and Incentive Contracts
Wu-Chun Chi Department of Accounting National Chengchi University
Taipei, TAIWAN, ROC (02)2939-3091 ext. 81031
Hung-Chao Yu*
Department of Accounting National Chengchi University
Taipei, TAIWAN, ROC (02)2939-3091 ext. 81010
Preliminary draft – Comments welcome
August 1, 2002
and Incentive Contracts
Abstract
Prior agency models examining the design of incentive contracts generally suffer two major deficiencies. First, they implicitly assume that performance measures are available for contracting between the principal and agent. In many situations, however, certain important variables are unobservable (or unverifiable) and, therefore, uncontractible. The assignment of property rights of uncontractible output can provide one efficient solution to this problem. Second, under a multitask environment, it is common that doing one task may not only affect its “major” output, but also positively or negatively influences the outputs of other tasks. However, these positive and negative
crossover effects have been overlooked in prior agency studies. The main purpose of this study is to
use a multitask principal-agent model to investigate the joint impacts of property rights and crossover effects on incentive design and the agent’s relative effort allocation decisions. We also provide a rule for determining the assignment of property rights.
JEL classification: D82, J33, L22, M41
1. INTRODUCTION
In his seminal paper of summarizing the contracting theory and accounting, Lambert (2001) indicates that much of the motivation for current accounting and auditing research relates to the control of incentive problems. In addition, Bushman and Smith (2001) also identifies that creating incentives to take actions is one of the three fundamental contracting roles for accounting information.1 In fact, during the past two decades, the design of incentive contract to motivate agent’s effort contributing to the success of a company has received much attention in managerial accounting.2 Many early incentive models have addressed various factors determining optimal compensation contract from a single task perspective (e.g., Arya, Fellingham, and Glover 1997; Balakrishman and DeJong 1993; Cohen and Loeb 1989; Datar and Rajan 1995; Kirby, Reichelstein, Sen, and Paik 1991; Kwon 1989; Magee 1988; Villadsen 1995). As indicated by Holmström and Milgrom (1991), however, if there are multiple tasks facing the agent, incentive pay not only serves the dual function of allocating risks and motivating work effort, but also serves to direct agent’s allocating his attention among various duties. Since multitask is common in the business world, an examination of incentive design under a multitask environment should provide more valuable insights into our understanding of incentive problems.3
Based on Holmström and Milgrom’s (1991) multitask setting, Feltham and Xie (1994) explores the role of multiple performance measures in influencing the direction and intensity of an agent’s effort allocation among different tasks. Feltham and Wu (2000) provides a general characterization of the relative weights assigned to two performance measures (i.e., accounting earnings and market price) in an optimal linear contract in a two-task setting. In contrast, Indjejikian and Nanda (1999) presents a
1The other two contracting roles are the filtering of common noise from other performance measures and the rebalancing of
managerial effort across multiple activities.
2See Gibbon (1998), Lambert (2001), Murphy (1999) and Prendergast (1999) for prior reviews of aspects of empirical and
theoretical incentive contract literature.
3Hemmer (1995) analyzes incentives and job design using a two-stage production process. Under a similar multistage
setting, Hemmer (1998) examines the interactions among responsibility assignment, performance measures, and incentive schemes. Different from multistage, which involves agent’s efforts toward sequential tasks (i.e., the second operation is conditioned on the output generated from the first operation), multitask involves agent’s doing several tasks simultaneously.
two-period multitask dynamic model to show that the use of more aggregate performance measures and greater consolidation of responsibility helps mitigate the ratchet effect. Since empirical evidence has shown that incentive contracts do affect agent’s behavior (Prendergast 1999) and an understanding of how incentive scheme should be designed constitutes a cornerstone of the theory of the firm (Baker, Jensen, and Murphy 1988), additional research should continue to shed light on unsolved issues related to principal’s design of optimal incentive contracts.
In general, most of prior agency studies in the area of incentive design have two distinct features. First, these studies focus on the role of performance measures in an optimal contracting framework under conditions of information asymmetry.4 The key characteristic is the informativeness of the performance measure about the agent’s action.5 In particular, Banker and Datar (1989), Lambert and
Larcker (1987), and Lambert (2001) emphasize that the informativeness of a performance measure should be a function of its sensitivity to the agent’s actions and its noisiness. Second, these studies implicitly assume that performance measures are available for contracting between the principal and agent (e.g., Feltham and Xie 1994; Feltham and Wu 2000). This later feature is problematic in capturing the salient features of the real world because in many situations some certain important variables are either unobservable (Prendergast 1999) or unverifiable by a third party (Hart and Moore 1988). For example, the condition of a productive asset is unobservable to the principal and, therefore,
4Prior empirical studies have documented that profitability measures are extensively used in annual bonus plans and in
long-term performance plans for corporate executives (e.g., Benston 1985; Murphy 1985). The literature also finds an important trend of using accounting numbers in contracts between pre-IPO entrepreneurs and venture capital firms in the U.S. (e.g., Kaplan and Stromberg 1999). However, over the past three decades, accounting profitability measures have become relatively less important in determining cash compensation of top executives (Bushman et al. 1998). Finally, there is a growing use of alternative performance measures (for example, the residual income performance measures) for executive compensation contracts (Wallace 1997). See Bushman and Smith (2001) for a comprehensive review of executive compensation research.
5The “informativeness” criterion is attributed to Holmström (1979). This criterion states that any performance measure that
is marginally informative about an agent’s actions, given other available performance measures, should be included in the incentive contract. However, field and empirical studies such as Merchant (1987) and Maher (1987) do not find empirical evidence to support this informativeness criterion. Yim (2001) reconciles the inconsistency between the theory and practice by abandoning the conventional full-commitment assumption. With the possibility of renegotiation, a performance measure‘s usefulness in incentive contracting depends on its information quality, not simply on whether the performance measure is informative. In particular, Yim (2001) finds that a performance measure is useful when its information quality is either sufficiently poor or sufficiently rich.
is not contractible. Also, the reputation of a company may be observable to its shareholders but is unverifiable by the court. Hence, shareholders can not use firm’s reputation as a performance measure to sign incentive contract with the manager because this contract is not enforceable. Unfortunately, prior agency-based incentive studies are silent in this problem. In essence, the restriction of contracts to observable variables constrains the principal to an inefficient set of choices (Magee 2001).
