Theoretical Economics Letters, 2013, 3, 22-30
http://dx.doi.org/10.4236/tel.2013.35A2005 Published Online September 2013 (http://www.scirp.org/journal/tel)
State-Dependence and Conditional Audit Policy
Ruey-Ji Guo, Yenpao Chen, Chun Chen Lee
Department of Accounting, Soochow University, Taipei, Taiwan
Received July 30, 2013; revised August 30, 2013; accepted September 9, 2013
Copyright © 2013 Ruey-Ji Guo et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
This paper investigates how the state-dependence between two periods affects the optimal audit decision. Using a prin-
cipal-agent model, based on a state-dependent assumption, we find that it is desirable to use a conditional (punitive)
audit mechanism when the agent’s under-declaration benefit is significant and the principal’s audit cost is moderate. In
that case, the audit policy for the current period will be contingent on the audit result in the preceding period.
Keywords: Information Asymmetry; Agency Problem; Audit Policy
1. Introduction
To alleviate the problem resulting from information
asymmetry, the principal tends to use audit mechanism to
abate the related agency costs, or to maximize the net
revenues concerned. The past studies have paid conside-
rable attention to the audit issues, including Antle [1],
Baron and Besanko [2], Demski and Sappington [3], and
Baiman, Evans and Noel [4]. Variables of particular in-
terest are the audit probability and the conditions upon
which it depends. Using the data from the American pulp
and paper industry, Helland [5] found that firms which
are discovered in violation of environmental regulations
experience a one- or two-quarter period of more frequent
inspections from regulators. Placing higher inspection
probability on agents with the record of incompliance
appears to be a scheme preferred by the regulators.
Audit rules conditional on compliance history are
broadly adopted by researchers as well. For instance,
Landsberger and Meilijson [6] proposed that an agent’s
current audit results affect its probability of future audits.
By targeting audits based on current audit outcomes,
They demonstrated that conditional auditing is more
cost-effective than random auditing and that agencies
could increase tax revenue given the same enforcement
budget and fine scheme. The use of such endogenous and
dynamic enforcement mechanisms that use information
obtained through prior audit to assign the agent’s prob-
ability of future audits are extended to the literature of
pollution regulation and tax compliance (e.g. Greenberg
[7], Harrington [8], Friesen [9], and Stafford [10]).
In this paper, we present a two-period model with a
state-dependent scenario, i.e. the nature state in period
two is correlated with that in period one, to address the
principal’s audit decision and study the potential effect of
state-dependence between different periods. Essentially,
audit mechanism is associated with penalty system. As
Landsberger and Meilijson [6] note that penalty sys-
tems usually include two elements of both penalty func-
tion and probability of detection, while from the practical
point of view, the latter more than the former is under the
control of the authorities. Accordingly, this paper focuses
on the issue of optimal audit probability (probability of
detection), and aims to identify the desirability of condi-
tional audit.
By controlling the state-dependent degree of return,
we demonstrate that a conditional audit policy can make
the audit implementation more efficient only when the
auditee’s benefit of under-declaring return is larger than
the expected penalty under complete audit and the audit
cost is relatively moderate. In that case, by using condi-
tional audit mechanism, the principal can concentrate
costly audit resources on the auditees with higher prob-
ability to under-declare their actual returns. Additionally,
we find that the state-dependent degree of return indeed
plays a prominent role in whether to use conditional audit
policy or not. This paper points out it may not be desir-
able for the principal to implement any conditional audit
especially in a situation that the returns in two periods
are totally independent.
2. The Model
In this paper, we use a principal-agent hierarchy, inclu-
ding a principal, an auditor and a manager, to address the
issue of conditional audit in a two-period scenario. It is
C
opyright © 2013 SciRes. TEL

