C 0 and C 00 to be the first and second
derivatives of the C : function with respect to its argument. We
also assume that C 0 and C 00 functions are positive everywhere for
both auditors and both efforts types.
(iii) Audit Fees
We assume that the cost of audits to the client firm is fixed for
each method (A and B) and are equal to kA and kB . Adequate
supply of both types of auditors and perfect competition among
them would ensure that both types of auditors break even across
firm types. The fees kA and kB , thus, must allow each type of
auditor to recover their expected costs. The fee differential
between the high quality and low quality audits is a sum of the
operating cost differential, the expected liability cost differential
and the cost differential arising out of the loss of business due to
type I errors. Unless the higher quality audit commands higher
fees, incentives to supply higher quality audits would be
questionable. Therefore, we require that in equilibrium, the
high quality auditors are paid higher fees and the corresponding
fees kA must be greater than kB . We are primarily concerned with
the differential in the fees for the two audit services, denoted by
K where K kA ÿ kB . To the extent high quality audits are
offered by the bigger firms as we mentioned earlier, kA > kB is
consistent with the existing empirical evidence.
(iv) Audit Report and Audit Accuracy
After carrying out an audit program, the audit firm which provides
the higher quality audit service (A) comes up with a finding(f)
about the client firm which can be either `high' (HA) or `low'
(LA). Thus, f "fHA, LAg F . Similarly, the audit firm which
supplies the lower quality audit service (B) also issues its finding
(g) after its audit which again can be either `high' (HB) or `low'
(LB). Thus g "fHB, LBg G. The auditors' reports are seen by all
parties in this scenario. The probability of a `high' assessment by
an auditor of quality level m; m 2 fA; Bg, for a client firm of type
t; t 2 f1; 2g, is denoted by qtm ; Prob[High |t, m] qtm . We can
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view q1m and q2m as indicators of the level of audit accuracy i.e., the
ex-ante probability that an auditor makes a correct report, for
audit method m; m A; B. Specifically, a high q1m indicate a low
type 1 error and a low q2m indicate a low level of type II error.
Given that the auditing standards ensure a basic level of auditor
efficiency, we view qtm as being determined by the auditors' efforts
e. Specifically, q1m em and q2m em are the two error levels chosen
by the auditors through their effort choices. We assume qtm em to
be such that accuracy increases with effort but at a decreasing rate.
0
00
0
00
Formally, q1m
> 0; q1m
< 0 but q2m
< 0; q2m
> 0, where 0 and 00
respectively denote the first and second derivatives of the
respective functions. Further, q1m 1 1 and q2m 1 0. This
implies that perfect audit accuracy is impossible to achieve and
that audit failures will occur. For the sake of notational
convenience, henceforth we suppress the argument em from q1m
and q2m . For the equilibrium analyzed in the demand section, we
need that the high and low quality audits are well defined and
require that q1A > q1B and q2A < q2B . As we show below, this is
exactly what happens. Also for the low quality audit to have value,
for the equilibrium level of effort eB , we assume that q1B > q2B .
(v) Consequences of Audit Failure
An audit failure occurs when auditors report H L findings for a
type 2 (1) client firm. If a type 1 firm is assigned L, an auditor of
type m risks loss of business, denoted by lm . If, on the other hand,
a type 2 firm is assigned H, the auditor risks being sued by the
investors in the firm (and found guilty of audit failure by the
court) who claim to have been misled by the audit findings. The
expected cost of this (type II) error is represented by the penalty
function Pm .24 If 1 2 proportion of the type 1 (type 2) client
firms engage the type A auditors, the probability of a loss 1A 1B
must be 1 1 ÿ q1A (equivalently, 1 ÿ 1 1 ÿ q1B ). Similarly,
the probability of a loss PA PB must be 2 q2A (equivalently,
1 ÿ 2 q2B . There is some evidence that courts are increasing the
auditors' common law tort liability and are awarding increasingly
higher damages for audit failures (Beneish and Chatov, 1993; and
Gormley, 1988). At the time of a lawsuit, the auditor is often one
of the few solvent parties worth pursuing (Kothari et al., 1988; and
Minow, 1984). Thus, increasing high damage awards increases the
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probability that auditors with deeper pockets will face a higher
expected penalty for an audit failure than do other audit firms. If
the larger and more established audit firms are also the ones with
deep pockets, and since the size of an audit firm and audit quality
are correlated (DeAngelo, 1981), it follows that auditors of higher
quality may face higher expected costs due to type II errors.
Therefore, we assume that for non-zero 2 in equilibrium,
independent of efforts, the higher quality auditor is exposed to
a higher level of expected penalty, i.e., 2 :PA 1 ÿ 2 :PB .25 At
first we assume that Pm is exogenous. An assumption of
exogenous Pm seems to be consistent with a standard of strict
liability. Though the existing laws do not specify a standard of
strict legal liability, the existing rule of due-diligence defense is
becoming, in many respects, equivalent to that of a strict liability
(Nelson, Ronen and White, 1988). Minow (1984), Chatov (1987),
Gormley (1988) and Kripke (1988) among others have
documented the evolution of an increasingly popular strict
liability interpretation of auditors' liability by courts, especially
within the Securities Act of 1934. Later, we relax this assumption
and consider alternative regimes where Pm is determined by some
audit outcome, and thus, is partially controllable by the auditor
through increased effort. Comparison with a fixed and exogenous
Pm then serves as a benchmark.
(vi) The Auditors' Maximization Problem
The supply of audit quality is determined by the auditors'
maximization problem. In our model, both types of auditors are
assumed to be risk neutral. The type A auditors maximize their
income in the following way:26
Max
kA ÿ CA eA ÿ 1 1 ÿ q1A lA ÿ 2 q2A :PA :
e
A
Similarly, the type B auditors maximize their income in the
following way:
Max
kB ÿ CB eB ÿ 1 ÿ 1 1 ÿ q1B lB ÿ 1 ÿ 2 q2B :PB :
e
B
Assume that the parameters of the problems are such that an
interior solution exists and can be characterized by the following
first order conditions:27
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0
0
CA0 eA 1 q1A
lA ÿ 2 q2A
:PA 01 q1A lA ÿ 02 q2A :PA ;
0
0
:PB ÿ 01 q1B lB 02 q2B :PB :
lB ÿ 1 ÿ 2 q2B
CB0 eB 1 ÿ 1 q1B
The above first order conditions can be interpreted as equating
the marginal operating costs with the marginal benefit from
reduction of penalty and business losses after considering the
impact on change in demand.28
Although 1 , 2 , 01 and 02 are decided in equilibrium, we know
that 1 2 1 is the maximum values of 1 and 2 and
therefore, in the equilibrium if 1 2 1, both 01 and 02 must
be equal to zero. For the sake of tractability, we assume that both:
@ 2 1
@eA @eB
and:
@ 2 2
@eA @eB
are also zero.
We now note two important requirements with respect to the
supply of the audit services that must be satisfied in any proposed
equilibrium. The first is that the optimal solution of the auditors
effort choice problem must satisfy eA > eB implying that,
motivated by self interest, the more efficient auditors supply a
high quality audit service. If so, a meaningful inference about
audit quality can be derived from the knowledge of auditor
efficiency by the market participants. The second requirement is
that the fees of a type A auditor will be greater than the fees of
the type B auditor. Both these conditions form part of the
equilibrium conditions developed in the following section.
3. THE ECONOMIC ENVIRONMENT AND DEMAND FOR AUDIT SERVICES
(i) Basic Information Structure
The outside investors have common prior knowledge that
proportion of all client firms are type 1 and 1 ÿ proportion
are type 2. The insiders know their own firm types, but any direct
disclosure of firm types is prone to adverse selection problems.
