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Principal-agent models in multi-dimensional settings Tan, Patricia M. S. 1994

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PRINCIPAL-AGENT MODELS IN MULTI-DIMENSIONAL SETTINGSByPatricia M.S. TanB. Accountancy, National University of Singapore, 1984A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF COMMERCE AND BUSINESS ADMINISTRATIONWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAOctober 1994© Patricia M.S. Tan, 1994In presenting this thesis in partialfulfilment of the requirements foran advanceddegree at the University of BritishColumbia, I agree that the Libraryshall make itfreely available for reference and study. I further agree that permissionfor extensivecopying of this thesis for scholarlypurposes may be granted by thehead of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)________________Department of CourcQ QThe University of British ColumbiaVancouver, CanadaDate Oro\erDE-6 (2188)AbstractThe thesis applies the Principal-Agent models to the following two settings:1. The agent is employed to work on a multi-stage project;2. The agent is responsible for multiple tasks.The method of analysis is an analytical one.Part I studies the multi-stage problem in which periodic applications of effort by the agent arerequired. The agent also obtains private information as the project evolves and he decides if the projectshould be abandoned or continued. We show that the agent’s decision to continue is not always alignedwith the principal’s desire. The result provides an economic rationale for the sunk cost phenomenon.There also exist conditions under which the agent chooses to prematurely abandon the contract.Part II studies the effort allocation problem and provides insight with respect to the job designproblem. When the agent is responsible for more than one task, the principal simultaneously studiesthe incentive problem for all the tasks and decides on the task grouping and assignment. The relativeprecision of the performance measures of the agent’s effort in each task affects the cost to the principalof extracting high effort levels from each of the task. The principal should not settle for costlesslyavailable but highly noisy information. Rather, the management accounting system of each firm shouldbe designed to be consistent with the technology of the firm, its product strategy, and its organizationstructure. This allows the principal to more efficiently induce desired levels of effort.11Table of ContentsAbstractList of Figures ViiAcknowledgement Viii1 Introduction and OverviewPARTI 42 Incentive Contracts with Continuation Decisions 42.1 Introduction2.1.1 The Incentive Problem2.1.2 Severance Pay2.2 The Abandonment Problem2.3 R & D as an Example2.4 Agency Literature Review2.5 Conclusion3 The Model and Two Single-Stage Settings3.1 Introduction3.2 The Model3.2.1 General Characteristics3.2.2 The Problem3.3 Second-stage Moral Hazard, No First-stage3.3.1 The Model3.3.2 First-best Solution3.3.3 Second-best Solution446781012Moral Hazard1313• • . 14• 14• . 20• . 20• . . 2022231113.4 First-stage Moral Hazard, No Second-stage Moral Hazard3.4.1 The Model3.4.2 First-best Solution3.4.3 Second-best Solution3.5 Example3.5.1 No First-stage Moral Hazard Problem3.5.2 No Second-stage Moral Hazard Problem29293031363737Appendix 3A4 Analysis of the Multi-Stage Setting4.1 Benchmark Case4.1.1 The Model4.1.2 Analysis of the Problem4.2 The General Model4.2.1 First-best Solution4.2.2 Analysis of the Second-best Problem4.3 Communication4.4 Example4.4.1 Expected Compensation Cost, Wage Levels and4.4.2 Review of Results4.5 Literature Review4.6 Implications and ConclusionsAppendix 4AAppendix 4BAppendix 4CBibliography43858810811053535354565658717373798183Cutoff PointsivPART II5 Effort Allocation and Job Design5.1 Introduction5.2 Tasks with Long-term and Short-term Impact .5.3 Objectives of the Model5.4 Agency Literature Review5.4.1 Multitask Literature5.4.2 Job Design5.4.3 Investment in Monitoring Technology .5.5 Conclusion6 Single-Task Principal-Agent Model with Costly6.1 Introduction6.2 The Model6.3 Quadratic Cost Setting6.3.1 First-best Setting6.3.2 Use of Monitoring Technology6.3.3 Comparative StaticsMonitoring Technology 122122122125126126127Appendix 6A 1291321321331331341361371371391411431121121121141161181181201201217 Multitask Principal-Agent Model with Costly Monitoring Technology7.1 Introduction7.2 The Model7.2.1 General Characteristics7.2.2 Interdependency in either V(t) or 11(t)7.2.3 First-Best Setting7.3 Analysis of the Second-Best Setting7.3.1 General Solution for Costless Performance Measures7.3.2 Costless Performance Measure Available for Only Task 17.3.3 A Costly Monitoring Technology7.4 Quadratic Cost SettingV7.4.17.4.27.4.37.4.47.4.57.4.67.4.77.4.8Appendix 7A143145146150153155163170174174176178Appendix 7BAppendix 7CBibliography198199200IntroductionFirst-best SolutionSecond-best Solution - Costless Noisy Performance Measures .Comparative Statics - Costless Noisy Performance Measures . .Costless Performance Measure Available for Only Task 1Second Best Solution - Costly Monitoring TechnologyComparative Statics - Costly Monitoring TechnologyThe Value of Monitoring7.5 Some Implications7.5.1 Job Design, Organization Structure and Incentive Plans . .7.5.2 Investment in Monitoring TechnologyviList of Figures3.1 Decision Tree Depicting the Agent’s Sequential Decision Problem3.2 Comparison of * with y (no first-stage moral hazard)3.3 Comparison of with y* (no second-stage moral hazard)3.4 Behavior of Expected Compensation Cost (no first-stage moral hazard)3.5 Behavior of Wage Levels3.6 Behavior of Optimal Cutoff with XH3.7 Behavior of Expected Compensation Cost (no second-stage moral hazard)3.8 Behavior of Wage Levels3.9 Behavior of Optimal Cutoff with XH4.10 Behavior of Expected Compensation Cost (The General Model)4.11 Behavior of Wage Levels4.12 Behavior of Optimal Cufoff with H7.13 1447.14 1487.15 1497.16 1597.17 1617.18 1667.19 1687.20 1687.21 1697.22 1697.237.24182835393940414142767778Behavior of V(t’1,t2) with changes in t2Behavior of total certainty equivalent with 112 (ii < 0)Behavior of total certainty equivalent with 122 (v > 0)Determining the optimal level of monitoring (ii < 0)Determining the optimal level of monitoring (zi > 0)Behavior of h2, lj and ä as ii variesBehavior of 122, t and â as r variesBehavior of 122, lj and â as r variesBehavior of 122, t1 and ã as r variesBehavior of 122, t and â, as r variesBehavior of 122, lj and â, as r varies 170Behavior of total certainty equivalent with r 173viiAcknowledgementI am most grateful to my supervisor Gerald A. Feltham for his illuminating guidance and continuoussupport during the course of this dissertation. I also wish to thank the members of my thesis committee,Ken Hendricks and Vasu Krishnamurthy for their invaluable suggestions and comments. I have benefittedfrom discussions with Jack Hughes, Paul Fischer and Young Kwon. My special thanks go to my husbandDennis who has been most supportive, understanding and encouraging throughout the course of my Ph.Dprogram. Finally, I am very grateful to God for seeing me through the thesis.Financial support from Nanyang Technological University is gratefully acknowledged.viiiChapter 1Introduction and OverviewIn the principal-agent relationship, the principal delegates to the agent the responsibility of managingpart of the firm’s operations. A major assumption of agency models is that individuals are motivatedsolely by self-interest. The agent’s aversion to effort and the agent’s private information result in tensionin the relationship between the principal and the agent. Thus, the interests of the principal and theagent are unlikely to be aligned. The agent wishes to expend as little effort as possible in order tomaximize his personal utility, thus his choice of effort level is unlikely to lead to a maximization ofthe principal’s profit. Research in this area focuses on the optimal contractual relationships betweenthe ljrincipal and the agent. It examines the relationship between the firm’s information system andits employment contracts. The principal-agent model provides a coherent and useful framework foranalyzing and understanding managerial accounting procedures. The purpose of the thesis is to extendthe agency model to consider multidimensional aspects of the principal-agent problem. We specificallylook at the following two problems in an agency setting:1. The agent is employed to work on a multi-stage project;2. The agent is responsible for multiple tasks.The thesis consists of seven chapters. This chapter (Chapter 1) provides an introduction to the thesis.The remaining six chapters are grouped into two parts. Part I studies the multi-stage problem whilepart II examines the multiple task problem.Part I consists of Chapters 2, 3 and 4. In the multi-stage problem, periodic applications of effortby the agent are required and the agent also obtains private information as the project evolves. A key1Chapter 1. Introduction and Overview2feature of the model is that the project is subject to abandonment after the initial stage. In particular,we derive the optimal incentive contracts for a setting in which the agent isemployed to undertake atwo-stage project which may be subject to abandonment after the first stage. Effort is required in bothstages, but there is only one outcome at the end of the project. After the first stage, the agent receivesprivate information and decides if the project should be continued or abandoned. We show that undercertain conditions, the agent chooses to continue although under first-best conditions, the project isabandoned. The agent’s selection of the cutoff point is ex ante efficient butex post inefficient. Thisresult provides an economic rationale for the sunk cost phenomenon. Conversely, we show that thereexist conditions under which the agent chooses to prematurely abandon the project.Chapter 2 is an introduction to Part I. It discusses the incentive problems when agents are employedin risky multi-stage projects. In Chapter 3, we set up the general model and examine two special cases.The results provide us with useful insights when we analyze the general model in Chapter 4. We alsoexamine a benchmark case in which information is publicly observable butnot contractible.Part II consists of Chapters 5, 6 and 7. It deals with the effort allocation problem and providesinsights with respect to the job design problem. The agent is responsible for two tasks and his attitudetowards performing the two tasks determines his personal cost of effort. We examine how changes in theagent’s attitude affects the optimal effort levels. We vary the precision of the performance measure in thesecond task and explore how this affects task assignment and optimal effort levels, In the extreme case,we consider what happens if there is no costlessly available performance measure on the second task.The principal explores the option of investing in a costly monitoring technology to extract a signal thatcan be used in the compensation contract. We determine the factors that affect the optimal monitoringlevel.The analysis indicates that when an agent is responsible for more than onetask, incentive issuesshould not be addressed task by task. The principal should simultaneously studythe incentive problemChapter 1. Introduction and Overview 3for all the tasks. While a good incentive plan is critical for motivating performance, the issue of effectivejob design should not be ignored. A good job design and a well-designed incentive plan are bothnecessary to motivate the agent to exert the optimal effort levels at the minimum cost. Also, sincethe precision of the performance measures affects the cost to the principal of extracting high effortlevels, the principal should not just settle for costlessly available but highly noisy information. Rather,as Johnson and Kaplan (1987) advocate, the management accounting system of each firm should bedesigned to be consistent with the technology of the firm, its product strategy, and its organizationstructure. The provision of such an information system may be costly but it allows the principal to moreefficiently induce desired levels of effort. When the benefits outweigh the cost of information collection,the principal invests in an accounting system that provides more congruent and less noisy performancemeasures.Chapter 5 is an introduction to Part II. In Chapter 6, we examine the principal’s monitoring decisionin a single task model. This provides us with useful insight when we examine the two-task model inChapter 7. With two tasks, task grouping and effort allocation are critical and we analyze the principal’smonitoring decision in such a setting.Chapter 2Incentive Contracts with Continuation Decisions2.1 Introduction2.1.1 The Incentive ProblemThis section of the dissertation examines the incentives of risk- and work-averse agents to work onprojects with the following characteristics:• risky — probability of failure is high, but if successful, the returns can be extraordinary;• long-term and multi-stage; and• subject to abandonment.Motivating agents to take up such projects is different from motivating them in traditional type of work,like sales and manufacturing. Firms not only seek to provide incentives to induce the agents to take upsuch risky investments and work hard at them, but also seek to provide incentives for them to abandonthe project if the profit prospect is low.1The market value of the agents’ human capital may depend on their past performance. By undertaking a risky investment, agents put their human capital at risk. A good outcome may help to increasethe reputation value of the manager. On the other hand, a bad outcome, including abandonment, doesnot reflect well on the manager’s talent and his value on the market may be adversely affected. Kanter(1989, p. 310) states that professional careers (i.e., careers defined by skill) are produced by projects,1 In our analysis, we do not examine the incentive problem of motivating the agents to select from among alternativeprojects.4Oh apter 2. Incentive Contracts with Continuation Decisions 5with reputation as the key variable in success. Each project adds to the value of a reputation as it issuccessfully completed. On the other hand, project abandonment or failure blights the agents’ careerchances, destroys their merit increases and limits their scope to take risks again (Twiss and Goodridge,1989, p. 51). Peters (1989) suggests that in innovative and highly risky work, managers should supportfailure instead of penalizing the agent. Otherwise, the agents will become afraid to take risk, or theywill be reluctant to terminate a project they began even though the profitability prospect is low.Such long-term projects usually involve the acquisition of firm/project-specific human capital. Milgrom and Roberts (1992, p.363) define firm-specific human capital as “knowledge, skills, and interpersonal relationships that increase workers’ productivity in their current employment, but are useless ifthe workers leave to join other firms”. If the project is abandoned and the agent’s employment withthe firm is terminated, then the market value for the agent’s services upon reentering the job marketis not higher than before he joined the firm and undertook the project. From the firm’s perspective,these skills are also difficult and expensive to replace. Hence, it is in the interest of both parties thatthe employment relationship lasts at least for the duration of the project. In R & D work, skills mayalso be project-specific for legal reasons. If the skill is tied to the trade secret of the firm, the agent isnot free to leave the firm and continue with the project on his own or with any other firm.Williamson (1985, p. 259) argues that agents accepting employment of a firm-specific kind willrecognize the risks of “one labor power and one job” and insist upon surrounding such jobs with protectivegovernance structures. In the absence of such governance structures, Williamson (1985, p. 272) predictsthat agents must be paid a wage premium to accept such employment.We suggest that one type of protective governance structure in employment which involves firm- andproject-specific skills is the reliance on long-term contracts which commit the firm to future compensation, and severance pay should the project be abandoned because newly-received information indicatesthat it is no longer profitable. The difficulty arises when the agent has built up a substantial amount ofChapter 2. Incentive Contracts with Continuation Decisions 6firm-specific and project-specific skill and knowledge which has little or no value for other firms, shouldthe agent seek employment elsewhere. Therefore, an agent, who could choose between jobs which buildup general skills and jobs which build up firm- and project-specific skills, will be reluctant to choosethe latter types unless he is compensated should the project be abandoned. The termination paymentprovided for in an efficient contract acts as a contractual safeguard for the agent and it would encouragethe continued investment in firm- and project-specific skill. Williamson (1985, pp. 33—34) states that,in general, transactions which require special purpose technology and which do not enjoy any protectivesafeguards are unstable contractually. They are either replaced by general purpose technology or somekind of contractual safeguards will be introduced to encourage the continued use of the special purposetechnology.2.1.2 Severance PayIf the project is abandoned, the agent’s firm-specific skill is no longer productive and the market doesnot value these skills. Therefore, it will be optimal to offer the agent a contract which has a severancepay. Project abandonment should not be treated like a failure as this would result in the agent becomingtoo risk averse in taking up investments, and at the same time, if the project is started, it will result inthe agent being very unwilling to stop it even if profitability is low.The use of severance pay when projects are abandoned is similar to the use of golden parachutesin takeovers. Just as golden parachutes help deter executives from resisting takeover attempts that arepersonally costly for them but beneficial to the shareholders, providing severance pay to executives in theevent of project abandonment would help deter executives from resisting dropping projects that are nolonger profitable. Also, providing for severance pay helps provide motivation for the agent to undertakerisky projects which may be subject to abandonment.Chapter 2. Incentive Contracts with Continuation Decisions 72.2 The Abandonment ProblemThe agent is usually most informed about the value of the project as he is directly involved in it. Atthe same time, some projects are very technical and may be beyond the understanding of the principal.Therefore, in most instances, the principal must rely on the agent to make any decisions about theproject. At each stage of the project, the agent assesses the development of the project and decides if itis profitable to continue with it. Twiss (1992) lists a number of factors which he claims, cloud the issueof project abandonment. They include the following:1. A sunk cost mentality - stick with the existing project because of the investment already made.2. New project euphoria - abandon the old in favor of the new.Our analysis