Different from prior agency studies, the main purpose of this study is to investigate the joint impacts of property rights and crossover effects6 on incentive design. This issue is important for two reasons. First, while the property rights theory of the firm has received much attention in economics over the past few decades (e.g., Che and Qian 1998; Chiu 1998; Coase 1960; Grossman and Hart 1986; Hart and Moore 1990; Joskow 1988; Meza and Lockwood 1998),7 it has not been formally examined
in the managerial accounting research. More important, property rights provide one efficient solution to the principal when certain important variables are unobservable or unverifiable (Hart and Moore 1988). In economic terms, property rights are different from other forms of incentive contract only where contracts determining specific decision rights over an asset and assigning its returns are imperfect and incomplete. Then, property rights may be identified with the right to exercise residual
control where the contract is silent about decision rights, or with the right to receive any residual returns that remain after contractual obligations are fulfilled.8 As Milgrom and Roberts (1992) points out, property ownership is “clearly the most common and effective means to motivate people to create, maintain, and improve assets, and its importance in practical business life would be hard to overstate (p.
6The term “crossover effects” was first introduced by Holmström and Milgrom (1991, p. 982) to describe the situation in
which a good job coordinating the response to a customer complaint would save costs for the manufacturer but typically also enhance future sales. However, they sidestepped this possibility and did not formally incorporate crossover effects into their multitask model setting. We will explain the basic notion of crossover effects in more details later in this section and section 2.
7See Barzel (1989) and Milgrom and Roberts (1992, chapter 9) for comprehensive discussions about the property rights
theory in economics.
8According to Milgrom and Roberts (1992), residual control represents the right to make any decisions concerning the
asset’s use that are not explicitly controlled by law or assigned to another by contract. On the other hand, residual returns denote the net income produced by an asset that is left over after everyone else has been paid. Prior economic analyses of property rights have concentrated on these two issues. Tying together residual control and residual returns is the key to the incentive effects of property rights.
321)”. Since economic literature has demonstrated that the distribution of property rights over productive assets (which are not contractible) determines agent’s motivation and incentive of doing tasks, an incorporation of the concept of property rights should provide significant contribution to the managerial accounting literature in the analysis of incentive design.
Second, under a multitask environment, it is ubiquitous that in many situations doing one task may not only affect its own output directly, but also influences the outputs of other tasks indirectly. For example, a medical school professor’s clinical experience should be positively helpful to his teaching performance. On the contrary, an overly high production volume may negatively harm the value of the productive asset. Theoretically speaking, these positive and negative crossover effects generate externality problem that may affect the determination of optimal incentive contract and agent’s effort allocation decision. However, few attempts, if any, have ever been made to examine the impacts of crossover effects on principal’s incentive design. Therefore, a rigorous consideration of multitask crossover effects together with property rights provides a good opportunity to investigate the overall incentive design in a more realistic and rich setting, which in turn improves the relevance of managerial accounting research to the decision makers. This study intends to provide an initial step in addressing this issue.
To illustrate how property rights and crossover effects may jointly influence the incentive design, we present an agency model in which a risk-neutral principal contracts with a risk-averse agent to perform two tasks in a single period. Our focus is limited to two aspect of incentive design, namely the determination of optimal incentive intensity and agent’s relative effort allocation. To illustrate the impacts of crossover effects on agent’s relative effort allocation, we consider a firm’s problem of motivating a single agent to perform two types of activities (e.g., production and maintenance) that may generate one positive crossover effect and one negative crossover effect to two outputs. To illustrate how the assignment of property rights may affect the principal’s design of optimal incentive contract, we assume that one output is contractible but the other one is uncontractible. This setting
allows the property rights to play a significant role to the determination of optimal compensation contract. Without formally considering crossover effects and implicitly assuming that the principal owns the property rights of the uncontractible output, Holmström and Milgrom (1991) concludes that, since there are many tasks facing the agent, a higher incentive to a contractible output may provide the agent with disincentives to other important tasks whose outputs are uncontractible. Different from their paper, we incorporate property rights and crossover effects into the model and show that Holmström and Milgrom’s (1991) conclusion is valid only under certain conditions. In addition, we also examine the agent’s relative effort allocation decision under different property rights scenarios and provide a rule for determining the assignment of property rights. Overall, our analysis extends the managerial accounting literature to a wider class of incentive design issues such as property rights and crossover effects.
The remainder of this paper is organized as follows. A basic two-task principal-agent model is presented in section 2 to illustrate the role of property rights and crossover effects in incentive compensation and agent’s relative effort allocation. In section 3 we describe a special case in which only a short-term contract is feasible and the crossover effects are trivial. A rule of determining the property rights between the principal and agent is also discussed. A final summary and conclusion is followed in section 4.
2. THE MODEL AND ANALYSES 2.1 Basic Incentive Model:
Following Holmström and Milgrom’s (1991, 1994) multitask agency models, suppose an agent has two different tasks which require effort levels e1 and e2,9 where {e1 , e2 } ∈R , and produce two +2
9The Holmström and Milgrom models are more general with respect to the number of tasks. We limit the number of tasks to
two so as to focus on the joint impacts of crossover effects and property rights on incentive design. We can assume that the measurement units of e1 and e2 are different (e.g., hour vs. minute) to reflect the fact that the agent’s per-unit-cost of doing
output values ϖ1 and ϖ2. Suppose we have the following two production functions f and g: 1 2 1 1 ( , ) ε ϖ = f e e + and ~ (0, 2) 1 1 σ ε N , (1) 2 2 1 2 ( , ) ε ϖ =g e e + and ~ (0, 2) 2 2 σ ε N , (2)
where cov(e1,e2)=0. As depicted in equations (1) and (2), a given task effort level will affect more than one output value. In particular, task effort level ei has a direct effect toϖiand a crossover effect to
j
ϖ , for i≠ . In a typical manufacturing setting, for example, we can regard j ϖ1 as the production output and regard ϖ2 as the value of the productive asset, given the same production task effort e1
and maintenance task effort e2. In this example, production effort e1 is beneficial to ϖ1 (the direct
contribution), but is harmful to ϖ2 (the negative crossover effect). In contrast, maintenance effort e2
may contribute to both ϖ1 (the positive crossover effect) and ϖ2 (the direct contribution). In mathematical terms, these direct and crossover effects can be denoted by ∂E(ϖi) ∂ei >0 and
0 )
( ∂ ≠
∂Eϖi ej (for i≠ j ), respectively, and ∂E(ϖi) ∂ei > ∂E(ϖi) ∂ej .