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To communicate the true financial position of their firm to the
rest of the market, the insiders may engage either of two kinds of
available audit services, high quality (A) (offered by the more
efficient auditors) and low quality (B) (offered by the less
efficient auditors), described in the previous section. This choice
transmits a message m "fA; Bg
. An audit is mandatory;
therefore no audit is not a possible message. Since these two
types of audit services are provided by two distinct classes of audit
firms, this message (audit method) can be observed by the rest of
the market.
(ii) The Outsiders' Role
Besides the informed managers and the auditors, the other
players in this game are the outside investors who decide on the
market values of the client firms. The market beliefs and market
values in our model are conditioned by both what the client firms
convey through their choice of the audit method and what the
audit firms report through their findings. This feature makes our
structure somewhat different from a standard signaling game.
The market value of a client firm from a high quality audit VA
following the release of the audit report is a function of the
managers' message of choosing high quality auditing (A) and the
audit findings (H or L). Thus, VA "fVHA ; VLA g. Similarly, the
market value of the firm from a low quality audit is
VB "fVHB ; VLB g. Given that the audit services A and B are
imperfectly efficient, neither VHA nor VLA if the firm chooses A,
and neither VHB nor VLB if the firm chooses B, may exactly reflect
the true or full-information values which are, as stated before,
V a1 and V a2 respectively. Both types of firm pay exactly the
same fee for a given audit method. Therefore, there is no typespecific cost that may help outsiders to understand the firm type
(unlike in a standard signaling game). The outside investors
operate in a competitive market, with the result that they decide
on the market values in such a manner as to break even across all
client firms. In other words, regardless of the actions the client
firms choose, their total market value always adds up to their true
value given by V a1 1 ÿ V a2 .
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(iii) Optimization Program of Client Firms
We assume that all players in the game are risk neutral. The
objective function of the insider-manager of a client firm in this
scenario consists of a weighted average of the true value of the
firm and its market value.29 Let V t; m denote the expected
market value of a client firm of type t at the start of the game
(prior to the release of the audit report). Then, for m A; B:
V t; A qtA VHA 1 ÿ qtA VLA ÿ K ;
and:
V t; B qtB VHB 1 ÿ qtB VLB ;
where K kA ÿ kB represents the differential between the audit
fees required for the two audit services and qtm 's are obtained
from the audit firm's optimization program (see Section 2
above).
If is the weight attached to the market value and 1 ÿ to
the true value of the firm by the insider-manager, the objective
function, W V t; m, is:
W V t; m V t; m 1 ÿ V at ; m"fA; Bg; t"f1; 2g:
The insider-manager chooses m or the type of audit to maximize
the above objective function.
Figure 1 presents the game tree described above. Although so
far we have presented the game in terms of pure strategies, A or
B, our analysis of the game allows for mixed strategies as well.
Denote by t the probability with which client firms of type t send
a message m A. Intuitively, t can be viewed as the proportion
of type t firms that choose a high quality audit.
(iv) Nash Sequential Equilibrium (NSE)
We use the concept of sequential equilibrium developed by
(Kreps and Wilson, 1982) and adopt the definition in our setting
in the following way. A Nash-Sequential Equilibrium (NSE) , is an
ordered set t ; ; eA ; eB ; kA ; kB satisfying the following
conditions:
C1. Client firms choose t 2 argmax fW V t; t g; t 2 0; 1;
t 2 T.
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DE AND SEN
Figure 1
Game Tree
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C2. Audit firms choose their efforts e such that a type A audit
firm chooses:
eA 2 argmax fkA ÿ CA ea ÿ 1 1 ÿ q1A lA ÿ 2 q2A :PA g
and a type B audit firm chooses:
eB 2 argmax fkB ÿ CB eB ÿ 1 ÿ 1 1 ÿ q1B lB ÿ
1 ÿ 2 q2B :PB g:
C3. kA ; kB are such that both types of audit firms break even
in equilibrium.
C4. Quality and prices of the two audit supplies are well
defined; i.e., eA > eB and kA > kB .
C5. Outside investors, in a perfectly competitive market,
choose V t; t so that they break even across the
unobserved types of the client firms, implying:
X
V t; m; 1 ; 2 V a1 1 ÿ V a2 :
t
C6. If m; m A; B, is an equilibrium action, market posterior
beliefs following an H finding tjHm; 1 ; 2 , and
market posterior beliefs following an L finding
tjLm; 1 ; 2 , are determined by Baye's rule in the
following manner:
1jHm; 1 ; 2
q1m 1m
;
q1m 1m q2m 2m 1 ÿ
2jHm; 1 ; 2
q2m 2m 1 ÿ
:
q1m 1m q2m 2m 1 ÿ
1jLm; 1 ; 2
1 ÿ q1m 1m
;
1 ÿ q1m 1m 1 ÿ q2m 2m 1 ÿ
2jLm; 1 ; 2
1 ÿ q2m 2m 1 ÿ
1 ÿ q2m 2m 1 ÿ 1 ÿ q1m 1m
where: tm t if m A and 1 ÿ t if m B.
Further, qtm 's follow properties outlined in Section 2(iv)
and are obtained from the audit firm's effort choice
outlined in C2.
C7. If A or B is an off-equilibrium action, tj:; :; :"0; 1.
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4. DEMAND FOR AUDIT SERVICES: EQUILIBRIUM RESULTS
(i) Some Basic Issues
Before we present our equilibrium results, we intuitively discuss a
few technical issues and results that we need for developing our
equilibrium results.
For any well defined equilibrium, we require that eA > eB and
kA > kB , which is guaranteed by the supply side results developed
in Section 2. For all eA > eB the properties of the qtm : function
would assure that 1 > q1A > q1B and q2B > q2A > 0. Further, for a
type B audit to be informative, we require that q1B > q2B .
Combining the restrictions and allowing for weak inequality, we
have:
1 > q1A q1B > q2B q2A > 0; . . . . . . (A1)
where at least one of the two inequalities holds strictly. The above
accuracy structure implies two conditions. First, both auditors A,
and B, are generally efficient but not perfect in that both auditors
are more likely to report a `high' finding for a type 1 firm than
for a type 2 firm. Second, type A auditors are more accurate than
type B auditors. We also denote K kA ÿ kB , the fee differential
in an equilibrium, which plays a prominent role in our analysis.
In Section 3, VHm and VLm are defined as the market value of a
client firm conditional on the firm drawing H and L findings
respectively, given an audit method m; fm 2 A; Bg. When 1
proportion of type 1 firms and 2 proportion of type 2 client
firms choose m, define VHm 1 ; 2 and VLm 1 ; 2 as the
corresponding values. Risk-neutrality on the part of the capital
market participants implies that:
VHm 1 ; 2 V a1 1jHm; 1 ; 2 V a2 2jHm; 1 ; 2
and:
VLm 1 ; 2 V a1 1jLm; 1 ; 2 V a2 2jLm; 1 ; 2 :
Note that V a1 > VHm 1 ; 2 and VLm 1 ; 2 > V a2 .
Similarly, define V t; m; 1 ; 2 as the expected outcome for a
client firm of type t; t 1; 2, if it opts for an audit method m,
given that 1 proportion of type 1 firms and 2 proportion of type
2 firms choose m. Further, define W V t; m; 1 ; 2 as the
corresponding value of the firm's objective function. As the
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posterior probability functions embedded in the values indicate,
given t and m the values depend on three sets of parameters:
ft ; t 1; 2g, ft ; t 1; 2g and fqtm ; m A; Bg. While the first
set is drawn by the nature so to speak, the other two are not.
Changes in the parameters may affect the type-specific values
differently. This raises the following questions: whether, and
under what conditions, are the expected outcomes from the same
audit method different for the two different types of client firms?
Further, whether, and under what conditions, are the expected
outcomes of a client firm of a given type different under the two
different audit methods? These questions are important and are
addressed through a series of technical observations that are
necessary for the proofs of the propositions and detailed in the
Appendix.
