Description
While audit action is a prominent measure for reducing agency cost, it remains inconclusive how the
previous audit experience has an effect upon current audit decision. Guo et al. (2005) conclude that it is
unnecessary for the principal to use conditional audit in two-period audit policy, suggesting that the
optimal audit policy in current period is tomaintain the same audit probability as that in previous period.
However, their analyses are based on punitive conditional audit. This paper modifies their model by
introducing an incentive conditional audit regime. We find that, provided both audit cost and underdeclaring
benefit are moderate, the incentive conditional audit policy will dominate the punitive conditional
audit policy and the conditional audit mechanism will be a desirable solution.
Incentive vs. punitive conditional audit policy
Ruey-Ji Guo
a, *
, Ming-Chin Chen
b
a
Department of Accounting, Soochow University, Taipei, Taiwan
b
Department of Accounting, National Chengchi University, Taipei, Taiwan
a r t i c l e i n f o
Article history:
Received 13 December 2012
Accepted 11 February 2015
Available online 10 June 2015
Keywords:
Asymmetric information
Agency problem
Conditional audit
a b s t r a c t
While audit action is a prominent measure for reducing agency cost, it remains inconclusive how the
previous audit experience has an effect upon current audit decision. Guo et al. (2005) conclude that it is
unnecessary for the principal to use conditional audit in two-period audit policy, suggesting that the
optimal audit policy in current period is to maintain the same audit probability as that in previous period.
However, their analyses are based on punitive conditional audit. This paper modi?es their model by
introducing an incentive conditional audit regime. We ?nd that, provided both audit cost and under-
declaring bene?t are moderate, the incentive conditional audit policy will dominate the punitive con-
ditional audit policy and the conditional audit mechanism will be a desirable solution.
© 2015 College of Management, National Cheng Kung University. Production and hosting by Elsevier
Taiwan LLC. All rights reserved.
1. Introduction
In practice, there are a variety of agency structures, and each one
has its speci?c principal and agent(s). Because of the existence of
private information held by the agent(s) and divergence of interest
between the principal and the agent(s), agency cost has long been a
major concern in setting a principal-agent relationship. Hence, it is
a common phenomenon for the principal to take some kind of audit
action upon the agent(s) to reduce the agency cost. Once the
principal hires an auditor to audit the agent(s), it forms a three-tier
agency (i.e., principal-auditor-agent) relationship.
The economics of a principal-supervisor (or auditor)-agent
relationship has been broadly studied by prior research for a long
time. Modeling an auditor as the one maximizing expected utility,
Antle (1982) studies agency problems resulting from the agency
structure of an owner-manager-auditor hierarchy. In the agency
hierarchy of consumer-regulator-?rm (regulated), Baron and
Besanko (1984) analyze how the regulator makes optimal audit
decisions on the regulated ?rm and sets the related pricing policy
to maximize total social welfare of the parties concerned, including
consumers as well as the regulated ?rm. Demski and Sappington
(1987) examine the regulation problems between a self-
interested regulator and a self-interested ?rm under a setting
where consumers (or Congress) can instruct the regulator's action
and the latter can supervise the monopoly ?rm's operation.
Meanwhile, based on a principal-agent model, Baiman, Evans, and
Noel (1987) provide an insight into how a principal uses the in-
formation from the agent's report and hires a utility-maximizing
auditor to mitigate the inef?ciency caused by information
asymmetry.
Furthermore, a number of studies also take into account the
agency relationship that allows for the possibility of collusion be-
tween the auditor and the audited agent (Baiman, Evans, &
Nagarajan, 1991; Kofman & Lawarree, 1993; Laffont & Martimort,
1999; Tirole, 1986). Tirole (1986) examines the effects of bribes in
a hierarchical contract involving a principal, a supervisor and an
agent. Baiman et al. (1991) address the issue of collusion between
managers and auditors. Kofman and Lawarree (1993) propose an
optimal audit contract when both an internal and an external
auditor are available, assuming the internal auditor may collude but
the external one not. In exploring the simultaneous use of two
collusive supervisors, Laffont and Martimort (1999) show that
competition between regulators will relax collusion-proof con-
straints and improve social welfare when regulators make collusive
offers that are accepted by the agent.
In addition to the literature involving single-period audit model,
there are a number of research examining the principal's auditing
behavior in multiple-period scenarios (e.g., Chen, 2006; Chen &Liu,
2007; Greenberg, 1984; Landsberger & Meilijson, 1982). In the
analysis of multiple-period audit policy, it is supposed to be
* Corresponding author.
E-mail address: [email protected] (R.-J. Guo).
Peer review under responsibility of College of Management, National Cheng
Kung University.
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Asia Paci?c Management Review 20 (2015) 234e240
reasonable thinking for the policy maker to use conditional audit
regime. A few laboratory experiments contribute to empirical evi-
dence regarding the performance of audit rules used for multiple
periods. For instance, Clark, Friesen, and Muller (2004) conduct an
experiment to compare Past-Compliance Targeting (Harrington
1988) and Optimal Targeting (Friesen, 2003) with random audit-
ing. They ?nd Optimal Targeting generates the lowest inspection
rates, but random auditing the highest compliance. Cason and
Gangadharan (2006) undertake a laboratory experiment to
analyze the conditional audit rule proposed by Harrington (1988),
in which participants move between two inspection groups that
differ in the probability of inspection and severity of ?ne. They ?nd
compliance behavior does not change as sharply as the model
predicts. Because of the complexity and abundance involved in
multiple-period audit models, it is not only dif?cult but impossible
for any single research to explore and compare all possibilities.