10 For notation
presentation, let fi =∂f ∂eiand gi =∂g ∂ei , for i = 1 or 2. Suppose the agent has the following
increasing and convex quadratic cost function:11
) ( 2 1 ) , ( 2 2 2 1 2 1 e e e e C = + . (3)
This form of cost function reflects the fact that agent is effort-averse and agent’s effort in two tasks is substitutable.12 In our model, the principal is assumed to be risk-neutral and the agent is
10Our crossover effects are different from the substitution (or complementary) effect, which involves the change in input
factor due to the relative price change in another input factor.
11Similar to Feltham and Xie (1994), Feltham and Wu (2000), and Lambert (2001), we restrict agent’s cost function to this
simple form to ensure that (a) closed form interior solutions exist, and (b) the level of effort equals its marginal cost. Including a fixed component or an interaction term to C(e1,e2) shall not change our analysis.
12This assumption is different from Holmström and Milgrom (1991), which assumes that working is pleasant to the agent
up to some limit (in their terms, the agent’s cost function satisfies two criteria:C'(t)≤0 for t≤ and t C(t)=0, where t denotes the limit beyond which incentives are required to encourage agent’s work), and finds that agent’s optimal incentive under the single task setting is higher than that under a multitask setting. We do not appeal to this assumption because, if t is large enough, there is no agency problem at all. If t is small, on the other hand, Holmström and Milgrom (1991) does not provide guideline for determining the optimal incentive beyond t . Furthermore, Holmström
risk-averse with a negative exponential utility function U(W)=−exp(−rw), where r > 0 denotes
agent’s risk aversion coefficient and W denotes his payoff (net of possible effort cost). The important
feature of this utility function is that it exhibits constant absolute risk aversion. This means that the agent’s wealth doe not affect his risk aversion and therefore does not affect the agent’s incentive. The principal offers the following linear contract:13
2 2 1 1 2 1, ) (ϖ ϖ =α+βϖ +β ϖ S (β1andβ2 ≥0), (4)
where α is a fixed salary, and β1 and β2 are the commission rates (which are measures of wage
incentive intensity) on output values ϖ1 and ϖ2, respectively. We restrict β’s to be nonnegative to
avoid encouraging agent to conceal his performance.
If the principal and agent can sign an incentive contract based on both ϖ1 and ϖ2, the principal’s incentive design problem can be formulated by the following Program I:
Program I:
{
f g f g}
e e, + − − 1⋅ − 2⋅ , , ,1 2 1 2 max α β β β β α s.t. + + − + − − ∈ 2 2 2 2 2 1 2 1 2 2 2 1 2 1 2 1 2 1 2 1 ) ( 2 1 max arg ) , (e e α β f β g e e rβ σ rβ σ , U r r e e g f ≥ + + − + − − 2 2 2 2 2 1 2 1 2 2 2 1 2 1 2 1 2 1 ) ( 2 1 β σ β σ β β α ,where the first and second constraints are the agent’s incentive constraint (IC) and individual rationality constraint (IR), respectively, and U is the agent’s reservation utility. Note that the IC
and Milgrom (1991) supports their assumption using instructor’s teaching higher-think skills as an example (see footnote 9 on page 32). We believe this example is restrictive to the description of agent’s behavior because instructors generally enjoy more rewards from education effort than pure monetary incentive. This may not be the case for other professions such as auditors and manufacturing workers. Therefore, we restore the classical effort-averse assumption and see how an optimal incentive scheme can be designed. Baron and Kreps (1999) and Lambert (2001) also adopts this effort-averse assumption in their introduction of agency models.
13To simplify the following analyses, we assume a linear contract. Prior studies such as Holmström and Milgrom (1991,
1994) and Feltham and Xie (1994) also restrict their analyses to linear contracts. Holmström and Milgrom (1987) and Banker and Dartar (1989) analyze conditions under which linear contracts are optimal. See Lambert (2001, 29-30) for discussions of justifications for the common use of linear contract in principal-agent models.
constraint implies that the agent’s reaction functions with respect to principal’s incentive strategy are 1 2 1 1 1 f g
e =β +β and e2 =β1f2 +β2g2. Similar to Holmström and Milgrom (1991, 1994), we use the first-order conditions of agent’s IC constraint and the total certainty equivalent (TCE) approach to rewrite Program I as follows:
Program II: + − + − ( + ) 2 1 ) ( 2 1 max 2 2 2 2 2 1 2 1 2 2 2 1 , 2 1 σ β σ β β β f g e e r s.t. e1 =β1f1+β2g1, 2 2 2 1 2 f g e =β +β .
