Discussion Paper No. 106
Managerial Retention Cost, Manager Specific Effort
and the Use of Leading Indiactors

Yasuhiro Mazda
April 2012TOHOKU MANAGEMENT & ACCOUNTING RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND
MANAGEMENT TOHOKU UNIVERSITY
KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANManagerial Retention Cost, Manager Speciﬁc Eﬀort
and the Use of Leading Indiactors

Yasuhiro Mazda∗
Graduate School of Economics and Management
Tohoku University
Sendai, Japan
ymazda@econ.tohoku.ac.jp∗

Tel: +81-22-217-6280; Fax: +81-22-217-6280.

Managerial Retention Cost, Manager Speciﬁc Eﬀort and the Use of Leading
Indiactors

Abstract

This study examines the role of leading indicators in a managerial contracting setting. For the
purpose of incentive provision when the outcome of the manager’s activities are in the long run, theuse of leading indicators such as customer satisfaction is often observed. In the two-period contract
setting in this paper, I focus on the setting where those activities are manager-speciﬁc and the ﬁrm
can gain the outcome of the manager’s activities only when the manager stays at the ﬁrm in thesecond period. Under this setting, I show the equilibrium cost to induce the manager to stay atthe ﬁrm and describe the role of leading indicators. Furthermore, I show the relationship between
manager-speciﬁc eﬀort exerted in the ﬁrst period and the eﬀort incentive in the second period. I
show, in equilibrium, complementarity between the two kinds of eﬀorts above does not necessarily
help improve the incentive of the manager-speciﬁc eﬀort in the ﬁrst period.

Key Words:

Leading Indicators; Retention cost; Manager-speciﬁc eﬀort;
Moral hazard; Short-term contracts; Eﬀort substitution

1 Introduction
When the owner of a ﬁrm wants to motivate a manager from the long-term perspective, use ofleading indicators is often observed in practice as in Balanced Scorecard (e.g., Kaplan and Norton, 2006). The purpose of this paper is to show the roles of leading indicators and describe theequilibrium level of eﬀorts under the sequence of short-term contracts.
Christensen, Feltham and Sabac [2003] analyzed the sequence of short-term contracts andshowed that there is no pure strategy equilibrium the manager stays at the ﬁrm in the second period unless the intertemporal correlation between performance measures used in the contract for
performance evaluation is zero.
In this paper, I consider the setting where the manager can access the labor market after the ﬁrst
period. Therefore I analyze the contract regime is the sequence of short-term contracts, where theowner oﬀers one period contract at the beginning of each period. Under the sequence of short-term
contract setting, it is natural to think that it is costly to induce the manger to work for the same
company in the second period as well. In order to analyze the relevant cost and ineﬃciency causedby the manager’s access to the labor market, I introduce the notion of retention cost, cost to induce
the manager to stay at the ﬁrm in the second period.
In order to analyze the ineﬃciency caused by the manager’s access to labor market, I introduce
the retention cost, the cost to induce the manager to stay in the second period, and thereby analyze
the equilibrium where the ﬁrm gains the outcome of the manager’s ﬁrst period activity only whenthe manager stays in the ﬁrm in the second period.
In classic moral hazard problems, the focus is on the tradeoﬀ between incentive provision of
the agent and imposing risk on him (e.g., Holmstrom, 1979, 1982). Indjejikian and Nanda [1999]
analyzed the sequence of short-term contracts where performance measures in each period haveinter-temporal correlation. Under the sequence of short-term contracts scheme, Indjejikian andNanda [1999] showed that there is ratchet eﬀect, a negative eﬀect of use of nonﬁnancial performance measures on ﬁrst period eﬀort. Sliwka [2002] also analyzed the weight on the performancemeasures, especially on the leading indicators under the similar setting and showed that the weighton leading indicators can be negative. As Christensen Feltham and Sabac [2003] pointed out, In-1