shows that these two responses may be rational in the second-best world when theagent’s effort is not observable and the agent makes the abandon/continue decision. In comparing theagent’s abandon/continue decision with the principal’s decision when effort is observable, we obtain thefollowing two possibilities:1. The agent continues the project even when information indicates it is no longer profitable to doso. We term this overcontinuation.2. He prematurely abandons the project, which we term overabandonment.What is commonly perceived as the ‘sunk cost mentality’ or escalation behavior may be explainedby our result on overcontinuation, and what is perceived as ‘new project euphoria’ may be explained byour result on overabandonment. In escalation behavior, the agent adheres to and increases his earliercommitment even when new information indicates that continuing the earlier commitment will result inworse consequences. Such behavior has been generally termed as irrational and as evidence that decisionmakers do not ignore sunk costs. Kanodia, Bushman and Dickhaut (1989) provide an explanation forChapter 2. Incentive Contracts with Continuation Decisions 8such behavior based on reputation. They show that when the agent has private information about hishuman capital, a desire for reputation-building may lead the agent to demonstrate escalation behavior.My model provides an alternative explanation based on induced moral hazard, a term introduced byDemski and Sappington (1989). Unlike the overcontinuation problem, very little has been said aboutthe overabandonment problem. With the benefit of hindsight, it is easy to spot the overcontinuationproblem and conclude that an unsuccessful project should have been abandoned earlier. This is notpossible with the overabandonment problem, because it results in missed opportunities which are muchmore difficult, if not impossible, to spot.2.3 R & D as an ExampleOne example of a multi-stage project that is subject to abandonment is R D type work. Holmstrom(1989) remarks that “the agency costs associated with innovation are likely to be high” •2 Features of R& D projects which cause contracting to be particularly demanding are:• risky — probability of failure is high, but if successful, the returns can be extraordinary;• labor intensive — substantial human effort is required at each stage;• idiosyncratic — R & D projects are not easily compared with other projects; and• long-term and multi-stage — projects are subject to termination notwithstanding efforts previouslyexpended. Sometimes, the project is abandoned after many years of work, due for example, toinformation that competitors are far ahead in the research or that the research has little value. Ofparticular importance is that companies have often terminated R & D programs “for reasons thathave nothing to do with the research quality.” (Schneiderman, 1991)2Thjs suggests that finding ways to reduce these costs is a worthwhile endeavor.Chapter 2. Incentive Contracts with Continuation Decisions 9Gibson (1981, pp.320—326) discusses three major phases in an R & D project. He states that as thesephases are executed, “there should be a steadily decreasing risk caused by increased knowledge”. At theinitial selection point, risk is the highest. He also states that the cost of an R & D project is minimal atthe beginning and rises with the state of certainty regarding the success of the project.The first phase is called the intuitive/heuristic phase. The R & D idea is submitted and a feasibilitycheck is performed. A tentative budget is set. If the initial indicators are positive, then the projectproceeds into the critical phase. As the project progresses, new information on the project is receivedand the research scientist revises the probability of success of the project. At the same time, theexpenditure on the project is increasing, and this is the point at which the scientist must decide if theproject should be continued or abandoned. Gibson states that a decision to stop the project is hard tomake since the scientist has an investment in the success of the project. The scientist tends to be overlyoptimistic and rarely objective. We propose that compensation packages which include a provision forseverance pay (or a similar measure) helps motivate the agent to abandon the project if it is no longersufficiently profitable. In the final phase, the project enters the commercial marketing phase.Gibson’s description of the major phases of the R & D project fits the characteristics of the projectsexamined (as discussed in Section 2.1) in this part of the dissertation. Since R & D projects are oftencritical to the earnings growth of a company, it is important that the research personnel be properlymotivated to undertake such activities. Kanter (1987) observes that a controversy has been brewingin recent years over how to best and most fairly compensate those from whose efforts originate newproducts or technology. Proper compensation packages must be designed to attract, retain, and motivatethe scientists and research engineers who undertake such projects. This part of the dissertation helpsprovide some insight into the nature of efficient compensation packages for this group of personnel.Chapter 2. Incentive Contracts with Continuation Decisions 102.4 Agency Literature ReviewTwiss (1992) states that project selection and project abandonment are two critical and difficult decisionareas in technology management. Project abandonment is important because of the high proportion ofprojects that are discontinued before their development is completed. Yet, hardly any work on multistage projects with project abandonment has been done, although there has been some work in thearea of project selection. We review the work in this area as it is closely related to the idea of projectabandonment. Lambert (1986) examines the incentives of an agent to invest in a risky single-stageproject. His alternative is to invest in a safe project. The agent works to acquire private informationabout the risky project. Lambert derives conditions under which underinvestment or overinvestment inthe risky project occurs. He concludes that underinvestment occurs when the risky project is a priorimore profitable than the safe project. Balakrishnan (1991) examines a similar model with the additionalfeature that the agent has precontract private information on the agent’s skill. This information isrelevant because the ex ante probability of success in the risky project is strictly increasing in theagent’s skill. By looking at the default project, i.e., the project that would have been chosen if the agentdoes not work to acquire information on the risky project and instead uses his precontract informationalone, Balakrishnan shows that for the set of agents whose default project is the risk free project,overinvestment in the risky project occurs.Banker, Datar and Gopi (1989) consider the project selection strategy of the agent, given that hehas private post-contract information. A necessary condition for underinvestment and overinvestmentis that the agent is risk-averse, and underinvestment and overinvestment occur as a result of a trade-offbetween risk premium cost and suboptimal project selection cost.Another relevant area of literature is that on the value of communication of predecision information.Penno (1984) examines if there is strict value to communication of the agent’s private pre-decisioninformation, denoted by in a one-period moral hazard setting. There are two crucial assumptions inChapter 2. Incentive Contracts with Continuation Decisions 11his model:1. There exists private information which informs the agent that effort is ineffectual; and2. In response to this information, the agent has the option of reducing effort to a level of zero.Penno shows that there exists a cutoff o such that effort level equals zero for < o and effort levelis strictly positive for > He demonstrates that a strict improvement can always be generated bychoosing communication, where the message space is either or > o. If o, the agent receivesa constant wage level. The gain from communication is achieved by allowing the risk-neutral principalto absorb risk from the risk-averse agent. Our model is, in some sense, similar to Penno’s model. Thetermination/continuation decision is equivalent to a form of communication, and Penno’s analysis showsthat allowing for this decision is strictly valuable. Our analysis takes the model further by examining ifboth the principal and the agent agree on the same cutoff (which is denoted by o in Penno’s model).Melumad (1989) examines a one-period model in which the agent acquires private post-contract predecision information and he is allowed to breach the contract by paying the principal predetermineddamages. If the agent continues with the contract, he selects his effort level which is subject to themoral hazard problem. Melumad concludes that it is never optimal to include a severance payment inthe compensation contract. This result is driven by the fact that the agent’s market value after thebreach of contract is the same as that before he joins the firm. Melumad does not examine whether theagent’s choice of breach of contract is in line with the interest of the principal.Demski and Sappington (1987) model an agent who is responsible for two activities, planning andimplementation, and the latter entails no disutility to the agent. This means that in isolation, there isno moral hazard concerning the implementation activity. They show that it is sometimes optimal tocreate motivation concerns (termed as induced moral hazard) in the implementation activity in orderto be more efficient in motivating the agent in the planning activity. We see a similar result in ourmodel. Although the abandonment/continuation decision entails no disutility to the agent, we obtainChapter 2. Incentive Contracts with Continuation Decisions 12overabandonment and overcontinuation as optimal outcomes in the second-best world.2.5 ConclusionIn this part of the dissertation, we examine the incentives of risk- and work-averse agents to work onmulti-stage projects which are subject to abandonment. We obtain overcontinuation and overabandonment as the possible outcomes. The circumstances leading to each situation are determined. With thebenefit of hindsight, the overcontinuation behavior has generally been called irrational. Our results showthat far from being irrational, they are optimal choices under the particular set of circumstances.In the next chapter, we set up the general model and examine two special cases of the general model.These provide us with useful insights when we analyze the general model. Chapter 4 looks at the generalmodel and a benchmark case in which information is publicly observable but not contractible.Chapter 3The Model and Two Single-Stage Settings3.1 IntroductionThe principal in our model faces a two-stage project. He seeks to attract an agent to join the firm andundertake the project. Working on the project will build up project-specific skills. A key feature of themodel is that the agent receives private information after the first stage, from which he decides whetherthe project is to be abandoned or continued. The principal offers a long-term contract to the agent, whichincludes a provision for severance pay should the project be abandoned. The level of the severance payplays a critical role in ensuring that the project will be terminated if it is no longer sufficiently profitable.Since the probability of success of the project is higher with higher effort level, the principal wants tomotivate the agent to work hard. At the same time, the principal seeks to motivate the agent to makethe abandon/continue decision in the principal’s favor. A moral hazard problem exists in the effort levelchoice. The agent experiences no disutility from the abandon/continue decision, which is observableby the principal. Thus, viewed in isolation, there is no motivation concern in the abandon/continuedecision, and the agent’s incentive is aligned with the principal’s.Our results show that the moral hazard problem in the effort level choice leads to a motivationconcern in the abandon/continue decision. Such concern is termed induced moral hazard by Demski andSappington (1987). As a result, the principal may prefer to motivate the agent to choose to continue theproject even when it does not appear profitable to do so. This explains why some firms appear reluctantto terminate their projects even when information received indicates that the probability of success is13Chapter 3. The Model and Two Single-Stage Settings 14low. Such escalation behavior has been generally termed as irrational, and as evidence that decision-makers do not ignore sunk costs. Kanodia, Bushman and Dickhaut (1989) provide an explanation forsuch behavior based on reputation. They show that when the agent has private information about hishuman capital, a desire for reputation-building may lead the agent to demonstrate escalation behavior.Our model provides an alternative explanation based on induced moral hazard.While escalation is one possibility, our results also indicate that when the return from a successfulproject is relatively high, the induced moral hazard problem may lead the principal to prefer to motivatea higher cutoff point. Thus, at times, firms may appear too hasty in terminating their project.We set up the general model in section 3.2. In sections 3.3 and 3.4, we discuss two special caseswhich will provide us with useful insights when we analyze the general model. We examine a numericalexample in section 3.5. All proofs are provided in the appendix, In the next chapter, we analyze thegeneral case.3.2 The Model3.2.1 General CharacteristicsWe consider a three-date economy, i = 0, 1 and 2. The principal has a project, and he employs an agentto undertake the work. The project has two stages. The completion of the first stage coincides with date= 1. The returns of the project are realized at the end of the second stage, which occurs at date i = 2.The agent takes tim at stage 1 and t at stage 2 (if there is no abandonment), where m, n E {h, 1}, withh and 1 corresponding to high effort and low effort respectively. If the project is carried to completion,there are two possible cash flows: XH represents a favorable outcome and XL represents an unfavorableoutcome. We assume that if the agent takes either or t2j, the project crashes with probability one,i.e., Pr(xLItim, t2) = 1, if either mn or n equals 1.After the agent has implemented his first-stage effort, he privately observes information signal y.Chapter 3. The Model and Two Single-Stage Settings 15This signal allows him to update his probability assessment of a high cash flow, given the project hasnot crashed. In particular, let Pr(xHItlh, t2h, y) = y. We assume that the signal y is generated fromthe uniform distribution over the interval zero to one. Also, we assume that the agent is unable tocommunicate the information signal to the principal, because of the excessive cost of communication.To utilize this information, we assume that the principal provides for the possibility of projectabandonment after the first stage. Since the information is privately observed by the agent, the abandonment/continuation decision must be delegated to him.1 After the agent has observed y and updatedthe probability of obtaining XH, he decides if the project should be continued or abandoned. The agentexperiences no direct disutility from making this decision. If the project is abandoned, the agent’semployment is terminated and he enters the job market.We assume the principal and the agent enter into a two-stage contract. The principal wants tomotivate the agent to choose tlh, and if the project is continued, the principal wants to motivate theagent to choose t2. The principal can commit to hire the agent for both stages, unless the project isabandoned. We assume it is in the principal’s interest to commit to the contract for the duration ofthe project, since the project-specific skill of the agent is difficult and expensive to replace. The agent,on the other hand, cannot commit to remain with the firm for both stages. In particular, since thecontract provides for project abandonment and subsequent termination of the agent’s employment, theagent is free to leave the firm after the first stage concomitant with deciding that the project should beabandoned.In this setting, the compensation package serves both to motivate the agent to choose tlh and t2h, andto induce an abandon/continue decision that is in the principal’s interest. The compensation packageconsists of two components:‘Project abandonment may make the principal better off. For example, the project may require the principal to invest$4 million in the first stage. If the project is continued, an additional investment of $10 million is required. By providingfor abandonment, if a very bad signal is received, the principal can avoid the further investment by allowing the agent toabandon the project.Chapter 3. The Model and Two Single-Stage Settings 161. If the project is carried on to completion, the agent earns a fixed wage, in1, paid at the end of stage1, and an amount tv (xj) contingent on the cash flow xj, j = H, L realized, at the end of stage 2.2. If the project is abandoned at the end of stage 1, the agent’s employment is terminated and he isgiven severance pay, to3. The agent enters the job market, earns a net wage of ink in return foreffort level t21.Thus, the long-term compensation contract {[w1,w2(x1)],w5} includes the following provision. At i = 1,if the agent decides that the project is to be abandoned, then his employment will be terminated andhe receives severance pay to3. Otherwise, if the project is to be continued, the agent will be paid in1 inthe first stage andw2(x1) in the second stage.The principal is risk-neutral while the agent is risk-averse. The agent also experiences a pecuniaryprivate cost with effort supply, i.e., we assume that the direct impact of the agent’s effort on his utility isrepresented as a “financial” cost to him. This cost might represent an opportunity cost of the time spenton the project. We assume that the agent’s utility function exhibits constant absolute risk aversion, r,and is represented as follows:w2, tim,t20) = — exp[—r(wi + W2 — tim — t2n)],where wi is the first-stage aggregate income and w2 is the second-stage aggregate income. Hence,H(wi , W2 tim,t20) = U(W)V(tim)V(t20),where U(w) = —exp(--rw),V(tjm) = exp(rtjm),andw = Wi+W2.In effect, the agent is only concerned with aggregate consumption.2 Thus, we need only solve for the2Thus, the analysis is not influenced by a lack of banking. The model is equivalent to a single-consumption date model,but there are multiple sequential acts.Chapter 3. The Model and Two Single-Stage Settings 17total compensation w = w1 +w2. If the project is abandoned after the first stage, w1 = w3 and w2which is determined by the job market.The utility value is non-positive everywhere. We assume v(ti) = 1, i = 1,2, and v(t13) = V(t2h) =V(th). The agent experiences a higher pecuniary cost with higher effort, so that v(t3) > v(t) = 1. Hisreservation utility level is K, K < 0 and the equivalent wage level is iii, i.e., UQth) = K.The time-line of the game is as follows:• At i = 0, the principal and the agent enter into a two-stage compensation contract.