R.-J. GUO ET AL. 23
assumed that, in a vertical structure owned by the princi-
pal, the manager operates a business unit and holds pri-
vate information concerning its realized returns. The
principal can obtain the true return information only by
assigning the auditor to undertake audit task. For sim=
plicity, it is assumed that all parties are risk neutral, and
the auditor is absolutely independent.
In the two-period scenario, nature is assumed to be the
only one factor affecting the realized return, i.e. high
return (RH) or low return (RL). In period one, the prob-
ability of high return is p (and 0 < p < 1). However, un-
der the assumption of state-dependence, the probability
of high return occurring in period two will be contingent
on the realized return in period one. If the realized return
in period one is high (low), the probability of high return
occurring in period two will be ph(pl). It is assumed that
. While the realized return in either
period one or period two is the manager’s private infor-
mation, the probabilities p, ph and pl can be estimated
objectively according to the past operating results, and
regarded as common information. In the paper, the prin-
cipal will take the information into account while plan-
ning the related audit policy. Specifically, we incorporate
the exogenous variables p, ph and pl into the principal’s
objective function to develop the optimal two-period
audit policy. At the end of each period, the manager is
required to declare a return level (either high return (RH)
or low return (RL)) to the principal, and transfers a por-
tion (α) of the return to the latter. The transferring agree-
ment brings about an incentive for the manager to un-
der-declare the return.
0
lh
ppp1
This paper allows the principal to consider a two- pe-
riod conditional audit mechanism to deter the manager’s
possible under-declaration of return. At the end of each
period, the principal needs to determine whether to em-
ploy the auditor at cost C to audit the low return declared
by the manager. If the auditor finds the under-declaration
of return, the manager will be required to pay a penalty
of
P
. Following the previous literature (e.g. Malik [11]),
the penalty of
P
is set as a maximal possible amount to
minimize the expected audit cost, and it can be regarded
as a legally specified limit on liability. Under a condi-
tional audit mechanism, the audit policy for the second
period is likely to be dependent on the audit result in the
first period. In period one, the probability to audit the
low return declared by the manager is assumed to be A.
However, if the under-declaration of return in period one
is found and revealed by the auditor, the audit probability
for the second period will be enhanced up to

'
Furthermore, the auditor’s audit quality is defined as a
probability, r that the manager’s under-declaration can be
found by the auditor. In other words, there remains a
probability, 1r

that the auditor will be unable to dis-
close the manager’s dishonest behavior even if the for-
mer has accomplished the related audit task. In this paper,
we exclude the possibility of blackmail or collusion be-
tween the auditor and the manager. Both C and r are as-
sumed to be the common information of all parties in-
volved. The following is the timing on the relevant
events.
1) The principal and the manager achieve an agree-
ment that the latter will transfer a certain portion (α) of
the return to the former.
2) Nature determines the realized return in period one,
which is a high return (RH) with the probability of p or a
low return (RL) with the probability of . 1p
3) The manager declares the return in period one ()
to the principal and will transfer to the latter.
1
ˆ
R
1
4) The principal assigns the auditor to undertake audit
task at cost C with an audit probability A if the manager
declares a low return in period one (i.e.
ˆ
R
 
1
ˆ
L
RR).
5) The auditor presents an audit report. If the un-
der-declaration of return is disclosed, the manager will
have to pay the principal a penalty of P, assumed to be
larger than


H
L
RR   for compensation and pun-
ishment. Also, the manager’s dishonest record will be
kept for reference in next period.
6) Nature determines the realized return in period two
once more. If the realized return in period one is RH(RL),
the probability of high return occurring in period two will
be ph(pl), where lh
ppp

.
7) The manager declares the return in period two, ,
and will transfer
2
ˆ
R
2
ˆ
R


to the principal.
8) The principal sends the auditor at cost C with a
probability A if the manager was not found under-de-
claring the return in period one and declares a low return
in period two (i.e. 2
ˆ
L
RR), but with a probability
A

if the manager was found under-declaring the return in
period one and declares a low return in period two, where
A
A


.
9) The auditor presents an audit report, and the man-
ager will be required to pay a penalty of P if the un-
der-declaration of return is disclosed.
10) Transfer takes place.
Based on the self-interested and rational assumption, if
the outcome in either period one or period two is a low
realized return, the manager will necessarily choose to
declare a low return to the principal. However, if the
outcome is a high realized return, the manager will be
likely to truthfully declare a high return or dishonestly
declare a low return, depending on the result of cost and
benefit analysis. In the first period, if the realized return
is a high one, the manager will need to evaluate the dif-

A
Aa provided the manager declares a low return
once more in period two; otherwise, the audit probability
will remain to be A, where 01A

, , and 01aA
A
A . That is so-called “conditional audit” used in this
paper.