djejikian and Nanda [1999] and Sliwka [2002] implicitly assumed the manager’s commitment tostay, i.e., the manager is committed to stay at the ﬁrm in the second period even under a short-term
contract setting. Christensen, Feltham and Sabac [2003] argues that there are three kinds of relevant
commitment researchers should consider, commitment to the content to the contract, fairness ofcontract1 and stay commitment. In this paper, I consider the setting where there is no stay commitment but the owner have the opportunity to give the manager an oﬀer that induces the manager tostay at the ﬁrm. Kaplan and Norton [2006] emphasizes the use of nonﬁnancial performance measures such as leading indicators in Balanced Scorecoard. Christensen, Feltham and Sabac [2003]
showed that, under the sequence of short-term contracts, the mamager stays at the ﬁrm after the ﬁrst
period unless intertemporal correlation equals zero.
Christensen, Feltham and Sabac [2003] analyzed a moral hazard problem in a two-period setting
and showed that there is no pure strategy equilibrium where the manager stays in the owner’s ﬁrm
in the second period unless performance measures in two periods have non-zero correlation.
This paper is organized as follows. In the next section, I describe the basic model and characterize the retention cost. In the third section, I examine the incentive and delegation problem
of pre-contract investment by adding the opportunity to make an investment decision before contracting. In the forth section, I introduce the new assumption, the complementarity of eﬀorts, andanalyze the relationship between retention cost and complementarity of eﬀorts. In the sixth section,
I make concluding remarks.

2 Basic model
There are two players in this model, the risk-neutral owner of the ﬁrm and the risk-neutral manager.
The owner owns the assets to produce goods or service, but needs someone else, the manager, togain the outcome from those assets. The owner hires the manager for this purpose for as manyas two periods and the manager works for the manager. The owner’s objective from the contract
relationship with the manager is maximization of expected outcome net expected payment to the
manager.

1

For fairness, see Baron and Besanko [1987] for example.

2

The manager exerts the two kind of eﬀorts: strategic eﬀort in period one and operational eﬀort
in period two, respectively2 . I assume that the outcome of strategic eﬀort cannot be delivered to theﬁrm unless the manager stays at the ﬁrm in the second period as well.
Now denote e s ∈ R is the outcome of strategic eﬀort incurring cost of eﬀort 21 e2s , which the
manager chooses and pays in the ﬁrst period. The outcome of strategic eﬀort e s realizes at the endof the second period. I assume that e s is unobservable. Importantly, strategic eﬀort e s is speciﬁcto the manager hired in period 1. That is, the outcome of strategic eﬀort is not delivered to theﬁrm when the manager leaves the ﬁrm after the ﬁrst period and the owner hires another manager
for the second period. Let us denote eo ∈ R operational eﬀort in period 2. Operational eﬀort
is unobservable and incurs cost of eﬀort 12 e2o . Operational eﬀort is chosen by the manager in thesecond period and realizes at the end of the second period.
The outcome of two kinds of eﬀort, which is delivered to the ﬁrm at the end of the second
period, is jointly observed as follows: x = k · e s + eo + ε x , where k ≥ 0 is the productivity parameterof strategic eﬀort e s and ε x is a random variable subject to normal distribution with mean zeroand variance σ2 . The outcome x is assumed to be observable and veriﬁable and therefore can becontracted upon. Notice that the eﬃcient eﬀort levels are given by e s = k, eo = 1.
Additionally, the performance measure s = e s + ε s realizes in the ﬁrst period and is observableand veriﬁable. The term ε s is a random variable subject to normal distribution with mean zero andvariance σ2 . This performance measure conveys the information regarding the strategic eﬀort but
the outcome of strategic eﬀort is delivered to the ﬁrm in the second period, so I call this performancemeasure the leading indicator. I assume that there can be correlation between the noise terms in theoutcome and the leading indicator: τ = corr(ε x , ε s ), where −1 ≤ τ ≤ 1. The correlation between
outcome and the leading indicator corresponds to the characteristics of prospective performancemeasures such as customer satisfaction3 .
At the beginning of each period, the owner oﬀers the manager a one-period linear contract asa take-it-or-leave-it oﬀer. Let w1 = f1 + v1 · s denote the ﬁrst period contract, where f1 is ﬁxed
compensation and v1 is the coeﬃcient for the leading indicator s. Similarly, let w2 = f2 + v2 · x
denote the second period contract, where f2 is ﬁxed compensation and v2 is the coeﬃcient for the2
3

The terminology follows Sliwka [2002]
See Kaplan and Norton [1996] for detailed description on custmer satisfaction measures.

