Legal Liabilities Audit Accuracy and the

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

ß_{Blackwell Publishers Ltd. 2002, 108 Cowley Road, Oxford OX4 1JF, UK}

* 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, SUNY-Buffalo, 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.

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).

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

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 refinements

8

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.

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

Description of Clients for Big Six and Non Big Six Audit Firms, 1982±1987

Number of Clients Total Sales Audited Number of Companies Audited

($billions) 1±25 mill.c 26±50 mill. 51±100 mill. 101±250 mill. 250 mill and over

Year B6a NB6b B6 NB6 B6 NB6 B6 NB6 B6 NB6 B6 NB6 B6 NB6

1987 6942 1336 3664 111 2943 911 833 119 765 135 855 88 1546 73

1986 6716 1457 3123 105 2553 862 815 156 776 121 839 94 1433 78

1985 6543 1417 3525 104 2691 986 770 135 791 112 794 104 1488 78

1984 6488 1452 3465 104 2694 912 772 137 764 116 779 107 1479 71

1983 6173 1572 3170 104 2566 1097 766 171 699 115 765 109 1377 71

1982 5727 979 3290 96 1879 815 684 102 638 88 788 39 1468 75

Yearly

Average 6430 1369 3373 104 2554 930 773 137 739 115 689 90 1465 74 Average

% 82 18 97 3 73 27 85 15 87 13 88 12 95 5

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 ofWho Audits America, The Data Financial Press.

DE

AND

SEN

ß

Blackwe

ll

Publi

shers

Ltd

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).

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

exam-ined 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.

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 commodities

15

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

"f

1

;

2

g ˆ

T

denotes the two types such that

type 1 is the better of two types. Under full information, the

market would value

a

₁

and

a

₂

as

V

…a

1

†

and

V

…a

2

†

19

such that

V

…a

1

†

>

V

…a

2

†

. Without full information, the market must use

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

increase with effort. The quality differential is characterized by

the difference function

…e† ˆ

C

B

…e† ÿ

C

A

…e†

such that

0

…e†

>

0

and

…

0

† ˆ

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

k

A

and

k

B

. 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

k

A

and

k

B

, 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

k

A

must be greater than

k

B

. We are primarily concerned with

the differential in the fees for the two audit services, denoted by

K

where

K

ˆ

k

A

ÿ

k

B

. To the extent high quality audits are

offered by the bigger firms as we mentioned earlier,

k

A

>

k

B

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

"f

HA, LA

g

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

"f

HB, LB

g

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

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

ˆ

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

q

tm

as being determined by the auditors' efforts

e. Specifically,

q

1m

…

e

m

†

and

q

2m

…

e

m

†

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.

Formally,

q

_1m0

>

0;

q

_1m00

<

0 but

q

_2m0

<

0;

q

_2m00

>

0, where

0

and

00

respectively denote the first and second derivatives of the

respective functions. Further,

q

1m

…1† ˆ

1 and

q

2m

…1† ˆ

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

q

2m. 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

e

B

, 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

l

m. 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

P

m.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

P

A

…

P

B

†

must be

2

q

2A

(equivalently,

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

:

_P

_A

₁

_ÿ

₂

_†

:

_P

_B

_.

25

_At

first we assume that

P

m

is exogenous. An assumption of

exogenous

P

m

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

P

m

is determined by some

audit outcome, and thus, is partially controllable by the auditor

through increased effort. Comparison with a fixed and exogenous

P

m

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

_e

k

A

ÿ

C

A

…

e

A

† ÿ

1

…1

ÿ

q

1A

†

l

A

ÿ

2

q

2A

:

P

A

:

A

Similarly, the type

B

auditors maximize their income in the

following way:

Max

_e

k

B

ÿ

C

B

…

e

B

† ÿ …1

ÿ

1

†…1

ÿ

q

1B

†

l

B

ÿ …1

ÿ

2

†

q

2B

:

P

B

:

B

C

_A0

…e

A

† ˆ

1

q

1A0

l

A

ÿ

2

q

2A0

:P

A

‡

01

q

1A

l

A

ÿ

02

q

2A

:P

A

;

C

_B0

…eB

† ˆ …1

ÿ

₁

†q

_1B0

lB

ÿ …1

ÿ

₂

†q

_2B0

:PB

ÿ

0₁

q

1BlB

‡

0₂

q

2B

:PB

:

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

ˆ

2

ˆ

1 is the maximum values of

1

and

2

and

therefore, in the equilibrium if

₁

ˆ

₂

ˆ

1, both

0₁

and

0₂

must

be equal to zero. For the sake of tractability, we assume that both:

@

2

₁

@e

_A

@e

_B

and:

@

2

₂

@e

_A

@e

_B

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

e

A

>

e

B

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

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 ˆ

. 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

…V

A

†

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,

V

A

"

fV

HA

;

V

LA

g. Similarly, the

market value of the firm from a low quality audit is

V

B

"

fV

HB

;

V

LB

g

. Given that the audit services

A

and

B

are

imperfectly efficient, neither

V

HA

nor

V

LA

if the firm chooses

A

,

and neither

V

HB

nor

V

LB

if the firm chooses

B

, may exactly reflect

(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

ˆ

A

;

B

:

V

…

t

;

A

†

q

tA

V

HA

‡ …1

ÿ

q

tA

†

V

LA

ÿ

K

;

and:

V

…

t

;

B

†

q

tB

V

HB

‡ …1

ÿ

q

tB

†

V

LB

;

where

K

ˆ

k

A

ÿ

k

B

represents the differential between the audit

fees required for the two audit services and

q

tm

'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

†Š, is:

W

‰

V

…

t

;

m

†Š

‰

V

…

t

;

m

†Š ‡ …1

ÿ

†

V

…

a

t

†;

m

"f

A

;

B

g;

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

ˆ

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

; ;

e

A

;

e

B

;

k

A

;

k

B

†

satisfying

the

following

conditions:

C1. Client firms choose

_t

2

argmax

f

W

‰

V

…

t

;

t

†Šg;

t

2 ‰0;

1Š;

Figure 1

C2. Audit firms choose their efforts

e

such that a type

A

audit

firm chooses:

e

_A

2

argmax

fk

A

ÿ

C

A

…e

a

† ÿ

1

…

1

ÿ

q

1A

†l

A

ÿ

2

q

2A

:

P

A

g

and a type

B

audit firm chooses:

e

_B

2

argmax

fkB

ÿ

CB

…eB

† ÿ …

1

ÿ

₁

†…

1

ÿ

q

1B

†lB

ÿ

…

1

ÿ

2

†q

2B

:

P

B

g

:

C3.

k

_A

;

k

_B

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.,

e

_A

>

e

_B

and

k

_A

>

k

_B

.

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

t

V

…t

;

m

;

₁

;

₂

† ˆ

V

…a

₁

† ‡ …

1

ÿ

†V

…a

₂

†

:

C6. If

m

;

m

ˆ

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:

…

1

jHm

;

₁

;

₂

† ˆ

q

1m

_1m

q

1m

1m

‡

q

2m

2m

…

1

ÿ

†

;

…

2

jHm

;

₁

;

₂

† ˆ

q

2m

_2m

…

1

ÿ

†

q

1m

1m

‡

q

2m

2m

…

1

ÿ

†

:

…

1

jLm

;

₁

;

₂

† ˆ

…

1

ÿ

q

1m

†

_1m

…

1

ÿ

q

1m

†

1m

‡ …

1

ÿ

q

2m

†

2m

…

1

ÿ

†

;

…

2

jLm

;

₁

;

₂

† ˆ

…

1

ÿ

q

2m

†

2m

…

1

ÿ

†

…

1

ÿ

q

2m

†

2m

…

1

ÿ

† ‡ …

1

ÿ

q

1m

†

1m

where:

tm

ˆ

t

if

m

ˆ

A

and

ˆ

1

ÿ

t

if

m

ˆ

B

.

Further,

q

tm

's follow properties outlined in Section 2(

iv

)

and are obtained from the audit firm's effort choice

outlined in C2.

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

e

_A

>

e

_B

and

k

A

>

k

B

, which is guaranteed by the supply side results developed

in Section 2. For all

e

_A

>

e

_B

the properties of the

q

tm

…

:

†

function

would assure that 1

>

q

1A

>

q

1B

and

q

2B

>

q

2A

>

0. Further, for a

type

B

audit to be informative, we require that

q

1B

>

q

2B

.