R.-J. GUO ET AL. 24
ference of transferring amounts (

H
L
RR  ), the ex-
pected penalty (
A
rP ), and the unfavorable effect on the
audit probability in period two (possibly adjusted from A
to
A
) in order to make an optimal declaring decision.
In the second period, if the outcome is a high realized
return, whether the manager chooses under-declaration or
not will be contingent on the comparison between two
factors, i.e. the benefit from the difference of transferring
amounts (

H
L
RR  ) and the loss from the expected
penalty (
A
rP or '
A
rP depending on the previous audit
result).
Since the realized return is the manager’s private in-
formation, the principal’s audit policy will be dependent
on the return declared by the manager. On the basis of
cost and benefit, the principal will take audit action only
when the manager declares a low return. It is assumed
that the auditor is required to present some evidence to
support her audit report on the under-declared return, and
the evidence cannot be falsified. Hence, the audit result
will necessarily be a low return if the realized return is
low, but if the realized return is high, the audit result will
be subject to the effect of the audit quality (r).
3. The Analyses
3.1. The Agent’s Strategy
In this section, we first characterize the manager’s possi-
ble strategies under the variety of parameter combina-
tions. As the aforementioned, on the basis of the
self-interested and rational assumption, the manager will
consistently declare low return to the principal if the re-
alized return in either period one or period two is low.
Moreover, the factors influencing the manager’s declara-
tion behavior include the transferring ratio of return de-
clared (α), the penalty (P), the audit probabilities (A and
A
) and the audit quality (r). Using the relative relations
among those parameters, we infer three possible strate-
gies that the manager will take into account. They are
shown in Lemmas 1 to 3, respectively. To simplify the
denotation, we let
H
L be in the latter analy-
ses, and
RRR
R

 denotes the manager’s maximal possible
benefit while choosing to under-declare the return.
Lemma 1:
If RArP   , the manager will honestly declare the
return in each period. That is, when the realized return in
period one (or in period two) is high (i.e. 1
H
RR

or
2
H
RR), the manager will consistently declare high
return to the principal (i.e. 11
ˆ
H
RRR or
22
ˆ
H
RRR).
[Proof] See Appendix 1.
Lemma 2:
Under the condition of
A
rPRArP 
  , when the
realized return in period two is high (i.e. 2
H
RR), the
manager will under-declare the return if she has no un-
der-declaration record; but she will honestly declare high
return if she was found under-declaring return in period
one. Nevertheless, when the realized return in period one
is high (i.e. 1
H
RR

), the manager will choose to un-
der-declare the return in period one.
[Proof] See Appendix 2.
Lemma 3:
Under the condition of RArP


 , when the rea-
lized return in period one (or in period two) is high (i.e.
1
H
RR

or 2
H
RR

), the manager will choose to un-
der-declare the return to the principal (i.e. 11
ˆL
RR R


or 22L
ˆ
RRR

).
[Proof] See Appendix 3.
3.2. The Principal’s Strategy
On the other hand, while facing the manager’s possible
strategy responses, the principal first needs to consider if
it is economically desirable to use audit measure as an
incentive mechanism. The precondition concerned is that
the expected payoff from auditing should exceed the re-
lated cost (i.e. rpP C). Otherwise, the audit mecha-
nism will not be conducted. Hence, we regard rpP C
as an implied assumption in the latter analysis.
To highlight the difference between A and
A



A
a

 and search for the optimal value of a, we re-
write the preconditions in the Lemmas aforementioned
as:
01RArP RrPA 

 (1)
() 0ArPRAar PARr PAa  1

 
(2)
where 01aA


() 0RAarPAAa RrP    (3)
where 0A1

 and 01aA

 .
Intuitively, if RrP
  (or 1RrP  ), there can
exist the Situations (1), (2) or (3). Specifically, the Situa-
tion (1) can be called as a “deterrent audit policy” since
the audit policy will effectively deter the manager’s un-
der-declaring behavior. The Situation (2) can be referred
to as a “punitive conditional audit policy” in that the au-
dit probability in period two will be adjusted up to a de-
terrent (higher) audit probability from a normal (lower)
audit probability if the manager was found under-de-
claring the return in period one. Finally, the Situation (3)
can be regarded as a “laissez-faire audit policy,” in which
the manager will choose to under-declare the return due
to no sufficient audit influence. It is assumed that the
principal can obtain the expected payoffs 1, 2 and
3 under Situations (1), (2) and (3), respectively. In
Lemmas 4 and 5, it will be shown that the audit policy
from situation (1) will dominate that from either situation
(2) or Situation (3). Hence, provided
π π
π
RrP  , the au-