3

outcome. I assume that wi is the alternative payoﬀ for the manager in period i ∈ 1, 2.
As I mentioned above, Christensen, Feltham and Sabac[2003] showed that the manager does notstay at the ﬁrm without a special case under the setting of sequence of short-term contracts. Withproduction technology and short-term contracts, it is optimal for the manager to gain the payoﬀ
above the alternative payoﬀ in the ﬁrst period and leave the ﬁrm after the ﬁrst period4 ．Now, I
assume that the owner can commit to pay the retention cost c s as follows. When the owner oﬀers
the ﬁrst period contract, the owner oﬀers the retention cost c s , which is paid to the manager atthe beginning of the second period if the manager accepts the second period contract 5 . On theother hand, if the manager rejects the second period contract and leaves the ﬁrm, c s is not paid
to the manager. Additionally, I assume that the second period contract must cover the manager’s
alternative payoﬀ in the second period, w2 . Finally, the owner can precommit to the payment ofretention cost c s 6 .
Introduction of the retention cost is justiﬁed as follows. First, the strategic eﬀort is speciﬁc tothe manger. That is, the owner does not gain the outcome of the strategic eﬀort if the manager
leaves the ﬁrm. Therefore, it is natural to assume that the owner wants the opportunity to oﬀer theretention cost c s to induce the manager to stay at the ﬁrm in the second period. Second, I focus
on the setting where the manager can access the labor market after the ﬁrst period. If I analyze thelong-term contract with or without commitment, it will be diﬃcult to study the cost to induce themanager to stay at the ﬁrm in the second period. For this purpose, I need the contract regime to be
the sequence of short-term contracts.
The sequence of events is as follows. First, the owner oﬀers the ﬁrst period contract and retention cost to the manager. Second, the manager decides to accept or reject the owner’s oﬀer. If the
manager accept, the two parties enters the contract relationship. If the manager rejects, the relationship immediately ends. Third, the manager exerts the strategic eﬀort. Forth, the leading indicatorrealizes and the compensation is paid to the manager according to the ﬁrst period contract. Fifth,
The model of Christensen et al. [2003] corresponds to the case of k = 0 in this paper. In their model (especiallywhen τ > 0), the manager can gain the expected payoﬀ w1 + 12 v22 τ2 by choosing the lower strategic eﬀort than prescribedby the owner. This is due to the fact that ﬁxed payment covers the cost of strategic eﬀort the owner wants the manager toexert.
5
Notice that the ﬁrst period contract does not contain any portion of the second period contract, so the manager has
to decide to accept or reject the ﬁrst period contract without any commitment or promise regarding the second period
contract.
6
Of course, the choice of stay or leave is veriﬁable.
44

the owner oﬀers the second period contract to the manager. Sixth, the manager decides to accept
or reject the owner’s oﬀer. If the manager accept, the two parties enters the contract relationshipin the second period and retention cost is paid to the manager according to the arrangement on theretention cost in the ﬁrst period. If the manager rejects, the relationship ends. Seventh, the manager
exerts the operational eﬀort. Eighth, the outcome is delivered to the owner and the compensation is
payed to the manager according to the second period contract and contract relationship ends.