Differentiating the objective function with respect to β1 and β2 gives the following two first-order conditions: 0 2 1 1 1 2 2 1 1 1 1 2 2 1 1 1 1 2 2 1 1 1 ∂ − = ∂ ⋅ − ∂ ∂ ⋅ − ∂ ∂ ⋅ + ∂ ∂ ⋅ + ∂ ∂ ⋅ + ∂ ∂ ⋅ βσ β β β β β β r e e e e e g e g e f e f , (5) 0 2 2 2 2 2 2 2 1 1 2 2 2 2 1 1 2 2 2 2 1 1 ∂ − = ∂ ⋅ − ∂ ∂ ⋅ − ∂ ∂ ⋅ + ∂ ∂ ⋅ + ∂ ∂ ⋅ + ∂ ∂ ⋅ β σ β β β β β β r e e e e e g e g e f e f . (6)
Plugging equations (5), (6), and constraints e1 =β1f1+β2g1 and e2 =β1f2 +β2g2 into the
objective function gives:
2 1 A C B E B C D − ⋅ ⋅ − ⋅ = β , (7) 2 2 B C A D B E A − ⋅ ⋅ − ⋅ = β , (8) where ( 2) 1 2 2 2 1 f rσ f A= + + , )B=(f1g1+ f2g2 , )( 2 2 2 2 2 1 g rσ g C= + + , 2 1 1 2 2 2 2 1 f f g f g f D= + + + , and 2 1 1 2 2 2 2 1 g f g f g g
E= + + + . Therefore, if ϖ1 and ϖ2 are both contractible, the principal will design the incentive intensities at the levels specified by equations (7) and (8).
In many practical situations, however, ϖ2 may not be economically observable or its noise
component 2 2
both parties. In this situation, equations (7) and (8) can be reduced to the following (9) and (10): A D = ∞ → 1 2 2 lim β σ , (9) 0 lim2 2 2 = ∞ → β σ . 14 (10)
Equations (9) and (10) indicate that, if the value of the productive asset (i.e., ϖ2) cannot be
contracted upon and the principal owns the property rights of the productive asset, she will design the optimal incentive scheme with respect to ϖ1 using equation (9). By comparing equations (7) and (9),
we can explore how ϖ2’s contractability may affect the magnitude of optimal β1:
Equation (7) – Equation (9) = ) ( ) ( 2 B AC A BD AE B − ⋅ + ⋅ − .
Obviously, the sign of the difference between equations (7) and (9) is indeterminable, depending on the specifications of the production functions f and g. We focus on the case where f1 > 0 and g1≤ 0
(which means the manufacturing effort is beneficial to ϖ1 but is harmful to ϖ2) and f2 ≥ 0 and g2 > 0
(which means the maintenance effort is beneficial to both ϖ1 and ϖ2) because it captures the salient
features of a real manufacturing environment. Based on this assumption, we formulate the production functions using the following separable and linear form:
1 2 1 1 1 π κ ε ϖ = e + e + , (π1 ,κ≥0), (11) 2 2 2 1 2 π ε ϖ =−qe + e + , (0≤q≤π1 ,π2 ≥0), (12) where πi measures the per-effort-unit contribution of ei to output values ϖ1 and ϖ2 (hereafter called the contribution coefficient of ei), κ measures the per-effort-unit benefit of maintenance to manufacturing (hereafter called the positive crossover effect), and q measures the per-effort-unit harm
14If ϖ
2 is not available for signing incentive contract, equation (4) will reduce to S(ϖ)=α+β1ϖ1, implying that equations
(6) and (8) will also vanish. Therefore, β2 becomes trivial and we should only focus on β1 specified in equation (9).
of manufacturing to the value of productive asset (hereafter called the negative crossover effect). It should be noted that the negative crossover effect q (a) is greater than zero because we want to capture the harm feature of e1 on ϖ 2, and (b) is less than π1 because, if q > π1, task e1 should not be
undertaken as it is undesirable to both the principal and agent, no matter who owns the property rights of ϖ2. Also, we assume π1 > κ because, for output ϖ1, e1’s direct effect is usually larger than e2’s
crossover effect. Similarly, we assume π2 > q. In the following analyses, we assume that ϖ1 is
contractible but ϖ2 is uncontractible.
2.2 The Joint Impacts of Crossover Effects and Property Rights on Incentive Intensity:
Using equations (11) and (12), we can compare β1’s under two different settings: (a) only ϖ1 can
be contracted upon and the property rights of ϖ2 belongs to the principal (denoted by P
1
β ) and (b) similar to setting (a) except that the property rights of ϖ2 belongs to the agent (denoted by A
1
β ). These lead to the following Proposition 1:
PROPOSITION 1: The principal’s optimal β1 to the contractible output ϖ1 is determined as follows:
(1) If the principal owns the property rights of ϖ2, then 2
1 2 2 1 2 1 2 2 1 1 π κ σ κ π π κ π β r q P + + + − + = . In
addition, the comparative statics show that 1 <0 ∂ ∂ q P β , 1 >0 ∂ ∂ κ βP , and 1 0 2 > ∂ ∂ ∂ q P κ β .
(2) If the agent owns the property rights of ϖ2, then 2
1 2 2 1 2 2 1 1 π κ σ κ π β r A + + + = . In addition,
the comparative statics show that 1 =0 ∂ ∂ q A β , 1 >0 ∂ ∂ κ βA , and 1 0 2 = ∂ ∂ ∂ q A κ β . (3) ( 1 1 )<0 ∂ − ∂ q A P β β , ( 1 1 ) 0 > ∂ − ∂ κ β βP A , and ( 1 1 ) 0 2 > ∂ ∂ − ∂ κ β β q A P . Proof: See Appendix.
principal’s optimal incentive design of another task. Several important implications of this result deserve further discussions. First, part (1) of Proposition 1 shows the comparative statics that
0 1 ∂ > ∂βP κ , 0 1 ∂ < ∂βP q , and 0 1 2 ∂ ∂ >
∂ βP κ q . Because q and κ denote the indirect effects of e
1
and e2 on ϖ1 and ϖ2, respectively, an increase in κ implies that effort e2 is more valuable in a
sense that it not only provides an indirect contribution to ϖ1, but also provides a direct contribution to
2
ϖ . Since the principal owns the residual claims of ϖ1 and the property rights of ϖ2, she has high incentive to set P
1
β at a higher level to motivate the agent to exert more effort on e2 (i.e.,
0
1 ∂ >
∂βP κ ). On the other hand, an increase in q implies that effort e
1 becomes less favorable to the
principal because it impairs the value of ϖ2, which is owned by the principal. Therefore, she will provide a weaker P
1
β to discourage the agent to put more effort on e1 (i.e., ∂ 1 ∂q<0
P
β ).When both
q and κ increase simultaneously, however, the principal will tend to increase P
1
β because, even though a higher P
1
β will motivate the agent to exert more effort on e1, which in turn incurs an extra
cost q to ϖ2 and an incremental contribution π1 to ϖ1, a higher P
1
β will also induce the agent to put more effort on e2, which in turn generates incremental contributions of π2 to ϖ2 and κ to ϖ1.