• The agent chooses either low or high first-stage effort level lm, m C {1, h}.• At i = 1, a signal, y on the viability of continuing the project is privately observed by the agent.If the agent has chosen t13, he revises the probability of high cash flow, and accordingly decides ifthe project is to be continued or abandoned.• If the project is abandoned, the agent’s employment is terminated, he is paid his severance pay w,and he enters the job market and earns a wage of wk. The principal faces alternative investmentwith return B.3• If the project is continued, the agent is paid his first-stage wage, w1. He chooses the second-stageeffort level, incurring additional disutility v(t2m).• At i = 2, the outcome x, j = L, H is publicly observed.• The agent is paid his second-stage wage,w2(x).Refer to Figure 3.1 in which we use a decision tree to depict the agent’s sequential decision problem.3For example, some of the capital equipment purchased for the project could be sold and the money reinvested at theassumed interest rate of zero. The principal considers abandonment only if XL < B + tot + th.;IJItLIyu,v(2t)9uA.J(2t1)u(lt1)F%R3’c5’AGEScco,iDSTPGEs‘tossVofFHIGH(t21bIGHCONIN.*;FF0P[f(tg.€cEWEDB XLj:0XL Buv(t)UA ujijFigure3.1:DecisionTreeDepictingtheAgent’sSequentialDecisionProblemChapter 3. The Model and Two Single-Stage Settings 19The principal seeks to motivate the agent to take t15 and choose some cutoff point , such that ifhe observes y < , the agent abandons the project, while if he observes y > , he continues with theproject. The agent’s abandon/continue decision constitutes a level of communication to the principalof the former’s private information. An abandonment (continuation) implies that p < (>)U. Theagent’s effort is not observable, though his decision of whether to continue is observable. However, theprincipal cannot determine why the abandonment/continuation decision was made. For example, whenthe principal observes an abandonment decision, he cannot determine whether the agent has shirked inthe first stage or if the information signal received indicates unfavourable conditions.If the agent takes t11, the possible cash flows are B if the project is abandoned, and XL if the projectis continued. On the other hand, if the agent takes {tlh,t25jif continue}, there are a total of threepossible gross cash flows, namely, B, XH and XL. The probability of obtaining each cash flow dependson the cutoff point . Let PBS, PHS and PLS denote the probability of obtaining cash flow B, XH andXL, respectively, given the agent picks {tis,t2slifcontinue} and .PBS =flPH1z= J P(XHII1S,t25,y)f(y)dyS2— p1PLS = f [1— P(XHItlh,t25,y)]f(y)dy==- 2 (3.1)Chapter 3. The Model and Two Single-Stage Settings 203.2.2 The ProblemThe principal’s problem is to choose a compensation contract and cutoff point that induces the agent tochoose high effort and the desired investment. Following Grossman and Hart (1983), we decompose theprincipal’s problem into two parts: (i) the Contract Choice Problem in which the principal identifiesthe optimal compensation contract for inducing high effort for each feasible cutoff E (0, 1); (ii) theCutoff Point Selection Problem in which the principal identifies the cutoff point that maximizeshis expected net profits. Assume that E (0, i). Letu = U(wi + W2(XL))= U(wi + W2QCH)) =Ud = U(w5+wk)=U(wd).Observe that the contract can be represented by either (WI, wh, w3) or (UI, uh, ud) with Wp =U1(u) =p = 1, h, d.Before we analyze the general problem, we consider the following two special cases:1. No first-stage moral hazard.2. No second-stage moral hazard.3.3 Second-stage Moral Hazard, No First-stage Moral Hazard3.3.1 The ModelWe consider a simple scenario in which the agent chooses effort level only once, after he has observedthe signal and decided if the project should be continued or abandoned. The time-line of the game is asfollows:4The cases for = 0 and = are not interesting. If = 0, no abandonment is provided for in the contract. If 1,the principal will never take up the project in the first place, since he will definitely be abandoning it after the first period.Chapter 3. The Model and Two Single-Stage Settings 211. At i = 0, the principal and the agent enter into a compensation contract.2. A signal, y on the viability of continuing the project is privately observed by the agent. He revisesthe probability of high cash flow, and accordingly decides if the project is to be continued orabandoned.3. If the project is abandoned, the agent’s employment is terminated and he is paid w3.5 The agententers the job market and earns a wage of w,,. The principal faces alternative investment withreturn B.4. If the project is continued, the agent chooses the effort level, incurring disutility V(tm).5. At i = 1, the outcome x1, j = L, H is publicly observed.6. The agent is paid his wage wfr).The principal seeks to motivate the agent to choose some cutoff point Q, such that if he observesy < , he abandons the project, while if he observes y> , he continues with the project. If the projectis continued, the principal seeks to motivate the agent to choose the high effort level.This case is related to the literature dealing with post-contract information. In this strand of literature, it is usually assumed that the agent is committed to the firm. Even when the information receivedis not favorable, there is no provision in the contract to allow the agent to leave the firm. Melumad(1989) permits the agent to quit the contract if the information is unfavorable, upon the payment ofdamages and shows that allowing for a breach results in a Pareto improvement. However, he does notexamine whether the agent’s incentive to quit is aligned with the principal’s incentive.5Note that to3 may either be positive, i.e., the agent receives severance pay; or negative, i.e., the agent pays a penaltyto withdraw front the contract.Chapter 3. The Model and Two Single-Stage Settings 223.3.2 First-best SolutionFirst, we consider a setting where the agent’s effort is observable; thus, there is no incentive problem.The principal offers a compensation package to the agent such that the agent is indifferent betweencontinuing or abandoning the project. We assume that when the agent is indifferent, he decides inthe principal’s best interest. If the project is abandoned, the transfer payment is w3. If the project iscontinued, the agent receives compensation wj. Thus, the agent is indifferent between abandoning orcontinuing the project when U(w8 + wk) = U(wf)v(th). The participation constraint requiresU(w8 + wk) + U(wf)v(th)(1—=Consequently, the first-best compensation package is:= WWk,Wf = ?Ji+th.Whether w is positive or negative depends on the level of wk.The principal’s expected payoff, for a given cutoff , isSubstituting for w3 and Wf, the first-order condition with respect to is:B—ti’+wk—xH—(1—)xL+tE’+th=O (3.2)Note that the second-order condition on is —(XH—XL), which is negative. This implies that theprincipal’s objective function is strictly concave, thus the first-order condition is necessary and sufficientto obtain the optimal cutoff. The first-best cutoff is0= B+wk±t,-LifxL <B+wk+th <XH1 ifB+wk+th>XH.Chapter 3. The Model and Two Single-Stage Settings 23To provide an intuitive explanation for y, we rewrite the first-order condition evaluated at y* in thefollowing manner:B — [z — wk] = J3 + (1——[z + t5].The left-hand side of the equality is the return to the principal if the project is abandoned, while theright-hand side of the equality gives the expected profitability if the project is continued. Thus, fory (, < y*) it is optimal to continue with (abandon) the project.3.3.3 Second-best SolutionWe now consider the setting in which the agent’s effort is not observable. The principal wants to motivatethe agent to choose some cutoff and to take th if the project is continued. If the agent is offered thecontract (ui, uh, ud), and he takes effort th, then his expected utility with cutoff is:EHh = yUd + (1 —2)u5v(t + (1 —The agent’s first-order condition on is:— Ytthv(th) — (1 — )uiv(t5) = 0.Hence, the agent will choose0 ifud<ulv(th)=if ujv(h) <d < UhV(th)1 ifuduhv(th).We note that for an interior solution, the agent’s second-order condition on is also satisfied.The agent’s first-order condition with respect to implies the following:• For y < , the agent is better off abandoning the project than continuing it at effort level th.• For y > , the agent is better off continuing the project at effort level th than to abandon it.Chapter 3. The Model and Two Single-Stage Settings 24However, the principal must ensure that for y < ‘, the agent will prefer to abandon the project ratherthan continue it with effort level t1. Also, for y , the principal must ensure that the agent prefersto continue the project with effort level th than with effort level t1. This is achieved by imposing thefollowing two constraints:Ud ‘UI,Y’UhV(h) + (1 — > uj.Since the agent’s first-order condition on implies that ud = yuhv(th) + (1 — )Zlv(th), one of these twoconstraints is redundant. For subsequent analysis, we use the constraint ua U!.The Contract Choice ProblemThe contract choice problem for each feasible cutoff E (0, 1) is given as follows:[P3.1] min{Ud,Uh,U,} PB(h(Ud) — wk) + PHh(tLh) + pLh(u!)s,t. PB’Ud + PHUhV(th) + PLUIV(th) > K,Ud > Uj,and Ud — YUhV(th) — (1 — )ujv(th) = 0.The first constraint is the participation constraint and ensures that the agent’s expected utility fromjoining the firm is at least as high as his reservation utility level. The second constraint ensures that theagent will weakly prefer to abandon the project rather than continue it with effort level t1 if he observesthat y < . The last constraint is the agent’s first-order condition on , and it requires that for eachvalue of the information signal y, the specified abandonment/continuation decision is optimal for theagent. The second and third constraints together also ensure that the agent weakly prefers to continuethe project with effort level th than to continue with effort level t1 if he observes that y > .Chapter 3. The Model and Two Single-Stage Settings 25Let .i, p and ji be the Lagrange multipliers of the first, second and third constraints respectively.Using first-order conditions, we obtain the following characterizations of the optimal compensationpackage.h’(i2d) =h’(h) =____h’(111) = )1v(th)—-—1(1—)v(th).Lemma 3.1: At the optimal solution, all three constraints are binding, with )i > 0, p’ > 0 and 77 <0.The optimal expressions for d, u and uh are given as follows:6- - 2IUd—Ul —2v(th) — (vQh) — l)(1 + ñ2)’2K{iv(th) — v(th) + 1)and Uh = . (3.3)V(th)[21V(th) — (vQh)— 1)(1 + p2)]Note that ftd < K = Uth). Recall that ud U(w + Wk). Thus, w5 < til — wk. wk is the market’semployment alternative if the agent abandons the project after he receives the information on the project.w may be positive (i.e., the principal pays the agent a severance pay for termination of the contract) ornegative (i.e., the agent pays the principal a penalty to withdraw from the contract), and this partiallydepends on the level of wk. If wk = zi’, then at the optimal solution, w8 is negative. This is consistentwith Melumad (1989). He assumes that wk = til, i.e., the agent’s employement alternative before andafter he obtains the private information on the project is unchanged. He proves that it is never optimalfor the principal to pay the agent a severance pay for termination of the contract.Also, as increases, Üd = ii increases, while fzh decreases, i.e., the spread between ii and jdecreases.7 To provide incentive for the agent to choose high effort if the project is to be continued, the6See appendix 3A for details.7See appendix 3A for details.Chapter 3. The Model and Two Single-Stage Settings 26principal imposes risk on the agent, who is then compensated for bearing this risk in the form of a riskpremium. A high cutoff implies that the project is to be continued and effort is required to be exertedonly when very good information is received. Thus, the higher is , the lesser is the amount of risk thatis needed to be imposed on the agent and the smaller is the spread between h and iij.On the other hand, a low cutoff implies that effort is required to be exerted even when informationis not too favorable. A greater amount of risk is needed to be imposed on the agent. Lemma 3.2 givesthe lower bound of the cutoff for a solution to exist.Lemma 3.2: A necessary condition for a solution to exist is that > (1—(‘ — the last constraint cannot be satisfied. The cost of motivating the agent becomesinfinitely high and no feasible wage contract exists.The Cutoff Point Selection ProblemTo consider the implication of i < 0, we examine the full principal’s problem.max{Ud,Uh,Ul} PB[B — h(ud) +wk]+pH[XH — h(uh)] +pL[XL — h(ui)]s.t. PBd + PHUhV(th) + pLulv(th) K,Ud Ui,and— YUhV(th) — (1 — )ulv(th) = 0.Taking the principal’s first-order condition with respect to and substituting in the agent’s first-ordercondition for , we obtainB — h(ud) + wk ——h(uh)1 — (1—— h(Uj)]— 1V(th)[Uh — uj] = 0. (3.4)Chapter 3. The Model and Two Single-Stage Settings 27Since ‘uh > uj, thenSign (qi) = Sign {B — h(ud) + Wk — — h(Uh)] — (1 — —Lemma 3.1 states that ‘ji < 0. This implies that at the optimal cutoff point 9,B — h(nd) + Wk <*[ h(u5)] + (1 — — h(uj)].The left-hand side of the inequality is the return to the principal if the project is abandoned, wbile theright-hand side gives the expected profitability conditioned on * if the project is continued. Thus, at*, the principal is not indifferent between abandoning and continuing the project. He strictly prefersthat the project be continued at * and is indifferent at a cutoff point lower than From the pointof view of the principal, the second-best contract motivates the agent to overabandon the project. Itis important to note that this conflict of interest refers to the abandonment/continuation decision afterthe agent observes the value of p.The agent does not put in any effort before the abandon/continue decision and the principal needsto motivate the agent to choose high effort only if the project is continued. If p > is observed andthe project is continued, the probability of obtaining the high wage payment associated with the goodoutcome is p if the agent works hard. Therefore, as increases, this probability increases and the costof motivating the agent to work hard decreases. While the principal desires a lower cutoff, to keep theexpected compensation cost low, he settles for a higher cutoff.Next, we compare the second-best optimal cutoff point with the first-best cutoff point.Proposition 3.1: The first-best cutoff point is lower than the second-best optimal cutoff point, i.e.,y* <12*.Chapter 3. The Model and Two Single-Stage Settings 28Figure 3.2: Comparison of y with y (no first-stage moral hazard)R: RevenueEC(FB): Expected Compensation Cost (first-best)EC(SB): Expected Compensation Cost (second-best)In the first-best case, the expected compensation cost isEC(FB) = — w) + (1 — )(ü + th).Thus, the slope of the expected compensation cost is given by —(wk + th). At the second-best optimalcutoff point, the slope of the expected compensation cost is strictly more negative than that for first-best.8 As Figure 3.2 shows, this implies that yThus, there exists a range of information signals yE (y,) in which the agent abandons the projecteven though the principal would have chosen to continue if he could observe the agent’s effort. Themoral hazard problem with the effort level choice leads the principal to prefer to motivate a higher cutoffpoint.See appendix 3A for proof0y yE C.’ (F &)E C.’ (SB)Chapter 3. The Model and Two Single-Stage Settings 293.4 First-stage Moral Hazard, No Second-stage Moral Hazard3.4.1 The ModelWe consider another simple scenario in which the agent chooses his effort level oniy once, but in thiscase, the choice is made before he observes the signal and decides if the project should be continued orabandoned. The time-line of the game is as follows:1. At i = 0, the principal and the agent enter into a compensation contract.2. The agent chooses effort, incurring disutility v(tm).3. A signal, y on the viability of continuing the project is privately observed by the agent. He revisesthe probability of high cash flow, and accordingly decides if the project is to be continued orabandoned.4. If the project is abandoned, the agent’s employment is terminated and he is paid w.9 The agententers the job market and earns a wage of wk. The principal faces alternative investment withreturn B.5. If the project is continued, at i = 1, the outcome xj, j = L, H is publicly observed.6. The agent is paid his wage w(z).The principal seeks to motivate the agent to choose the high effort level and some cutoff point ,such that if he observes y < , he abandons the project, while if he observes y > , he continues withthe project.Dye (1983) analyzes the value of communication when the agent receives private information after hehas chosen his effort level. In his setting, the information is not used for any decision-making purposes.9Note that ws may either be positive (i.e., the agent receives severance pay,) or negative (i.e., the agent pays a penaltyto withdraw from the contract).Chapter 3. The Model and Two Single-Stage Settings 30Any value of communication derives from improved risk sharing in the compensation contract, and theagent’s communication constitutes a choice from among a menu of compensation contracts that arecontingent on the outcome. In my model, the agent’s continuation decision is a form of communicationwith a coarse message space, i.e., abandonment implies that y < while continuation implies that y> .My model is similar to Dye (1983) in that information is received and communication takes place afterthe agent has chosen his effort level. However, unlike Dye, the information in my model has decision-making value and the expected gross returns to the principal is different depending on the message. InDye’s model, the private signal received by the agent is correlated with the output but the message fromthe agent has no impact at all on the output.3.4.2 First-best SolutionWhen the agent’s effort is observable, there is no incentive problem. The principal offers a compensationpackage to the agent such that the agent is indifferent between abandoning or continuing the project.If the project is abandoned, the transfer payment is w,. If the project is continued, the agent receivescompensation Wf. Thus, the agent is indifferent between abandoning or continuing the project whenU(w3 + wk) = U(wf). The participation constraint requiresU(w3 + Wk)V(th) +U(w1)v(ts)(1— ) =Consequently, the first-best compensation package is:U)8 W+thWk,U)f W+th.The first-best cutoff point in the present single-stage moral hazard problem is lower than that for the nofirst-stage moral hazard problem. In the present problem, compensation for the agent’s effort is a sunkcost at the time of the continuation decision, thus it is not relevant in the determination of the cutoffChapter 3. The Model and Two Single-Stage Settings 31point. On the other hand, in the no-first-stage moral hazard problem, at the time of the continuationdecision, compensation for effort is a relevant cost in the determination of the cutoff point. The first-bestcutoff point in the present problem is given as follows:0 ifB+wk<XL= B+w,,-1. if XL <B + Wk <XH1 ifB+wk>XH.At the first-best cutoff,B — [th + t1 Wk] = YXH + (1 — y)XL—[u + thJ. (3.5)The left-hand side of the equality is the return to the principal if the project is abandoned, while theright-hand side of the equality gives the expected profitability if the project is continued. Thus, at y y”,the principal is indifferent between abandoning and continuing the project, and for y> y*(y < y*) heprefers to continue with (abandon) the project.3.4.3 Second-best SolutionThe Contract Choice ProblemThe contract choice problem for each feasible cutoff is given as follows:[P3.2] min{Ud,,Ul} PB(h(Ud) — wk) + pHh(Uh) + pLh(ul)s.t. [pBud + PHUh + PLUI]V(th) K,[PBttd +PHUh +PLUIJV(th) > Ud,[PB’ud +PHUh +PL’Ul]V(th) uj,and Ud — yuh — (1 — )ui = 0.The first constraint is the participation constraint and ensures that the agent’s expected utility fromjoining the firm is at least as high as his reservation utility level. The second constraint ensures that theChapter 3. The Model and Two Single-Stage Settings 32agent will weakly prefer to choose the high effort level and cutoff rather than choose effort level t1 andabandon the project always. The third constraint ensures that the agent will weakly prefer to choose thehigh effort level and cutoff rather than choose effort level t1 and continue the project always. The lastconstraint is the agent’s first-order condition on , and it requires that for each value of the informationsignal y, the specified abandonment/continuation decision is optimal for the agent. To motivate theagent to work and choose , must be greater than ud, otherwise the agent will never work but willchoose to abandon the project always. Also, must be less than ud, otherwise the agent will neverabandon the project. This implies that the third constraint is never binding, thus the Lagrange multiplierfor the constraint is zero. For subsequent analysis, we ignore the third constraint.Let 2, 2 and ‘q be the Lagrange multipliers of the first, second and fourth constraints respectively.Using first-order conditions, we obtain the following characterizations of the optimal compensationpackage.h’(id) = V(th)[)2 + 1L2] — +PB PB7)2Yh’(Üh) = v(th)[.A2 + P2] —= v(th)[2 + P2] —Lemma 3.3: At the optimal solution, )‘2 > 0, P2 > 0 and )2 > 0.The optimal expressions for ud, uz and u, are given as follows:ttd = K,-______________= ..