R.-J. GUO ET AL. 25
a maximal expected payoff, .
*
1
π dit policy from situation (1) will be the principal’s opti-
mal strategy in light of maximizing her expected payoff. As Proposition 1 shows, if the manager’s benefit of
under-declaring return is less than the expected penalty
under complete audit (i.e. RrP   ), the optimal audit
policy will be the deterrent audit policy (from Situation
(1)). In that case, the principal will implement random
audit action (i.e.
A
RrP   ) to induce the manager to
honestly declare her return, and to totally deter the un-
der-declaring behavior. Obviously, since the manager
necessarily chooses the honest declaration in period one,
it is unnecessary to use a conditional audit probability in
period two.
Lemma 4:
If RrP   , then the principal’s maximal expected
payoff in the Situation (1), , will be larger than that in
the Situation (2), .
*
1
π
*
2
π
[Proof] See Appendix 4.
Lemma 4 indicates that, as the incentive from un-
der-declaring is insignificant (i.e. RrP
  ), the deter-
rent audit policy (from Situation (1)) will be better than
the punitive conditional audit policy (from Situation (2))
such that the principal can obtain a larger expected pay-
off. The deterrent audit policy may be not desirable ex post
since audit is costly, but it reduces the expected audit
cost and maximize the principal’s expected payoff ex
ante. Although it is doubtful that the principal would be
able to adhere to her announced audit policy, the reputa-
tional forces could contribute toward inducing the prin-
cipal to make such a commitment, especially in a re-
peated game setting such as royalty or tax collection.
Hence, if the probability of random audit doesn’t change
dramatically over time, the manager could resort to in-
ferences according to the principal’s previous behavior.
Lemma 5:
If RrP   , then the principal’s maximal expected
payoff in the situation (1), , will be larger than that in
the Situation (3), .
*
1
π
*
3
π
[Proof] See Appendix 5.
Similarly, as the incentive from under-declaring is in-
significant, Lemma 5 points out that the deterrent audit
policy (from Situation (1)) will dominate the laissez-faire
audit policy (from Situation (3)) as well. Accordingly,
we can infer the following result in Proposition 1.
3.3. Deterrent Audit Policy As for when the manager’s benefit of under-declaring
return is not less than the expected penalty under
complete audit (i.e. RrP   ), the optimal audit policy
will need a different consideration. Firstly, we present a
few technical definitions in the following Lemma 6.
Proposition 1:
If the benefit of under-declaring return is relatively in-
significant (i.e. RrP   ), the principal’s optimal audit
policy will be conducting random audit with the audit
probability of
A
RrP   , provided audit is cost-ef-
fective (i.e. CrpP), and will have no effect on the
manager’s declaration of the return in period two.
a
Lemma 6:
Let

 
111 2 lh
CrpPprpP rrppPrp

 

,
[Proof] By the Lemmas 4 and 5, it’s straightforward
that the principal can maximize the expected payoff by
using the audit policy from the Situation (1) and achieve

 
211 2 lh
CrpPprpP rrppPrp



,

 


2
12112 hlhh
A
rpp PCrpCrpPprpPrrpp Prprp PC




 




and
  


1 1
2112
lhh
arpCrpPprpPrrppPrprpPCA


 


1
under RrP   and l, we have
h
ppp 12 CC

.
Also, if 1
CC , both and
21
0AC 111 0a


are assured.
Using the related denotation, we characterize the princi-
pal’s optimal audit policies in Proposition 2a.
Proposition 2a:
Following the denotation in Lemma 6, as the benefit of
under-declaring return is sufficiently large (i.e.
RrP   ) and there is a state-dependent return in pe-
riod two (i.e. lh ppp

), the principal’s optimal audit
policy will be conducting,
[Proof] See Appendix 6.
3.4. Conditional Audit Policy
In Lemma 6, 1 and 2 C are used to set up the trigge-
ring points for different audit policies in light of audit
cost. Additionally, 1
C
A
denotes a certain random audit
probability, and is used as a punitive audit factor.
1
a
1) uniformly complete audit (i.e. 1
A
A