Even in a two-period audit, there still are lots of variants of audit
models.
Using a two-period audit model, Guo, Tsay, and Liu (2005)
analyze a punitive conditional audit regime and conclude that
conditional audit mechanism may be unnecessary in achieving the
principal's optimal audit policy. They argue, as nature states be-
tween two periods are mutually independent, the optimal audit
policy for the second period can be independent of the audit result
in the ?rst period. However, their argument is essentially derived
from punitive conditional audit, and it remains uncertain whether
their conclusions hold under other conditional audit regime. Hence,
in this paper, we extend the research on conditional audit by
incorporating an incentive mechanism in our analytical model. Our
analyses show that the incentive conditional audit policy will
dominate the punitive conditional audit policy and the conditional
audit mechanism will be a desirable solution, provided both audit
cost and under-declaring bene?t are moderate.
In next section, we characterize the model used in this paper.
Section 3 presents the related analyses and results. Section 4 con-
cludes our ?ndings and discusses the implications of this research
as well as its application in designing management mechanisms.
2. The model
In this paper, we examine a three-tier hierarchy comprising a
principal, an auditor and a manager. The principal owns the vertical
structure; the manager runs an operating unit with private infor-
mation about its realized return; the auditor collects information
for the principal. Following Tirole (1986), it is assumed that the
principal lacks either the time or the knowledge necessary to su-
pervise the manager, and that the auditor also lacks either the time
or the resources required to run the vertical structure. It is further
assumed that all players are risk neutral. Also, the auditor is
considered to be independent and will not collude with the man-
ager. In two-period audit scenario, nature is assumed to be the only
one factor affecting the realized return, i.e., high return (R
H
) or low
return (R
L
). In the paper, high (low) return is used to denote high
(low) output, good (poor) operating income, or large (small) sales
revenue, depending on transferring agreement (or regulation).
Based on previous experience, the probability that the manager
realizes a high return is p, and he realizes a low return with a
probability of 1 À p. The probability of high (low) return is the same
in either period one or period two, and the realized return in period
two is independent of that in period one. While the probability of
high return is common information, the ?nal realized return is the
manager's private information. That is, the principal cannot learn
the manager's realized return unless the former takes an action
such as audit.
According to some kind of contract or regulation, the manager
has to transfer a certain portion (a) of the return declared by him to
the principal. In other words, the manager can reserve only the
1 À a portion of the return. The mechanism brings about an
incentive for the manager to under-declare the return. To deter the
under-declaring behavior, the principal can employ an auditor at
cost C to audit the return declared by the manager provided the
latter declares a low return. If the auditor ?nds the under-
declaration of return, the manager has to pay the principal a pen-
alty of P. Similar to Kofman and Lawarree (1993), we assume P is an
exogenously given number, which may be interpreted as, for
instance, a legally speci?ed limit on liability. Also, P is assumed to
be larger than a(R
H
À R
L
) for compensation and punishment. In the
paper, the audit capability (or audit quality) of the auditor is
denoted by the probability, r, that the under-declaration of return
can be found by the auditor, and there remains a probability of 1 Àr
that the auditor will fail to ?nd the under-declaration of return.
Both C and r are common information of all parties involved.
Meanwhile, we exclude the possibility of collusion between the
auditor and the manager. If the model allows the possibility of the
collusion, the principal will likely need to employ either a second
supervisor or a “bounty-hunter” mechanism to ensure the effec-
tiveness of the audit policy. In that case, the principal will have to
increase monitoring cost to ensure the effectiveness of the audit
policy. We expect that, under the model considering the collusion
between the auditor and the manager, it will bring about a negative
effect on the principal's net expected revenues. Due to the
complication involved with collusion, we exclude the possibility of
collusion in the model and focus on the comparison of the incentive
conditional audit policy with the punitive one.
In a two-period audit decision, the audit policy for the second
period can be dependent on the audit result in the ?rst period; i.e.
there exists the possibility of “conditional audit.” It is assumed that
the audit probability for the ?rst period is A if the manager declares
low return, but the audit probability for the second period will
depend on the audit result in the ?rst period. Conceptually, the
conditional audit policy may be in a punitive or an incentive way.
With respect to the punitive conditional audit, the principal can
adopt different punitive ways. For example, upon discovering the
manager's under-reporting in the prior period, the principal may
increase audit probability in the current period or adjust up the
under-reporting penalty. In contrast, the principal also can adopt
incentive conditional audit in the way either reducing audit prob-
ability in the current period or decreasing possible penalty upon
the manager without under-reporting record in prior period.
Guo et al. (2005) analyze the effect of the punitive conditional
audit policy in the way of increasing audit probability in the current
period upon the manager with under-reporting record in prior
period. In their setting, if the under-declaration of return in period
one is found and revealed by the auditor, the audit probability for
the second period can be enhanced up to A
þ
(?A þ a where a 0)
provided the manager declares low return once again in period
two. As a result, Guo et al. (2005) conclude that, given the manager
declares a low return, the principal will adopt random audit (i.e.