(ii) Equilibrium Results
Since firms of each type can choose either A or B or randomize
between the two strategies, there are nine type-strategy
combinations that are candidates for an equilibrium outcome.
At first we show that unlike Melumad and Thoman (1990), no
separation of types is possible in our model.
Proposition 0. There is no separating NSE. Specifically, neither
of the following two situations is NSE.
(I) Type 1 chooses high quality auditing 1 1 and type 2
chooses low quality auditing 2 0.
(II) Type 2 chooses high quality auditing 2 1 and type 1
chooses low quality auditing 1 0.
Proof. See Appendix.
It is easy to see that the two separating equilibrium candidates
1 1; 2 0 and 1 0; 2 1 do not satisfy NSE
conditions. Audit fees are fixed, and there are no other
dissipative costs, type-related or otherwise, in this model. Thus,
non-mimicry, an essential prerequisite for separation, cannot be
guaranteed. Before we present our main equilibrium result, in
Table 2, our findings for each type-strategy combination are
summarized in the corresponding cell. We now discuss the
intuition underlying the main results (cells (1) and (3)). The
main findings of this study are that in equilibrium:
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Table 2
Strategies
Type 1 chooses
high quality
auditing : 1 1
Type 1 chooses
low quality
auditing : 1 0
Type 1 chooses a
mixed strategy :
0 < 1 < 1
Type 2 chooses
high quality
auditing : 2 1
NSE exists for
`low' K only :
robust to
refinements (1)
No NSE (4)
No NSE (7)
Type 2 chooses
low quality
auditing : 2 0
No NSE (2)
NSE possible;
not robust to
refinement (5)
No NSE (8)
Type 2 chooses
a mixed strategy
: 0 < 2 < 1
NSE exists for
`higher' K (3)
No NSE (6)
No NSE (9)
(1) type 1 firms strictly prefer high quality auditing 1 1 if
type 2 firms prefer high quality auditing or randomize
between high quality and low quality auditing
0 < 2 1; and
(2) type 2 firms choose only high quality auditing 1 if
the differential between the fees of the two audit
methods, given by K where K kA ÿ kB , is sufficiently
low; randomize between the two pure strategies
0 < 2 < 1 if K is higher; and avoid high quality auditing
altogether 2 0 if K is very high (as we shall see below,
K will have to be unrealistically high to make this
happen).
The statements in (1) and (2) above together imply that, for
the most part, in equilibrium 1 1 and 0 < 2 1. In these two
cases the quality and prices of audit supply are well defined. In
equilibrium, given q1A q1B q2B q2A with a strict inequality
holding in at least one case, type 1 firms benefit more from
higher efficiency associated with high quality auditing than from
low quality auditing. If type 2 firms expect the same outcome
from the two audit methods, A and B, inducing them to
randomize between the two strategies, type 1 firms will strictly
prefer A with its higher expected outcome. Of course, if type 2
firms prefer A to B, it follows that type 1 firms will prefer it even
more so. This explains the preference of a type 1 firm for high
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quality auditing. As for a type 2 firm, it expects less from A than
from B when 1 2 1, A being more informative than B.
However, if it switches from A to B, it is liable to fetch V a2 or its
true value; since type 1 firms do not choose B, any firm choosing
B is deemed a type 2 firm and is valued accordingly. If the
differential in audit fees are sufficiently low, its expected
outcome from A may still exceed V a2 . This results in the
equilibrium strategy combination 1 2 1.
If audit fees are higher, the expected outcome net of costs for a
type 2 firm from high quality auditing may be no higher than
V a2 , making it indifferent, and consequently randomize,
between the two strategies at the margin. This results in the
equilibrium strategy combination 1 1; 0 < 2 < 1. It is also
true that given the demand pattern, audit firms' supply decision
of audit quality and the break-even condition results in higher
fees for a type A audit. In this equilibrium the firms which choose
high quality auditing consist of all type 1 and some type 2 firms.
On the other hand, all firms which opt for low quality auditing
are type 2. This equilibrium is semi-separating in that sense.
How high should the differential audit fees be so that no type 2
firm chooses A, making 2 0? For this to happen, V a2 must
exceed the expected outcome from high quality auditing net of
fees which in this case is V a1 ÿ K (if no type 2 firm chooses A,
any firm which chooses A commands the valuation appropriate for
a type 1 firm). However, if V a2 V a1 ÿ K , it is not worthwhile
even for type 1 firms to employ this method. In other words, audit
fees must be so high as to make all firms reject a high quality audit,
an implication which contradicts what is observed empirically (see
Table 1). Given no demand for high quality auditing, in this case,
it is also not clear whether the audit fees and quality will be
differentiated in our model. The only two possible equilibrium
outcomes in our model are (1) a differentiated pooling (DP)
equilibrium where all firms opt for A or high quality auditing if the
fee differential between the two services is below a certain level; (2)
a differentiated semi-separating (DSS) equilibrium, if the
differential is above that level but not too high. In the two
equilibria, the A-pool is differentiated in the sense that a type 1
firm, at the start of the game, has a higher expected outcome from
high quality auditing than a type 2 firm. The above intuition is
formally presented in Proposition 1.
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Proposition 1. (1) For 0 K K , where q2A VHA 1 2 1
1 ÿ q2A VLA 1 2 1 ÿ V a2 K , the DP eq
Legal Liabilities, Audit Accuracy
and the Market for Audit Services
Sankar De and Pradyot K. Sen*
1. INTRODUCTION AND OVERVIEW
In recent years, liability lawsuits against auditors seem to have
reached an epic proportion. The liability claims against the big
six audit firms in the US is estimated to be about $30 billion,
which exceeds their total partner's capital by a sizable amount
(Balachandran, 1993). Several large jury decisions and out of
court settlements often get reported in the financial press. The
seriousness of the problem is appreciated by academics and
practitioners alike.1 Since the early 1990s, the practitioners felt
that the existing liability regime put an ever increasing burden
on the audit firms and their clients (Weinbach, 1993; Cook, 1993;
and Freedman, 1993). The litigation phenomenon is not limited
to the US alone. US audit firms routinely face lawsuits for work
done outside the US.2 Although the US remains the preferred
* The authors are respectively from the Center for Professional Development in Finance,
Berkeley, California and the Haas School of Business, University of California, Berkeley;
and the University of Cincinnati. They wish to thank Bala Balachandran, Robert Chatov,
Srikant Datar, Jere Francis, Angela Gore, Bob Hagermann, Jack Hughes, Bjorn Jorgensen,
Ken Klassen, Rick Lambert, Bob Magee, Nahum Melumad, V. G. Narayanan, Zoe-Vonna
Palmrose, Victor Pastena, Stefan Reichelstein, Kevin Sachs, Toshi Shibano, Brett
Trueman, the workshop participants at University of California at Berkeley, SUNYBuffalo, University of Iowa, Center for Applied Mathematics ± New Jersey Institute of
Technology, New York University, University of Wisconsin-Madison, European
Econometric Society Annual Meetings and European Economics Association Annual
Meetings, several anonymous referees and the editor of this journal for their comments
on different versions of this paper. Part of the work was done when both the authors
visited the Indian Institute of Management Calcutta, India. Typing assistance of Rona
Velte and Kathy McCord is greatly appreciated. (Paper received May 2000, revised and
accepted January 2001)
Address for correspondence: Pradyot K. Sen, Department of Accounting and Information
System, College of Business, 302 Lindner Hall, University of Cincinnati, Cincinnati OH
45211-0211, USA.
e-mail: [email protected]
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DE AND SEN
forum for litigation, audit litigation has begun to become
prevalent in the common law countries such as England,
Caribbean, Canada, Australia, New Zealand, and in many Asian
countries such as Hong Kong and Singapore. Even in countries
such as Japan and Germany, audit related litigation has become
more common.3 The Bank of Credit and Commerce
International (BCCI) litigation, perhaps involving the most
amount of money in any litigation, was filed in both Luxembourg
and the UK. It is believed that in countries such as New Zealand
and Australia, the rate of litigation, considering the size of the
economy, far exceeds professional malpractice litigation
anywhere in the world (Grobstein and Liggio, 1993). Given the
differences in the liability levels and regimes in different parts of
the world, it is important to understand how a rational auditor
would respond to changing liability levels and regimes and
modify her efforts level and fees and how such choices would
influence the supply of audit accuracy and the demand for audit
services. This paper is an attempt to study this problem through a
model, where in a market characterized by asymmetric
information between the firm-managers and the shareholders,
firms demand auditing to signal their private information and,
anticipating the demand and responding to various liability levels
and regimes, rational auditors of different operating efficiency
choose the level of care (effort).