(1 — y)v(th)and Uj = . (3.6)(1— y)2v(th)As increases, 11h increases at an increasing rate, while i decreases at an increasing rate, i.e, the spreadChapter 3. The Model and Two Single-Stage Settings 33between €th and ii increases at an increasing rate.10 To motivate the agent to exert effort, the principalneeds to impose risk on the agent and then compensate him in the form of a risk premium. Here, effortis exerted before information is received. At the point of effort selection, the probability of obtaining afavorable outcome is — 2), which decreases as increases. Thus, the higher is , the lower is theprobability that outcome is informative of the agent’s effort and a compensation contract with a biggerspread is necessary to motivate effort. Lemma 3.4 gives the upper bound on for a solution to exist.Lemma 3.4: Necessary conditions for a solution to exist are:1. < and2. vQh) < 2.if > [2)(t)] ith becomes positive and no feasible wage contract exists. Here, the agent needs to exerteffort before the abandon/continue decision. If the probability of abandoning the project is very high,the cost of motivating the agent to work hard in the first stage is too excessive.The Cutoff Point Selection ProblemTo consider the implication of ij2 > 0, we examine the full principal’s problem.max{Ud,h,U} PB[B — h(ud) + Wk] +PH[XH — h(uh)] +pL[XL — h(ui)]s.t. [PBUd +PHUh +PLU1JV(th) > K,[PBud + PHUC + PLUI]V(th) Ud,and — yUh — (1—= 0.‘°See appendix 3A for details.Chapter 3. The Model and Two Single-Stage Settings 34Taking the principal’s first-order condition with respect to and substituting in the agent’s first-ordercondition for , we obtainB— h(ud) + Wk — ,*[x — h(uh)] — (1 — — h(uj)] — 712[tth — uj] = 0. (3.7)Since nh > uj, thenSign (2) = Sign {B — h(ud) + Wk — y*[XH — h(uh)] — (1 — *)[XL —Lemma 3.3 establishes that 72 > 0. This implies that at the optimal cutoff point ,B— h(ud) + Wk > *[xH — h(uh)] + (1 — — h(u)].The left-hand side of the inequality is the return to the principal if the project is abandoned, while theright-hand side gives the expected profitability if the project is continued. Thus, at , the principalis not indifferent between abandoning and continuing the project. He strictly prefers that the projectbe abandoned at * and is indifferent at a cutoff point greater than . From the point of view ofthe principal, the second-best contract motivates the agent to overcontinue the project. It is importantto note that this conflict of interest refers to the abandonment/continuation decision after the agentobserves the value of y.In this setting, the agent puts in the effort before the abandon/continue decision. The principal needsto motivate the agent to work hard and then choose cutoff . At the point of choosing effort input, theprobability of obtaining the high wage payment is (1 — p2). As increases, this probability decreasesand the cost of motivating the agent to work hard increases. While the principal desires a higher cutoff,he settles for a lower cutoff to keep the expected compensation cost low.Unless the agent has invested high effort level in the project, he would not choose to continue theproject. Overcontinuation in the project increases the probability that the project outcome is informativeabout the agent’s effort level. Thus, a compensation contract with a smaller spread is enough to motivateeffort, and the savings inherent in a smaller risk premium to the agent offset the cost of overcontinuation.Chapter 3. The Model and Two Single-Stage Settings 35Figure 3.3: Comparison of with y (no second-stage moral hazard)R: RevenueEC(FB): Expected Compensation Cost (first-best)EC(SB): Expected Compensation Cost (second. best)Next, we compare the second-best optimal cutoff point with the first-best cutoff point.Proposition 3.2: The first-best cutoff point is higher than the second-best optimal cutoff point, i.e.,y5 >?.The slope of the expected compensation cost for the first-best case is —wk, and for the second-bestcase, the slope at the optimal cutoff point is strictly greater than —wk.’1 Thus, as Figure 3.3 indicates,this implies that y > y.Thus, there exists a range of information signals y E (y , y) in which the agent continues the projecteven though the principal would have chosen to abandon it if he could observe the agent’s effort. This canbe related to the sunk cost phenomenon, in which firms appear reluctant to abandon their projects evenwhen the information received indicates that the probability of a good outcome is low. The moral hazardproblem with the effort level choice results in an induced moral hazard problem in the abandon/continue11See appendix 3A for details.!I Y_c.(FS)Chapter 3. The Model and Two Single-Stage Settings 36decision which leads the principal to prefer to motivate a lower cutoff point.This result is consistent with Balakrishnan (1991) who examines a single-stage (first-stage) moralhazard problem with precontract information asymmetry on the agent’s types. For a set of agent’stype, the agent chooses a risk free project if he does not work to acquire information. This is termed thedefault project, i.e., the project that would have been chosen with the precontract information alone. iledemonstrates that for the set of agent types whose default project is the risk free project, overinvestmentin the risky project occurs. In our model without precontract information, the default option for theagent if he does not work hard is to terminate the project (the risk free option) and we show thatovercontinuation of the project (the equivalence of overinvestment in the risky project) occurs.3.5 ExampleUsing the following numerical values, we show how the expected compensation costs and the optimalwage levels vary as the cutoff point varies:r=l= 0.36a = 0.693 U(zD) = —0.5ti, = 0.2 =‘ V(ts) = 1.2214Figures 3.4, 3.5 and 3.6 relate to the no-first-stage moral hazard problem, while figures 3.7, 3.8 and 3.9relate to the no-second-stage moral hazard problem.Chapter 3. The Model and Two Single-Stage Settings 373.5.1 No First-stage Moral Hazard ProblemFigure 3.4 shows that the expected compensation cost decreases as the cutoff point increases for boththe first-best and second-best cases. While the rate of decrease is constant for the first-best case, therate of decrease for the second-best case is decreasing in 9. Figure 3.5 shows the optimal wage levels forthe second-best case as the cutoff point varies. Observe that tv is decreasing in 9, while both tv and w3are increasing in 9. We also note that when the cutoff point is very low, U)6 is negative, i.e., the agentpays the principal a penalty to withdraw from the contract.We expand the example by varying the levels of XH, while keeping the values of B and XL constant.B = -5XL = —30.Figure 3.6 shows how the optimal cutoff points vary as Xjj varies. We observe that y” < 9*. Also, thedeviation from the first-best cutoff is larger at higher levels of s, when the optimal cutoff is lower. Werecall that the rate of decrease of the second-best expected compensation cost decreases as 9 increases,while the rate of decrease of the expected compensation cost in the first-best case is constant. This implies that when the principal’s desired cutoff is low, greater savings in expected cost result from movingto a higher cutoff than when the principal’s desired cutoff is high. Thus, we observe greater deviationfrom the first-best cutoff when the optimal cutoff is lower.3.5.2 No Second-stage Moral Hazard ProblemFigure 3.7 shows the expected compensation costs as the cutoff point varies. In the first-best case, thecost decreases at a constant rateas the cutoff point increases, while in the second-best case, the cost isconvex. It decreases and subsequently increases at an increasing rate in 9. Figure 3.8 shows the optimalChapter 3. The Model and Two Single-Stage Settings 38wage levels for the second-best case as the cutoff point varies. Observe that wh is increasing in , whilewi is decreasing in .Figure 3.9 shows how the optimal cutoff points vary as XH varies. We observe that y’ > Also,the deviation from the first-best cutoff is larger at lower levels of ZH, when the optimal cutoff is higher.We recall that in the first-best case, the expected compensation cost decreases at a constant rate, whilein the second-best case, the expected cost is convex, and at higher , the expected cost increases at anincreasing rate in . This implies that when the principal’s desired cutoff is high, greater savings inexpected cost result from moving to a lower cutoff than when the desired cutoff is low. Thus, we observegreater deviation from the first-best cutoff when the optimal cutoff is higher.Chapter 3. The Model and Two Single-Stage Settings 39No First-Stage Moral Hazard CaseFigure 3.4Behavior of Expeded Compensation Cost1.4 -1.3 -1.20C-)C90lf)CE0.9-V0.8-a ssU0.7 -0.6 -0.5- I •0.2 0.25 0.3 0 35 0.4 0.45 0.5 0.55 0.6Cutoff PointFigure 3.5Behavior of Wage Levets2.8-2.62.42.221 .81.61.4-J120’01-0.8 -0.60.40.20—0.2- I I0.2 0 25 0.3 0 35 0.4 0.45 0.5 0.55 0.6Cutoff PointChapter 3. The Model and Two Single-Stage Settings 40Figure 3.6Behavior of Optimal Cutoff with X0.52-0.5 -0.48 -0.46 -0.44 -0.42 -0.4-00.38-2 0.36-0.3400.320.30.280.260.24 -0.22 -0.2 -0.18—20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100chapter 3. The Model and Two Singl&Stage Settings 41No Second-StagtMoral Hazard CaseFigure 3.7Behavior of Expected CompensottOfl Cost1.20.6 • . I • I • I • I • I0 2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6Cutoff PointFigure 3.8Behovior of Wage Levels0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6Cutoff PointChapter 3. The Model and Two Single-Stage Settings 42Figure 3.9Sehovior of Optimol Cutoff Point X052 -0.50.480.46 -0.44 -0.42 -0.4-00.38-2 0.36-0 0.34-0.32- U0.3-0.28-0.26 -0.24 -0.22 -0.2 -0.18—20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100Chapter 3. The Model and Two Single-Stage Settings 43Appendix 3A(I) No First-Stage Moral Hazard(1) Proof of Lemma 3.1:From Proposition 2 of Grossman Hart (1983), we know that the participation constraint is binding.It is obvious that at least one of the second and third constraints of problem [P3.1] is binding.• Second constraint is binding: The proof is by contradiction. Suppose not. Then p’ = 0 and= A +PB/ 1iV(th)h (uh) = .1v(th) —________h’(ui) = Alv(th)— PHIf ii > 0, the agent prefers to abandon the project always.If ii = 0, a fixed wage contract results and the agent has no incentive to choose the high effort.If ‘ji < 0, the second constraint is violated.Therefore, at the optimal solution, the second constraint is binding and ui > 0.• Third constraint is binding and i <0: The proof is by contradiction. Suppose that ji > 0. Thenh’(ud) > )1,h’(’uh) Alv(th),h’(ui) < )1v(th).The agent will strictly prefer to abandon the project always. Therefore, at the optimal solution,7ji <0.(2) Derivation of d, h and iSince the second constraint is binding (Lemma 3.1), d = iii. Using the agent’s first-order condition onChapter 3. The Model and Two Single-Stage Settings 44v (t,. )2,., ud = = The agent’s expected utility if he takes th is as follows:EH = 2d + —2)2hv(th) + —UhV(th)[v(th) — {v(th) - 11(1 + p2)]— v(th)—v(th)+1Since EH = K,K[iv(th) — v(th) + 1]=v(th){v(th) — [v(th) — 1J(1 + 2)]2K[v(th) — vQh) + 11= v(th)[2v(th) — [v(th) - 11(1 + 2)Jdiih — 2K(v(th) — 1)[v(th)(1 + 2) — 2(v(th)— 1)1d— — [vQh) — 11(1 + 2)]2< 0.By substitution, we obtain the expressions for 12d and for a given .2KUdu1 =[2iV(th) — [V(th) — 11(1 + p2)]düj — —2K(v(th) — — 2)— [2v(th) — [vQh)— 1](1 + 2)]2> 0.(3) To show that Ud < K2Let d = çbK, where = We prove that > 1. Suppose not. Then2v(th)—(v(th)—1)(1+ 2(vQh)1)[1--2—2k] 0(v(th)—1){1—]2 < 0,which cannot hold. Therefore, q’> 1 and 11d <K.Chapter 3. The Model and Two Single-Stage Settings 45(4) Proof of Lemma 3.2Since iLd = i (Lemma 3.1), the third constraint of [P3.1] can be expressed as follows:V(th)uh = Ud[yv(h) — v(th) + 1].For the constraint to hold, > (1—(5) Proof of Proposition 3.1:At the optimal cutoff point (see equation (3.4)),(B — h(Ud) + Wk) — *(xH — h(uh)) — (1 — *)(x — hQuj)) — 111[UhV(th) — U1V(th)] = 0For purpose of this proof, we redefine some variables:U(wh)v(th) = Uh= wj = h(Üh)+th.U(wj)v(th) == wj =Then the agent’s first-order condition on is:ZLd = YUh + (1—Rewriting the principal’s first-order condition on , we have:(B—hQud) + Wk) — j*( — h(h) — th) — (1 — *)(x — h(Lj) — t) — 1[Üh — ni] = 0.It can be rewritten as follows:B + wk + th — lix — (1 —*)= h(ud) — *h(Üh) — (1 — *)h() + j1[fih—flu.Chapter 3. The Model and Two Single-Stage Settings 46Since Ud = YUh + (1 — )u1, and the agent is risk averse, therefore, h(ud) < *h(fih) + (1 —Lemma 3.1 states that Ji <0 and for h > j, this implies thatB+wk+th_*XH_(1_*)xL<0. (3.8)Recall that the principal’s first-order condition with respect to the first-best cutoff point y* is given by(equation (3.2)):B + wk + th YXH — (1 — y*)z = 0.From section 3.3.1, the principal’s objective function under first-best is strictly concave and is given bythe following expression:EP = [B ± Wk + th] + (1 — )xH + (1 —)2xL — W — th. (3.9)The derivative of EP with respect to is given by:EP’ = B + Wk + th — YXH — (1 — )xL. (3.10)We note that EP’ = 0 at y and EP’ <0 at . This implies that > y.(6) Slope of Expected Compensation CostIn the first-best case, the expected compensation cost is:EC(FB) = (W—Wk)+(1—y)(W+th).Thus, EC’(FB) =— — — th= —(wk-i-th).Let R() denote the expected gross return at cutoff , Then R’ = B—— (1—)xL. In the secondbest case, at the optimal cutoff R’ = EC’(SB). The proof of Proposition 3.1 (see (3.8)) indicates thatR’ < —(wk + th). This implies that at the second-best optimal cutoff, EC’(SB) < —(wk + th). Thus,EC’(SB) <EC’(FB).Chapter 3. The Model and Two Single-Stage Settings 47(II) No Second-Stage Moral Hazard(1) Proof of Lemma 3.3:• Second constraint of problem [P3.2] is binding: The proof is by contradiction. Suppose not. Then0 andh’(ud) =PB/ ?)2Yh (uh) = )‘.2v(th) —h’(u) = 2V(th)- 2(1 )If 72 > 0, the agent will not work and prefers to abandon the project always.If = 0, a fixed wage contract results and the agent has no incentive to choose the high effort.If ‘72 <0, the agent will never abandon the project.Therefore, at the optimal solution, the second constraint is binding and p > 0.• Fourth constraint is binding and 2 > 0: The proof is by contradiction. Suppose that < 0.Thenh’(ud) < vQh)[.\2 + P2],h’(uh) v(th)[)i2 + P2],> v(t)[. + P2].The agent will strictly prefer to continue the project always. Therefore, at the optimal solution,72 > 0.(2) Derivation of ud, h and i1:From the agent’s first-order condition on , ü = The agent’s expected utility if he chooses thChapter 3. The Model arid Two SingleStage Settings 48is as follows:EHh = [d + —2)h + —)2Ü1]V(th)== v(th)[(1 + üd + (1 —Since EH = = K (Lemma 3.3),(1—2h =K[2—v(th)(1+)]4’ U/j =(1 — y)v(th)di — —2K[v(th) — 11dQ — (1—> 0.d2iih> 0.dyBy substitution, we obtain the expression for j for a given .K__U =(1—y) l—y)— K[v(th)(1 + 2) — 2]— (1—)2v(th2K[v(th)— 1I(1 + )(1—< 0.d2u11< 0.dy2(3) Proof of Lemma 3.4:— K[2—v(th)(1+)]Uh —(1—y)v(th)Chapter 3. The Model and Two Single-Stage Settings 49Since ‘uh is negative, thusv(th)(1+) < 22—v(th)=.y<V(th)For to exist, V(th) < 2.(4) Proof of Proposition 3.2:The expected compensation cost at is:EC() = [h(ud) Uk] + -2)h(uh) +Totally differentiating EC() with respect to , we obtain the following:EC’ = h(ud) — h(uh) — (1 — i)h(u1 — Uk+ h’(ud) + -2)h’(uh) + —At the optimal cutoff, R’ = EC’. Thus,B + Wk —Y5H — (1 —*)= h(ud) — i*h(uh) — (1 —+ *h(ud) + (1 —*2)h(uh) + (1 —The optimal compensation contract is (equation (3.6)):Ud = K,K{2—v(th)(1+)]Uh(1— y)v(th)K[v(th)(1 + 2) — 2]and Uj =(1—Also, h(u,) = — ln(—u). Through substituition, differentiation and rearrangement of the terms, weobtain the following:h(ud) - *h(u) — (1- *)h(u) +1(-*2)hI(u ) + (1 -Chapter 3. The Model and Two Single-Stage Settings 50— 1 2(v(th)— 1)2(1 +*)(1 —in— r ‘[vQh)(1 + *2) — 2*][2 — v(th)(l + *)] — v(th)(l + *2) — 2*invQh)(l + *2) — 2*- [2-v(th)(1+?)](1-)(3.11)Let2(v(th) — 1)2(1 + ) (1 —_____________________________— in[v(th)(1 +2) _.2][2 vQh)(1+)] vQh)(1+2)— 2vQh)(1 + 2) — 2(3.12)yin[2- v(th)(1 + )](1-Using the foiiowing steps, we prove that for 1 < v(th) <2 and 0 < < 2—v(th) P > 0.v(t,)1. When v(Ih) = 1,(1_)2inP = 0_[in1 )2+Y (i_)21=0.2. For each feasible cutoff , let P denote the derivatives of P with respect to v(th).— 4(v(th) — 1)[v(th)3(v(th — 2) +2(2v(2th) — 3v(th) + 4) + v(th)(v(th) — 4) + v(th)jPv—v(th)[v(th)2— 2 +v(th)][v(th) + v(th) — 212= M * [v(th)3( — 2) +2(2v(2th) — 3v(th) + 4) + v(th)(v(th) — 4) + v(th)],where M = 4(v()—1)v(t,)[v(t,)2 2+v(h)12[v (th)+V(h) 212 which is strictly greater than zero.3. LetQ = v(th)3(v(t — 2) +2(2v(2th) — 3v(th) + 4) + V(th)(V(th) —4) + v(th)= v(2th)[ + 2 + 1] — vQh)[23+ 32 + 4 — ] + 42Q is convex in v(th) and reaches a minimum at a vaiue of v(th) < 1. This impiies that for v(th) 1,Q is monotonicaliy increasing.4. When v(th) = 1, Q = (1 — )3 > 0. This implies that for v(th) 1, Q > 0.Chapter 3. The Model and Two Single-Stage Settings 515. Recall that P,, = M * Q. We conclude that for v(th) 1, P > 0.6. If P is monotonically increasing for v(th)> 1 and P = 0 for v(th) = 1, then P > 0 for v(th) > 1.Therefore, at the optimal cutoff ,h(ua) — *h(u) (1 *)h(u) + (1 —*2)h!