A ¼ A
þ
¼ aðR
H
À R
L
Þ=rP and a ¼ 0) in period one or in period two
(regardless of previous audit record) provided the manager's
bene?t of under-declaration is less than the expected penalty under
complete audit; otherwise, the principal will consistently take
complete audit (i.e., A ¼ A
þ
¼ 1 and a ¼ 0) in both period one and
period two. In either of the two cases, there is no application of
conditional audit.
Extending the related issue, this paper considers an incentive
conditional audit policy to examine the possibility of conditional
audit. In period two, we allow the audit probability on the decla-
ration of lowreturn can be reduced to A
À
(?A À a where a 0) if the
R.-J. Guo, M.-C. Chen / Asia Paci?c Management Review 20 (2015) 234e240 235
manager was not found under-declaring the return in period one.
Otherwise, the audit probability will remain to be A.
The timing on the relevant events is presented as follows:
(1) Nature determines the realized return in period one, R
1
. R
1
is
either high return (R
H
) or low return (R
L
).
(2) The manager declares the return in period one,
b
R
1
, and will
transfer a$
b
R
1
to the principal.
(3) The principal sends the auditor at cost C with probability A if
the manager declares low return in period one (i.e.,
b
R
1
¼ R
L
.)
(4) The auditor presents an audit report. If the under-declaration
of return is disclosed, the manager will be required to pay the
principal a penalty of P. Also, the principal will maintain a
record on the manager's dishonesty.
(5) Nature determines the realized return in period two, R
2
. R
2
is
either high return (R
H
) or lowreturn (R
L
), and is independent
of R
1
.
(6) The manager declares the return in period two,
b
R
2
, and will
transfer a$
b
R
2
to the principal.
(7) The principal sends the auditor at cost C with probability A if
the manager was found under-declaring the return in period
one and declares low return in period two (i.e.,
b
R
2
¼ R
L
.), but
with probability A
À
if the manager was not found under-
declaring the return in period one and declares low return
in period two.
(8) The auditor presents an audit report, and the manager will
have to pay the principal a penalty of P if the under-
declaration of return is disclosed.
(9) Transfer takes place.
After nature determines the realized return in each period, the
manager will choose to declare high return or low return to the
principal. If the outcome is high realized return (with probability p),
the manager can either truthfully declare high return or dishon-
estly declare low return. However, if the outcome is low realized
return (with probability 1 À p), the manager will declare only low
return to the principal based on the assumption of self-interested
and rational behavior.
If the realized return in period one is high, whether manager
chooses to under-declare return will depend on the difference be-
tween transferring amounts (a(R
H
À R
L
)), the expected penalty
(ArPÞ, and the possible effect on the audit probability for period two
(A or A
À
). However, in the second period, whether to under-declare
return will depend on both the difference between transferring
amounts (a(R
H
À R
L
)) and the expected penalty (i.e., ArP or A
À
rP,
depending on the previous audit result).
Since the principal cannot observe the realized return, his audit
policy will only depend on the return declared by the manager, and
he will consider taking audit action only when the manager de-
clares a low return. With respect to audit quality, it is assumed that
the audit result will be low return if the realized return is low, but
the audit result will be subject to the effect of the audit quality (r) if
the realized return is high. It's assumed that the auditor has to
present a speci?c evidence to support his audit report concerning
under-declared return, and the evidence cannot be falsi?ed.
3. The analyses
To analyze strategy equilibriums between the principal and the
manager, we ?rst characterize the manager's optimal declaring
strategies under possible parameter combinations. In the model of
this paper, the factors in?uencing the manager's declaring behavior
include transferring ratio of return declared (a), penalty (P), audit
probabilities (A and A
À
) and audit quality (r). On the basis of the
self-interested and rational behavior, the manager will consistently
declare low return to the principal if the realized return in either
period one or period two is low. Besides, according to the relative
relations among the parameters, we can infer the following three
possible strategies, corresponding to lemmas 1 to 3, to be taken by
the manager. To simplify the denotation, we let DR ? R
H
À R
L
in the
following analyses.
Lemma 1. If aDR A
À
rPð ArPÞ, the manager will honestly
declare the returnwhether in period one or in period two. That is, if
the realized return in either period one or period two is high (i.e.,
R
1
¼ R
H
or R
2
¼ R
H
), the manager will consistently and honestly
declare high return to the principal (i.e.,
b
R
1
¼ R
H
¼ R
1
or
b
R
2
¼ R
H
¼ R
2
).
[Proof] If the realized return in period two is R
H
, under the
condition of aDR A
À
rPð ArPÞ, the expected penalty will be too
signi?cant for the manager to under-declare the return. Hence, the
manager will truthfully declare high return. By the same token, if
the realized return in period one is R
H
, the manager will also choose
to declare high return since his decision in period one will have no
effect on his declaration decision and expected payoff in period
two.
Lemma 2. Under the condition of A
À
rP 0Þ, A
À
þ a 1 and
a 0, we ?nd that only the optimal combination of
A ¼ A
À
¼ aDR=rP and a ¼ 0 can satisfy all of the Kuhn-Tucker
conditions. Thus, we obtain the maximal expected payoff
Fig. 2. Optimal policy combination under incentive audit policy.