Since the net benefit is still positive to the principal, she has strong incentive to set P
1
β at a higher level (i.e., ∂2β1P ∂q∂κ >0). These results are valuable to the managerial accounting literature because it suggests that, if the value of the productive asset is important to the principal, she can (a) ex ante, employ independent verifier to examine the asset’s physical condition or lower down the incentive for the manufacturing output, and (b) ex post, use non-financial performance measures such as machine breakdown frequency to alleviate the agency problem. This result provides a theoretical foundation to support Sears, Roebuck, and Co.’s dropping the by-the-job rate system and paying mechanics hourly salary to balance both quantity and quality (Horngren, Datar, and Foster 2002, 803).
Second, part (2) of Proposition 1 shows the comparative statics of A
1
β with respect to q and κ when the agent owns the property rights of ϖ2. Although an increase in κ only provides an indirect contribution to ϖ1, the principal still prefers to set A
1
on e2. Therefore, we have ∂β1 ∂κ >0
A . In contrast, since the agent owns the property rights, the
principal will never take q into consideration in designing her incentive scheme. Thus, we have 0
1 ∂ =
∂βA q and 0
1
2 ∂ ∂ =
∂ βA q κ . The managerial implication of these results is that, even though the
principal can ignore q because the property rights of ϖ2 belongs to the agent, she may need to give the agent some kind of premium to compensate the agent’s losses due to q so that he will exert more effort on e1. For example, let e1 and e2 denote working and leisure activity, respectively, and define ϖ1
and ϖ2 as the production output value and the agent’s family life, respectively. In this example, the agent will share A
1
β portion of the production output ϖ1 but will possess his own family life ϖ2 (which is not contractible). Obviously, the more the e1 the agent exert, the higher the harm to the
agent’s family life. Therefore, part (2) of Proposition 1 implies that the principal may motivate the agent to put more effort on e1 not by providing a higher incentive intensity β1A, but by giving extra
premium or subsidy. This example can explain why many Taiwanese hi-tech companies afford “oversea” compensations, cars, and local lodging services (rather than an increase in β) to management staffs and workers who are dispatched to their subsidiaries in Mainland China.
Third, we can further explore the effects of property rights on principal’s optimal incentive design by calculating the difference between P
1 β and A 1 β : 2 1 2 2 1 1 2 1 1 π κ σ π κ π β β r q A P + + − = − . (13)
Equation (13) indicates that q and κ not only affect the sign of P A
1
1 β
β − , but also affect the magnitude of the difference between P
1
β and A
1
β . We first discuss the negative crossover effect q. Since (a) P A
1
1 β
β − is positive when q equals zero, (b)∂β1P ∂q<0, and (c) 0
1 ∂ =
∂βA q , part (3) of
Proposition 1 implies that the line P A
1
1 β
β − has a positive intercept and a negative slope (i.e., 0
) ( 1 − 1 ∂ <
∂ βP βA q ). As depicted by the dot line in panel A of Figure 1, when the negative crossover
effect q approaches a certain q* =(π2/π1)κ, for any given κ, from either direction, the property rights of ϖ2 become less relevant to the principal’s determination of ϖ1’s incentive intensity. We define
this situation as property rights irrelevance (PRIR). Under this situation, the principal may tend to ignore the property rights effect in designing her optimal incentive intensity on ϖ1. On the other hand, if q moves away from q* in both directions, the distance between P
1
β and A
1
β increases. Therefore, the property rights of ϖ2 will become more relevant to the principal’s incentive design (see the dot line in panel B of Figure 1). We call this situation property rights relevance (PRR). Under this situation, the principal will emphasize more on the property rights’ effect in designing her optimal incentive intensity on ϖ1.
[Insert Figure 1 here]
Different from the comparative statics analysis of q, parts (1) and (2) of Proposition 1 indicate that the principal’s optimal incentive intensity β1 is increasing in κ (i.e., ∂β1P ∂κ >0 and
0
1 ∂ >
∂βA κ ), no matter who owns the property rights. However, since an increase in κ will generate
both direct and indirect contributions to the principal when she owns the property rights of ϖ2, but will only provide indirect contribution to the principal when the agent owns ϖ2, the effect of an
increase in κ on principal’s determination of optimal incentive intensity on ϖ1 is stronger when the principal owns the property rights of ϖ2. Due to this reason, we have ∂(β1P−β1A) ∂κ >0 in part (3) of proposition 1. Based on the facts that P A
1
1 β
β − is negative when κ equals zero and 0
) ( 1 − 1 ∂ >
∂ βP βA κ , we know that the line P A
1
1 β
β − has a negative intercept and a positive slope. As depicted by the dot line in panel A of Figure 2, when the positive crossover effect κ approaches a certain κ* =(π1/π2)q, for any given q, from either direction, the property rights of
2
ϖ become less relevant to the principal’s determination of incentive intensity on ϖ1. On the contrary, the property rights of ϖ2 will become more relevant when κ moves away from κ* in both directions (see the
dot line in panel B of Figure 2).
[Insert Figure 2 here]
Finally, the marginal effects of increasing both q and κ on the difference between P
1
β and A
1
β
should be examined through two PRR scenarios (i.e., the original P
1
strictly larger than A
1
β ) and one PRIR scenario (i.e., the original P
1
β is equal to A
1
β ). Under the first PRR case, in which the original P
1
β is strictly smaller than A
1
β , a simultaneous increase in q and κ will decrease the relevance of the property rights of ϖ2 to principal’s incentive design on β1. In contrast, under the second PRR case (in which the original P
1
β is strictly larger than A
1
β ) and the PRIR case (in which the original P
1
β is equal to A
1
β ), a simultaneous increase in q and κ will
increase the relevance of the property rights to principal’s incentive design. Therefore, we have 0 / ) ( 1 1 2 − ∂ ∂ > ∂ βP βA q κ .