(u) + (1 — * )2h1 (U ) > 0.This implies thatB+wk Y XH_(1y*)XL >0.Recall from equation (3.5) that at the first-best cutoff,B+wk yx _(1_y*)XL =0.The principal’s objective function under first-best is given by the following expression:(3.13)The derivative of EP with respect to is given by:EP’ = B + wk — YXH — (1 — )xL. (3.14)The second-order condition on is negative, thus EP is strictly concave in . We note that EP’ = 0 atand EP’ > 0 at . This implies that * <*(5) Slope of Expected Compensation CostIn the first-best case, the expected compensation cost is:EC(FB) = :;(zi + th — Wk) + (1 — )(ti + th).Thus, EC’(FB) == Wk.Chapter 3. The Model and Two Single-Stage Settings 52In the second-best case, as the proof for Proposition 3.2 indicates, at the optimal cutoff point,EC’(SB) > Wk.Therefore, EC’(SB) > EC’(FB).Chapter 4Analysis of the Multi-Stage SettingWe analyze the general model where there are two stages of moral hazard. In section 4.1, we introdncea benchmark case in which information is publicly observable but not contractible. In section 4.2, boththe first-best and the second-best cases for the general model are analyzed. We examine if there is anyvalue to communication of the specific value of the information in section 4.3. We provide a numericalexample in Section 4.4. All proofs are provided in the appendix.4.1 Benchmark Case4.1.1 The ModelConsider as a benchmark case, the cutoff that would be employed if the agent’s information is publiclyobservable but not contractible. Thus, the principal makes the abandon/continue decision based on y,but y cannot be used as an argument in the agent’s compensation. We assume that if the agent takes tij,the information signal p = 0 is observed, while if the agent takes tlh, the information signal p e (0, 1)is observed. Since p is not contractible, the agent will not be penalized if p = 0 is observed and theprincipal abandons the project.53Chapter 4. Analysis of the Multi-Stage Set ting 544.1.2 Analysis of the ProblemThe Contract Choice ProblemFor a given cutoff , the principal selects the compensation contract to solve the following:[P4.1] min{Ud,Uh,U,} PBh(h(Ud) — wk) + pHhh(uh) + pLhhQul)s.t. PBhUdV(th) + PHhUhV(2th) + PLhU1V(2th) > K,PBhUdV(th) + PHhUhV(2th) + PLhU1V(2th) Ud,and YUhV(th) + (1 — I)uzv(th) > u,where we recall from Chapter 3, equation (3.1) that PBh = th PHh = (1 — p2), and PLh (1The first constraint is the participation constraint and ensures that the agent’s expected utility fromjoining the firm is at least as high as his reservation utility level. The second constraint is the incentiveconstraint; it requires that the agent’s expected utility from working hard in both periods weakly exceedsthe utility level he could obtain by taking tjj. Recall that if the agent takes t11, information signal y = 0is observed and the principal abandons the project. The third constraint ensures that if the project iscontinued, the agent will weakly prefer to take t2h than t21.For a given , there are three constraints and three unknowns. All three constraints are binding,thus for a given , the optimal contract (id, uh, ii1) is uniquely defined as follows:= K,2K[1 — (1 — )v(t5][1 —Uh =—)[1 + — vQh)(1—2K[1—v(t5)] /and ut = . 4.1)—y)[l + y — v(th)(1—Chapter 4. Analysis of the Multi-Stage Setting 55As 9 increases, ilj increases and the spread between ilk and ü1 decreases. Intuitively, the abandon/continue decision made by the principal will determine if the agent will face a riskless or riskycompensation package. If the project is abandoned, the agent obtains a riskless payment, ild. If theproject is continued, the agent receives a risky compensation package. If the probability that the agentreceives a risky pay package is low, the agent will require a lower risk premium. Thus, if the principalchooses a high cutoff point, then the probability that the agent receives a risky pay package is low,and the agent will require a lower risk premium as incentive to choose {tlh, t2hlif continue}. Thus, as 9increases, the spread between ilk and ñj decreases. Also, the probability that the agent must be compensated for his second-stage effort decreases. Therefore, the expected compensation cost is monotonicallydecreasing in 9.Lemma 4.1 gives the upper and lower bounds of 9 for a feasible solution to exist. From the twosingle-stage moral hazard problems in chapter 3, we know that the second-stage moral hazard problemresults in the lower bound (Lemma 3.2) while the first-stage moral hazard problem determines the upper bound (Lemma 3.4). When 9 is too low, motivating the agent to work hard given poor information(second-stage moral hazard problem) is very costly. On the other hand, when 9 is too high, motivatingthe agent to work hard when the chances of abandonment is very high (first-stage moral hazard problem)is too excessive.Lemma 4.1: Necessary conditions for a solution to exist are:1. (1— uz) < U < v(ts)’ and2. V(th) < 2.If 9 —.y, the second constraint cannot be satisfied, while if U (1 — V(th))’ the third constraint‘See appendix 4A for details.Chapter 4. Analysis of the Multi-Stage Setting 56cannot be satisfied. Since > (1 — for ñ to exist, V(th) < 2.The Cutoff Point Selection ProblemUsing the solutions from the Contract Choice Problem, the principal chooses the cutoff point thatmaximizesPBh[B — Wd(!J) + We] + pHh{XH — Wh(Y)] + kh[XL —The relative levels of B, XH and XL determine what cutoff point the principal seeks to implement. IfXH is relatively high, the principal prefers a lower cutoff point, while a relatively low XH implies thatthe agent prefers a higher cutoff point. Let V denote the optimal cutoff in the benchmark case.4.2 The General ModelWe now analyze the general model in which information is privately observed by the agent. Here, thecompensation contract seeks to motivate the agent to do two things:1. Choose tlh, and t2h if the project is to be continued.2. After choosing tlh and observing the information signal p privately, make the abandon/continuedecision which is in the principal’s best interest.In the benchmark case, the compensation contract only needs to motivate the agent to do the former.4.2.1 First-best SolutionFirst, we consider a setting where the agent’s effort is observable, and thus there is no incentive problem.The principal offers a compensation package to the agent such that the agent is indifferent betweencontinuing or abandoning the project. We assume that when the agent is indifferent, he decides in theprincipal’s best interest. The agent receives severance pay w, if the project is abandoned, and a totalChapter 4. Analysis of the Multi-Stage Setting 57pay package of W1 if the project is continued. Let Ud = U(w3 + wk) and Uf = U(wj). Thus, the agent’sutility if he abandons the project is given by udv(th), while his utility if he continues the project isufv(2th). The agent is indifferent between continuing or abandoning the project if udv(th) = v(2th).The first-best cutoff point is denoted y. The participation constraint requirestldV(th)Y + ufv(2th)(1—=Consequently, the first-best compensation package is:Ws = W—Wk+th,= W+2th.The principal’s expected payoff, for a given cutoff , is(B_wS)+(1_2)xH+ (1_)2xL_(1_)wf.Substituting for w3 and w1, the first-order condition with respect to is:B—t+wk—th—xH—(1—)XL+ti+2th = 0= B + Wk + th — YXH — (1 — )xL = 0. (4.2)Note that the second-order condition on is —(XH — XL), which is negative. This implies that theprincipal’s objective function is strictly concave, thus the first-order condition is necessary and sufficientto obtain the optimal cutoff. The first-best cutoff is0= B+wk+—L ifZL <B+wk+th <XH1 ifB+wk+thXH.The first-best cutoff for the general model is identical to that for problem [P3.1], the no-first-stagemoral hazard case, and is higher than that for problem [P3.2], the no-second-stage moral hazard problem.Chapter 4. Analysis of the Multi-Stage Setting 58We note that past effort does not determine the cutoff level. Only future effort is relevant. To provide anintuitive explanation for y, we rewrite the first-order condition evaluated at y in the following manner:The left-hand side of the equality is the return to the principal if the project is abandoned, while theright-hand side of the equality gives the expected profitability if the project is continued. Thus, for,> * ( < y*), it is optimal to continue with (abandon) the project.24.2.2 Analysis of the Second-best ProblemWe now consider the setting in which the agent’s effort is not observable: The principal wants to motivatethe agent to take tlh and choose some cutoff point . Assume that E (0, 1). If the agent is offered thecontract (ui, uj,, ud), and he takes effort {tlh,t2j,Iif continue}, then his expected utility with cutoff is:EHh = YUdV(th) + (1 —2)uhv(2th) + (1 —)2ujv(2th).The agent’s first-order condition on is:UdV(th) — yUhV(2th) — (1 — )u1v(2th) = 0. (4.3)Hence, the agent will choose0 ifuduzv(th)= if ujv(t) < U < UhV(th)1 ifud>uhV(th).We note that for interior solutions, the agent’s second-order condition on is also satisfied.The agent’s first-order condition with respect to implies the following:2We are unable to establish formally the relationship between the benchmark cutoff * and the first-best cutoff y. Theexample in section 4.4 indicates the following relations:• When xH is relatively high, and y is low, then * > *•• When XH is relatively low, and is high, then <Chapter 4. Analysis of the Multi-Stage Setting 59• For y <, the agent is better off abandoning the project than continuing it at effort level 2h•• For y , the agent is better off continuing the project at effort level t2h than to abandon it.However, the principal must ensure that for y < , the agent will prefer to abandon the project ratherthan continue it with effort level t21. Also, for y , the principal must ensure that the agent prefersto continue the project with effort level t2h than with effort level i21. This is achieved by imposing thefollowing two constraints:d Uj,yUhV(tJ-,) + (1 — ñ)uzV(th) uj.Since the agent’s first-order condition on implies that d = yuj,V(th) + (1— 0u1V(th), one of these twoconstraints is redundant. For subsequent analysis, we use the constraint ud > tg. This constraint alsoimplies that if the agent takes tj, he is better off abandoning the project at i = 1 than to continue withit.The Contract Choice ProblemFor a given cutoff , the principal selects a compensation contract which solves the following:[P4.2] min{UJ,Uh,Uf} PBS (hQud) — Wk) + prnzh(us) + PLhlz(tzz)s,t. PBhttdV(th) + pHhtLhV(2tj,) + PLhtult’(215) K,PBSUdVQS) + PHSUSV(2t5) + PLSU1V(2th) Ud,> Uj,and Ud — yzzhv(th) — (1 — = 0.The first constraint is the participation constraint and ensures that the agent’s expected utility fromjoining the firm is at least as high as his reservation utility level. The second constraint is the incentiveChapter 4. Analysis of the Multi-Stage Setting 60constraint and requires that the agent’s expected utility from working hard in both stages, and choosingas the cutoff point, weakly exceeds the utility level he could obtain by taking the lower effort leveland abandoning the project after the first stage. The third constraint ensures that the agent will weaklyprefer to abandon the project rather than continue it with effort level t21 if he observes that y < .The last constraint is the agent’s first-order condition on , and it requires that for each value of theinformation signal y, the specified abandonment/continuation decision is optimal for the agent.Let .\, , I2 and i be the Lagrange multipliers of the four constraints. The principal’s Lagrangianformulation is as follows:L = PBh(h(Ud) — wk) PHhh(Uh) —pLhh(u)+ .\[pBhUdV(th) + PHhhV(2t )+ pLhulv(2th) — K]+ 1LI[ud{pBhv(th) — 1} + UhPHhV(2th) + UIPLSV(2th)]+ p2[UdUI]+ 77[ud — iflhV(th) — (1 —Using first-order conditions, we obtain the following characterizations of the optimal compensationpackage, denoted (d,uih,),h’(72d) =PBh PBh PBhh’(üh) = v(2th)( + P1)—_____PHhh’(21) = v(2th)( + p1) —P2 — j(1 —)v(th)PLh PLhLemma 4.2 establishes the signs of the Lagrange multipliers for the two incentive constraints.Lemma 4.2: At the optimal solution, pi 0 and p2 > 0 with at least one of p’ and P2 strictly greaterthan zero.Chapter 4. Analysis of the Multi-Stage Setting 61The sign of ij, the Lagrange multiplier on the first-order condition on the agent’s choice of cutoff isdetermined later, and as we shall see, it depends on the sign of ii and /2. We note that in the principal’sproblem, for a given , there are four constraints and three unknowns. At the optimal solution, oneof the four constraints is redundant. The participation constraint is always binding. We consider thefollowing three cases:• CaseA:pi>Oandp2>O.• CaseB:j1>Oand=O.• Case C:1=Oandp2>O.Subsequent analysis shows that the value of the optimal cutoff determines which one of the last threeconstraints is redundant, thus which one of the above three cases applies. Lemma 4.3 gives us the valueof that results in case A.Lemma 4.3: The first three constraints are binding if, and only if, = A, where- — V(th) + 1 — /(2v(th) + 1)YA —v(th)Furthermore, in this case, d = ud(A), j =1(A) and ih = uh(A), whered(A) = i1(A) = K,/ — K[2 — /(2vQh) + 1)]and uh.A) — . 4.4v(th) + 1— /(2v(th) + 1)The fourth constraint is redundant, since given the above contract, the agent will find it optimal toselect iA as the cutoff point.The cutoff point A is solely determined by v(tj). As v(th) increases, both A and uh(A) increase.Chapter 4. Analysis of the Multi-Stage Setting 62In case B, the binding incentive constraint is that the contract must motivate the agent to choose{tlh, t2hlif continue} compared to choosing t11 and always abandoning the project. The level of ia isindependent of and is given by ha = K. We derive the following expressions for hh and hi1 for a given— K[2—v(th)(1+)]Uh— (1—)V(2th)K[v(th)(1+ 2)— 2]and uj = . (4.5)(1—)2v(2th)We observe that the optimal expressions for ud, u and uh are similar to those for problem [P3.2], theno-second-stage moral hazard case. (See Chapter 3, equation (3.6).) The moral hazard problem in thefirst stage drives the results of case B.As increases, uh increases at an increasing rate, while hi decreases at an increasing rate. Thus, thespread (hij,—uij) increases at an increasing rate as increases. To motivate the agent to exert effort, theprincipal needs to impose risk on the agent and then compensate him in the form of a risk premium.The binding incentive constraint is that which motivates him to work hard in both stages, and at thatpoint of effort selection in the first stage, the probability of obtaining a favorable outcome is (1 — 2)which decreases as increases. The higher is , the lower is the probability that outcome is informativeof the agent’s effort and a compensation contract with a bigger spread is necessary to motivate effort.In case C, the binding incentive constraint is that if the project is continued, the contract mustmotivate the agent to choose t2h versus choosing t. For a given ,2Kd —UIv(th)[2V(th) — (v(th)—i(i + p2)]’2K[v(th)—v(th)+1]and Uh = . (4.6)v(2th)[2v(th) — (v(th)—1)(1 + p2)]We observe that the optimal expressions for ud, i and uh are similar to those for problem [P3.1], theno-first-stage moral hazard case. (See Chapter 3, equation (3.3).) The moral hazard problem in thesecond stage drives the results of case C.Chapter 4. Analysis of the Multi-Stage Setting 63As increases, fh decreases and il = ud increases, i.e., the spread between fi and iij decreases. Inthis case, the binding incentive constraint is that which motivates the agent to work hard in the secondstage. A high implies that the project is to be continued and effort is required to be exerted only whenvery good information is received. Thus, the higher is , the lesser is the amount of risk that is neededto be imposed on the agent to motivate him to work hard and the smaller is the spread between Gh andUj.Proposition 4.1 demonstrates that to motivate a cutoff point other than A, either case B or case Capplies.Proposition 4.1:1. When the principal wants to motivate a cutoff point = QA, case A applies and rn > 0, p > 0and ,j = 0.2. When the principal wants to motivate a cutoff point > , case B applies and p > 0, P2 = 0and ‘> 0.3. When the principal wants to motivate a cutoff point < , case C applies and p = 0, P2 > 0and ,j < 0.The relative levels of XH, XL and B determine the cutoff point the principal would want the agentto select. If XH is relatively high, the principal will prefer a relatively low cutoff point, whereas if XH isrelatively low, the principal will prefer a relatively high cutoff point.As in the single-stage moral hazard problems, for a feasible solution to exist, there are upper andlower bounds to . From chapter 3, Lemma 3.2, the second-stage moral hazard problem results in theChapter 4. Analysis of the Multi-Stage Setting 64lower bound. When is too low, motivating the agent to work hard given poor information (secondstage moral hazard problem) is very costly. From Lemma 3.4, the first-stage moral hazard problemdetermines the upper bound. When is too high, motivating the agent to work hard when the chancesof abandonment are very high (first-stage moral hazard problem) is too expensive. Lemma 4.4 gives theupper and lower bounds of for a feasible solution to exist in this two-stage moral hazard problem.Lemma 4.4: Necessary conditions for a solution to exist are:3v(t)-1 < < 2-v() and2. v(th) <Outside these bounds, Iih becomes infinitely high and no feasible wage contract exists.Next, we examine the behavior of the required risk premium if the project is continued, as varies.If the project is continued, the principal motivates the agent to choose t2h. Also, the principal seeks tomotivate the agent to choose tlh in stage 1, otherwise there is no gain from continuing the project. Theprincipal provides the necessary incentive for the agent to choose high effort by imposing some risk onhim, and the agent is then compensated for bearing this risk in the form of a risk premium. In Equation(4.7), ir() represents the risk premium and the compensation for effort.U[h(uh) + (1 — )h(ui) — r()j = [uh + (1 — )uh]v(th). (4.7)In the subsequent discussion, for simplicity, we call ir() the required risk premium if the project iscontinued. Besides, it is the element of risk premium that is the critical factor in our results. Let ()(()) denote the required risk premium in the second-best (benchmark) case if the project is continued.31n appendix 4C, we show that if abandomnent of the project is not allowed, the principal will not employ the agent toundertake the project if v(t) >Chapter 4. Analysis of the Multi-Stage Setting 65Lemma 4.5:1. In the benchmark case, the required risk premium if the project is continued is decreasing in ü,i.e., ii’() < 0.2. In the second-best case:• If < VA, the required risk premium if the project is continued is decreasing in , i.e.,*‘(n) <0.