R.-J. Guo, M.-C. Chen / Asia Paci?c Management Review 20 (2015) 234e240 238
p
*
1
¼ 2aR
L
þ2paDR À2aDRð1 À pÞC
rP:
On the other hand, in situation (ii), the principal's expected
payoff is
p
2
¼ p
n
aR
H
þ p
h
aR
L
þ A
À
À
rP À C
Á
i
þ ð1 À pÞ
aR
L
À A
À
C
o
þð1 À pÞ
È
aR
L
À AC þ p
h
aR
L
þ A
À
À
rP À C
Á
i
þð1 À pÞ
À
aR
L
À A
À
C
ÁÉ
¼ 2aR
L
þ paDR þ A
À
À
rpP À C
Á
À ð1 À pÞ
A
À
þ a
C
(6)
To maximize p
2
, subject to 0 A
À
0 and 3/0) can satisfy all of
the Kuhn-Tucker conditions. Hence, the maximal expected payoff
becomes
p
*
2
¼ 2aR
L
þ paDR þ
À
aDR
rP À 3
ÁÀ
rpP À C
Á
À ð1 À pÞaDRC
rP;
Meanwhile,
p
*
1
Àp
*
2
¼ 2aR
L
þ2paDR À2aDRð1 À pÞC
rP À
Â
2aR
L
þ paDR
þ
À
aDR
rP À 3
ÁÀ
rpP À C
Á
À ð1 À pÞaDRC
rP
Ã
¼ paDR ÀaDRð1 À pÞC
rP À paDR þaDRC
rP
þ 3
À
rpP À C
Á
¼ paDRC
rP þ 3
À
rpP À C
Á
>0
(7)
Appendix 4. (Proof of lemma 5)
In situation (iii)
a
, the principal's expected payoff is
p
3a
¼ p
n
aR
L
þ A
À
rP À C
Á
þ p
h
aR
L
þ A
2
r
À
rP À C
Á
þð1 À ArÞðA À aÞ
À
rP À C
ÁÃ
þ ð1 À pÞ
Â
aR
L
À A
2
rC
Àð1 À ArÞðA À aÞC
ÃÉ
þ ð1 À pÞ
È
aR
L
À AC
þp
Â
aR
L
þ ðA À aÞ
À
rP À C
ÁÃ
þ ð1 À pÞ½aR
L
À ðA À aÞC?
É
¼ 2aR
L
þ
h
2A
À
þ a þ
A
À
þ a
arp
i
À
rpP À C
Á
:
(8)
To maximize p
3a
, subject to A
À
þ a 0:
(9)
Appendix 5. (Proof of lemma 6)
In situation (iii)
b
, the principal's expected payoff is
p
3b
¼ p
n
aR
H
þ p
h
aR
L
þ A
À
À
rP À C
Á
i
þ ð1 À pÞ
aR
L
À A
À
C
o
þð1 À pÞ
È
aR
L
À AC þ p
h
aR
L
þ A
À
À
rP À C
Á
i
þð1 À pÞ
À
aR
L
À A
À
C
ÁÉ
¼ 2aR
L
þ paDR þ A
À
À
rpP À C
Á
À ð1 À pÞ
A
À
þ a
C:
(10)
To maximize p
3b
subject to A
À
þ a ð2 À pÞC, only the optimal combination of A ¼ aDR=rP À 3 ,
A
À
¼ aDR=rP À 3
2
, and a ¼ 3
1
satis?es all of the Kuhn-Tucker con-
ditions, where 3
2
¼ 3 þ 3
1
, 3
1
¼ 3 P=ðpaDR À rpP 3 Þ, 3 > 0 and 3 /0.
Thus, we obtain the maximal expected payoff:
p
*
3b
¼ 2aR
L
þ paDR þ
À
aDR
rP À 3
2
ÁÀ
rpP À C
Á
À ð1 À pÞ
À
aDR
rP À 3
Á
C:
Hence,
p
*
1
À p
*
3b
¼ 2aR
L
þ 2paDR À 2aDRð1 À pÞC
rP
À
Â
2aR
L
þ paDR þ
À
aDR
rP À 3
2
ÁÀ
rpP À C
Á
Àð1 À pÞ
À
aDR
rP À 3
Á
C
Ã
¼ aDRpC
rP À 3 ð1 À pÞC þ 3
2
À
rpP À C
Á
¼ aDRpC
rP þ 3
1
À
rpP À C
Á
þ 3
ÂÀ
rpP À C
Á
Àð1 À pÞC
Ã
> 0
À
qrpP > ð2 À pÞC
Á
:
(11)
Appendix 6. (Proof of proposition 2)
Since aDR>rP leads to
aDR
rP
>1, there is only the possibility of the
situation (iii)
a
or (iii)
b
. In situation (iii)
a
, the principal's expected
payoff is
p
3a
¼ p
n
aR
L
þ A
À
rP À C
Á
þ p
h
aR
L
þ A
2
r
À
rP À C
Á
þð1 À ArÞðA À aÞ
À
rP À C
ÁÃ
þ ð1 À pÞ
Â
aR
L
À A
2
rC
Àð1 À ArÞðA À aÞC
ÃÉ
þ ð1 À pÞ
È
aR
L
À AC
þp
Â
aR
L
þ ðA À aÞ
À
rP À C
ÁÃ
þ ð1 À pÞ½aR
L
À ðA À aÞC?
É
¼ 2aR
L
þ
h
2A
À
þ a þ
A
À
þ a
arp
i
À
rpP À C
Á
:
(12)
To maximize p
3a
, subject to A
À
þ a 1, ðA
À
þ aÞrP
ð1 þ arpÞ ð1 þ rpÞrP, the constraint of ðA
0
þ aÞrPð1 þ arpÞ
While audit action is a prominent measure for reducing agency cost, it remains inconclusive how the
previous audit experience has an effect upon current audit decision. Guo et al. (2005) conclude that it is
unnecessary for the principal to use conditional audit in two-period audit policy, suggesting that the
optimal audit policy in current period is tomaintain the same audit probability as that in previous period.