2.3 The Joint Impacts of Crossover Effects and Property Rights on Relative Effort Allocation: Proposition 1 has demonstrated the combined consequences of property rights and crossover effects on principal’s determination of optimal incentive design and how these consequences may change when individual crossover effect changes. As indicated by Lambert (2001), however, in a multitask setting, the emphasis shifts from motivating the intensity of the agent’s effort to the allocation of his effort among tasks. Accordingly, in this section, we will take a further step to explore the joint impacts of property rights and crossover effects on agent’s effort allocation between e1 and e2.
We first look at the case in which the agent owns the property rights of ϖ2. From equations (A1) and (A2) in the Appendix, we have:
2 1 2 2 1 2 1 2 2 1 2 2 1 1 1 1 1 ) ( ) ( σ κ π σ κ π κ π π π β r r q q eA A + + + + ⋅ − + ⋅ = − = , (14) 2 1 2 2 1 2 2 1 2 1 2 2 1 2 1 2 2 ) ( ) ( σ κ π κ π κ σ κ π π κ β π r r eA A + + + ⋅ + + + ⋅ = + = . (15)
Therefore, the agent’s effort allocation can be represented by the effort ratio ERA eA eA
1 2 / ≡ : ) ( ) ( ) ( ) ( 2 1 2 2 1 2 2 1 1 2 2 1 2 1 2 2 1 2 σ κ π κ π π κ π κ σ κ π π r q r ERA + + − + + + + + ≡ . (16)
In contrast, if the principal owns the property rights, then from the constraints of Program II and equations (11) and (12), we have:
2 1 2 2 1 2 1 2 2 1 1 2 1 1 1 ) ( σ κ π κπ π κ π π β β π r q q eP P P + + + − + ⋅ = − = , (17) 2 1 2 2 1 2 1 2 2 1 2 2 1 2 ) ( σ κ π κπ π κ π κ β π κβ r q eP P P + + + − + = ⋅ + = (18)
Therefore, the agent’s effort allocation can be represented by the effort ratio ERP eP eP
1 2 / ≡ : 1 π κ ≡ P ER . (19)
Equation (19) indicates that, when the principal owns the property rights, the agent will ignore the production coefficients of ϖ2 (i.e., q and π2) but solely focus on the production coefficients of ϖ1 (i.e., π1 and κ). Since equation (11) has shown that the production function of ϖ1 involves two linearly substitutable input factors e1 and e2, the agent’s optimal effort ratio will be the ratio of π1 and
κ.15 On the other hand, when the agent owns the property rights, the agent will also take the production coefficients of ϖ2 into his effort allocation decision. Therefore, equation (16) shows that the agent’s optimal effort ratio is a function of all four production coefficients.
To further examine the effects of property rights on agent’s relative effort allocation, we can take the difference between ER and P ER : A
)] )( ( [ ) )( ( 2 2 1 1 2 1 1 2 1 2 2 1 2 1 κ π π σ π σ κ π κ π π + − − ⋅ + + + = − q qr r q ER ERP A . (20)
Since the nominator in equation (20) is always positive, the sign of ERP−ERA depends on the
sign of the denominator, which is determined by the bracket term [ ( )( 2 2)]
1 1 2 1 π π κ σ − −q + qr .
15In fact, equation (19) is exactly the effort ratio obtained in Feltham and Wu (2000) when there is only one performance
Obviously, the bracket term is positive only when the following condition holds: + + + ⋅ > 2 1 2 2 1 2 2 1 1 π κ σ κ π π r q . (21)
From part (2) of Proposition 1, we can see that the term inside the bracket of equation (21) is right the A
1
β . Therefore, equation (21) can be reduced to q A
1 1 β
π ⋅
> . Since π1 measures the per-effort-unit contribution of e1 to ϖ1 and
A
1
β denotes the portion of ϖ1 the agent can share when the agent owns the property rights of ϖ2, A
1 1 β
π ⋅ represents the marginal benefit the agent can earn from exerting one more unit of effort e1. Recall that q measures the per-effort-unit harm of e1 to ϖ2.
Hence, the agent’s relative effort allocation decision under different property rights situations is determined by comparing the marginal benefit (from ϖ1) and marginal cost (from ϖ2) of employing an extra unit of e1, given the agent owns the property rights of ϖ2. In particular, if marginal cost is
larger than marginal benefit, ER tends to be greater than P ER . That is, if the negative crossover A
effect q is high enough (i.e.,q A
1 1 β
π ⋅
> ), the agent will put “relatively” more (less) effort on e2 rather
than e1 when the principal (agent) owns the property rights.16 Note that β1P plays no role to agent’s
relative effort allocation decision because equation (19) has indicated that the agent’s effort ratio ER P
is not a function of q.
3. SHORT-TERM CROSSOVER EFFECTS – A SPECIAL CASE
In our manufacturing case, a more difficult task facing the principal is how to design an incentive contract within a “relatively short” period in which the harm of production effort e1 to ϖ2 is severe
(i.e., q is large, which implies that the productive asset tends to be “overused,” given a certain effort level e1), but the benefit of maintenance effort e2 to ϖ1 is negligible (i.e., κ is small). This situation
is of particular interest to the principal because, when q is large but κ is small, the adverse
consequences of an improper incentive contract on TCE-maximization will be even stronger. The reasons are as follows. First, since the harm of e1 on ϖ2 is unobservable during a short time period,
the principal cannot write an enforceable short-term contract based on ϖ2. This “externality” becomes a real problem to the principal when q is large. Second, the principal may mitigate the “externality” problem if she can induce the agent to exert a higher level of e2. However, because the principal can
only design a contract based on ϖ1 and the agent’s cost function is convex in e1 and e2, if κ is too
small, the principal cannot induce a satisfactory level of e2 through a high incentive intensity on ϖ1.
17
Clearly, these problems will become trivial when a long-term contract is feasible between the principal and agent (e.g., the length of the enforceable contract is longer than the useful life of the machine). In this section, we examine this short-term effect using an extreme situation in which κ equals zero.