Combining the restrictions and allowing for weak inequality, we

have:

1

>

q

1A

q

1B

>

q

2B

q

2A

>

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

ˆ

k

A

ÿ

k

B

, the fee differential

in an equilibrium, which plays a prominent role in our analysis.

In Section 3,

V

Hm

and

V

Lm

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;

f

m

2

A;

B

g

. When

1

proportion of type 1 firms and

₂

proportion of type 2 client

firms choose

m, define

V

Hm

…

1

;

2

†

and

V

Lm

…

1

;

2

†

as the

corresponding values. Risk-neutrality on the part of the capital

market participants implies that:

V

Hm

…

1

;

2

† ˆ

V

…

a

1

†

…

1

j

Hm

;

₁

;

₂

† ‡

V

…

a

₂

†

…

2

j

Hm

;

₁

;

₂

†

and:

V

Lm

…

1

;

2

† ˆ

V

…

a

1

†

…

1

j

Lm

;

₁

;

₂

† ‡

V

…

a

₂

†

…

2

j

Lm

;

₁

;

₂

†

:

Note that

V

…

a

1

†

>

V

Hm

…

1

;

2

†

and

V

Lm

…

1

;

2

†

>

V

…

a

2

†

.

Similarly, define

V

…

t;

m

;

₁

;

₂

†

as the expected outcome for a

client firm of type

t;

t

ˆ

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

†Š

as the

posterior probability functions embedded in the values indicate,

given

t

and

m

the values depend on three sets of parameters:

f

t

;

t

ˆ

1

;

2

g

,

f

t

;

t

ˆ

1

;

2

g

and

fq

tm

;

m

ˆ

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

ˆ

1†

and type 2

chooses low quality auditing

…

2

ˆ

0†.

(II) Type 2 chooses high quality auditing

…

₂

ˆ

1

†

and type 1

chooses low quality auditing

…

1

ˆ

0†.

Proof.

See Appendix.

(1) type 1 firms strictly prefer high quality auditing

…

1

ˆ

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

…

_ˆ

₁

_†

_if

the differential between the fees of the two audit

methods, given by

K

where

K

ˆ

k

A

ÿ

k

B

, is sufficiently

low; randomize between the two pure strategies

…0

<

2

<

1†

if

K

is higher; and avoid high quality auditing

altogether

…

2

ˆ

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

ˆ

1 and 0

<

2

1. In these two

cases the quality and prices of audit supply are well defined. In

equilibrium, given

q

1A

q

1B

q

2B

q

2A

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

quality auditing. As for a type 2 firm, it expects less from

A

_than

from

B

_when

₁

_ˆ

₂

_ˆ

_1,

A

_{being more informative than}

B

_.

However, if it switches from

A

_to

B

_{, it is liable to fetch}

V



a

₂

_†

_{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



a

₂

_†

_{. This results in the}

equilibrium strategy combination

…

1

ˆ

2

ˆ

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



a

₂

_†

_{, making it indifferent, and consequently randomize,}

between the two strategies at the margin. This results in the

equilibrium strategy combination

…

₁

ˆ

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

ˆ

0? For this to happen,

V

…

a

2

†

must

exceed the expected outcome from high quality auditing net of

fees which in this case is

V



a

₁

_{† 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



a

₂

_†

V



a

₁

_{† 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}

Proposition 1.

(1) For 0

K

K

, where

q

2A

V

HA

…

1

ˆ

2

ˆ

1

†

‡…1

ÿ

q

2A

†

V

LA

…

1

ˆ

2

ˆ

1† ÿ

V

…

a

2

† ˆ

K

, the DP equilibrium

…

₁

ˆ

₂

ˆ

1

†

is an NSE and robust to Universal Divinity. (2)

For

K

>

K

>

K

, where

V

…

a

1

† ÿ

V

…

a

2

† ˆ

K

, the DSS

equi-librium

…

₁

ˆ

1

;

0

<

₂

<

1†

is viable and is unique.

Proof

: See Appendix.

In the second equilibrium, both actions,

A

and

B

, are observed.

Proposition 1 demonstrates that the first equilibrium is robust to

the strong Universal Divinity refinement due to Banks and Sobel

(1987) and, therefore, survives weaker refinements such as

Cho-Kreps Intuitive criterion. The proof of uniqueness is

straight-forward though tedious. Instead, we present below a graphical

representation of the equilibrium condition in Figure 2 that

brings out the intuition behind the uniqueness property.