However, their analyses are based on punitive conditional audit. This paper modifies their model by
introducing an incentive conditional audit regime. We find that, provided both audit cost and underdeclaring
benefit are moderate, the incentive conditional audit policy will dominate the punitive conditional
audit policy and the conditional audit mechanism will be a desirable solution.
Incentive vs. punitive conditional audit policy
Ruey-Ji Guo
a, *
, Ming-Chin Chen
b
a
Department of Accounting, Soochow University, Taipei, Taiwan
b
Department of Accounting, National Chengchi University, Taipei, Taiwan
a r t i c l e i n f o
Article history:
Received 13 December 2012
Accepted 11 February 2015
Available online 10 June 2015
Keywords:
Asymmetric information
Agency problem
Conditional audit
a b s t r a c t
While audit action is a prominent measure for reducing agency cost, it remains inconclusive how the
previous audit experience has an effect upon current audit decision. Guo et al. (2005) conclude that it is
unnecessary for the principal to use conditional audit in two-period audit policy, suggesting that the
optimal audit policy in current period is to maintain the same audit probability as that in previous period.
However, their analyses are based on punitive conditional audit. This paper modi?es their model by
introducing an incentive conditional audit regime. We ?nd that, provided both audit cost and under-
declaring bene?t are moderate, the incentive conditional audit policy will dominate the punitive con-
ditional audit policy and the conditional audit mechanism will be a desirable solution.
© 2015 College of Management, National Cheng Kung University. Production and hosting by Elsevier
Taiwan LLC. All rights reserved.
1. Introduction
In practice, there are a variety of agency structures, and each one
has its speci?c principal and agent(s). Because of the existence of
private information held by the agent(s) and divergence of interest
between the principal and the agent(s), agency cost has long been a
major concern in setting a principal-agent relationship. Hence, it is
a common phenomenon for the principal to take some kind of audit
action upon the agent(s) to reduce the agency cost. Once the
principal hires an auditor to audit the agent(s), it forms a three-tier
agency (i.e., principal-auditor-agent) relationship.
The economics of a principal-supervisor (or auditor)-agent
relationship has been broadly studied by prior research for a long
time. Modeling an auditor as the one maximizing expected utility,
Antle (1982) studies agency problems resulting from the agency
structure of an owner-manager-auditor hierarchy. In the agency
hierarchy of consumer-regulator-?rm (regulated), Baron and
Besanko (1984) analyze how the regulator makes optimal audit
decisions on the regulated ?rm and sets the related pricing policy
to maximize total social welfare of the parties concerned, including
consumers as well as the regulated ?rm. Demski and Sappington
(1987) examine the regulation problems between a self-
interested regulator and a self-interested ?rm under a setting
where consumers (or Congress) can instruct the regulator's action
and the latter can supervise the monopoly ?rm's operation.
Meanwhile, based on a principal-agent model, Baiman, Evans, and
Noel (1987) provide an insight into how a principal uses the in-
formation from the agent's report and hires a utility-maximizing
auditor to mitigate the inef?ciency caused by information
asymmetry.
Furthermore, a number of studies also take into account the
agency relationship that allows for the possibility of collusion be-
tween the auditor and the audited agent (Baiman, Evans, &
Nagarajan, 1991; Kofman & Lawarree, 1993; Laffont & Martimort,
1999; Tirole, 1986). Tirole (1986) examines the effects of bribes in
a hierarchical contract involving a principal, a supervisor and an
agent. Baiman et al. (1991) address the issue of collusion between
managers and auditors. Kofman and Lawarree (1993) propose an
optimal audit contract when both an internal and an external
auditor are available, assuming the internal auditor may collude but
the external one not. In exploring the simultaneous use of two
collusive supervisors, Laffont and Martimort (1999) show that
competition between regulators will relax collusion-proof con-
straints and improve social welfare when regulators make collusive
offers that are accepted by the agent.
In addition to the literature involving single-period audit model,
there are a number of research examining the principal's auditing
behavior in multiple-period scenarios (e.g., Chen, 2006; Chen &Liu,
2007; Greenberg, 1984; Landsberger & Meilijson, 1982). In the
analysis of multiple-period audit policy, it is supposed to be
* Corresponding author.
E-mail address: [email protected] (R.-J. Guo).
Peer review under responsibility of College of Management, National Cheng
Kung University.
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Asia Paci?c Management Review 20 (2015) 234e240
reasonable thinking for the policy maker to use conditional audit
regime. A few laboratory experiments contribute to empirical evi-
dence regarding the performance of audit rules used for multiple
periods. For instance, Clark, Friesen, and Muller (2004) conduct an
experiment to compare Past-Compliance Targeting (Harrington
1988) and Optimal Targeting (Friesen, 2003) with random audit-
ing. They ?nd Optimal Targeting generates the lowest inspection
rates, but random auditing the highest compliance. Cason and
Gangadharan (2006) undertake a laboratory experiment to
analyze the conditional audit rule proposed by Harrington (1988),
in which participants move between two inspection groups that
differ in the probability of inspection and severity of ?ne. They ?nd
compliance behavior does not change as sharply as the model
predicts. Because of the complexity and abundance involved in
multiple-period audit models, it is not only dif?cult but impossible
for any single research to explore and compare all possibilities.