2.1 Solution to the basic model

Solution is fulﬁlled by backward induction.Before solving for equilibrium retention cost c s , I givethe description of the owner’s second period problem. The owner’s problem at the beginning of thesecond period is as follows7 :max
subject tok · ê s + e∗o + τ(s − eˆs ) − E[ f2 + v2 · x]1e∗o ∈ arg max f2 + v2 · [k · e s + eo + τ(s − e s )] − e2oeo21f2 + v2 · [k · e s + e∗o + τ(s − e s )] − [e∗o ]2 ≥ w2 .
2

(IC2)
(PC2)

Note that hat means the conjecture throughout the paper. The ﬁrst constraint (IC2) is the incentive compatibility constraint. This constraint means that the operational eﬀort is chosen optimallyby the manager. The second constraint (PC2) is the participation constraint. This constraint ensuresthe manager’s expected payoﬀ is not less than w2 , his alternative payoﬀ in the second period.
Because the manager is risk-neutral, the owner’s problem above involves no other tradeoﬀ thanincentive provision of eﬀort exertion. Therefore equilibrium eﬀort and slope coeﬃcient are givenby e∗o = 1, v∗2 = 1.
Next, I turn to the solution to retention cost. For preparation, let us focus on the manager’s
reaction to the ﬁrst period contract ( f1 , v1 ). The manager’s expected payoﬀ from the contract rela-

7
Note that, at the beginning of the second period, strategic eﬀort e s is the conjecture to the owner but is private
information to the manager who has chosen e s in the ﬁrst period.

5

tionship with the owner is as follows:
1 ∗[e]2 + c s + w2
2 s,stay1
E[ f1 + v1 · s | e∗s,leave ] − [e∗s,leave ]2 + w2 ,2
E[ f1 + v1 · s | e∗s,stay ] −(if stay)(if leave)

where e∗s,stay and e∗s,leave indicates the manager’s optimal choice of strategic eﬀort if the manager
stays or leaves the ﬁrm after the ﬁrst period, respectively. Hence I get the optimal choice of thestrategic eﬀort in each case:e∗s,stay = v1 + v∗2 (k − τ)

(if stay)e∗s,leave = v1 .

(if leave)

Therefore, the manager optimally chooses to stay when the following inequality holds.
E[ f1 + v1 · s | e∗s,stay ] −

1
1 ∗
[e
]2 + c s ≥ E[ f1 + v1 · s | e∗s,leave ] − [e∗s,leave ]2 .
2 s,stay2(1)

This inequality reduces to:
cs ≥

1 ∗ 2
[v ] (k − τ)2 .
2 2

(2)

In equilibrium, the owner wants to minimize the retention cost c s . Therefore the optimal retention cost is written as:
c∗s =

1 ∗ 2
[v ] (k − τ)2
2 2

(3)

Additionally, incentive provision of strategic eﬀort in the ﬁrst period only requires eﬀort cost. Then
I get the following equilibrium strategic eﬀort level and the ﬁrst period slope coeﬃcient:v∗1 = τ(4)e∗s = k(5)

Hence I get the following proposition regarding the equilibrium retention cost and optimal slope
coeﬃcients.
Proposition 1. Equilibrium retention cost is characterized by c∗s = 12 [v∗2 ]2 (k − τ)2 . Equilibriumeﬀort levels are eﬃcient and the slope coeﬃcients in the contract are given by v∗1 = τ and v∗2 = 1.

6

2.2 Stay or leave?: the owner’s optimal choice

When c s is too large, it may be optimal not to hire the same manager in both period. Therefore,
under the setting of this model where there is no other beneﬁt to the owner from the manager’s
stay, it is optimal for the owner to induce the manager to stay when the following inequality holds.
For simplicity, I assume that the alternative payoﬀ of the manager is zero (w1 = w2 = 0) in later
sections.
1
1
1
k · e∗s − [e∗s ]2 + e∗o − [e∗o ]2 − c∗s ≥
2
2
2

(6)

The equality holds in equilibrium, so the following is given.
Corollary 1.1. When k ≥ 2τ holds, it is optimal for the owner to induce the manager to stay at theﬁrm in the second period.

The corollary says that this constraint does not matter when k is large enough. In the later
section, I show that large k is not necessarily desirable.