In the stylized market characterized by us, on the supply side of
the market (in Section 2), two different types of audit firms
(differing in their operating cost structures, which is well known
in the market) decide optimally their level of care (characterized
as high quality or low quality audit services), based on the
(anticipated) demand, the legal liabilities arising from audit
failures and other considerations such as potential loss of
business. We characterize a high quality service by lower type I
and type II errors.4 The identity of auditors and, therefore their
efficiency, as well as the audit findings are public information
whereas the (two) client firm types (referred to as `good' and
`bad' or equivalently, type 1 and type 2) are known only to the
firm-managers. While the marginal benefits to auditors are
represented by the audit fees offered for the services, the
marginal audit costs typically have three components; operating
cost structures which vary across the audit firms, potential costs of
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loss of business arising from a type 1 audit error, and potential
costs of legal liabilities arising from a type 2 audit error. In our
model, some audit firms are more efficient than others, due to
their operating cost structure, and this advantage in operating
cost structure of some firms is a common knowledge in the
market. However, all auditors are generally efficient in the sense
that they all generate favorable findings for a good client firm
with a higher probability than for a bad client firm. However, no
audit service achieves perfect audit accuracy (zero type I and type
II errors) because the costs involved may be prohibitive. In this
scenario, if more efficient auditors are subjected to a higher level
of liability and business loss arising from type I error, we
characterize an equilibrium where more efficient auditors also
provide a higher quality audit service and receive higher audit
fees.5
On the demand side of the market (in Section 3), the potential
client firms and outside investors who cannot directly observe
firm types, engage in a game of market valuation under
asymmetric information. Since direct communication of the
private information (unobserved types) to the outside investors is
prone to adverse selection problems, the client firms signal their
type through their choice of audit quality in order to generate
favorable market valuations. In equilibrium, given that the
available audit services are heterogeneous in terms of accuracy
and required fees,6 in an asymmetrically informed market, choice
of a particular type of audit service (more efficient or less
efficient) by a client firm together with the audit report on the
firm provide additional value relevant information. Because of
the way investors use audit information in their evaluation of a
firm, good client firms demand high quality auditing because
they have a high probability of being identified as good firms.
Bad client firms, on the other hand, would prefer a low
probability of correct identification and would like to choose a
low quality audit service. However, if the market sees through this
incentive and perceives that no good client firm is likely to opt
for a low quality audit service, they run the risk of being singled
out through their action. For the bad firms, it is essentially a
choice between two unfavorable options. The choice is decided
by the differential in fees for the two audit services. The (bad)
client firms, therefore, strategically consider the possible reaction
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of the market, the audit fees, and decide whether to demand
high quality or low quality audit services.7
In this model, we show (in Section 4) that under fairly general
conditions, depending on the level of the differential in fees for
the two audit methods, three Nash Sequential Equilibrium (NSE)
outcomes, which are robust to refinements8 are possible:
1. For cost differential below a certain level, a pooling
outcome in which all firms choose high quality auditing
(DP).
2. For the differential exceeding that level, a semi-separating
outcome in which all good firms opt for high quality
auditing while bad firms randomize between the two audit
methods (DSS).
3. If the differential is too high, a third possibility exists where
both types forego high quality auditing.
We focus on the second equilibrium where all good and either
some or all bad client firms demand high quality audit services in
equilibrium, unless the differential in fees for high and low
quality services is too much. On the other hand, client firms who
choose low quality audit services are always bad themselves. The
intuition behind the second equilibrium is that as more and
more bad firms choose a high quality audit, the expected value of
such a firm in the pool is diluted and, given sufficiently high fees,
eventually no higher than what could be obtained with the help
of a low quality audit. As a result, in the second equilibrium, on
the margin, the bad firms become indifferent to the choice of
audit method and consequently, randomize between the two
audit methods. Interestingly, the proportion of bad quality firms
which choose high quality auditing in this equilibrium is unique
given the parameters of the system and, further, higher for a low
differential between the respective costs of the two audit
methods.
So long as the market has a high proportion of good client
firms, our finding suggests perhaps an overwhelmingly high
demand for the high quality audits. If big audit firms offer high
quality audit services, our result that all good and all or some bad
client firms opt for high quality auditing in equilibrium in spite
of the higher fees offers an explanation why a high proportion of
all US firms use the services of the big six (big eight) audit firms
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for audit purposes. Table 1 shows that, during 1982±1987, 82% of
all audited firms in the US used big eight audit firms.9 In the
higher annual sales range of $250 million and more, this
proportion was 95%. In terms of value, the big eight audit firms
audited about 97% of the total sales of all US audited firms in this
period, though the existing empirical evidence suggests that the
bigger firms also charge higher fees. Though several reasons have
been suggested for this overwhelming preference for the services
of the big audit firms in spite of their higher fees, there is no
formal model of demand for audit services that explains this
phenomenon in the existing literature on auditing.10
Using this scenario, the study offers a number of insights into
the impact of legal liabilities on the demand for and supply of
audit services. First of all, it shows that if there are auditors with
(two) different skill levels and their marginal costs of providing
the same level of service systematically differ, and if the more
efficient firms are subjected to a `higher' level of liability (in a
sense to be specified later) for audit failure, audit services with
(two) different accuracy levels could be simultaneously available
in the market where the more efficient auditors exert higher
efforts leading to higher accuracy and command higher fees.
Second, the differential in fees for the different services and the
degree of their audit accuracy play a crucial role in determining
the demand of the client firms for such services and the extent of
revelation of their unobserved types in equilibrium. An increase
in legal liability induces an auditor to increase her efforts level
and improve her accuracy. However, we show (in Section 5) that,
for all client firms, good and bad, together, demand for a high
quality audit is inversely related to its fees. It is comforting to note
that the inverse relationship between audit fees and demand,
argued in the price theoretic analyses (e.g. Benston, 1985), holds
in a more complex game-theoretic setting. More interestingly, we
show that the demand for a high quality audit is also inversely
related to its accuracy as well (Proposition 3). The inverse
relationship of audit demand with audit accuracy goes somewhat
counter to the prevailing wisdom where a higher quality is
assumed to increase demand. The importance of this result is as
follows. An increase in legal liabilities (a) increases auditor's cost
and hence the fees; leading to increased marginal cost to client
firms, (b) increases audit accuracy or quality through increased
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Table 1
Description of Clients for Big Six and Non Big Six Audit Firms, 1982±1987
Number of Clients Total Sales Audited
($billions)
Year
NB6
b
1±25 mill.c
26±50 mill.
51±100 mill.
101±250 mill.
250 mill and over
B6
NB6
B6
NB6
B6
NB6
B6
NB6
B6
NB6
B6
NB6
6942
6716
6543
6488
6173
5727
1336
1457
1417
1452
1572
979
3664
3123
3525
3465
3170
3290
111
105
104
104
104
96
2943
2553
2691
2694
2566
1879
911
862
986
912
1097
815
833
815
770
772
766
684
119
156
135
137
171
102
765
776
791
764
699
638
135
121
112
116
115
88
855
839
794
779
765
788
88
94
104
107
109
39
1546
1433
1488
1479
1377
1468
73
78
78
71
71
75
6430
1369
3373
104
2554
930
773
137
739
115
689
90
1465
74
82
18
97
3
73
27
85
15
87
13
88
12
95
5
ß Blackwell Publishers Ltd 2002
Notes:
a
B6 represents big six
b
NB6 represents non big six.
c
This represents the range of company size audited in terms of million dollar (US).