 and
0a

) if 1 CC

;
2) some kind of conditional audit (i.e. 1
A
A, 1 aa

,

R.-J. GUO ET AL. 26
and ) if , or;
11 12
3) no audit (i.e. ) if .
1AAa

A
CCC
02 CC
[Proof] See Appendix 7.
Provided the benefit of under-declaring return is con-
siderably significant (i.e. RrP   ), based on Lemma 3,
the manager necessarily chooses to under-declare her
return when a high return is realized. As Proposition 2a
shows, to respond to the manager’s strategy, the principal
will implement an uniformly complete audit policy (i.e.
1
A
A 
CC
CC
CCC
) in each period if audit cost is relatively neg-
ligible (i.e. 1). But if audit cost is relatively larger
(i.e. 2), no audit action (i.e. ) will be a de-
sirable policy. In contrast, as the audit cost is moderate
(i.e. 12 ), the optimal policy will be a condi-
tional audit policy (i.e. 1 and
0A
1AA 1
A
A
 ). In
this case, the principal will first take a random audit ac-
tion in period one, and then decide whether to adjust up
the audit probability in period two, depending on the
audit result in previous period.
Since 1 corresponds to the punitive effect
of conditional audit policy, a larger 1 implies the prin-
cipal can conduct a random audit in period one with a
lower audit probability, and then implement a complete
audit in period two when the manager has a un-
der-reporting record and declares a low return once more.
In Proposition 2b, we examine the impacts of audit cost,
audit quality, and under-reporting penalty on the punitive
effect of conditional audit policy.

1
1a

A
a
Proposition 2b:
As conditional audit policy is the principal’s optimal
choice, ceteris paribus, an increase in audit cost results in
a larger punitive effect of conditional audit (i.e.
10aC). However, a rise in audit quality or penalty
results in a decrease in the punitive effect of conditional
audit (i.e. 10ar and 10aP

 ). The former
leads to a lower random audit probability in period one
(1
A
), but the latter leads to a higher one (since
11
1
A
a ).
[Proof] See Appendix 8.
As conditional audit policy is an optimal one, an in-
crease in audit cost will be more unfavorable to audit
action and result in a lower random audit probability in
period one (a more punitive effect). Conversely, as audit
quality or penalty grows, the expected benefit of random
audit in period one will increase, and it leads to a higher
random audit probability in period one (a less punitive
effect). In Proposition 2c, we further investigate how the
degree of return dependence in two consecutive periods
influences the punitive effect of conditional audit policy.
Proposition 2c:
Let hh and ll
. Following the
preconditions in Proposition 2b, if the realization of high
return in period one brings about a much more chance of
realizing a high return in period two (i.e. a larger
pppppp
h
p

),
the principal will reduce the punitive effect in conditional
audit policy (i.e. a smaller 1). Nevertheless, if the re-
alization of low return in period one results in a much
more chance of realizing a low return in period two or a
much less chance of realizing a high return in period two
(i.e. a larger l
a
p

), the principal will enlarge the punitive
effect in conditional audit policy (i.e. a larger 1). The
former leads to a higher random audit probability in pe-
riod one (1
a
A
), but the latter leads to a lower one (since
11
1
A
a

).
[Proof] See Appendix 9.
As Proposition 2c shows, h and l have oppo-
site impacts on the punitive effect of conditional audit
policy. Intuitively, since h
p
p
p

grows means that a high
return in period one will lead to a much more chance of
realizing a high return in period two, it become more
favorable to increase the frequency of audit in period one
and results in a higher random audit probability in the
period (a less punitive effect). In contrast, if a low return
in period one leads to a much less chance of realizing a
high return in period two, it will be less favorable to
enlarge the audit in period one and results in a lower
random audit probability in the period (a more punitive
effect).
3.5. No Conditional Audit Scenario
Finally, we are interested in how the optimal audit policy
will be if the return-dependence in two periods vanishes
totally. Is there any possibility for conditional audit pol-
icy to be desirable? As shown in the following proposi-
tion, the answer seems negative in light of the assump-
tions in the paper.
Proposition 3:
Using the denotation in Proposition 2a, if the benefit
of under-declaring return is relatively significant (i.e.
RrP   ), and the return in period two is independent
of that in period one (i.e. lh ), it will be not
desirable for the principal to conduct any conditional
audit, and the optimal audit strategy will be either
ppp
1) complete audit policy (i.e. 1
A
A  and 0a

)
provided CrpP , or
2) no audit policy (i.e. ) provided
0ACrpP.
[Proof] Following the proof of Proposition 2a, under
l
pp h p