The equilibrium pool valuation expected by a type 2 firm can

be expressed as a quadratic expression in

₂

and can be

graphically represented as the downward sloping curve. The

equilibrium would exist when a type 2 firm becomes indifferent

between this value and its true value

V

…

a

2

†

, which it will receive

upon correct identification by the market. That level of

2

is

represented by the point of intersection of the horizontal line

(representing the level of

V

…

a

2

†) and the downward sloping (in

pool) valuation line. As the graph indicates, this value is always

unique. The formal proof of uniqueness is available from the

authors upon request.

30

(iii) Other Equilibrium Candidates

criterion.

31

This should be contrasted with the equilibrium result

in Proposition 1 which, as we have noted above, is robust to the

much stronger Universal Divinity refinement.

Finally, we note that whether (i)

₂

ˆ

_{1, or (ii)}

₂

ˆ

_{0, or (iii)}

0

<

₂

<

_{1, it is never optimal for a type 1 firm to employ a mixed}

strategy 0

<

₁

<

_{1, ruling out (7), (8) and (9) in Table 2.}

Further, if type 2 randomizes, type 1 strictly prefers

A

. This rules

out (6) in the table. To see the intuition behind cases (i) and (ii)

above, note that when type 2 firms choose a pure strategy, say

A

, a

type 1 firm can have itself identified as indeed a type 1 firm by

choosing the other strategy (

B

in this case) and command a

higher market value. It will, therefore, never be indifferent

between the two strategies. The intuition behind case (iii) above

is straightforward. Given

q

1A

q

1B

q

2B

q

2A

with a strict

inequality holding in at least one case, type 1 firms stand to

benefit more from higher efficiency associated with high quality

Figure 2

auditing than do type 2 firms. If type 2 firms expect the same

outcome from the two audit methods,

A

and

B

,

32

inducing them

to randomize between the two strategies, type 1 firms will strictly

prefer

A

with its higher expected outcome Note that this rules

out both (6) and (9) in Table 2.

5. IMPACT OF ACCURACY AND FEES ON DEMAND FOR AUDITING

(i) Social Value of Auditing

In our setting, the social value of auditing exists because both

types of auditing have some discriminating power and a type 1

firm expects to gain from the valuation induced by the audit

report.

33

Allocational distortions are totally eliminated if

q

1

ˆ

1

and

q

2

ˆ

0, i.e., if the auditing is perfect. In this case,

…

1

jH

† ˆ

…

2

jL† ˆ

1. In this situation, similar to Melumad and

Thoman (1990) a separating equilibrium in which type 1 firms

choose a high quality audit while type 2 firms choose the

minimum cost alternative is feasible because, with perfect audit

efficiency, type 2 firms can no longer hope for mis-classification.

Thus, if auditing is mandatory, type 2 firms choose the low quality

audits. In this case type 1 firms will expect a market value of

V

…a

1

† ÿ

K

A

while type 2 firms will expect

V

…a

2

†

. As we show in the

following Proposition 2, the gain from auditing is an increasing

function of the accuracy

q

1

and a decreasing function of the type

2 error

q

2

. Since

q

1

…q

2

†

is increasing (decreasing) in auditors'

efforts

e

, it follows that the gains from auditing is increasing in

the auditors' efforts

e

.

34

Proposition 2.

The gain from both types of auditing,

G

m

, to a

type 1 firm in equilibrium is an increasing function of

q

1m

and

a decreasing function of

q

2m

:

@Gm

@q

_1m

>

0 and

@Gm

@q

_2m

<

0

;

where

G

m

‰q

1m

V

Hm

…

1

ˆ

1

;

2

†

‡…

1

ÿ

q

1m

†V

Lm

…

1

ˆ

1

;

2

† ÿ

K

m

Š ÿ ‰V

…a

1

† ‡ …

1

ÿ

†V

…a

2

†Š:

Proof

. See Appendix.