Source: Computed using data from various issues of Who Audits America, The Data Financial Press.
DE AND SEN
1987
1986
1985
1984
1983
1982
Yearly
Average
Average
%
B6
a
Number of Companies Audited
THE MARKET FOR AUDIT SERVICES
359
auditor's efforts; and therefore, provides incremental value to the
client firms. So far, the literature on audit markets seems to have
concluded that the joint effect of legal liabilities, as manifested
through these two factors, is at best, an empirical issue and may
even offset each other in determining the demand for audit
services (see for example, Lys, 1993).
Our result that an increase in both fees and accuracy reduces
the overall demand of high quality audits, raises serious questions
to the arguments used in the empirical auditing literature. Since
legal liabilities play a crucial role in providing incentives to
auditors, we examine (in Section 6) the role of the legal system,
through assessment of penalties consequent to audit failures, in
providing efficiency-improving incentives to the auditing
profession. We consider two regimes; (1) a `verifiable due care
regime' when it is possible for the litigation process to observe
the (type II) error level being allowed by the auditor and (2) a
`non-verifiable due care regime' where, although the (type II)
errors themselves could not be verified, the value of the firm,
were it to receive a `low' report, could be inferred from the
market value of other firms and similar other sources. We find
that though legal penalties for audit failures increases auditor's
effort, and therefore, accuracy, a penalty regime that relates the
penalty to the type II error (verifiable due care regime) induces
an effort level that is less than that induced by an exogenous
penalty regime (the benchmark case). The present legal practice
in the US of awarding damages based on the actual losses
suffered by the plaintiffs, is consistent with the non-verifiable due
care regime. According to our results, it induces an effort level
that is higher than that induced by the (benchmark) exogenous
penalty regime. We are also able to conclude that a proportional
liability regime induces a higher effort than a joint and several
liability regime where the level of awards may be independent of
the auditors care level or the effort. If the more established and
larger audit firms are the ones who are more wealthy and
therefore, believed to have `deep pockets', our results suggest
that the practice of `deep pocket' awards in the US, would
penalize more established and larger audit firms more severely
than the smaller firms for the same kind of audit failures. This
differential penalty may be undesirable from the point of view of
promoting efficiency in the smaller firms.
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The role of legal liabilities and operating cost structure has
been addressed by Nelson, Ronen and White (1988) in a supply
side model that uses similar characterization to our Section 2. As
Demski (1988) notes, there is no demand for auditing per se in
their model. More recently, Gigler and Penno (1995) have
addressed the problem of demand for auditing in a model where
different auditors have different cost structures. However, their
model is essentially different from ours in that the auditors in our
model have fixed types and are identified unambiguously with
their types. In Gigler and Penno, the auditors switch their types
randomly in different time periods in a Markovian manner, thus
making any type-identification impossible. More recently,
Hillegeist (1999) studied the role of different liability rules on
audit quality choice. Though Hillegeist uses a similar game
theoretic model and asks some of the same questions, the focus
of his study and ours is fundamentally different. In the Hillegeist
model, there is only one type of auditor, who makes a uniform
quality choice. Thus, quality differentiated audit does not exist in
the Hillegeist model and some of the issues we raise regarding
the differential impact of legal liabilities on different auditors
cannot be addressed through the Hillegeist model. We do not
explicitly model financial reporting and the litigation process
that Hillegeist does, and therefore, cannot address some of the
interesting reporting related issues addressed by Hillegeist. The
audit technology used in Hillegeist is also less general than ours
in that in the Hillegeist model, there is never an audit failure with
a high type firm, or when a low type reports low. Thus, the
auditor in the Hillegeist model never pays a cost for a type II
error. Since auditors in our model can commit type II errors,
they also suffer a cost associated with business loss when type II
errors occur, and must trade off that cost when deciding on the
audit quality. In the Hillegeist model, after the firm type is
revealed (exogenous) bankruptcy can occur to the owner.11 A
deep pocket award, thus transfers wealth from the owner to the
auditor. In our model, we do not explicitly model such a
bankruptcy and we focus on the inefficiency a deep pocket award
may cause by differentially affecting the different types of
auditors. In spite of these differences, Hillegeist arrives at
fundamentally the same conclusions as ours, in that an exposure
to a higher liability level leads to a higher audit quality.12
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However, since any penalty paid by the auditor can possibly
reduce the penalty to be paid by the insider, auditor penalty has
an insurance role in the Hillegeist model and adversely affects
the owner's disclosure behavior. We do not explicitly model any
penalty to the insiders. Taken together, our study and the
Hillegeist study provide some confirming and complementary
findings on the issue of audit quality choice and auditor liability.
Other studies that have addressed the issue of demand for
auditing, particularly in special contexts such as new issues
markets, include Titman and Trueman (1986), Datar, Feltham
and Hughes (1990), Bachar (1989) and Sarath and Wolfson
(1989). Though they contribute useful insights, the scope of these
studies is different from ours.13 In an insightful study, Melumad
and Thoman (MT) (1990) address some of the same issues examined in the present study. However, there are important
differences between the two models as well as the results derived
from them. In MT, the manager can disclose the true type apart
from selecting the auditor. In our model, adverse selection
problems rule out direct disclosure of type. Rather, the audit
findings together with the choice of the auditor constitute an
(imperfect) signal of firm type. Further in MT, the auditor makes
no error while auditing good client firms. Thus, a `low' report
unmistakably identifies a bad client firm, leading to a separating
result in their scenario.14 This is different from and less general
than our model where the audit is imperfect for both types. We
have shown that, if the probability of an error in auditing good
client-firms is non-zero, though small, a separating equilibrium
cannot exist. Most importantly, in either of the two viable
equilibrium outcomes with a high quality audit pool in our study
(named DP, for Differentiated Pooling and DSS, for Differentiated
Semi-Separating), the firms of different types in the same pool ex
ante expect different market outcomes which are imperfectly
related to their true firm types. As a result, for sufficiently high
audit costs, though the same for both types of client firms, high
quality auditing may be economical for good client firms but
uneconomical for bad client firms. This feature of our framework
enables us to eliminate a number of un-intuitive equilibrium
outcomes. Unlike our model, there is no intuitive correspondence
between audit costs across different equilibria in MT.
In Section 7, we offer some concluding remarks.
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2. THE NATURE AND SUPPLY OF AUDIT SERVICES
In this section we sketch a model of auditor's maximization
problem that is consistent with the observed reality. It is well
recognized that audit services are heterogeneous commodities15
and that some audit firms, perhaps the larger ones, provide
higher quality audit services than others, achieving lower error
rates (higher accuracy).16 Further, as Feltham, Hughes and
Simunic observe:
. . . the literature on audit quality . . . has only demonstrated the existence of
two or perhaps three distinct auditor quality levels (1992, p. 377).
Thus, we model the difference in audit quality by assuming the
existence of two separate types of audit firms, who differ in their
operating efficiencies. We show that the economic incentives of
the audit firms leads to supply of two levels of audit qualities
corresponding to their respective efficiency levels. Thus, an audit
firm does not merely assert that it is offering a `high' or a `low'
quality service. Given their efficiency types, their incentives are
well understood and the quality of the services they supply are
rationally anticipated by the market participants. For the
signaling equilibrium analyzing the demand side, we focus on
the fact that two audit services of different qualities, which are
priced differentially, exist simultaneously in the market. Both
audit services are informative, one more so than the other in a
sense to be made precise below. Neither of them, however, is
perfectly informative.17
(i) The Audit Firm's Role
The audit firm in our scenario provides a third-party evaluation
of the financial position of the client firms.18 There are n firms
which are potential audit clients. The insiders of each firm know
its true financial position which, however, is not directly
observable to the outside market. The financial position for any
client firm, denoted by a, can be either type 1 (good) or type 2
(bad). The types t "f1; 2g T denotes the two types such that
type 1 is the better of two types. Under full information, the
market would value a1 and a2 as V a1 and V a2 19 such that
V a1 > V a2 . Without full information, the market must use
the audit report for firm valuation.