• If > VA, the required risk premium if the project is continued is increasing in , i.e.,k’(Q) > 0. Also, the rate of increase of the required risk premium is increasing in .In the benchmark case, the principal makes the abandon/continue decision. When the project iscontinued and if the agent is motivated to work hard when he observes , then he is motivated to workhard when he observes p > . Hence, is a key determinant of the incentive contract and increasingreduces the risk premium imposed on the agent.In the second-best case, if < jj, the severance pay w5 is low and the agent is strictly better offchoosing {t15,t2alif continue} compared to choosing tij and abandon always. The binding incentiveconstraint is that if the project is continued, the contract must motivate the agent to choose t2h versuschoosing t2. At i = 1, as increases, the probability of obtaining the high wage payment associated withthe good outcome increases, the cost of motivating the agent to work hard decreases, and the requiredrisk premium decreases.If > y, the binding incentive constraint is that the contract must motivate the agent to choose{tlh, t2hlif continue} compared to choosing t11 and always abandoning the project. At i = 0, the probability of obtaining the high wage payment is (l — 2) and it decreases as increases. The cost ofChapter 4. Analysis of the Multi-Stage Setting 66motivating the agent to work hard increases and the required risk premium increases as Q increases.The next proposition compares the second-best optimal compensation contract with the benchmarkoptimal contract, for a given .Proposition 4.2: To implement a given ñ, the following relations between the benchmark and the second-best compensation contracts hold.YYA lJ>YA Y<YAtIdtUdI< UduIdK ttd<ttdK (4.8)Uh>tth Uh>UhuiI<iil uI>ulFor a given , the expected compensation cost in tbe second-best case is weakly higher than that in thebenchmark case, and they are equal at =If ii> A, the severance pay for the second-best case equals that for the benchmark case, and thelevel is the same for all ji. The spread between h and i is greater in the second-best case. Asincreases, the spread in the second-best case increases at an increasing rate, while the spread in thebenchmark case decreases.If < I/A, the severance pay in the second-best case is lower than that in the benchmark case,which remains constant at K. The former decreases as decreases. The spread between tLh and Uj inthe second-best case is less than that in the benchmark case. As Q increases, the spread in both thesecond-best and the benchmark cases decreases.Recall that when information is privately observed by the agent, the compensation contract mustmotivate the agent to do two things:Chapter 4. Analysis of the Multi-Stage Setting 671. Choose tlh, and t2h if the project is to be continued.2. After choosing tlh and observing the information signal y privately, make the abandon/continuedecision which is in the principal’s best interest.On the other hand, when information is publicly observable and the principal chooses the cutoff point,the compensation contract only needs to motivate the agent to choose tlh, and t2h if the project is to becontinued. Therefore, for a given 9, the expected compensation cost in the second-best case is weaklyhigher than that in the benchmark case, and they are equal at 9 = ÜA Given the optimal contract at 9A,it is in the agent’s interest to select 9A as the cutoff. The higher compensation cost in the second-bestcase reflects the cost arising from the induced moral hazard problem with the abandon/continue decisionbecause information is not publicly observable.The Cutoff Point Selection ProblemUsing the solutions from the Contract Choice Problem, the principal chooses the cutoff point thatmaximizesPBS{B — Wd(y) + Wk] + PHh{XH — Wh(y)] + l3Lh[xi. —Let 9* represent the optimal cutoff point in the second-best case. The following proposition comparesthe second-best cutoff with the benchmark cutoff.Proposition 4.3:•1f9*9A then y=y=y• If 9* <YA, then 9 <9 <th and ri <0.• If 9* > YA, then 9A <9* <9* and ij >.Chapter 4. Analysis of the Multi-Stage Setting ‘ 68The sign of ,j for 9 tells us if 9* > 9 or 9* <9* and its relation to 9A.We examine the behavior of the expected compensation cost to help us understand why the proposition holds, As 9 increases, two factors simultaneously determine the behavior of the expected compensation cost:1. The probability that the agent’s employment will be terminated after the first stage increases, thusthe probability that the piincipal incurs second-stage compensation cost decreases. This causesthe expected compensation cost to decrease.2. The required risk premium if the project is continued may either increase or decrease, causing theexpected compensation cost to either increase or decrease respectively.If the required risk premium is decreasing in 9, then the expected compensation cost will be decreasingin 9. If the required risk premium is increasing, then the net effect on the expected cost can eitherbe decreasing or increasing, depending on which is the dominant factor. In the benchmark case, fromLemma 4.5, the required risk premium is decreasing in 9; therefore, the expected compensation cost ismonotonically decreasing in 9.We consider the second-best case. When 9 < 9, Lemma 4.5 tells us that the required risk premiumis decreasing in 9. Therefore, the expected compensation cost is monotononically decreasing in 9. Forany 9, the expected compensation cost in the benchmark case is lower than that in the second-best case,and they are equal at 9 = 9A The proof for Proposition 4.3 indicates that for any 9, the slope of theexpected compensation cost in the second-best case is steeper than that in the benchmark case. Sincethe expected gross return is concave in 9, this implies that 9* < 9* < ÜAWhen 9 > 9., Lemma 4.5 tells us that the required risk premium is increasing in 9. If the first factordominates, the expected compensation cost decreases as 9 increases. However, if the second factorChapter 4. Analysis of the Multi-Stage Setting 69dominates, the expected cost increases as increases. The proof for Proposition 4.3 indicates that forany , the slope of the expected compensation cost in the second-best case is strictly greater than thatin the benchmark case. Since the expected gross return is concave in , this implies that * > * > .if * > *, the principal would prefer a lower cutoff point than * if he could observe y and make theabandon/continue decision. There exists a range of information signals y e (*, ) in which the agentabandons the project even though it appears optimal to continue with it. In fact, in the second-bestcase, the moral hazard problem with the effort level choice results in an induced moral hazard problemin the abandon/continue decision, so that the principal prefers to motivate a higher cutoff point. Thisresult indicates that there can arise cases when firms may appear too hasty in abandoning their projects.On the other hand, when < * the principal would prefer a higher cutoff point than if hecould observe y and make the abandon/continue decision. Therefore, there exists a range of informationsignals y (*, *) in which the agent continues the project even though it is not optimal to do so. Infact, in the second-best case, the induced moral hazard problem in the abandon/continue decision leadsthe principal to prefer a lower cutoff point. In the abovementioned range of information signals, theprincipal is better off allowing the agent to continue with the project.Twiss (1992) studies the causes of successes and failures in technological innovation. An examinationof projects which fail leads him to the conclusion that these projects should never have been initiatedor should never have been allowed to proceed thus far in the first place. This is because when he looksback at the information available at the time of project selection or evaluation, he concludes that properuse of this information would have avoided these failures. However, if such information was private tothe agent at the time of project evaluation and the agent makes the abandon/continue decision, thenour results indicate that this phenomenon may not be avoidable in a principal-agent relationship with afirst-stage moral hazard problem. The induced moral hazard problem in the abandon/continue decisionleads the principal to prefer to motivate a lower cutoff point.Chapter 4. Analysis of the Multi-Stage Setting 70Next, we compare the second-best cutoff point with the first-best cutoff.Proposition 4.4:• If YA, theny* <* YA.• ff > YA, then YA <* <y*.The moral hazard problem in the effort level choice leads to a different optimal cutoff point. Comparedto the first-best cutoff point, the agent tends to overabandon the project when XH is relatively high and<A and tends to overcontinue the project when ZH is relatively low and * > jlA. This relation issimilar to that between the optimal benchmark cutoff and the second-best cutoff.The results in the two specific cases in Chapter 3, the no-first-stage and no-second-stage moralhazard problems, provide us with insights into this problem. Compared to the first-best cutoff, in theno-first-stage moral hazard case, overabandonment results, while in the no-second-stage moral hazardcase, overcontinuation results. In the general model where there are two stages of moral hazard, weconclude that:• when * < YA, the second-stage moral hazard problem dominates the first-stage moral hazardproblem, resulting in overabandonment; and• when * > A, the first-stage moral hazard problem dominates the second-stage moral hazardproblem, resulting in overcontinuation.From the numerical example in section 3.5, we observe that in the no-first-stage moral hazard problem,the deviation of the second-best cutoff from the first-best cutoff is greater when the first-best cutoff islow. On the other hand, in the no-second-stage moral hazard problem, the deviation is greater whenthe first-best cutoff is high. It is thus not surprising that in this two-stage moral hazard problem, weChapter 4. Analysis of the Multi-Stage Setting 71observe that when y is low, the second-stage moral hazard problem dominates, and when y” is high,the first-stage moral hazard problem dominates.4.3 CommunicationNext, we examine if there is value to communication of the agent’s signal to the principal. In the analysisabove, although there is no communication of the signal itself, the abandon/continue decision serves asone level of communication. If the agent chooses abandonment, he is in effect communicating to theprincipal that the signal is below the cutoff. On the other hand, the decision to continue implies thata signal above the cutoff has been received. Our results indicate that the first-best cutoff cannot beachieved in this setting. If ZH is relatively high and the first-best cutoff is low, then the second-bestcutoff is higher than the first-best. There is an overabandonment of the project. On the other hand,if XH is relatively low and the first-best cutoff is high, then the second-best cutoff is lower than thefirst-best. There is overcontinuation of the project.There are two potential benefits if the agent is able to communicate the specific value of y to theprincipal:1. Planning function value: a smaller distortion of the cutoff from first-best, which implies a lesssevere overabandonment or overcontinuation problem; and2. Control function value: a lower cost of motivating the agent to work hard (this derives from alower risk premium that needs to be imposed on the agent to induce him to work hard).We assume that the principal announces and precommits to a menu of contracts, (ud, {uh(y), u,(y)}).After the agent chooses first-stage effort and observes the signal y, he decides whether to continue ornot and which contract to select. If the project is to be continued, communication of the specific valueof p has no planning value, and Proposition 4.5 shows that communication of the specific value of p hasno control function value too.Chapter 4. Analysis of the Multi-Stage Setting 72Proposition 4.5: Communication of the specific value of y has no valueOur proof indicates that to satisfy the truth-telling constraints, the compensation contract cannot bemade contingent on communication. Thus, the least cost contract is the original contract under nocommunication, i.e., (id, {‘t%h, f}). Since the compensation contract cannot be made contingent oncommunication, this implies that communication of the specific value of y cannot be used to solve theoverabandonment and overcontinuation problem. We see similar results in Lambert (1986).Lambert (1986) examines a single-stage moral hazard problem. The agent chooses between a riskyproject or a risk-free project. Before the project selection, the principal seeks to motivate him towork to acquire private information on the risky project. Lambert derives conditions under whichunderinvestment or overinvestment in the risky project occurs. When communication is introduced,he demonstrates that the underinvestment problem disappears. Balakrishnan (1991) examines a similarmodel to Lambert’s with the additional consideration that the agent has precontract information. He alsodemonstrates that strict value to communication arises only if, absent communication, underinvestmentresults. However, both Lambert’s and Balakrishnan’s results depend critically on the assumption thatthe risky project does not require any more effort than does the risk-free project. Thus, we see that inour model, the presence of the second-stage moral hazard problem implies that the overabandonmentproblem does not disappear when communication is introduced. Lambert (1986) states that it is unclearif the overinvestment problem disappears when communication is introduced.Chapter 4. Analysis of the Multi-Stage Setting 734.4 Example4.4.1 Expected Compensation Cost, Wage Levels and Cutoff PointsUsing the following numerical values, we show how the expected compensation costs vary as the cutoffpoint varies:r=1Wk = 0.36iii = 0.693 = UQth) = —0.5= 0.2 V(th) = 1.2214Figure 4.10 shows how the expected compensation cost varies as the cutoff point varies for the threecases, benchmark, first-best, and second-best. In both the benchmark and the first-best cases, expectedcompensation is monotonically decreasing as the cutoff point increases. The probability of compensatingthe agent for t2h is decreasing as the cutoff point increases. Also, recall that in the benchmark case,the required risk premium is also decreasing in . In the second-best case, the expected compensationcost is convex in and reaches a minimum at > ÜÁ• We denote the cutoff point where the expectedcost is minimum as Üm• Note that at the cutoff point 9A, the expected compensation for the benchmarkcase is equal to that in the second-best case. This is consistent with our analysis. In the second-bestcase, if the principal seeks to implement ñ = I/A, he does not need to motivate the agent to select 9A•Given the optimal compensation contract to motivate the agent to choose {tih,i2eIif continue}, it is inthe agent’s interest to select DA. For subsequent discussion, we partition the range of cutoff points intothree regions (see Figure 4.10):• Region 1: < I/A,• Region 2: I/A I/rn, andChapter 4. Analysis of the Multi-Stage Setting 74Region 3: > .In the cutoff point selection problem, the principal chooses the optimal cutoff point to maximize hisexpected gross return less the expected compensation costs. R() is the expected gross return givencutoff point .1 2 1 2R(y)=yB+(1—y )XH+(1—y) XL.The expected gross return given cutoff point is concave in . At the optimal cutoff point, marginalexpected gross return equals marginal expected compensation cost.In both regions 1 and 2, the expected compensation cost in the second-best case is decreasing in .In region 1, the rate of decrease in the second-best case is greater than that for the benchmark case, thus> In region 2, the rate of decrease in the second-best case is smaller than that in the benchmarkcase, therefore < In region 3, expected compensation cost for the second-best case is increasing,while that for the benchmark case is decreasing. Therefore, < This is consistent with Proposition4.3 in section 4.2.2. Figure 4.11 shows the optimal wage levels as the cutoff point varies for the twocases, benchmark and second-best.We expand the example by varying the levels of XH, while keeping the the values of B and XLconstant.B=-5XL = —30.Figure 4.12 shows how the optimal cutoff points vary as XH varies for the three cases, benchmark,first-best and second-best. We observe that when > , then VA < <. When * < VA, thenVA > Y* >?.Chapter 4. Analysis of the Multi-Stage Setting 75Also, Figure 4.12 indicates that the optimal cutoff point decreases as XH increases. Recall that theexpected gross return isThus,R’()=B—xH—(1—)xL.As XH increases, holding B and XL constant, R’() decreases for a given . In the benchmark case, theexpected compensation cost is decreasing in at a decreasing rate. At the optimal cutoff point, marginalexpected gross return equals marginal expected compensation cost. Thus, as ZH increases, the optimalcutoff point decreases. In the second-best case, the expected compensation cost is convex in with aminimum at m• Thus, the optimal cutoff point also decreases as ZH increases.Figure4.10BehaviorofExpectedCompensationCostRt6’ON3Co4I2.II.1IIIII(I) 0 U 0 0 a) ci E 0 U 0 a) 0 a) Q. x w1.61.51.41.31.2 1.1 10.90.80.7-I Ise0.20.3I—II0.40.50.6VACutoffPoint0)paWageLevelsN)(-nC-)C0-4’-o0DpN)ppc-flp010I‘1CDLLuags-w’ivpsrstjujFigure4.12BehaviorofOptimalCutoffwithx0.540.52nc0.480.460.44_0.42O’40 00.3800.360.340.32oVA0.30.280.260.240.220.20.18406080100XH2000Chapter 4. Analysis of the Multi-Stage Setting 794.4.2 Review of ResultsIn the second-best case, the desired cutoff is ex ante efficient from the principal’s viewpoint and exante incentive compatible from the agent’s viewpoint. However, it appears inefficient from the principal’sviewpoint ex post. We continue with the above example and examine two cases for specific values of5H.OvercontinuationLet XH = 40, then y = .365 and = .356. To implement as the cutoff, the optimal compensationcontract is tih = 1.7208, ti’j = 0.6222 and 0.333. The expected compensation cost is 0.9989 and theexpected returns (after deducting the compensation cost) to the principal is 8.4652. On the other hand,if the principal were to implement y as the cutoff, the optimal compensation contract is ?.ljh = 1.73936,= 0.6094 and tI = 0.333. The expected compensation cost is 0.9982 and the expected return (afterdeducting the compensation cost) to the principal is 8.4641. Thus, the principal will not choose toimplement yK but is better off implementing *Next, we consider what happens if at i = 1, y = 0.36 is privately observed by the agent. Given thecompensation contract, the agent is strictly better off continuing with the project and he chooses to doso. However, the principal’s expected return (after deducting the expected compensation cost) is higherif the project is abandoned as shown below:• If the project is abandoned, the net gain to the principal is:Net gain = B—tZ= —5—.333= —5.333.