3.1 Short-term Effects of Property Rights on Incentive Intensity:
Plugging 0κ = into parts (1) and (2) of Proposition 1 gives ( )/( 2)
1 2 1 1 2 1 0 , 1 π π π σ βPκ= = − q +r and )/( 2 1 2 1 2 1 0 , 1 π π σ βAκ= = +r , which implies , 0 0 1 0 , 1 − < = = κ κ β
βP A . In other words, under the situation
in which the benefit of maintenance activity e2 to ϖ1 is negligible (i.e., κ =0) and only a short-term
contract is feasible, the incentive intensity to ϖ1 will always be weaker when the principal possesses the ownership of ϖ2 than when the agent owns ϖ2. This conclusion is consistent with Holmström and Milgrom’s (1991) two-task example, which shows that, since there are two tasks facing the agent, a relatively higher incentive to one task may provides disincentive to another important task which can not be contracted by output performance measure. Note that Holmström and Milgrom’s (1991) two-task setting is equivalent to our model in which κ equals zero and the principal owns the property rights of ϖ2. In fact, Proposition 1 has indicated that, when κ is large enough (i.e., κ is bigger than κ* in Figure 2), the incentive intensity on
1
ϖ can be stronger when the principal owns
17Theoretically speaking, a high incentive intensity on
1
ϖ will induce the agent to put more effort on either e1 or e2. For a
the property rights of ϖ2. This reflects the fact that crossover effect κ plays an important role to principal’s determining the relative magnitude of P
1
β and A
1
β . If we differentiate the difference between P
1
β and A
1
β with respect to q, we have 0 / ) ( , 0 1 0 , 1 − ∂ <
∂ βPκ= βAκ= q . This result implies that, if there is no other alternatives available to the
principal’s incentive design (e.g., non-financial performance measurement or the agent’s long-term reputation), then when κ =0 and the principal owns the property rights of ϖ2, the incentive intensity P
1
β will be far more weaker than A
1
β as q increases. However, if κ is large enough, Proposition 1 indicates that this result shall be reversed.
3.2 The Determination of Property Rights:
So far, we have explored the effects of property rights on principal’s incentive design and agents relative task allocation. We now turn our attention to the determination of property rights. In particular, we intend to identify conditions under which the property rights of ϖ2 should be assigned to the
principal or the agent. We first formulate the following Lemma:
LEMMA: If we define TCEP (or TCEA) as the total TCE when the principal (or agent) owns the
property rights of ϖ2, then based on β1P and
A
1
β specified in Proposition 1 and equations (9) and (10), we have: (1) ) ( 2 ) ( 2 1 2 1 2 1 2 1 σ π π π r q TCEP + − = and
[
]
− ⋅ + − − + = 2 1 2 1 2 2 1 2 2 2 ) ( 2 2 1 q TCE r q q TCEA P π π σ π π .(2) Define Ω=TCEA −TCEP, then the sign of Ω is equivalent to the sign of −q 2−Z
1 ) (π , where 2 1 2 2 2 1 2 1 2 1 2 2 2 2 2 1 2 1( ) ( ) σ π π σ π σ π π σ r r r r Z = − + + − . (3) ∂Ω ∂r<0, 2 0 2 < ∂ Ω ∂ σ , ∂Ω ∂q<0, ( ) [ ( 2)] 0 2 2 2 1 2 1 1 ⋅ − − − ≥ ∂ Ω ∂ π qrσ π rσ π , and 0 2 > ∂ Ω ∂ π .
Proof: See Appendix.
Part (2) of the Lemma indicates that, since the sign of Ω determines the property rights assignment of of ϖ2, the term −q 2 −Z
1 )
(π can be regarded as the asset ownership rule for determining who should own ϖ2. More important, the first part of the rule 2
1 )
(π −q denotes the
square of direct net contribution (SDNC) of e1 on both ϖ1 and ϖ2, while the second part Z
constitutes an adjustment metric to the net contribution. Note that Z is not a function of the negative crossover effect q (recall that κ equals zero in our short-term analysis). Apparently, if Z is negative, it becomes an addition to the SDNC, resulting in a positive Ω. This implies that the agent should own the
2
ϖ . As shown in part (2) of the Lemma, a negative Z requires a small 2 2
σ , a small r, and a large π2.
That is, when the agent is not very risk sensitive, the uncertainty of the uncontractible output ϖ2 is low, and the per-effort-unit contribution of e2 to ϖ2 is high, the property rights of ϖ2 should be
assigned to the agent.18 These results are further supported by part (3) of the Lemma, which shows that ∂Ω ∂r<0, 2 0
2 <
∂ Ω
∂ σ , and ∂Ω ∂π2 >0. Note that, when Z is negative, the negative crossover effect q plays no role to the determination of optimal ownership right because π2 may be large enough
to cover q and the uncertainty threat to the risk-averse agent is negligible. The economic rule is: As
long as the adjustment Z is negative, then under the conditions that e1 adversely affects the
uncontractible output ϖ2, the social optimal property rights should belong to the agent, no matter how harmful e1 may be to ϖ2.
If Z is positive, on the other hand, the sign of Ω depends on the relative magnitudes of SDNC and adjustment Z. Since a positive Z requires a high uncertainty of ϖ2, a more risk-averse agent, and a low per-effort-unit contribution of e2 to ϖ2, the opportunity cost for agent to own ϖ2’s property rights
increases. If this opportunity cost is larger than SDNC, the property rights should be assigned to the risk-neutral principal. Note that the Lemma implies a rule: Given Z is positive, the higher the negative
crossover effect q, the higher the probability that the principal should own the property rights. This
18Since
2
ϖ becomes more valuable when π2 increases, assigning the property rights to the agent can better mitigate the
conclusion is further supported by the comparative static ∂Ω ∂q<0. The reason underlying this conclusion is that, if the agent owns the property rights when the harm of e1 on ϖ2 is high, the
principal will offer an overly high A
1
β to satisfy equation (18). This high A
1
β together with the property rights ownership provide three additional costs to the agent: (a) the ( A
1
β – P
1
β ) portion of production activity’s uncertainty 2
1
σ , (b) maintenance activity’s total uncertainty 2 2
σ , and (c) a higher harm q. If the principal owns the property rights, however, the agent will not suffer these three additional costs. Our analytical result enriches the managerial accounting and property rights literatures by explicitly emphasizing the importance of the negative crossover effect to the determination of asset ownership.