Even in a two-period audit, there still are lots of variants of audit
models.
Using a two-period audit model, Guo, Tsay, and Liu (2005)
analyze a punitive conditional audit regime and conclude that
conditional audit mechanism may be unnecessary in achieving the
principal's optimal audit policy. They argue, as nature states be-
tween two periods are mutually independent, the optimal audit
policy for the second period can be independent of the audit result
in the ?rst period. However, their argument is essentially derived
from punitive conditional audit, and it remains uncertain whether
their conclusions hold under other conditional audit regime. Hence,
in this paper, we extend the research on conditional audit by
incorporating an incentive mechanism in our analytical model. Our
analyses show that the incentive conditional audit policy will
dominate the punitive conditional audit policy and the conditional
audit mechanism will be a desirable solution, provided both audit
cost and under-declaring bene?t are moderate.
In next section, we characterize the model used in this paper.
Section 3 presents the related analyses and results. Section 4 con-
cludes our ?ndings and discusses the implications of this research
as well as its application in designing management mechanisms.
2. The model
In this paper, we examine a three-tier hierarchy comprising a
principal, an auditor and a manager. The principal owns the vertical
structure; the manager runs an operating unit with private infor-
mation about its realized return; the auditor collects information
for the principal. Following Tirole (1986), it is assumed that the
principal lacks either the time or the knowledge necessary to su-
pervise the manager, and that the auditor also lacks either the time
or the resources required to run the vertical structure. It is further
assumed that all players are risk neutral. Also, the auditor is
considered to be independent and will not collude with the man-
ager. In two-period audit scenario, nature is assumed to be the only
one factor affecting the realized return, i.e., high return (R
H
) or low
return (R
L
). In the paper, high (low) return is used to denote high
(low) output, good (poor) operating income, or large (small) sales
revenue, depending on transferring agreement (or regulation).
Based on previous experience, the probability that the manager
realizes a high return is p, and he realizes a low return with a
probability of 1 À p. The probability of high (low) return is the same
in either period one or period two, and the realized return in period
two is independent of that in period one. While the probability of
high return is common information, the ?nal realized return is the
manager's private information. That is, the principal cannot learn
the manager's realized return unless the former takes an action
such as audit.
According to some kind of contract or regulation, the manager
has to transfer a certain portion (a) of the return declared by him to
the principal. In other words, the manager can reserve only the
1 À a portion of the return. The mechanism brings about an
incentive for the manager to under-declare the return. To deter the
under-declaring behavior, the principal can employ an auditor at
cost C to audit the return declared by the manager provided the
latter declares a low return. If the auditor ?nds the under-
declaration of return, the manager has to pay the principal a pen-
alty of P. Similar to Kofman and Lawarree (1993), we assume P is an
exogenously given number, which may be interpreted as, for
instance, a legally speci?ed limit on liability. Also, P is assumed to
be larger than a(R
H
À R
L
) for compensation and punishment. In the
paper, the audit capability (or audit quality) of the auditor is
denoted by the probability, r, that the under-declaration of return
can be found by the auditor, and there remains a probability of 1 Àr
that the auditor will fail to ?nd the under-declaration of return.
Both C and r are common information of all parties involved.
Meanwhile, we exclude the possibility of collusion between the
auditor and the manager. If the model allows the possibility of the
collusion, the principal will likely need to employ either a second
supervisor or a “bounty-hunter” mechanism to ensure the effec-
tiveness of the audit policy. In that case, the principal will have to
increase monitoring cost to ensure the effectiveness of the audit
policy. We expect that, under the model considering the collusion
between the auditor and the manager, it will bring about a negative
effect on the principal's net expected revenues. Due to the
complication involved with collusion, we exclude the possibility of
collusion in the model and focus on the comparison of the incentive
conditional audit policy with the punitive one.
In a two-period audit decision, the audit policy for the second
period can be dependent on the audit result in the ?rst period; i.e.
there exists the possibility of “conditional audit.” It is assumed that
the audit probability for the ?rst period is A if the manager declares
low return, but the audit probability for the second period will
depend on the audit result in the ?rst period. Conceptually, the
conditional audit policy may be in a punitive or an incentive way.
With respect to the punitive conditional audit, the principal can
adopt different punitive ways. For example, upon discovering the
manager's under-reporting in the prior period, the principal may
increase audit probability in the current period or adjust up the
under-reporting penalty. In contrast, the principal also can adopt
incentive conditional audit in the way either reducing audit prob-
ability in the current period or decreasing possible penalty upon
the manager without under-reporting record in prior period.
Guo et al. (2005) analyze the effect of the punitive conditional
audit policy in the way of increasing audit probability in the current
period upon the manager with under-reporting record in prior
period. In their setting, if the under-declaration of return in period
one is found and revealed by the auditor, the audit probability for
the second period can be enhanced up to A
þ
(?A þ a where a 0)
provided the manager declares low return once again in period
two. As a result, Guo et al. (2005) conclude that, given the manager
declares a low return, the principal will adopt random audit (i.e.