• If the project is continued, the net gain to the principal is:Netgain =Chapter 4. Analysis of the Multi-Stage Setting 80= .36 * (40 — 1.7208) + (.64) * (—30 — .6222)= —5.8177.Therefore, at i = 1, given the second-best optimal compensation contract and that y = 0.36 is observed,the principal is better off if the project is abandoned. From the point of view of an external observerwho subsequently sees the realization of y = 0.36, it may appear that the agency has been unwilling toforego projects due to the investments made in the first stage. This is the sunk cost phenomenon and isfrequently hailed as irrational.The principal will not additionally compensate the agent to abandon the project. Otherwise, it willprovide the agent with incentive to always report an observed y between and y if he observes alower value of y. The principal would then be strictly worse off relative to implementing the second-bestoptimal cutoff.OverabandonmentIn this second case, we let XH = 80, then y’ = 0.23236 and = .255. To implement * as the cutoff, the optimal compensation contract is 1’h = 1.8582, z 0.6174 and = 0.2574. The expectedcompensation cost is 1.1057 and the expected returns (after deducting the compensation cost) to theprincipal is 26.69. On the other hand, if the principal were to implement y as the cutoff, the optimalcompensation contract is thh = 2.0784, tiij = 0.5636 and th = 0.2036. The expected compensation costis 1.1964 and the expected returns (after deducting the compensation cost) to the principal is 26.64.Thus, the principal will not choose to implement y but is better off implementing *•Next, we consider what happens if at i = 1, y = 0.24 is privately observed by the agent. Given thecompensation contract, the agent is strictly better off abandoning the project and he chooses to do so.However, the principal’s expected return (after deducting the expected compensation cost) is higher ifthe project is continued as shown below:Chapter 4. Analysis of the Multi-Stage Setting 81• If the project is abandoned, the net gain to the principal is -5.257.• If the project is continued, the net gain to the principal is:Netgain == .24 * (80 — 1.8582) + (.76) * (—30 — .6 174)= —4.5152.Therefore, at i = 1, given the second-best optimal compensation contract and that p = 0.24 is observed,the principal is better off if the project is continued. From the point of view of an external observer whosubsequently sees the realization of p = 0.24, it may appear that the agency is not rational and is toofast in dropping projects.The principal will not additionally compensate the agent to continue the project. Otherwise, itwill provide the agent with incentive to always report an observed p between p’ and? if he observes ahigher value of p. The principal would then be strictly worse off relative to implementing the second-bestoptimal cutoff.4.5 Literature ReviewHardly any work on multi-stage projects with project abandonment decision has been done. However,we often hear that decision-makers are reluctant to terminate projects when new information receivedindicates that the probability of success is low. Such escalation behavior has been generally termed asirrational. Kanodia, Bushman and Dickhaut (1989) provide an economic explanation for such behaviorbased on reputation. They show that when the agent has private information about his human capital, adesire for reputation-building may lead the agent to demonstrate escalation (overcontinuation) behavior.In their model, there is no possibility of overabandonment behavior. Our model provides an alternativeexplanation for escalation behavior based on induced moral hazard. We show that such behavior isChapter 4. Analysis of the Multi-Stage Setting 82ex ante efficient but ex post inefficient from the principal’s perspective. Our model also shows thatoverabandonment behavior may occur when the return from a successful project is relatively high.A closely related area of literature is project selection. The idea of overcontinuation (overabandonment) is similar to Lambert’s (1986) overinvestment (underinvestment). Lambert’s model is a first-stagemoral hazard problem. If the agent does not work, the probability of success in the risky project is 0.5.Thus, in Lambert’s model, both underinvestment and overinvestment can occur depending on whetherthe first-best cutoff is less than or greater than 0.5. In contrast, in our model, if the agent does notwork, the probability of success in the project is nil. If there is only a first-stage moral hazard problem,only overcontinuation can occur.Banker, Datar and Gopi (1989) also examine the project selection strategy of the agent. Theycompare the strategy of the agent when information is private to him and he makes the project selectionwith the case in which the principal himself receives the information and makes the selection. Thelatter case is similar to our benchmark case. When information is private to the agent, the principal’sobjective is to motivate the agent to choose the appropriate cutoff. If the project is to be undertaken,there is no uncertainty in the outcome and no moral hazard concern as a fixed outcome is expected.Thus, the compensation of the agent depends on only the invest/do not invest decision. In contrast, theprincipal in our model seeks to motivate the agent to work hard and choose the appropriate cutoff. If theproject is to be continued, there is uncertainty in the outcome and a moral hazard concern exists. In bothBanker, Datar and Gopi (1989) and our model, underinvestment (overabandonment) and overinvestment(overcontinuation) occur as a result of trade-off between the risk premium cost and suboptimal projectselection (continuation decision) cost. However, in Banker, Datar and Gopi (1989), underinvestmentprevails when project returns are relatively low and overinvestment prevails when project returns arehigh. The reverse occurs in our model where overabandonment prevails when returns are high andovercontinuation prevails when returns are low.Chapter 4. Analysis of the Multi-Stage Setting 834.6 Implications and ConclusionsThe results from our model show that the problem of overabandonment and overcontinuation is likelyto be less severe in small firms in which the principal plays a key role in the project. He is well informedand can closely monitor the work done. The first-best cutoff may be attainable. Malidique and Hayes(1987, p. 157) state that the ease of innovation in small firms has inspired both puzzlement and jealousyin larger firms. Also, many successful large technological firms recreate the climate of the small firmby divisionalization, with each division manager given much autonomy in running the division. Theliterature gives excellent communication and freedom from bureaucracy as the main factors for successin small firms. Our model provides a different explanation for the advantage small firms have over largefirms in the management of risky, multi-stage projects. In small firms, the principal is able to monitorthe project closely and be kept well-informed of the progress of the project. Thus, he is in the positionto make the abandon/continue decision that is in his best interest. On the other hand, in a large firmwhere the principal is very detached from the project, the abandonment/continuation decision has to bedelegated to the agent. Some welfare loss is suffered since the agent does not share the same objectiveas the principal.The current literature on project management focuses only on the overcontinuation (or sunk cost)problem. Our model shows clearly that both overcontinuation and overabandonment problem can actually happen. For projects with relatively high returns if they are successful, the overabandonmentproblem can occur. On the other hand, for projects with relatively low returns if successful, the overcontinuation problem can occur. The lack of focus on the overabandonment problem in the literature maybe because, in practice, it is very hard to pinpoint overabandonment. The problem of overabandonmentresults in missed opportunities, while the overcontinuation problem may result in failure. While a failureis glaringly obvious, missed opportunities are not so clearly seen. Our model shows that overabandonment is a problem too and should not be ignored. A firm may lose its competitive advantage due to suchChapter 4. Analysis of the Multi-Stage Setting 84missed opportunities. As discussed in the previous paragraph, we expect the problem to be less severein a small firm in which the principal is very much involved in the project. This may help explain thefinding of Scherer (1984), who indicates that small firms have been responsible for a disproportionateshare of innovations.Chapter 4. Analysis of the Multi-Stage Setting 85Appendix 4ABenchmark Case(1) To show that both the second and third constraints are binding:It is obvious that at least one of the second and third constraints of problem [P4.1] must be binding.• Second constraint is binding: Proof is by contradiction. We drop the second constraint and assumethat the optimal solution induces{t15,2Iifcontinue}. The principal only needs to impose risk onthe agent to motivate him to work hard in stage 2. If the project is abandoned, the agent’s utilityis Udv(th). If the project is continued, we let the agent’s expected utility be uv(2t5), where ü, isdefined using the participation constraint.YUdV(t) + (1 — = K== [(1 + )u + (1 —where the ratio of ‘u to uj satisfies the third constraint. To minimize the risk imposed on theagent, the principal sets udv(th) = fiv(2th). To satisfy the participation constraint, this impliesthat ud=> K and a = Since the participation constraint is binding, u > Kimplies that the second constraint is violated. Thus, at the optimal solution, the second constraintis binding.• The third constraint is binding: Proof is by contradiction. If the third constraint is dropped, theprincipal could set ui = = ii. To satisfy the second constraint, ü = But this impliesthat the agent will prefer to take t21 which violates the third constraint.Chapter 4. Analysis of the Multi-Stage Setting 86(2) Derivation of Uh and iii:Since both the first and second constraints are binding, ltd — K and from the third constraint,-_______________Uhyv(th)Substituting for ud and uh in the participation constraint, we obtain ij:— 2I[1— Üv(th)]v(t)(1 - )[1 + — v(th)(1 —andd21>0.dyBy substitution, we obtain Üh as follows:— 2K[1— vQh)(1 — )][1 — v(ta)]Uk— v(2th)(1 - )[1 + - v(th)(1 -To prove that the spread between and ü1 decreases as increases:We note that the ratio of to is—v(t)— [vQa)— 1]— v(t)This ratio increases as increases and, since uk and are negative values and is increasing in , thisimplies that the spread between Uk and U1 decreases.(3) Proof of Lemma 4.1:The second constraint can be rewritten as:ud{yv(th) — 1} + (1 —2)v(2t) + (1 —)2uzv(2th) 0.Since lid, ILk and Uj are negative values, for the constraint to hold,1Chapter 4. Analysis of the Multi-Stage Setting 87The third constraint can be rewritten as:YUhV(th) > uj{1 — v(th) + v(th)}.For the constraint to hold,1 — v(th) + v(th) > 0.This implies that1y> 1—For to exist, 1—< —f-y’ which implies that v(th) < 2.Chapter 4. Analysis of the Multi-Stage Setting 88Appendix 4BThe General Model(1) Derivation of the optimal compensation package:Since the principal’s objective function in [P4.2] is convex in ui,, while the constraints are linear inu, first-order conditions are sufficient to ensure optimality. The first-order conditions for the optimalcompensation contract are as follows:(1) — PBhh’(Ud) + \pBhV(th) + ,LL1PBhV(th) — IL’ + p +1) = 0= h’(u,) v(th).. + 111) — + + —p—.PBh PBh PBh(2)— PHhh(Uh) + )PHhv(2th) + p1Hhv(2th) — = 0h’(u) = v(2th)( + ‘)— v(t)PHh(3) — PLhh’(Ul) + APLhV(2th) + IL1PLhV(2th) — 112 — — )v(th) = 0h’(uj) = v(2th)(+p1)—112________PLh PLh(2) Proof of Lemma 4.2:For a given cutoff point , the principal needs to choose an optimal contract to motivate the agent towork hard. Given , the probabilities Pjm, j = B, H, L and rn = 1, Ii are fixed. Thus the principal’sproblem is a standard nonlinear programming problem of minimizing a convex function subject to afinite number of linear constraints. Thus, for any value of , the Lagrange multipliers on the inequalityconstraints are nonnegative.Suppose both p and 112 equal zero. The optimal contract is as follows:h’(ud) = Av(th) + L.PBIzh’(uh) = \v(2th)—v(th)PHhChapter 4. Analysis of the Multi-Stage Setting89h’(uj) = )v(2ih)7)(1 —)v(th)PLhIf q> 0, the agent prefers to abandon the contract whatever the valueof observed y.If ,j = 0, a fixed wage contract results and the agent will prefer to take t11.If 7) < 0, the third constraint ud> ul is violated.Therefore, at the optimal solution, at least one of P1 and P2 must be strictly greater than zero.(3) Proof of Lemma 4.3:If the first three constraints are binding, then 1d = = I<. Thus,PBhKV(th) + PHhUhV(2th) +I3LhKV(th) = K,— K[1 — PBhV(th) —z3Lhv(2thXlUh —PHhV(2th)Substituting forJ3Bh, PIll and Lh, we obtain:— K{2[1 — 1v(th)] — (1 —Uh— (1 —i2)v(2th)(4.9)The agent’s first-order condition for (equation (4.3)) implies thatK[1 — v(th) + ?v(th)]Uhyv(th)Equating the above two expressions for i2,, and by substitution and rearrangement, we obtain thefollowing expression for :v(th) + 1 ± /(2v(th) + 1)v(th)Since must be less than one, thus:v(th) + 1 — /(2v(th) + 1)v(th)By substitution, we obtain the following expression for i%h:—K[1 — v(th) + lv(th)]Uh —yv(th)Chapter 4. Analysis of the Multi-Stage Setting 90— K[2—/(2v(th)+1)]— v(th) + 1— /(2v(th) + 1)(4) Differentiation of A and tth(A):- — v(th)+1—/(2v(th)+1)YA —v(th)dA — V(th)[1 — (2v(th) + 1)_h/2] — [v(th) + 1 — ,/(2V(th) + 1)]dv(th) — v(2th)— vQh) + 1 — /(2v(th) + 1)— v(2th)/(2v(th) + 1)Since v(th) + 1 > ./(2V(th) + 1), this implies that > 0.K[2—V(2v(th)+1)]ui,(A)=duh(A) — —K{2/(2V(th) + 1) — v(th) — 2}dv(th) — {v(th) + 1 — /(2v(th) + 1)]2V(2v(th) + 1)Since2/(2v(th) + 1) > V(th) + 2, this implies that > 0.(5) Case B: Derivation of ih, and ‘ud:In case B, EHh = Ud (p1 > 0) and id > i (p = 0). Thus, ‘i2d = K while K — e, e> 0. We deriveexpressions for i2,, and i for a given . From the agent’s first-order condition on , i2i=The agent’s expected utility is as follows:EHh = Y1LdV(th) + (1 —2)I1hv(2th) + (1 —)2ulv(2th)= Ud[YV(th) + (1 - )v(th)] + h[(1 -2)v(2th) - ñ(1 - )v(2th)j= v(th)[(1 +d + (1— hv(th)]. (4.10)Since EH = lid = K,2K(1—)?2hv(th) =Vth)Chapter 4. Analysis of the Multi-Stage Setting 91— K[2—v(th)(1+)J—(1 — y)v(2th)— —2K[v(th) — 1]dñ — (1 —> 0.d2’uih—--> 0.dy2By substitution, we obtain the expression for li1 for a given .- K______ttl— Uh(1— y)V(th) (1—— K[v(th)(1 + 2) — 2]— (1—)2v(2thd11— 2I<[v(th)—1j(1+)d — (1—)3v(2th)< 0.d211< 0.dy2(6) Case C: Derivation of lih, fij and lid:In case C, EHh > (,ui = 0) and lid = li1 ([‘2 > 0). Thus, lI = = K — 6, 6 > 0. We deriveexpressions for lih and lij for a given . iij = li and from the agent’s first-order condition onUd = Uj =The agent’s expected utility is as follows:EHh YUdV(th) + (1 —2)lihv(2th) + (1 —= v(t)—v(th)+1[v(th) - {v(th) - 1](1 + 2)] (4.11)Let T represents a positive expression. It is generally the square of the denominator. Since EHh = K,— K{v(th)—v(th)+1]- v(2th)[ñv(th) - [v(th) - 11(1 + 2)]— 2K[,v(th) — v(th) + 1]— [v(th) — 11(1 +p2)]dlih — 2K[v(th) — 1]{v(th)(1 + 2) — 2(V(th)—1)]di — TChapter 4. Analysis of the Multi-Stage Setting 92< 0.By substitution, we obtain the expression for for a given .2KUI =v(th)[2v(th) — [v(th) — 11(1 + p2)]dil1 —2K[v(th)—1](1—d — T> 0.(7) Proof of Proposition 4.1:(a)Proof that ij = 0 in case A:In case A, ji > 0 and i2 > 0. From the first three binding constraints, we obtain the following expressionfor uh:- K{2[1— — (1 —(1—p )V(2th)If the principal wants to motivate the agent to select the cutoff point A, then substituting the value ofPA into ‘uh, we obtain the following:— K[2— /(2v(th) + 1)]Uh(A)—Given that ud = = K and th = Uh(A), the agent will find it optimal to select A as the cutoff point.This implies that the last constraint is not binding, thus = 0.(b) Case B applies when the principal wants to motivate a cutoff point > :In case B,— K[v(th)(1-f-U2)—2j— (1 — ç1)2v(2th)Since 11d = K and ftd > this implies thatChapter 4. Analysis of the Multi-Stage Setting 93[v(th)(1+2)—2j1(1 —)2V(2th)=‘. — 1] — 2,i[v(2th) — 11 + v(th)[v(th) — 1] < 0v(th) + 1 — /(2v(th) + 1)= Y> = YA•v(th)Next, we prove that j > 0 in case B:In case B, ii > 0 and ji2 = 0. Suppose that 0. Thenh’(ud) < v(th)(A+l),h’(’uh) > v(2th) + p1),h’(rtj) > v(2th)(.A + t1).Then the agent will strictly prefer to continue the contract no matter what value of y is observed. Thus,in case B, 7)> 0.(c) Case C applies when the principal wants to motivate the agent to select a cutoff point < :In case C,- 2KUd == v(th){2Üv(th) — (v(th) — 1)(1 + 2)]Since EH = K and EH > , thus,21— (V(th) - 1)(1 + p2)] >=. — 1] — 2[v(2t1)— 1] + v(th)[v(th) — 1] > 0v(th)+1—/(2vQh)+1) -= Y< — YA•V(th)Chapter 4. Analysis of the Multi-Stage Setting 94Next, we prove that j <0 in case C:In case C, j = 0 and /12 > 0. Suppose that > 0. Thenh’(ud) > )v(th),h’(uh) < v(2th),h’(ui) < )v(2th).Then the agent will strictly prefer to abandon the project no matter what value of y is observed. Thus,in case C, < 0.(8) Proof of Lemma 4.4:In case A, from (4.4),K[2 — /(2v(th) + 1)1Uh —vQh) + 1 — ./(2v(th) + 1)Since the denominator is clearly positive, then Üh <0 and K <0 imply that2— /(2v(th) + 1) > 0.Simplifying the equation, we obtain the upper bound on v(th), i.e., vQh) <In case B, from (4.5),— K[2—v(th)(1+)]ZLh— (1 — Ü)v(2th)Since the denominator is clearly positive, then h <0 and K < 0 imply that2—v(th)(1+) >0.Simplifying the equation, we obtain the upper bound of , i.e., <2—i()In case C, from (4.6),— 2K[v(th) — v(th) + 1]Uh— v(2th)[2v(th) - [v(th)—11(1 + p2)]2KUi= v(th)[2v(th) — [v(th)—1](1 + p2)]Chapter 4. Analysis of the Multi-Stage Setting 95ui < 0 and K < 0 imply that [2v(th) — [V(th) — 1J(1 + 2)J > 0. Therefore, the denominator of I’t, ispositive, and fih <0 and K <0 imply thatv(th) — v(th) + 1 > 0.Simplifying the equation, we obtain the lower bound of , i.e., y> v()51(9) Proof of Lemma 4.5:From (4.7),U[h(us) + (1 — )h(uj) — = [u + (1 —1. In both the benchmark case (binding third constraint) and the second-best case for < jj’i (agent’sfirst-order condition on and that =[u + (1 — )uj]v(t5= u.Therefore,== [h(n5)— h(uj)].Next, the ratio of ‘uh to uj in both the benchmark case and the general case are equal and is givenby:uh v(th) — v(th) + 1——exp{—rh(us)j—jjv(ts) — v(th) + 1—exp[—rh(rt1)j — Uv(th)h(u5) — h(tti)= 1[YV(th)—V(th) + 11.Chapter 4. Analysis of the Multi-Stage Setting 96Thus,ir= 1{[Yv(th)—v(th)+1r V(th)Tr’= 1{[Yv(th) — vQh) + 1]+V(th) — 1r V(th) iv(th) — V(th) + 11 v(th) — 1 V(th)_____________— ln[= V(th)-V(th)+1 v(th)-V(th)+11— 1{ V(th)—1—ln[v(th)— 1— v(th) — v(th) + 1 v(th) — v(th) + 1+ 1]}.Letf = a—ln(a+1),a>O.= f’ = a+1f=O ata=Oandf’ >Ofora>O. Thus, a>ln(a+1)fora> 0. Since > 0,— 1 V(th) — 1—ln[ +— v(th) + 1 v(th) — v(th) + 11] > 0.Thus, ir’ <0.2. In the second-best case for y’> A (agent’s first-order condition on and that lid = K):[uh + (1 — )uj]v(th) = K.Therefore,= h(uh)+(1—)h(ul)—h(K)= [h(uh) — h(uj)] + h(ui) — h(IC).Next,ith — [2—v(th)(1--)](1—)1 [2—v(th)(1-1-)](1—)h(uh)—h(uj) =rChapter 4. Analysis of the Multi-Stage Setting 97Thus,1 [2— v(th)(l +—j + ln[—uj] — ln[—K]}.= ——{1n[r1 [2— V(th)(1 + — + 2[v(t,) — 1][2v(th) — 2v(th) — v(tñ) + 2]= ——{ln[r v(th)(l+)-2 [2v(th)2v(th)+2][2v(th)2+v(th)]}2[v(t,) — 1][2v(th) — 2v(th) —v(th)+2] 2v — 2+vr [2v(th) - 2 - V(th) +2][v(th) — 2+ V(th)] - ‘[2 - v(th)(1 + )](1—2(V(th) — 1) 2[v(th) — 1j[2v(th) — 2v(th) — v(th) + 2]=r 2v(th) — 2 — v(th) + 2+—[2v(th) — 2 — v(th) +2][v(th) — 2 + v(th)]Note that 2(v(,)—1) > 0 and < 1.Letg = ln(b+1)—q!th.1= ——q.b+1g=Oatb=0. If--- >q5forb>O,theng>Oforb>0.If b — 2(v(h)—1)— u(t,)—2—v(f,)+2 > 0, then,1 2v(th)—2—vQh)+2E-i1 yV(th)—2+V(th)—<1, and > . Thus, [ln(b + 1) — bb] > 0 and ir’ > 0.— [2v(th)2+v(th)]=N*{34v(2th)+23(th)(v(t )—4)—4(v(th)—1)—29[v(th)[v(th —2]—2]+(v(2th)—4)},where N denotes a negative expression.Leth = 34v(2th) +23v(th)(v(th) —4)—42(v(th) —1)— 2[v(th)[v(th) —2]— 2] + (v(2th) —4).