ß Blackwell Publishers Ltd 2002
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Audit firms often perform more than their duties prescribed
under the Generally Accepted Auditing Standards (GAAS).
Cushing and Loebbecke (1986) observe that:
many improvements in auditing standards and techniques are reflected in
the pronouncements of the AICPA . . . some time after such improvements
are made in the practices of individual firms.20
Senior members of the big six firms have often asserted that their
firms observe a higher standard than that which is required by
law.21 Since the audit firms need only to conform to the
Generally Accepted Auditing Standards (GAAS) for their audit
methods, regulation in the audit market cannot be the source of
such product differentiation. To wit, consider the following
quote from Elliot (1993):
There is clear undeniable evidence of demand for audit services before
(security) laws were passed, and there's enormous demand today for audits
that aren't required by law. My own firm (KPMG Peat Marwick), for example,
does more voluntary audits than statutory required audits ± by an order of
magnitude.
Providing a higher quality audit must therefore be in the best
interests of the concerned audit firms. Our auditors are strategic
in that they anticipate demand from the particular firm types and
provide an optimal level of effort consistent with their best
interests.22 However, they are independent in that they report
their findings truthfully.23 Audit quality (defined more precisely
later) increases with effort.
(ii) Auditors' Cost Functions
Let A and B represent two different audit services or methods
m; m "fA; Bg, where A is the higher quality method. The
suppliers of high quality audits are more cost-efficient than the
suppliers of low quality audits in that for a given level of effort,
higher quality auditors produce lower error rates. Since any audit
firm in our scenario supplies only one type of service, we shall use
A or B to indicate, interchangeably, both an audit firm and the
corresponding service. The operating costs are driven by the
efforts (e) spent by the auditors, which control the type I and type
II errors. We assume that the operating cost, C e, of providing a
given level of audit effort, denoted by e, is higher for the low
quality auditor (B) for all e, and let the operating cost differential
ß Blackwell Publishers Ltd 2002
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DE AND SEN
increase with effort. The quality differential is characterized by
the difference function e CB e ÿ CA e such that 0 e > 0
and 0 0. Define C 0 and C 00 to be the first and second
derivatives of the C : function with respect to its argument. We
also assume that C 0 and C 00 functions are positive everywhere for
both auditors and both efforts types.
(iii) Audit Fees
We assume that the cost of audits to the client firm is fixed for
each method (A and B) and are equal to kA and kB . Adequate
supply of both types of auditors and perfect competition among
them would ensure that both types of auditors break even across
firm types. The fees kA and kB , thus, must allow each type of
auditor to recover their expected costs. The fee differential
between the high quality and low quality audits is a sum of the
operating cost differential, the expected liability cost differential
and the cost differential arising out of the loss of business due to
type I errors. Unless the higher quality audit commands higher
fees, incentives to supply higher quality audits would be
questionable. Therefore, we require that in equilibrium, the
high quality auditors are paid higher fees and the corresponding
fees kA must be greater than kB . We are primarily concerned with
the differential in the fees for the two audit services, denoted by
K where K kA ÿ kB . To the extent high quality audits are
offered by the bigger firms as we mentioned earlier, kA > kB is
consistent with the existing empirical evidence.
(iv) Audit Report and Audit Accuracy
After carrying out an audit program, the audit firm which provides
the higher quality audit service (A) comes up with a finding(f)
about the client firm which can be either `high' (HA) or `low'
(LA). Thus, f "fHA, LAg F . Similarly, the audit firm which
supplies the lower quality audit service (B) also issues its finding
(g) after its audit which again can be either `high' (HB) or `low'
(LB). Thus g "fHB, LBg G. The auditors' reports are seen by all
parties in this scenario. The probability of a `high' assessment by
an auditor of quality level m; m 2 fA; Bg, for a client firm of type
t; t 2 f1; 2g, is denoted by qtm ; Prob[High |t, m] qtm . We can
ß Blackwell Publishers Ltd 2002
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view q1m and q2m as indicators of the level of audit accuracy i.e., the
ex-ante probability that an auditor makes a correct report, for
audit method m; m A; B. Specifically, a high q1m indicate a low
type 1 error and a low q2m indicate a low level of type II error.
Given that the auditing standards ensure a basic level of auditor
efficiency, we view qtm as being determined by the auditors' efforts
e. Specifically, q1m em and q2m em are the two error levels chosen
by the auditors through their effort choices. We assume qtm em to
be such that accuracy increases with effort but at a decreasing rate.
0
00
0
00
Formally, q1m
> 0; q1m
< 0 but q2m
< 0; q2m
> 0, where 0 and 00
respectively denote the first and second derivatives of the
respective functions. Further, q1m 1 1 and q2m 1 0. This
implies that perfect audit accuracy is impossible to achieve and
that audit failures will occur. For the sake of notational
convenience, henceforth we suppress the argument em from q1m
and q2m . For the equilibrium analyzed in the demand section, we
need that the high and low quality audits are well defined and
require that q1A > q1B and q2A < q2B . As we show below, this is
exactly what happens. Also for the low quality audit to have value,
for the equilibrium level of effort eB , we assume that q1B > q2B .
(v) Consequences of Audit Failure
An audit failure occurs when auditors report H L findings for a
type 2 (1) client firm. If a type 1 firm is assigned L, an auditor of
type m risks loss of business, denoted by lm . If, on the other hand,
a type 2 firm is assigned H, the auditor risks being sued by the
investors in the firm (and found guilty of audit failure by the
court) who claim to have been misled by the audit findings. The
expected cost of this (type II) error is represented by the penalty
function Pm .24 If 1 2 proportion of the type 1 (type 2) client
firms engage the type A auditors, the probability of a loss 1A 1B
must be 1 1 ÿ q1A (equivalently, 1 ÿ 1 1 ÿ q1B ). Similarly,
the probability of a loss PA PB must be 2 q2A (equivalently,
1 ÿ 2 q2B . There is some evidence that courts are increasing the
auditors' common law tort liability and are awarding increasingly
higher damages for audit failures (Beneish and Chatov, 1993; and
Gormley, 1988). At the time of a lawsuit, the auditor is often one
of the few solvent parties worth pursuing (Kothari et al., 1988; and
Minow, 1984). Thus, increasing high damage awards increases the
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DE AND SEN
probability that auditors with deeper pockets will face a higher
expected penalty for an audit failure than do other audit firms. If
the larger and more established audit firms are also the ones with
deep pockets, and since the size of an audit firm and audit quality
are correlated (DeAngelo, 1981), it follows that auditors of higher
quality may face higher expected costs due to type II errors.
Therefore, we assume that for non-zero 2 in equilibrium,
independent of efforts, the higher quality auditor is exposed to
a higher level of expected penalty, i.e., 2 :PA 1 ÿ 2 :PB .25 At
first we assume that Pm is exogenous. An assumption of
exogenous Pm seems to be consistent with a standard of strict
liability. Though the existing laws do not specify a standard of
strict legal liability, the existing rule of due-diligence defense is
becoming, in many respects, equivalent to that of a strict liability
(Nelson, Ronen and White, 1988). Minow (1984), Chatov (1987),
Gormley (1988) and Kripke (1988) among others have
documented the evolution of an increasingly popular strict
liability interpretation of auditors' liability by courts, especially
within the Securities Act of 1934. Later, we relax this assumption
and consider alternative regimes where Pm is determined by some
audit outcome, and thus, is partially controllable by the auditor
through increased effort. Comparison with a fixed and exogenous
Pm then serves as a benchmark.