A ¼ A
þ
¼ aðR
H
À R
L
Þ=rP and a ¼ 0) in period one or in period two
(regardless of previous audit record) provided the manager's
bene?t of under-declaration is less than the expected penalty under
complete audit; otherwise, the principal will consistently take
complete audit (i.e., A ¼ A
þ
¼ 1 and a ¼ 0) in both period one and
period two. In either of the two cases, there is no application of
conditional audit.
Extending the related issue, this paper considers an incentive
conditional audit policy to examine the possibility of conditional
audit. In period two, we allow the audit probability on the decla-
ration of lowreturn can be reduced to A
À
(?A À a where a 0) if the
R.-J. Guo, M.-C. Chen / Asia Paci?c Management Review 20 (2015) 234e240 235
manager was not found under-declaring the return in period one.
Otherwise, the audit probability will remain to be A.
The timing on the relevant events is presented as follows:
(1) Nature determines the realized return in period one, R
1
. R
1
is
either high return (R
H
) or low return (R
L
).
(2) The manager declares the return in period one,
b
R
1
, and will
transfer a$
b
R
1
to the principal.
(3) The principal sends the auditor at cost C with probability A if
the manager declares low return in period one (i.e.,
b
R
1
¼ R
L
.)
(4) The auditor presents an audit report. If the under-declaration
of return is disclosed, the manager will be required to pay the
principal a penalty of P. Also, the principal will maintain a
record on the manager's dishonesty.
(5) Nature determines the realized return in period two, R
2
. R
2
is
either high return (R
H
) or lowreturn (R
L
), and is independent
of R
1
.
(6) The manager declares the return in period two,
b
R
2
, and will
transfer a$
b
R
2
to the principal.
(7) The principal sends the auditor at cost C with probability A if
the manager was found under-declaring the return in period
one and declares low return in period two (i.e.,
b
R
2
¼ R
L
.), but
with probability A
À
if the manager was not found under-
declaring the return in period one and declares low return
in period two.
(8) The auditor presents an audit report, and the manager will
have to pay the principal a penalty of P if the under-
declaration of return is disclosed.
(9) Transfer takes place.
After nature determines the realized return in each period, the
manager will choose to declare high return or low return to the
principal. If the outcome is high realized return (with probability p),
the manager can either truthfully declare high return or dishon-
estly declare low return. However, if the outcome is low realized
return (with probability 1 À p), the manager will declare only low
return to the principal based on the assumption of self-interested
and rational behavior.
If the realized return in period one is high, whether manager
chooses to under-declare return will depend on the difference be-
tween transferring amounts (a(R
H
À R
L
)), the expected penalty
(ArPÞ, and the possible effect on the audit probability for period two
(A or A
À
). However, in the second period, whether to under-declare
return will depend on both the difference between transferring
amounts (a(R
H
À R
L
)) and the expected penalty (i.e., ArP or A
À
rP,
depending on the previous audit result).
Since the principal cannot observe the realized return, his audit
policy will only depend on the return declared by the manager, and
he will consider taking audit action only when the manager de-
clares a low return. With respect to audit quality, it is assumed that
the audit result will be low return if the realized return is low, but
the audit result will be subject to the effect of the audit quality (r) if
the realized return is high. It's assumed that the auditor has to
present a speci?c evidence to support his audit report concerning
under-declared return, and the evidence cannot be falsi?ed.
3. The analyses
To analyze strategy equilibriums between the principal and the
manager, we ?rst characterize the manager's optimal declaring
strategies under possible parameter combinations. In the model of
this paper, the factors in?uencing the manager's declaring behavior
include transferring ratio of return declared (a), penalty (P), audit
probabilities (A and A
À
) and audit quality (r). On the basis of the
self-interested and rational behavior, the manager will consistently
declare low return to the principal if the realized return in either
period one or period two is low. Besides, according to the relative
relations among the parameters, we can infer the following three
possible strategies, corresponding to lemmas 1 to 3, to be taken by
the manager. To simplify the denotation, we let DR ? R
H
À R
L
in the
following analyses.
Lemma 1. If aDR A
À
rPð ArPÞ, the manager will honestly
declare the returnwhether in period one or in period two. That is, if
the realized return in either period one or period two is high (i.e.,
R
1
¼ R
H
or R
2
¼ R
H
), the manager will consistently and honestly
declare high return to the principal (i.e.,
b
R
1
¼ R
H
¼ R
1
or
b
R
2
¼ R
H
¼ R
2
).
[Proof] If the realized return in period two is R
H
, under the
condition of aDR A
À
rPð ArPÞ, the expected penalty will be too
signi?cant for the manager to under-declare the return. Hence, the
manager will truthfully declare high return. By the same token, if
the realized return in period one is R
H
, the manager will also choose
to declare high return since his decision in period one will have no
effect on his declaration decision and expected payoff in period
two.
Lemma 2. Under the condition of A
À
rP 0Þ, A
À
þ a 1 and
a 0, we ?nd that only the optimal combination of
A ¼ A
À
¼ aDR=rP and a ¼ 0 can satisfy all of the Kuhn-Tucker
conditions. Thus, we obtain the maximal expected payoff
Fig. 2. Optimal policy combination under incentive audit policy.