, both 1 and 2 converge to C CrpP , i.e.
12 . Hence, the result (2) in Proposition 2a
will vanish.
CCrpP
In Proposition 3, we let both l and h converge
to so that the returns in periods one and two are in-
dependent each other. In the specific case, we obtain a
result that a conditional audit policy is uneconomical and
inadequate if the realized return in period one has no
effect on the realization of return in period two. As a
result, the principal will choose either a complete-audit
policy or a no-audit policy, depending on the size of au-
p p
p

R.-J. GUO ET AL. 27
dit cost (i.e. CrpP or CrpP).
4. Conclusion
In a state-dependent audit situation, we find that the
principal’s optimal audit policy will be a deterrent audit
policy if the manager’s benefit of under-declaring return
is less than the expected penalty under complete audit
and audit cost is insignificant. Nevertheless, if the benefit
of under-declaring return is considerably significant, and
the manager necessarily chooses to under-declare the
return, the principal’s optimal audit policy will be de-
pendent on audit cost. Ceteris paribus, if audit cost is
relatively negligible, the principal will take a uniformly
complete audit policy; but if audit cost is relatively larger,
the principal will tend to totally abandon audit action in
consideration of costs and benefits related. More inter-
estingly, as the audit cost is moderate, we find that a
conditional audit policy will be an optimal choice. In that
case, while the principal’s audit action cannot deter the
manager’s under-declaring behavior, the conditional au-
dit policy indeed can make audit action more efficient
and cost-effective.
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[9] L. Friesen, “Targeting Enforcement to Improve Com-
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R.-J. GUO ET AL. 28
Appendix 1 (Proof of Lemma 1)
Under

RArPArP 
  

, if 2
H
RR, then
2
ˆ
H
RR since the expected penalty will be too large for
the manager to under-declare the return. By the same
token, if 1
H
RR

, then 1
ˆ
H
RR since any audit result
in period one will not affect her declaration decision as
well as expected payoff in period two, and

  


 
 
 

11 11
: :
ˆˆ
1
11 1
0if
HH LH
HHhHH hLLHL
hHHhLL hHHhL
LH
RRRR RRRR
RRpRR pRRRRArP
Arp RRpRRArp RRpRR
RRArPR ArPRArP
 
 
 

 

 


L
Appendix 2 (Proof of Lemma 2)
If the realized return in period two is
H
R, under the
condition of RArP
  , the manager will choose to
under-declare the return if she has a clean record; how-
ever, she won’t do that if she was found under-declaring
return in period one since RArP


 .
On the other hand, if the realized return in period one
is
H
R, the manager will be inclined to under-declare the
return since









11 11
ˆˆ
10and 1
HH LHLHhHL
hh
RRRRRRRRRRArPArp RRArP
RArPArpRArPRArPArpRAr PArp
 
 
  
 h
.
Appendix 3 (Proof of Lemma 3)
Under the condition of

RArP ArP 

 

, if the real-
ized return in period two is high (i.e. 2
H
RR), the
manager will be inclined to under-declare the return
whether she was found under-declaring return in period
one or not. Meanwhile, if the realized return in period
one is
H
R, the manager will be also inclined to un-
der-declare the return since



 




 

11 11
ˆˆ
0
HH LH
HH HL
hH LH L
hH LhH L
LH h
h
h
RRRR RRRR
RRRRArP
pRR ArPArpRRAarP
pRRArPArpR RArP
RRArPArpArPAarP
RArPArp arP
RAar PArPArparP







 


 


 
  

 

.
Appendix 4 (Proof of Lemma 4)
In Situation (1), the principal’s expected payoff is

 
 
 
1
π1
1
1
HhHh L
LlH
lL
pRpRpR AC
pRACpR
pRAC
 





 

 


To maximize , subject to
1
π


0ARrP  ,
and a, we find the only optimal combi-
nation of
1Aa 0
*
A
RrP   and

*
01aR  

rP ,
satisfying all of the Kuhn-Tucker conditions. Actually,
the will never be used in that case since the manager
will honestly declare her return in period one under
*
a
*
A
RrP   (
A

). Thus we get the maximal ex-
pected payoff

 

*
121 1
2
Lh l
lhl
Rp pRppR
ppppppRC rP


π

 
 .
On the other hand, in Situation (2), the principal’s ex-
pected payoff is

R.-J. GUO ET AL.29






  




 



2
π1
11
11
22 1
11
LhHL
hL L
LlL lL
Lhh h
hl
pR ArPCpArRArR ArPC
pArRAaCAr RAC
pR ACpRArPCpRAC
Rpp ArRACppArACppAraC
ppArApp ApArP