(vi) The Auditors' Maximization Problem
The supply of audit quality is determined by the auditors'
maximization problem. In our model, both types of auditors are
assumed to be risk neutral. The type A auditors maximize their
income in the following way:26
Max
kA ÿ CA eA ÿ 1 1 ÿ q1A lA ÿ 2 q2A :PA :
e
A
Similarly, the type B auditors maximize their income in the
following way:
Max
kB ÿ CB eB ÿ 1 ÿ 1 1 ÿ q1B lB ÿ 1 ÿ 2 q2B :PB :
e
B
Assume that the parameters of the problems are such that an
interior solution exists and can be characterized by the following
first order conditions:27
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0
0
CA0 eA 1 q1A
lA ÿ 2 q2A
:PA 01 q1A lA ÿ 02 q2A :PA ;
0
0
:PB ÿ 01 q1B lB 02 q2B :PB :
lB ÿ 1 ÿ 2 q2B
CB0 eB 1 ÿ 1 q1B
The above first order conditions can be interpreted as equating
the marginal operating costs with the marginal benefit from
reduction of penalty and business losses after considering the
impact on change in demand.28
Although 1 , 2 , 01 and 02 are decided in equilibrium, we know
that 1 2 1 is the maximum values of 1 and 2 and
therefore, in the equilibrium if 1 2 1, both 01 and 02 must
be equal to zero. For the sake of tractability, we assume that both:
@ 2 1
@eA @eB
and:
@ 2 2
@eA @eB
are also zero.
We now note two important requirements with respect to the
supply of the audit services that must be satisfied in any proposed
equilibrium. The first is that the optimal solution of the auditors
effort choice problem must satisfy eA > eB implying that,
motivated by self interest, the more efficient auditors supply a
high quality audit service. If so, a meaningful inference about
audit quality can be derived from the knowledge of auditor
efficiency by the market participants. The second requirement is
that the fees of a type A auditor will be greater than the fees of
the type B auditor. Both these conditions form part of the
equilibrium conditions developed in the following section.
3. THE ECONOMIC ENVIRONMENT AND DEMAND FOR AUDIT SERVICES
(i) Basic Information Structure
The outside investors have common prior knowledge that
proportion of all client firms are type 1 and 1 ÿ proportion
are type 2. The insiders know their own firm types, but any direct
disclosure of firm types is prone to adverse selection problems.
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To communicate the true financial position of their firm to the
rest of the market, the insiders may engage either of two kinds of
available audit services, high quality (A) (offered by the more
efficient auditors) and low quality (B) (offered by the less
efficient auditors), described in the previous section. This choice
transmits a message m "fA; Bg
. An audit is mandatory;
therefore no audit is not a possible message. Since these two
types of audit services are provided by two distinct classes of audit
firms, this message (audit method) can be observed by the rest of
the market.
(ii) The Outsiders' Role
Besides the informed managers and the auditors, the other
players in this game are the outside investors who decide on the
market values of the client firms. The market beliefs and market
values in our model are conditioned by both what the client firms
convey through their choice of the audit method and what the
audit firms report through their findings. This feature makes our
structure somewhat different from a standard signaling game.
The market value of a client firm from a high quality audit VA
following the release of the audit report is a function of the
managers' message of choosing high quality auditing (A) and the
audit findings (H or L). Thus, VA "fVHA ; VLA g. Similarly, the
market value of the firm from a low quality audit is
VB "fVHB ; VLB g. Given that the audit services A and B are
imperfectly efficient, neither VHA nor VLA if the firm chooses A,
and neither VHB nor VLB if the firm chooses B, may exactly reflect
the true or full-information values which are, as stated before,
V a1 and V a2 respectively. Both types of firm pay exactly the
same fee for a given audit method. Therefore, there is no typespecific cost that may help outsiders to understand the firm type
(unlike in a standard signaling game). The outside investors
operate in a competitive market, with the result that they decide
on the market values in such a manner as to break even across all
client firms. In other words, regardless of the actions the client
firms choose, their total market value always adds up to their true
value given by V a1 1 ÿ V a2 .
ß Blackwell Publishers Ltd 2002
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(iii) Optimization Program of Client Firms
We assume that all players in the game are risk neutral. The
objective function of the insider-manager of a client firm in this
scenario consists of a weighted average of the true value of the
firm and its market value.29 Let V t; m denote the expected
market value of a client firm of type t at the start of the game
(prior to the release of the audit report). Then, for m A; B:
V t; A qtA VHA 1 ÿ qtA VLA ÿ K ;
and:
V t; B qtB VHB 1 ÿ qtB VLB ;
where K kA ÿ kB represents the differential between the audit
fees required for the two audit services and qtm 's are obtained
from the audit firm's optimization program (see Section 2
above).
If is the weight attached to the market value and 1 ÿ to
the true value of the firm by the insider-manager, the objective
function, W V t; m, is:
W V t; m V t; m 1 ÿ V at ; m"fA; Bg; t"f1; 2g:
The insider-manager chooses m or the type of audit to maximize
the above objective function.
Figure 1 presents the game tree described above. Although so
far we have presented the game in terms of pure strategies, A or
B, our analysis of the game allows for mixed strategies as well.
Denote by t the probability with which client firms of type t send
a message m A. Intuitively, t can be viewed as the proportion
of type t firms that choose a high quality audit.
(iv) Nash Sequential Equilibrium (NSE)
We use the concept of sequential equilibrium developed by
(Kreps and Wilson, 1982) and adopt the definition in our setting
in the following way. A Nash-Sequential Equilibrium (NSE) , is an
ordered set t ; ; eA ; eB ; kA ; kB satisfying the following
conditions:
C1. Client firms choose t 2 argmax fW V t; t g; t 2 0; 1;
t 2 T.
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DE AND SEN
Figure 1
Game Tree
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C2. Audit firms choose their efforts e such that a type A audit
firm chooses:
eA 2 argmax fkA ÿ CA ea ÿ 1 1 ÿ q1A lA ÿ 2 q2A :PA g
and a type B audit firm chooses:
eB 2 argmax fkB ÿ CB eB ÿ 1 ÿ 1 1 ÿ q1B lB ÿ
1 ÿ 2 q2B :PB g:
C3. kA ; kB are such that both types of audit firms break even
in equilibrium.
C4. Quality and prices of the two audit supplies are well
defined; i.e., eA > eB and kA > kB .
C5. Outside investors, in a perfectly competitive market,
choose V t; t so that they break even across the
unobserved types of the client firms, implying:
X
V t; m; 1 ; 2 V a1 1 ÿ V a2 :
t
C6. If m; m A; B, is an equilibrium action, market posterior
beliefs following an H finding tjHm; 1 ; 2 , and
market posterior beliefs following an L finding
tjLm; 1 ; 2 , are determined by Baye's rule in the
following manner:
1jHm; 1 ; 2
q1m 1m
;
q1m 1m q2m 2m 1 ÿ
2jHm; 1 ; 2
q2m 2m 1 ÿ
:
q1m 1m q2m 2m 1 ÿ
1jLm; 1 ; 2
1 ÿ q1m 1m
;
1 ÿ q1m 1m 1 ÿ q2m 2m 1 ÿ
2jLm; 1 ; 2
1 ÿ q2m 2m 1 ÿ
1 ÿ q2m 2m 1 ÿ 1 ÿ q1m 1m
where: tm t if m A and 1 ÿ t if m B.
Further, qtm 's follow properties outlined in Section 2(iv)
and are obtained from the audit firm's effort choice
outlined in C2.
C7. If A or B is an off-equilibrium action, tj:; :; :"0; 1.
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4. DEMAND FOR AUDIT SERVICES: EQUILIBRIUM RESULTS
(i) Some Basic Issues
Before we present our equilibrium results, we intuitively discuss a
few technical issues and results that we need for developing our
equilibrium results.