3 Pre-contract investment
In this section, I consider the case where there is an opportunity to make an investment to the
productivity of the strategic eﬀort. When starting an new project or installing a new business unit,
pre-contract investment is needed to acquire or improve productivity. In other situations, somemanagers study in business school by ﬁnancial support by the ﬁrm in order to improve their businessskills. Pre-contract investment analyzed in this section corresponds to these situations, which arecommon to many organizations. Additionally, argument in this section is useful to understand theequilibrium roles of retention cost and help understand the argument in the next section. In thissection I show the condition the manager prefers the pre-contract investment.
I assume that there is an investment opportunity to raise the productivity from k to k + a, wherea > 0 is constant and the both parties know the value of a. This investment incurs the cost da ≥ 0 to
the ﬁrm (the owner). This investment decision is assumed to be observable.
Before introducing the equilibrium, I examine the manager’s preference over this investment
7

decision when the investment decision right is delegated to the manager. As shown in the above
proposition, the expected payoﬀ from the contract relationship is w1 + w2 + c∗s irrespective of the
productivity of strategic eﬀort k and equilibrium eﬀort levels are eﬃcient independent of k. Therefore, the manager makes precontract investment when the following inequality holds.
11(k + a − τ)2 ≥ (k − τ)2 .
2
2

(7)

a[a + 2(k − τ)] ≥ 0.

(8)This reduces to:Hence, the manager’s investment decision is described in the following corollary.
Corollary 1.2. When k ≤ τ, the manager prefers to make an investment in the domain 0 ≤ a ≤−(k − τ) and does not otherwise. On the other hand, when k > τ, the manager always prefers toinvest8 .

Let us consider the implication of this corollary. When k is suﬃciently large, the manager
prefers to make an investment because the pre-contract investment increases the retention cost.
When k is smaller than τ, however, the manager may prefer not to make an pre-coctract investment
because the retention cost decreases by the investment. This corollary is useful when understanding the various functions of the retention cost in equilibrium. Recall that the strategic eﬀort ismanager-speciﬁc, say, the owner cannot acquire the outcome of the manager’s strategic eﬀort. The
retention cost functions like hostage through the leaving opportunity of the manager. Additionally,
pre-contract investment changes the magnitude of the retention cost in various ways. Hence, theretention cost varies with pre-contract investment, and thereby the manager does not necessarily
accept the pre-contract investment even when the productivity before investment is high.

4 Interaction between strategic eﬀort and operational eﬀort
In this section, I introduce the additional assumption, the interaction between strategic eﬀort andoperational eﬀort. I modify cost of operational eﬀort as follows.
1 2[e + 2p · e s · eo ],2 o
8In the domain of a < 0 it is optimal not to invest, but this possibility is precluded by the assumption a > 0.

8(9)

where −1 < p < 1 is the degree of complementarity or substitutability9 between strategic eﬀort
e s and operational eﬀort eo . Generally, eﬀorts are called substitutes when p > 0 and complementp < 010 . Notice that cost of operational eﬀort above incurs in the second period. This corresponds
the case the cost of operational eﬀort varies with the strategic eﬀort exerted in the ﬁrst period. Thereis no other additional assumption.
Before describing the equilibrium, let us see the eﬃcient result. Eﬃcient eﬀort levels maximizing the joint surplus are given as follows.
ees = k − p(10)eeo = 1 − p · ees(11)

Expected joint surplus is given as follows.
1
k · e s + eo − [e2s + 2p · e s · eo + e2o ]2with
eo = 1 − p · e s

(12)
(13)

In this section, the manager’s optimal choice of operational eﬀort is eﬃcient as in the basicmodel. Therefore the owner’s problem at the beginning of the ﬁrst periodis as follows.
1
max k · e s + eo − [e2s + 2p · e s · eo + e2o ] − c s2∗subject toeo = 1 − p · e∗s
f1 ,v1 , f2 ,v2(IC2)e∗s ∈ arg max E[ f1∗ + v∗1 · s + f2∗ + v∗2 · x]es11
− [e∗s ]2 − [{e∗o }2 + 2p · e s · e∗o ]2
2
1
E[ f1∗ + v∗1 · s] − [e∗s ]2 ≥ w1
2
1∗
E[ f1 + v1 · s | e s,stay ] − [e∗s,stay ]2 + c s
2
1≥ E[ f1 + v1 · s | e∗s,leave ] − [e∗s,leave ]2
2

(IC1)
(PC1)

(stay)

The ﬁrst constraint (IC2) is incentive compatibility constraint for the choice of the manager’s
operational eﬀort in the second period. (IC1) is the incentive compatibility constraint for the manager’s strategic eﬀort, optimized for the maximization of his expected payoﬀ from the contract
910

The similar setting appears in Dikkoli et al. [2009]See Vives [1999] for details.