(ii) Impact of Audit Fees on Audit Demand

The equilibrium results established in Proposition 1 above

indicate that, as the differential in fees between a high quality

audit and a low quality audit, represented by

K

, increases,

demand for high quality auditing decreases. For

K

in the low

range

…

0

K

K

, where

K

ˆ

q

2A

V

HA

…

1

ˆ

2

ˆ

1

† ‡ …

1

ÿ

q

2A

†

V

LA

…

1

ˆ

2

ˆ

1

† ÿ

V

…

a

2

††

from Proposition 1 above, all type 1

and type 2 firms choose a high quality audit. For

K

in a higher

range

…

K

>

K

>

K

, where

K

ˆ

V

…

a

1

† ÿ

V

…

a

2

††

, though all type 1

firms continue to use high quality audit services, not all type 2

firms do so. However, if

K

is in the very high range

…

K

>

K

†

, all

firms dispense with high quality auditing.

The fact that an increase in high quality audit fees reduces

demand for high quality auditing relative to low quality auditing

is not surprising. To that extent, high quality auditing behaves

just like any other economic good. Note, however, that this

demand is defined over a domain in

K

which is bounded from

both above and below and is downward sloping over this range.

Though all type 1 firms demand high quality auditing over this

range, the proportion of type 2 firms that demand high quality

auditing over this range is low for high audit fees. To see this

intuitively, the equilibrium condition for type 2 in Proposition 1,

given by

q

2A

V

HA

…

:;

2

<

1

† ‡ …

1

ÿ

q

2A

†

V

LA

…

:;

2

<

1

† ÿ

V

…

a

2

† ˆ

K

,

is decreasing in

₂

, with the result that a higher

K

entails a lower

₂

(Figure 3).

(iii) Impact of Audit Accuracy on Audit Demand

underlying differential in audit fees,

K

, is such that demand for

high quality auditing may exist in the first place: in other words,

K

<

_K

ˆ

_V

…a

₁

† ÿ

_V

…a

₂

†

_.

Proposition 3.

When

K

<

_K

ˆ

_V

…a

₁

† ÿ

_V

…a

₂

†

_{, as the accuracy}

of high quality auditing increases, the demand for high quality

auditing decreases.

Proof

. See Appendix.

The above results have implications for the incentive of an audit

firm to provide high quality audit services. In our model, the level

of audit accuracy and the operating costs of providing accuracy

Figure 3

are both increasing in audit efforts, indicating that higher

accuracy requires higher investigation costs. This can also be

easily verified from the first-order condition for optimization by

the audit firm. On the other hand, we have just seen above that

the levels of fees as well as the level of accuracy are negatively

related to the demand for audit services. Surely, the negative

impact of audit fees and audit accuracy together on the demand

for their services could be a disincentive for the audit firms to

initiate costly efficiency-improving changes in their technology.

35

This is a problem, because higher audit accuracy is socially

desirable. It improves the expected outcome for type 1 firms while

reducing the outcome for type 2 firms and, therefore, reduces

allocational distortions. Given this problem for the audit firms,

the role of legal liabilities, that is supposed to provide incentives

for increased auditor efforts becomes especially important.

6. THE ROLE OF LEGAL LIABILITIES IN MOTIVATING AUDITORS

Our result from the demand side that demand declines in

accuracy suggests that perhaps individual firms, left to

themselves, may lack sufficient incentives to provide socially

optimal levels of effort and accuracy. In this context, we discuss

the potential effectiveness of the legal system in correcting audit

failures and improving auditor performance. We are here

concerned with instances of audit failure and liabilities as under

the Securities Acts of 1933 and 1934 and common law of torts

rather than audit fraud.

36

Recall that in our scenario the auditors

report their findings truthfully.

reality the bigger (and presumably better) auditors are subjected

to relatively higher penalty for the reason of `deep pocket' and

for the fact that they are often the only solvent party around

(Kothari et al., 1988). Thus it is interesting to examine the

impact of such a `deep pocket' award on auditors' effort choice

problem in a strategic auditing setting which we do next. We

then consider two alternative legal regimes; the negligent liability

regime with a specified `due care' standard where the liability is a

function of the level of care exercised during the auditing

process and a strict liability regime where the legal liability is

based on the loss suffered by the investors.