Using the above Lemma, we provide the following guideline for assigning the ownership of uncontractible output ϖ2:
PROPOSITION 3: Under the short-term situation (i.e., κ= 0), the assignment of the property rights of
the uncontractible output ϖ2 can be determined by the following rule:
(1) The property rights of ϖ2 should be assigned to the agent when: (a) the agent is
not very risk sensitive, (b) the uncertainty of ϖ2 is low, or (c) the per-effort-unit contribution of e2 to ϖ2 is high. Since these conditions lead to a negative
adjustment metric Z, the negative crossover effect q plays no role to the determination of optimal ownership right.
(2) The property rights of ϖ2 should be assigned to the principal when: (a) the agent
is very risk sensitive, (b) the uncertainty of ϖ2 is high, or (c) the per-effort-unit contribution of e2 to ϖ2 is low. Since these conditions lead to a positive
adjustment metric Z, the probability that the principal should own the property rights is increasing in the magnitude of the negative crossover effect q.
Together, Proposition 3 suggests that, when only a short-term contract is feasible, the principal should take the negative crossover effect into consideration in determining the optimal property rights ownership. The importance of the negative crossover effect depends on three variables: the uncertainty of ϖ2, the agent’s risk-averse attitude r, and the direct contribution of e2 to ϖ2. In particular, if these
three variables lead to a negative adjustment Z value, the negative crossover effect plays no role to the property rights attribution. However, this negative crossover effect does matter when the above three variables give rise to a positive adjustment Z.
4. SUMMARY AND CONCLUSION
The main purpose of this paper is to adopt the property rights theory of the firm to examine, under a multitask setting, the impacts of crossover effects on the design of optimal incentive contract. While ignoring the crossover effects and implicitly assuming that the principal owns the uncontractible output, Holmström and Milgrom (1991) concludes that a higher incentive to a contractible output may discourage the agent to exert more effort on other important tasks whose outputs are uncontractible. Different from their model setting, we incorporate both property rights and crossover effects into the model and show that Holmström and Milgrom’s (1991) conclusion is no longer valid. In particular, the analytical results from our model reveal that, ceteris paribus, when a positive crossover effect κ exists, the principal should provide a higher incentive intensity on the contractible output, no matter who owns the uncontractible output. In contrast, when a negative crossover effect q exists, the principal should provides a weaker incentive intensity on the contractible output when she owns the property rights of the uncontractible output; however, the principal will ignore q in determining the incentive intensity when the agent owns the property rights of the uncontractible output. Second, when both the positive and negative crossover effects approach their corresponding cutoff points, the property rights of the uncontractible output become less relevant to the principal’s determination of incentive intensity on the contractible output. On the other hand, if any one of the two crossover effects moves away from its corresponding cutoff point, the property rights of the uncontractible output will become more relevant to principal’s incentive design. In addition, the difference between the incentive intensities under two property rights scenarios (measured by P A
1
1 β
β − ) is increasing in the positive crossover effect κ and is decreasing in the negative crossover effect q. Third, if the negative
crossover effect q is high enough, the agent will put relatively more effort on the task which generates a direct effect on the uncontractible output (i.e., e2) when the principal owns the property rights of this
uncontractible output. Finally, the agent should own the property rights of the uncontractible output if (a) the uncertainty of the uncontractible output is small, (b) the agent is less risk-averse, and (c) the direct contribution on uncontractible output is large.
Several limitations of our model and future research directions should be recognized. First, Milgrom and Roberts (1992) indicates that the most important attribute of transactions for examining property rights is the asset specificity attribute. According to their definition, assets are specific to a certain use “if the services they provide are exceptionally valuable only in that use (p. 307).” Asset specificity is important because it leads to the hold-up problem.19 Especially, if an asset is specific to a particular use, the owner of the specific asset can be held up, leading to value-destroying consequences such as discouraging the owner to invest in highly specific assets. In our model, we rule out the possibility of asset specificity and assume that the uncontractible output is for general purpose so that the agent’s effort decision will not be affected by the hold-up problem. Second, the model is designed and analyzed in a one-period, two-task setting with linear incentive contract. Therefore, the effects of agent’s reputation and nonlinear contract on the principal’s design of optimal incentive scheme cannot be examined. As Demski and Dye (1999) points out, linear contracts may not independently direct the agent to act in the best interest of the principal. This raises the question about whether the analytical results found in our paper can be applied to situations with nonlinear contracts. A direct extension of our study would be to incorporate these other features into the model. Third, managerial accounting literature has indicated that communication is valuable to principal’s production, marketing, and capital budgeting decisions because agent’s reporting of his private information to the principal can reduce the costs of information asymmetry (Berg, Daley, Gigler, and Kanodia 1990; Christensen 1981, 1982; Lambert 2001; Melumad and Reichelstein 1987, 1989). This result suggests that future agency-based
19The hold-up problem describes a general business situation in which either the principal or the agent worries about being
forced to accept disadvantageous contract terms later, after it has sunk an investment, or worry that’s its investment may be devalued by the actions of others. See Baron and Kreps (1999) and Milgrom and Roberts (1992) for more detailed discussions about this hold-up problem and its influences on property rights.
contracting research should take the value of communication into consideration in determining the optimal incentive intensity. Finally, our model can also be extended to incorporate the role of multiple performance measures in influencing the direction and intensity of agent’s effort allocation decision and the determination of relative weights assigned to various performance measures.
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Figure 1
The Impacts of Negative Crossover Effect q on Property Rights Relevance
Panel A: Property Rights Irrelevance (PRIR)
Figure 2
The Impacts of Positive Crossover Effect κ on Property Rights Relevance Panel A: Property Rights Irrelevance (PRIR)