9relationship and cost of strategic eﬀort and operational eﬀort. Notice that cost of operational eﬀort
depends on the choice of strategic eﬀort, which is chosen in the ﬁrst period. The constraint (PC1)
is for the manager’s acceptance of the second period. As in the basic model, the owner needs toensure the manager his alternative payoﬀ. Thee last constraint (stay) ensures that the manager stays
at the owner’s ﬁrm in the second period.
The solution to the owner’s problem above is similar to that of the basic model. Hence, I get thefollowing proposition on equilibrium eﬀort levels and slope coeﬃcients.
Proposition 2. Equilibrium eﬀort levels are introduced as follows:e∗s =v∗1 + (k − τ − p)

1 − p2e∗o = 1 − p · e∗s .

,

(14)
(15)

Equilibrium slope coeﬃcients are given as follows.
τ − (k − p) · p2,
1 − p2 + p4v∗2 = 1.

v∗1 =

(16)
(17)

proof. The proof is by straight forward algebra as in the basic model.

The case of p = 0 is equivalent to the basic model in section 2. Similar to the basic model,
there is no other tradeoﬀ than eﬀort exertion. Therefore, equilibrium slope coeﬃcient in the second
period contract is v∗2 = 1. In this section, there is interaction between strategic eﬀort and operationaleﬀort because cost of operational eﬀort depends on the strategic eﬀort chosen in the ﬁrst period.
Strategic eﬀort e s is motivated both by v1 · s, the bonus in the ﬁrst period contract, and by
v2 · x, the bonus in the second period contract. As mentioned earlier in this section, eﬃcient level ofstrategic eﬀort is k − p. However, conditional expected outcome after observing ﬁrst period signal sis E[x | s] = k · e s + eo + τ(s − e s ). Therefore v1 is chosen to compensate for ratchet eﬀect of portionof the second period contract −v2 · τ.
From here in this section, I analyze the optimal slope coeﬃcient v∗1 , retention cost c∗s and expected owner’s payoﬀ.
10

1.4
1.2
1.0
0.8
0.6
-1.0

0.0

-0.5

0.5

1.0

p

Figure 1: Equilibrium strategic eﬀort

When p < 0 is negative, i.e., eﬀorts are complementary, strategic eﬀort reduces the cost ofoperational eﬀort. On the other hand, when eﬀorts are substitute (p is positive), strategic eﬀort
increases cost of operational eﬀort. So it is natural to guess that equilibrium strategic eﬀort decreases
in p. However, as shown in ﬁgure 1, the optimal strategic eﬀort in the case of τ = 0.5 and k = 1
reaches the peak at the intermediate value between −1 < p < 0, and is not monotonically decreasing
in p.
Observation 1. When τ = 0.5, k = 1, e∗s takes its maximum at −1 < p < 0.

This fact is in contrast to the intuition above. In order to examine the mechanism behind thisobservation, I analyze retention cost and the expected payoﬀ below.
For preparation of the analysis, let us see the slope coeﬃcient of the leading indicator, v∗1 , in the
proposition above.
v∗1 =τ − (k − p) · p2.
1 − p2 + p4

Straight forward algebra gives the following corollary on the sign of v∗1 .

11

(18)

3.0
2.5
2.0

Τ=-0.5
Τ=0

1.5
1.0

Τ=0.5

0.5
0.0
-1.0

0.0

-0.5

0.5

1.0

p

Figure 2: Equilibrium retention costCorollary 2.1. The sign of the equilibrium slope coeﬃcient in the ﬁrst period contract is:v∗1 < 0if

τ < (k − p) · p2

(19)v∗1 ≥ 0ifτ ≥ (k − p) · p2 .