For any well defined equilibrium, we require that eA > eB and
kA > kB , which is guaranteed by the supply side results developed
in Section 2. For all eA > eB the properties of the qtm : function
would assure that 1 > q1A > q1B and q2B > q2A > 0. Further, for a
type B audit to be informative, we require that q1B > q2B .
Combining the restrictions and allowing for weak inequality, we
have:
1 > q1A q1B > q2B q2A > 0; . . . . . . (A1)
where at least one of the two inequalities holds strictly. The above
accuracy structure implies two conditions. First, both auditors A,
and B, are generally efficient but not perfect in that both auditors
are more likely to report a `high' finding for a type 1 firm than
for a type 2 firm. Second, type A auditors are more accurate than
type B auditors. We also denote K kA ÿ kB , the fee differential
in an equilibrium, which plays a prominent role in our analysis.
In Section 3, VHm and VLm are defined as the market value of a
client firm conditional on the firm drawing H and L findings
respectively, given an audit method m; fm 2 A; Bg. When 1
proportion of type 1 firms and 2 proportion of type 2 client
firms choose m, define VHm 1 ; 2 and VLm 1 ; 2 as the
corresponding values. Risk-neutrality on the part of the capital
market participants implies that:
VHm 1 ; 2 V a1 1jHm; 1 ; 2 V a2 2jHm; 1 ; 2
and:
VLm 1 ; 2 V a1 1jLm; 1 ; 2 V a2 2jLm; 1 ; 2 :
Note that V a1 > VHm 1 ; 2 and VLm 1 ; 2 > V a2 .
Similarly, define V t; m; 1 ; 2 as the expected outcome for a
client firm of type t; t 1; 2, if it opts for an audit method m,
given that 1 proportion of type 1 firms and 2 proportion of type
2 firms choose m. Further, define W V t; m; 1 ; 2 as the
corresponding value of the firm's objective function. As the
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posterior probability functions embedded in the values indicate,
given t and m the values depend on three sets of parameters:
ft ; t 1; 2g, ft ; t 1; 2g and fqtm ; m A; Bg. While the first
set is drawn by the nature so to speak, the other two are not.
Changes in the parameters may affect the type-specific values
differently. This raises the following questions: whether, and
under what conditions, are the expected outcomes from the same
audit method different for the two different types of client firms?
Further, whether, and under what conditions, are the expected
outcomes of a client firm of a given type different under the two
different audit methods? These questions are important and are
addressed through a series of technical observations that are
necessary for the proofs of the propositions and detailed in the
Appendix.
(ii) Equilibrium Results
Since firms of each type can choose either A or B or randomize
between the two strategies, there are nine type-strategy
combinations that are candidates for an equilibrium outcome.
At first we show that unlike Melumad and Thoman (1990), no
separation of types is possible in our model.
Proposition 0. There is no separating NSE. Specifically, neither
of the following two situations is NSE.
(I) Type 1 chooses high quality auditing 1 1 and type 2
chooses low quality auditing 2 0.
(II) Type 2 chooses high quality auditing 2 1 and type 1
chooses low quality auditing 1 0.
Proof. See Appendix.
It is easy to see that the two separating equilibrium candidates
1 1; 2 0 and 1 0; 2 1 do not satisfy NSE
conditions. Audit fees are fixed, and there are no other
dissipative costs, type-related or otherwise, in this model. Thus,
non-mimicry, an essential prerequisite for separation, cannot be
guaranteed. Before we present our main equilibrium result, in
Table 2, our findings for each type-strategy combination are
summarized in the corresponding cell. We now discuss the
intuition underlying the main results (cells (1) and (3)). The
main findings of this study are that in equilibrium:
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Table 2
Strategies
Type 1 chooses
high quality
auditing : 1 1
Type 1 chooses
low quality
auditing : 1 0
Type 1 chooses a
mixed strategy :
0 < 1 < 1
Type 2 chooses
high quality
auditing : 2 1
NSE exists for
`low' K only :
robust to
refinements (1)
No NSE (4)
No NSE (7)
Type 2 chooses
low quality
auditing : 2 0
No NSE (2)
NSE possible;
not robust to
refinement (5)
No NSE (8)
Type 2 chooses
a mixed strategy
: 0 < 2 < 1
NSE exists for
`higher' K (3)
No NSE (6)
No NSE (9)
(1) type 1 firms strictly prefer high quality auditing 1 1 if
type 2 firms prefer high quality auditing or randomize
between high quality and low quality auditing
0 < 2 1; and
(2) type 2 firms choose only high quality auditing 1 if
the differential between the fees of the two audit
methods, given by K where K kA ÿ kB , is sufficiently
low; randomize between the two pure strategies
0 < 2 < 1 if K is higher; and avoid high quality auditing
altogether 2 0 if K is very high (as we shall see below,
K will have to be unrealistically high to make this
happen).
The statements in (1) and (2) above together imply that, for
the most part, in equilibrium 1 1 and 0 < 2 1. In these two
cases the quality and prices of audit supply are well defined. In
equilibrium, given q1A q1B q2B q2A with a strict inequality
holding in at least one case, type 1 firms benefit more from
higher efficiency associated with high quality auditing than from
low quality auditing. If type 2 firms expect the same outcome
from the two audit methods, A and B, inducing them to
randomize between the two strategies, type 1 firms will strictly
prefer A with its higher expected outcome. Of course, if type 2
firms prefer A to B, it follows that type 1 firms will prefer it even
more so. This explains the preference of a type 1 firm for high
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quality auditing. As for a type 2 firm, it expects less from A than
from B when 1 2 1, A being more informative than B.
However, if it switches from A to B, it is liable to fetch V a2 or its
true value; since type 1 firms do not choose B, any firm choosing
B is deemed a type 2 firm and is valued accordingly. If the
differential in audit fees are sufficiently low, its expected
outcome from A may still exceed V a2 . This results in the
equilibrium strategy combination 1 2 1.
If audit fees are higher, the expected outcome net of costs for a
type 2 firm from high quality auditing may be no higher than
V a2 , making it indifferent, and consequently randomize,
between the two strategies at the margin. This results in the
equilibrium strategy combination 1 1; 0 < 2 < 1. It is also
true that given the demand pattern, audit firms' supply decision
of audit quality and the break-even condition results in higher
fees for a type A audit. In this equilibrium the firms which choose
high quality auditing consist of all type 1 and some type 2 firms.
On the other hand, all firms which opt for low quality auditing
are type 2. This equilibrium is semi-separating in that sense.
How high should the differential audit fees be so that no type 2
firm chooses A, making 2 0? For this to happen, V a2 must
exceed the expected outcome from high quality auditing net of
fees which in this case is V a1 ÿ K (if no type 2 firm chooses A,
any firm which chooses A commands the valuation appropriate for
a type 1 firm). However, if V a2 V a1 ÿ K , it is not worthwhile
even for type 1 firms to employ this method. In other words, audit
fees must be so high as to make all firms reject a high quality audit,
an implication which contradicts what is observed empirically (see
Table 1). Given no demand for high quality auditing, in this case,
it is also not clear whether the audit fees and quality will be
differentiated in our model. The only two possible equilibrium
outcomes in our model are (1) a differentiated pooling (DP)
equilibrium where all firms opt for A or high quality auditing if the
fee differential between the two services is below a certain level; (2)
a differentiated semi-separating (DSS) equilibrium, if the
differential is above that level but not too high. In the two
equilibria, the A-pool is differentiated in the sense that a type 1
firm, at the start of the game, has a higher expected outcome from
high quality auditing than a type 2 firm. The above intuition is
formally presented in Proposition 1.
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Proposition 1. (1) For 0 K K , where q2A VHA 1 2 1
1 ÿ q2A VLA 1 2 1 ÿ V a2 K , the DP eq