R.-J. Guo, M.-C. Chen / Asia Paci?c Management Review 20 (2015) 234e240 238
p
*
1
¼ 2aR
L
þ2paDR À2aDRð1 À pÞC
rP:
On the other hand, in situation (ii), the principal's expected
payoff is
p
2
¼ p
n
aR
H
þ p
h
aR
L
þ A
À
À
rP À C
Á
i
þ ð1 À pÞ
aR
L
À A
À
C
o
þð1 À pÞ
È
aR
L
À AC þ p
h
aR
L
þ A
À
À
rP À C
Á
i
þð1 À pÞ
À
aR
L
À A
À
C
ÁÉ
¼ 2aR
L
þ paDR þ A
À
À
rpP À C
Á
À ð1 À pÞ
A
À
þ a
C
(6)
To maximize p
2
, subject to 0 A
À
0 and 3/0) can satisfy all of
the Kuhn-Tucker conditions. Hence, the maximal expected payoff
becomes
p
*
2
¼ 2aR
L
þ paDR þ
À
aDR
rP À 3
ÁÀ
rpP À C
Á
À ð1 À pÞaDRC
rP;
Meanwhile,
p
*
1
Àp
*
2
¼ 2aR
L
þ2paDR À2aDRð1 À pÞC
rP À
Â
2aR
L
þ paDR
þ
À
aDR
rP À 3
ÁÀ
rpP À C
Á
À ð1 À pÞaDRC
rP
Ã
¼ paDR ÀaDRð1 À pÞC
rP À paDR þaDRC
rP
þ 3
À
rpP À C
Á
¼ paDRC
rP þ 3
À
rpP À C
Á
>0
(7)
Appendix 4. (Proof of lemma 5)
In situation (iii)
a
, the principal's expected payoff is
p
3a
¼ p
n
aR
L
þ A
À
rP À C
Á
þ p
h
aR
L
þ A
2
r
À
rP À C
Á
þð1 À ArÞðA À aÞ
À
rP À C
ÁÃ
þ ð1 À pÞ
Â
aR
L
À A
2
rC
Àð1 À ArÞðA À aÞC
ÃÉ
þ ð1 À pÞ
È
aR
L
À AC
þp
Â
aR
L
þ ðA À aÞ
À
rP À C
ÁÃ
þ ð1 À pÞ½aR
L
À ðA À aÞC?
É
¼ 2aR
L
þ
h
2A
À
þ a þ
A
À
þ a
arp
i
À
rpP À C
Á
:
(8)
To maximize p
3a
, subject to A
À
þ a 0:
(9)
Appendix 5. (Proof of lemma 6)
In situation (iii)
b
, the principal's expected payoff is
p
3b
¼ p
n
aR
H
þ p
h
aR
L
þ A
À
À
rP À C
Á
i
þ ð1 À pÞ
aR
L
À A
À
C
o
þð1 À pÞ
È
aR
L
À AC þ p
h
aR
L
þ A
À
À
rP À C
Á
i
þð1 À pÞ
À
aR
L
À A
À
C
ÁÉ
¼ 2aR
L
þ paDR þ A
À
À
rpP À C
Á
À ð1 À pÞ
A
À
þ a
C:
(10)
To maximize p
3b
subject to A
À
þ a ð2 À pÞC, only the optimal combination of A ¼ aDR=rP À 3 ,
A
À
¼ aDR=rP À 3
2
, and a ¼ 3
1
satis?es all of the Kuhn-Tucker con-
ditions, where 3
2
¼ 3 þ 3
1
, 3
1
¼ 3 P=ðpaDR À rpP 3 Þ, 3 > 0 and 3 /0.
Thus, we obtain the maximal expected payoff:
p
*
3b
¼ 2aR
L
þ paDR þ
À
aDR
rP À 3
2
ÁÀ
rpP À C
Á
À ð1 À pÞ
À
aDR
rP À 3
Á
C:
Hence,
p
*
1
À p
*
3b
¼ 2aR
L
þ 2paDR À 2aDRð1 À pÞC
rP
À
Â
2aR
L
þ paDR þ
À
aDR
rP À 3
2
ÁÀ
rpP À C
Á
Àð1 À pÞ
À
aDR
rP À 3
Á
C
Ã
¼ aDRpC
rP À 3 ð1 À pÞC þ 3
2
À
rpP À C
Á
¼ aDRpC
rP þ 3
1
À
rpP À C
Á
þ 3
ÂÀ
rpP À C
Á
Àð1 À pÞC
Ã
> 0
À
qrpP > ð2 À pÞC
Á
:
(11)
Appendix 6. (Proof of proposition 2)
Since aDR>rP leads to
aDR
rP
>1, there is only the possibility of the
situation (iii)
a
or (iii)
b
. In situation (iii)
a
, the principal's expected
payoff is
p
3a
¼ p
n
aR
L
þ A
À
rP À C
Á
þ p
h
aR
L
þ A
2
r
À
rP À C
Á
þð1 À ArÞðA À aÞ
À
rP À C
ÁÃ
þ ð1 À pÞ
Â
aR
L
À A
2
rC
Àð1 À ArÞðA À aÞC
ÃÉ
þ ð1 À pÞ
È
aR
L
À AC
þp
Â
aR
L
þ ðA À aÞ
À
rP À C
ÁÃ
þ ð1 À pÞ½aR
L
À ðA À aÞC?
É
¼ 2aR
L
þ
h
2A
À
þ a þ
A
À
þ a
arp
i
À
rpP À C
Á
:
(12)
To maximize p
3a
, subject to A
À
þ a 1, ðA
À
þ aÞrP
ð1 þ arpÞ ð1 þ rpÞrP, the constraint of ðA
0
þ aÞrPð1 þ arpÞ