(20)

It can be said that v∗1 is negative when k is suﬃciently large. Similar to Sliwka [2002], the slope
coeﬃcient in the ﬁrst period contract can be negative. Second period contract faces no tradeoﬀ, so
the second period slope coeﬃcient v∗2 is set to induce the eﬃcient strategic eﬀort incentive minus
ratchet eﬀect accompanied. The ﬁrst period bonus functions to compensate for this ratchet eﬀect.
This corollary is not enough to explain the non-monotonicity of optimal strategic eﬀort in p in ﬁgure
1. Now I turn to the examination of retention cost.
In equilibrium, retention cost is given as follows:11c∗s = [v∗1 ]2 + [e∗s ]2 − v∗1 · e∗s ,22∗ + (k − τ − p)v
with e∗s = 1
1 − p2
τ − (k − p) · p2.
and v∗1 =1 − p2 + p4
Now look at ﬁgure 2, an example of retention cost (k = 1).
12

(21)
(22)
(23)

Observation 2. Equilibrium retention cost takes its maximum at some intermediate value −1 < p <0.

Observation of ﬁgure 2 tells that retention cost is maximized at some point in −1 < p < 0.
This corresponds to the fact that strategic eﬀort takes its maximum at some point in −1 < p < 0.
Basically, strategic eﬀort is magniﬁed by the degree of complementarity, say −p. However, when
p is suﬃciently close to −1, motivating strategic eﬀort is too costly in the sense that retention costis too large. Therefore, when the degree of complementarity is extremely large, the owner refrainsfrom fully motivating the strategic eﬀort.
Next turn to the consideration on the relationship between retention cost and intertemporal correlation τ. The ﬁgure 2 shows that equilibrium retention cost is decreasing in intertemporal correlation τ. General result on this fact is the following corollary.
Corollary 2.2. Equilibrium retention cost is decreasing in intertemporal correlation τ ∀k.



proof. Proof is by straightforward algebra.

Before concluding this section, let us see owner’s expected payoﬀ in the case of positive intertemporal correlation τ = 0.5. Positive correlation corresponds to usual cases such as customer
satisfaction, where it is often said that there is positive correlation between future performance anda current customer satisfaction measure.
Next observation describes the owner’s expected payoﬀ under positive intertemporal correlation.
Observation 3. When k = 1 and τ = 0.5 > 0, the owner’s expected payoﬀ is largest at negative
degree of complementarity or substitution, p.

As I have described above, equilibrium retention cost is larger when eﬀorts are more complement. Additionally, when eﬀorts are suﬃciently complement, the owner does not provide themanager with much incentive of strategic eﬀort, because retention cost is too large if she is to do so.
This tendency of low powered incentive of strategic eﬀort is greater the degree of complementarity
is the greater, i.e., p is the more close to −1.

13

1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-1.0

0.0

-0.5

0.5

1.0

p
Figure 3: Owner’s expected payoﬀ

5 Conclusion
In this paper I have described the equilibrium managerial retention cost when the stay commitment
mentioned in Christensen, Feltham and Sabac [2003] is absent and the owner can commit to thepayment of the managerial retention cost, which is payed to the manager if the manager stays at
the ﬁrm in the second period. Prior research treated the intertemporal correlation between the ﬁrst
period performance measures and the second period performance measures/outcome and analyzed
the equilibrium eﬀort incentives under the existence of what is called ratchet eﬀect. I introduced
into the prior models the complementarity or substitutability between eﬀorts, especially strategic
eﬀort exerted in the ﬁrst period and operational eﬀort exerted in the second period.
The main result of this paper is that the complementarity of eﬀorts does not necessarily improves
the expected payoﬀ of the owner of the ﬁrm. This result is in clear contrast to the intuition that eﬀort
complementarity improves the eﬀort incentives.
There may be some obstacles to the future research. The description in this model is too complex
to mathematically treat and apply to other settings. However, the function of the retention cost in

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this model is signiﬁcant and seems to have many fruitful extensions in the future. Therefore oneof the future objectives in this line of this research contains the construction of the foundation of
applied models using the notion of retention cost.

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