(i) Role of the Legal System: The Benchmark Case

What is the impact of raising the legal liability of an auditor? The

auditors

effort

choice

problem

in

the

semi-separation

equilibrium is characterized by the following two equations that

are obtained from the auditor's first order condition derived in

Section 2(

vi

):

C

_A0

…

e

A

† ˆ

q

1A0

l

A

ÿ

2

q

2A0

:

P

A

ÿ

02A

q

2A

:

P

A

;

C

_B0

…

eB

† ˆ ÿ…1

ÿ

2

†

q

2B0

:

PB

‡

02B

q2B

:

PB

:

From now on we introduce the notation

0_2A

and

0_2B

to

distinguish between the derivative of

2

with respect to

e

A

and

e

B

. Notice that demand for auditing from type 2 firm is denoted

2

for a type

A

auditor and

…1

ÿ

2

†

for a type

B

auditor. Our

result that increased accuracy, and hence effort, reduces the

demand for auditing implies that

0_2A

<

0 whereas

0_2B

>

0. The

effect of a change in legal liability on the auditor's marginal cost

can now be ascertained by differentiating the above two

equations, with respect to

PA

and

PB

, as follows:

@

C

_A0

…

e

A

†

@

P

A

ˆ ÿ

2

q

2A0

ÿ

02

q

2A

;

@

C

_B0

…

e

B

†

@

P

B

ˆ ÿ…1

ÿ

₂

†

q

_2B0

‡

0₂

q

2B

:

efforts for a type

A

auditor. For a type

B

auditor,

0_2B

is positive but

q

₂0_B

is negative and therefore, the effect of increased

PB

is to

increase the marginal cost of effort of type

B

auditors at the

optimum resulting in a higher level of effort.

(ii) Impact of `Deep Pocket' Awards

Though increase in legal liabilities can induce a higher level of

effort, in reality it may neither be institutionally feasible nor

desirable to increase the legal liability of all auditors. This will be

particularly so, if Type

B

auditors, are relatively smaller and have

less wealth. On the other hand, Type

A

auditors could also be the

bigger and more successful auditors with `deep pockets' who may

end up paying a disproportionate share of the total liability.

Thus, it may so happen that

PA

would increase keeping

PB

unchanged. First of all, increased liability will make the higher

quality (type

A) auditing more expensive which will drive its

demand down. That apart, the increased effort and thus,

increased accuracy by type

A

auditors will also drive the demand

for high quality auditing down. The question is, how will this

change in demand and change in liability for type

A

auditors

influence the action of a type

B

auditor? We focus on the

semi-separation equilibrium to check that, and take the derivative of

the type

B

auditor's marginal cost with respect to

PA

to find that:

@

C

_B0

…eB

†

@

PA

ˆ ÿ

@

₂

@

PA

q

0

2B

PB

:

We have argued that:

@

₂

@

PA

<

0

;

market may remain unchanged or may even go down because the

efforts and accuracy of the type

B

auditors will decrease. Thus,

the market may not become more informationally efficient

because of the increased legal liabilities to the type

A

auditors. It

can also be easily verified from the auditors' optimization

conditions that the larger the penalty, the larger is the

improvement in audit efficiency. It follows then that the larger

and more established audit firms with superior audit

tech-nologies will be driven harder by the legal system to improve

their performance than are the smaller audit firms. If anything, it

is the latter firms that need stronger corrective actions. The `deep

pocket' awards further widens the gap between the two groups in

efficiency and performance.

37

It should, however, be noted that

the connection posited between the auditors' wealth and types,

though may be consistent with empirically observed facts, is

purely conjectural in the above analysis and more research is

needed before a definitive conclusion is arrived at on the subject

of `deep pocket' awards.

(iii) Liabilities Under Different Negligent Liability Regimes

The essential idea behind the negligent liability rule is to be able

to distinguish between a business failure and an audit failure and

penalize only audit failure (Minow, 1984). To accomplish that

one needs to set up a `due care' standard and examine through

the litigation process whether such due care was exercised or not.

Thus, in terms of our model, it would require the courts to

determine

ex post

what levels of

q

1m

and

q

2m

the audit firm had

employed in a specific case. Our result in Proposition 2 above

also suggests that a determination of

q

1m

and

q

2m

and a penalty

scheme tied to audit failure alone could be socially desirable.

We argued in Proposition 2 that the gains to type 1 firms from

auditing (and hence, the social value of auditing) increases with

accuracy

q

1m

and decreases with the type II error

q

2m

and thus

with the auditors' efforts. Therefore, a social objective function

that increases the auditors' efforts thereby increasing the

accuracy

…q

_1m

†

and decreasing the type II error

…q

_2m

†

seems

desirable. Since auditors' efforts are not observable, an

acceptable alternative social welfare function would be the