 



 


 




 


To maximize 2, subject to π0
A
RrP   ,
1RrPAa   and , it is derived that only
the combination of
0a
*
ARrP

  (A

) and
*
a


(where 0

 and 0

) can satisfy all of the
Kuhn-Tucker conditions. Hence, the maximal expected
payoff becomes
    
  
2
*
2
π221
11
Lhh h
hl
RppA rRACppArCppA raC
ppArAppApArP

 
  

 


Hence, overlooking the terms involved with

we
have




**
12
ππ 11
hl
pp ACArpppAC



0
l
,
where 01ARrP   .
Appendix 5 (Proof of Lemma 5)
In Situation (3), the principal’s expected payoff is

3
π22
1
L
hlh
RACpAraCpArP
ppArPpp ArPppArarP
  
 .
To maximize 3, subject to π
A
aRrP0A   ,
and , we find that only the optimal combination of 0a
*rARP

  (where 0

 and 0

) and
can satisfy all of the Kuhn-Tucker conditions.
Hence, the maximal expected payoff become
*=0a




*
3
π22
Lhll
RACppppppAr  P
,
where
A
RrP   .
According to the proof of lemma 4,
 

*
1
π21 1
2
Lhl
lhl
Rp pArPppArP
pp ppppAC
  
 .
Hence, overlooking the terms involved with

,

**
13
ππ 10
hll
pppppAC



,
where 01ARrP   .
Appendix 6 (Proof of Lemma 6)
Since
  
  


 

12
211
211
12
12
1
0
lh
lh
hlh
hlh
hhh l
lh
CC
rprpPprp PrrppP
rprpPprp PrrppP
rpPrpp PprpPrp P
ppppp p
pp ppppp
ppp



 




 


 
 




,
12
CC

holds under RrP   .
On the other hand,




 

1
2
2
0
21
10
11
2,
hl
hh
lh
A
rppPCrpCrpPprpP
rrppPrpP C
rpPprpPrrppP
rp CCC


 
 
 

and




  
1
2
1
1
21
12
112
hl
hh
lh
A
rppPCrpCrpPprpP
rrppPrprpP C
rpPprp PrrppPrpC
CC


 


.
Hence, 1 01A

 if .
12
CCC
Meanwhile,

R.-J. GUO ET AL. 30
  
  
1
1
0
211
211
,
lh
lh
a
rpCrpPprp PrrppP
rpCrpPprp PrrppP
CC


 

0
.
and
  

  
1
2
1
211
2
211
l
h
lh
a
rpCrpPprpPrrppP
rprpPC
rpCrpPprp PrrppP
CC

 

 

h
Hence, if .
1
01a 12 CCC
Appendix 7 (Proof of proposition 2a)
According to the result of Lemma 6, as RrP   , the
principal’s expected payoff will be

2
22
1
Lh
hl
RArpPACAarpCArpp P
AarppPArpp P
  

To maximize , subject to , , and
, we find the following results. Firstly, as
2
π1Aa 0A
0a
1
CCC

, only the optimal combination of 1
*
A
A

,
1
*
aa

, and 11
*
1AAa

 can satisfy all of the
Kuhn-Tucker conditions. In that case, and
1
0A1
1
0a1


C
will also be assured by Lemma 6 (since
21
CC

). Secondly, as 1, the combination of CC
*1
A

and *0a

(i.e. *1
A
) will be the optimal
solution, satisfying all of the Kuhn-Tucker conditions;
but if 2, the optimal solution will be CC*0A

(and
has no effect). a
Appendix 8 (Proof of proposition 2b)

  
 

1
2
22
1
2221140if
hlhh
aC
rprp PCrprpCrpPprpPrrpp Prprp PCCC



 










 
 

1
2
22
1
2111
22 21140if
hlhh
hlhh
arrprpPCpCppPrppPrpp P
prp PCrpCrpPprpPrrpp PrprpPCCC




  







 
  



1
2
222
1
2111
221 140if
hhl
hlhh
aPrp rpPCrprpprp
rpprpCrpPprpPrrpp Prprp PCCC




  



.
Appendix 9 (Proof of proposition 2c)




 
 

22 2
1
2
22
1
21 22
11 40if
hh
lh h
aprpPrrp PCrpPrpCrpP
prpPrrppPrprpP CCC
  



 





112 lh
aprPprprpPC

 

0.