= 123v(2th) +62v(th)(v(th) — 4) —8(v(th) — 1) — 2[v(th)[v(th) — 2] — 2].Chapter 4. Analysis of the Multi-Stage Setting 98For 0 < < 2—v(t,.) h > 0, i.e., h is an increasing function in for 0 < < 2_(v(.) At = 0,h = v(2th) —4 < 0. At = 2-(t) h = 4(v(h)-2)(vth)-1)2<0. Therefore, since h is continuousand increasing, h < 0 for 0 < < 2—v() Thus, r” > 0.(10) Proof of Proposition 4.2:Recall that in the benchmark case (equation (4.1)),— 2K[1 — (1 — )v(th)][1 — v(th)]Uh—- 2K[1— v(th)]andu1 =v(th)(1 — )[1 + — v(th)(i—(a) At = YA:By substituting == v(th)+1— (v(t,,)+1) into iih and j, we obtain the following:== u11(A)=K.(b)If>A:— —— K[v(th) — 1][2v(th) — 2v(th) — 2 + v(th)]Uh Uh— v(2th)(1 - )[v(th) + - v(t) +11Within the feasible range of, [V(th)+—v(th)+1] > 0. For > I/A,[v(th)—2v(th)—2+v(th)J < 0.Thus, h > Uh.The participation constraints of both problems [P4.1] (benchmark case) and [P4.2] (general case) arebinding and since ud = d, we obtain the following:—2)ihv(2th) + (1 —)2ñv(2th) = (1 —2)iihv(2th) + (1 —)2ii,v(2th).Therefore,(1+)[2h—üh]=(1—)[iiz—iLl].Chapter 4. Analysis of the Multi-Stage Setting 99Since u > uh, the equality implies that ij > ftj. Thus, if y> A, for a given cutoff,Ud =‘Uh > Uh,and j < ij.Next, let (e) denote the spread between uh and u in the benchmark (second-best) case.- 2K(v(th)—1)(1 — v(th))= —_______________________— )[v(t) + y — V(th) + 112K(v(ts) — 1)S=UI,—U1 = —- 2v(2th)(1—y)— 4K(v(th)—1)>0—.“ > 0.Then,—— 2K(v(th) — 1)[2v(th) — 2lv(t) — 2-- v(th)J— v(2th)(1 — [v(t) + — v(th) + 11> 0.(c)If<A:—— 2K(v(th) — 1)(v(th) — v(th) + 1)[2v(th) — 2v(th) — 2 + v(ta)Jh Uh— v(2th)[(t ) — — 2v(th) + V(th) — 1](1 — )[v(th) + U — v(th) + 1],The expression [U2v(th) — 2Uv(th) — 2U + v(th)J = 0 at U = YA and [U2v(th) —2Uv(th) — 2U + v(th)j > 0for U < YA• Also, for U < YA, from (4.6),2KUUI= v(th)[2Uv(th) - [vQh)—11(1 + U2)]< 0 and K < 0 imply that[2Uv(th) — [v(th) — 1](1+U2)] > 0 or [U2v(th) — U2 — 2Uv(ta) +v(th) —1] < 0.Therefore, Uh > Uh.Chapter 4. Analysis of the Multi-Stage Setting 100We compare the ratio of uh to uj at a given for the second-best case with the benchmark case. Inthe second-best case,— ivQh) — V(th) + 1Similarly, in the benchmark case,— sv(t5)— V(th) + 1—Since uih > Uk, = ‘- implies that lij > ij. Therefore, if < , for a given cutoff,Ud < Ud=K,Uh > Uh,li1 > li1•- — — 2K(v(th) — 1)— v(2th)[ —2v(th) + 2v(th) — V(th) + 11.—— 2K(v(ts) —1)[v(t) — 2v(th) — 2 + v(th)]—< 0.(11) Proof of Proposition 4.3:1. If=:From Proposition 4.2, at = A, = li., p = d, h,l. If the optimal benchmark cutoff is A andin the general case, if the principal wishes the agent to select cutoff A, Lemma 4.4 states thatgiven the compensation contract in (4.4), the agent will find it optimal to select A as the cutoff.Therefore, if = YA, then Ij’ =2. If <yA:Let EC()[EC()] denote the expected compensation cost at for the second-best [benchmark]case. Totally differentiating the expected compensation cost with respect to , we obtain theChapter 4. Analysis of the Multi-Stage Setting 101following:EC’ = h(uld) — w7 + h’(uid) — — [h(üh) —+- + (1 -EC’ = h(Id) — wk + h’(Id) — h(i1) — — h(i)]+ (1 -2)h’(ih) + (1 -)2h’(I1.We prove that EC’ <EC’ for 0 < < .EC’ — PlC’ = h(i1)— h(d) + h’(d) + (1 — )[h’(ih) — h’(h)]= —[A],2(1— ‘V(th)where A = Inv(th)(1—y)(l — v(th) + yv(th) + y)— (v(th) —1)[3(3v(2th) — 1) —2(7v(2th) + 1) + f1(5v(2th) — 3) — (v(2th) — 1)1(1 — v(h) + 2v(th) —2v(th) +2)(i — v(th) + v(th) + )(1 —For any 0 < < , A reaches its minimum at V(th) = 1, with a value of zero. Thus, for v(th)> 1,A> 0. Therefore, we conclude that for 0 < < , PlC’ < PlC’. At the optimal cutoff, R’ = EC’.Since R’ is negatively sloping, this implies that < < Proposition 4.1 establishes thatj < 0 for 2 < iiA•3. If V >1/A:Proposition 4.2 states thatUd Ud,‘Uh > Uh,and fj < ij.There is more compensation risk in the second-best case and PlC > EC. From section 4.1.1, inthe benchmark case, the spread between i and itj decreases as increases. From section 4.2.2,Chapter 4. Analysis of the Multi-Stage Setting 102in the second-best case, the spread— ) increases at an increasing rate as increases. Thisimplies that as increases, the difference (.EJC — EC) increases, thus ETC’ > EC’. At the optimalcutoff, R’ = EC’. Since R’ is negatively sloping, this implies that * > > Proposition 4.1establishes that ij> 0 for > .(12) Proof of Proposition 4.4:We consider the full principal’s problem. Using a Lagrangian formulation, we write the problem asfollows:L PBh[B — h(ud) + Uk] +pHh[XH — hQua)] +I3Lh[XL — h(uj)]+ A[pBh’udv(th) + pHh’UhV(2ih) + PLhUIV(2th) — K]+ 1L1[ud{Bhv(th) — 1} + ‘UhPHhV(2th) + UIPLhV(2th)]+ [L2[UdU1]+ ‘q[ud — fruhv(th) — (1 —Differentiating the problem with respect to and using the agent’s first-order condition with respect to(equation (4.3)), we obtainB—hQud) + Uk ——h(uh)] — (1 — *)[ — h(ui)] — (th) [nh—= 0.If U/ > j, thenSign (i) = Sign {B — h(ud) + Uk — — h(uh)] — (1 — *)[XL — h(uj)]}For purpose of this proof, we redefine some variables:U(wh)v(th) UhWh = h(fih)Chapter 4. Analysis of the Multi-Stage Setting 103U(wj)v(th) == h(’ij)+th.Then the agent’s first-order condition on is:Ud = yUh + (1 —1. If üRewriting the principal’s first-order condition on , we haveB — h(ud) + Wk — — h(h) th] — (1 — — h(1)— thj — [Üh— i] = 0.It can be rewritten as follows:B + Wk + th — YXH (1 — = hQud) — *h(h) — (1 — *)h(u) + [fih—Since ud = ãh + (1 — )u1, and the agent is risk averse, therefore, h(nd) < h(h) + (1 —If * YA, we know from Proposition 4.1 that < 0. With 2h > j, this implies thatB + Wk + th — — (1 — *)x <0.Recall that in the first-best case, the principal’s first-order condition with respect to the cutoff(equation (4.2)) is given by:B + Wk + th — yX — (1 y*)x 0.From section 4.2.1, the principal’s objective function under first-best is strictly concave in andis given by the following:EP ==(4.12)Chapter 4. Analysis of the Multi-Stage Setting 104The derivative of EP with respect to is given by:EP’ = B—i+wk—th—-xH—(l—)xL++2thB+wk+th—ixH—(1—)xL. (4.13)We note that EP’ = 0 at y* and EP’ < 0 at . This implies that for , > y’.2. If* > I/A:The expected compensation cost at is:EC() = ñ[h(ud) — Wk] + (1 -2)[h(h) + th] + (1 —)2{h(1+h]= — wk] + (1 —2)h(h) + (1 -)2h(1+ (1-Totally differentiating EC() with respect to , we obtain the following:EC’ = h(ud) — h(üh) — (1 — )h(i1 — Wk — th+ 1l’(ud) + (1 -2)h’(h) + (1 -At the optimal cutoff, R’ = EC’. Thus,B+wk+th_*rH_(1_*)sL = h(ud)_*h(uh)_(1_*)h(u() (4.14)+ *hI(ud) + (1 —*2)hI() + (1 —For > I/A, the optimal compensation contract (equation (4.5)) is:• = K,- — K[2—v(th)(1+)]Uh —(1— y)v(th)- K[vQh)(1-f-)—2]and ui = (l—y) v(th)Also, h(u) = —.1n(—u).Chapter 4. Analysis of the Multi-Stage Setting 105Through substituition, differentiation and rearrangement of the terms, we obtain the following:h(ud) - - (1- *)h(u) + (1 —*2)h(u) + (1 —— .2(V(th)_1)(1+*) (1_*)2v(th— r t[v(th)(1+*2) _2*][2_v(th)(1+*)] v(th)(1 +*2)_ 2*— -* lnv(th)(1 + *2) — 2*(4 15)y [2_v(th)(1+*)](1_*)The right-hand side (RHS) of equation (4.15) is exactly the same as that of equation (3.11) ofChapter 3. In appendix 3A(II)(4), we prove that for 1 < v(th) < 2 and 0 < < 2—v() theRHS of (4.15) is strictly greater than zero. This implies that at the second-best optimal cutoff (seeequation (4.14)),B + Wk +h — YXH —(1— *) >0.From equation (4.2), at the first-best cutoff,B + Wk + th — YXH — (1 — y*)x = 0.We note that EF’ = 0 at y and EP’ > 0 at The principal’s objective function is a strictlyconcave function of , so this implies that for * > A, <Y.(13) Proof of Proposition 4.5:If a project is to be abandoned, there is no value to communication of the specific value of y. Theseverance payment cannot be made contingent on the agent’s message.Next, we prove that communication of the specific value of y given continuation has no value. Considera menu of contracts (ud, {uh(y), uj(y)}). At the cutoff , (ud, {uh(), u,()}) = (ad, {i2h, u}). Note thatno risky contract should strictly dominate another. This implies that if (uh(s), ui(s)) and (Uh(t), uzQ))are the risky contracts selected when y = s and y = t respectively, then uh(s) > uh(t) if and only ifui(s) <ul(). Also, uh(y) is a weakly increasing function of y and uz(y) is a weakly decreasing functionof y, i.e., as y increases, the relevant contract becomes more risky.Chapter 4. Analysis of the Multi-Stage Setting 106For a given y > , the principal seeks to find a contract [u5(y),u(y)] that minimizes the expectedcompensation cost. The problem can be represented as follows:mm yh[uh(y)] + (1 — y)h[u,(y)]s.t. + (1 — yu,(y) + (1 —YUh(Y) + (1 — y)u(y) yUj + (1—We prove that the cost-minimizing contract is uh(y) = fth and uj(y) = ij.Suppose not. Then uh(y) > its and uj(y) <uij andyh[u5(y)]+(1—y)h[ j(y)] <yh[Ii5]+(1—y)h[ii].Define r(y) and r(y, ) as follows:U[yh[’u5( )] + (1 — y)h[uj(y)j — r(y)] = yuh(y) + (1 — y)u(y)U[yh[iis] + (1 — y)h[iii] — r(y, )] = y11h + (1 —i-(y, ) represents the required risk premium when the agent observes y > but chooses a less riskycontract {it5, iti}. Therefore, r(y) > r(y, ).The expected compensation cost can be rewritten as follows:yh[u5(y)] + (1 — y)h[uj(y)] = h[yuh(y) + (1 — y)ui(y)] + r(y)yh[it] + (1 — y)h[i%j] = h[yu2 + (1 — y)itij + r(y, ).If yh[rt5(y)] + (1 — y)h[1(y)] < yh[1i5]+ (1 — y)h[it1]then h[yu5(y) + (1— y)u(y)J — h[y115 + (1— y)it] + r(y) — r(y,) < 0.T(y) > T(y, ) implies that to satisfy the inequality,h[yuh(y) + (1—y)u1(y)] < h[yit5 + (1—y)uij]= yu(y)+(1y)uj(y) < yus+(1—y)ui.Chapter 4. Analysis of the Multi-Stage Setting 107This violates the second truth-telling constraint. Therefore, we conclude that the prinicpal’s cost mmimizing contract for a given y > is given by uh(y) = ?Ih and uj(y) = üj. The principal does notcondition the contract on the message.Chapter 4. Analysis of the Multi-Stage Setting 108Appendix 4CNo Abandonment ProvidedWe consider the case when the principal does not provide for abandonment in the contract. In thisinstance, the receipt of private information by the agent at date i = 1 is not utilised. The principalwants to motivate the agent to take in both periods.Pr(xHItlh,t2h) = J yf(y)dyPr(zLt1h,2h)If the agent takes t1 in either period, the project crashes with probability 1.The principal’s problem is given as follows:1 1maxUk,U, — h(u)] + — h(u)]1 1s.t. ZLhV(2th) + jv(2t) > Kuv(2t) + ulv(2th) j.The first constraint is the principal’s participation constraint. The second constraint is the incentiveconstraint and it ensures that the agent is weakly better off choosing {tlh,t2} than choosing tj in eitherof the two periods. The second constraint can be rewritten as follows:UhV(2th) = uj[1 — v(2t)].Both uh and u must be strictly negative, as the agent’s utility function is negative exponential. Thisimplies that for the above equality to hold, [1 — v(2th)] > 0 = v(th) < Thus, the principal will nottake up the project if v(th) since the cost of motivating the agent to choose th is now extremelyhigh.Chapter 4. Analysis of the Multi-Stage Setting 109Since both constraints are binding, we can solve explicitly for uj and tLh.— K,K[2—v(2th)]Uk —v(2th)We note that Uh > Uj as 2—v(2h) < 1.Bibliography[1] Balakrishnan R, 1991, “Information Acquisition and Resource Allocation Decisions”, The Accounting Review, vol. 66, no. 1, January, pp. 120-139.[2] Demski J.S. and D.E.M. Sappington, 1987, “Delegated Expertise”, Journal of Accounting Research,vol. 25, no. 1, pp. 68—89.[3] Dye R.A., 1983, “Communication and Post-Decision Information”, Journal of Accounting Research,vol. 21, no. 2, Autumn, pp. 514—533.[4] Gibson J.E., 1981, Managing Research and Development, New York: John Wiley Sons.[5] Grossman S.J. and O.D. Hart, 1983, “An Analysis of the Principal-Agent Problem”, Econometrica,vol. 51, no. 1, pp. 7—45.[6] Holmstrom, B., 1989, “Agency Costs and Innovation”, Journal of Economic Behavior and Organization, vol. 12, pp. 305—327.[7] Jelinek M. and C.B. Schoonhoven, 1990, The Innovation Marathon: Lessons from High TechnologyFirms, United Kingdom: Basil Blackwell Ltd.[8] Kanodia C., R. Bushman and J. Dickhaut, 1989, “Escalation Errors and the Sunk Cost Effect: AnExplanation Based on Reputation and Information Asymmetries”, Journal of Accounting Research,vol. 27, no. 1, pp. 59—77.[9] Kanter R.M., 1985, The Change Masters: Corporate Entrepreneurs at Work, London: Unwin Paperbacks.[10] —, 1987, “From Status to Contribution: Some Organizational Implications of the Changing Basisfor Pay”, Personnel, vol. 64, no. 1, pp. 12—37.[11] —, 1989, When Giants Learn to Dance, London: Simon & Schuster.[12] Lambert R.A., 1986, “Executive Effort and Selection of Risky Projects”, Rand Journal of Economics,vol. 17, no. 1, pp. 77—88.[13] Lee J.Y., 1987, ManagerialAccounting Changes for the 1990s, California: McKay Business Systems.[14] Maidique M.A. and R.H. Hayes, 1987, “The Art of High-Technology Management”, in: E.B.Roberts, ed., Generating Technological Innovation, New York: Oxford University Press, pp. 147—164. Reprinted from Sloan Management Review, Winter 1984.[15] Melumad N.D., 1989, “Asymmetric Information and the Termination of Contracts in Agencies”,Contemporary Accounting Research, vol. 5, no. 2, pp. 733-53.[16] Milgrom P. and J. Roberts, 1992, Economics, Organization and Management, New Jersey: PrenticeHall.110Bibliography 111[17] Penno M., 1984, “Asymmetry of Pre-Decision Information and Managerial Accounting”, Journal ofAccounting Research, vol. 22, no. 1, Spring, pp. 177—191.[18] Scherer F., 1984, Innovation and Growth, 2nd ed., Cambridge: MIT Press.[19] Schneiderman H.A., 1991, “Managing R & D: A Perspective from the Top”, Sloan ManagementReview, Summer, pp. 53—58.[20] Twiss B., 1992, Managing Technological Innovation, 4th ed., London: Pitman Publishing.[21] Twiss B. and M. Goodridge, 1989, Managing Technology for Competitive Advantage: IntegratingTechnological and Organisational Development, From Strategy to Action, Great Britain: PitmanPublishing.[22] Williamson O.E., 1985, The Economic Institutions of Capitalism, New York: The Free Press.Chapter 5Effort Allocation and Job Design5.1 IntroductionThere are many tasks that have to be carried out in the running of an organization. These tasks normally include basic production activities, service activities, supervisory/training activities, and productand technology development activities. When the principal employs agents to help him manage theorganization, it is usual that each agent is made responsible for more than one task. In a single tasksituation, the optimal contract is designed to motivate the agent to work hard. However, in a multitasksituation, the optimal incentive contract is designed not oniy to motivate the agent to work hard, butalso serves to direct the agent to devote an optimal amount of effort to each of these activities. Currentresearch, in dealing mainly with single task situations, treats the problem of motivating current productive effort separately from that of motivating research/investment effort, for example. However, if theagent’s disutility from the two effort types are not additively separable (i.e., interactions between thetwo effort types exist in the agent’s disutility function), it is important that the two tasks be consideredtogether in determining the optimal incentive contract to properly motivate the agent.’ Holmstrom andMilgrom (1991) deal with a multitask situation. They comment that “when there are multiple tasks,incentive pay serves not only to allocate risks and to motivate hard work, it also serves to direct theallocation of the agents’ attention among their various duties” (p.25).However, the cost of motivating an agent to achieve a particular combination of effort levels in theif the agent’s disutility from the two effort types are additively separable, the two must be considered jointlysince the compensation is evaluated jointly.112Chapter 5. Effort Allocation and Job Design 113multiple tasks for which he is responsible is determined, to a large extent, by the availability and precisionof the performance measures of the agent’s effort in each task. Holmstrom and Milgrom (1991) suggestthat if one activity is impossible to observe and its outcome impossible to measure, and an agent controlsmultiple activities, then using incentive contracts based on the output of the measurable activities leadsthe agent to spend little or no time on the former activity. Thus, fiat wage contracts are used to ensurethat the agent works on the various activities. However, fiat wage contracts imply that the agent puts inminimal effort in the activities. This does not sound appealing in a highly competitive environment inwhich product improvement and development are very important. The principal should instead considerthe possibility of investing in a costly post-decision monitoring system to obtain performance measuresof the agent’s effort. If the benefits from higher effort levels outweigh the cost of obtaining and using theinformation, then the investment in monitoring is worthwhile. This part of the dissertation addresses theeffort allocation issue in a multitask setting and examines how changes in the precision of the performancemeasures of the agent’s effort in each task affects the optimal effort levels in the multiple tasks.In a multitask setting, the agent’s attitude towards performing a given set of tasks appears to influencethe optimal effort levels given the precision of the performance measures of the agent’s effort in each task.Since a different combination of tasks may have a different impact on the agent, one of the problems withwhich the principal needs to deal is how to efficiently group these tasks into individual jobs. In somecases, task grouping is straightforward due to skill requirements. In other situations, the principal hasmuch more flexibility in the task assignment and the grouping of tasks. We ignore the skill requirementissue in subsequent discussion. Assuming that the prinicpal has perfect freedom in the grouping of thetasks, he seeks an optimal grouping and assignment of tasks to the agents. Behavioral scientists haveoften asserted the value of job design (job enlargement and job enrichment) in motivating employees.However, the agency theory literature has given only limited attention to job design. In fact, in mostof the models examined, the agent’s action space is single dimensional. In this part of the dissertation,Chapter 5. Effort Allocation and Job Design 114while we do not directly model the job design problem, we believe the analysis of the effort allocationissue provides useful insights with respect to that problem.5.2 Tasks with Long-term and Short-term ImpactIn this section, we examine a setting in which the agent is responsible for both current operations andinnovation activities, which differ in their impact on the firm’s profit position. Incentive pay serves notonly to motivate hard work, it also serves to direct the allocation of the agent’s effort among these tasks.If the incentive plan uses current year’s profit as the sole criterion for evaluation, this may result inthe agent concentrating on tasks with a short-term impact, and foregoing projects that bring long-termbenefits but hurt the short-term results. Rappaport (1982) suggests that one reason for the low researchand capital spending in the United States leading to a slowdown in the long-term growth of the economyis that firms have been preoccupied with short-term results, and this is in part due to the poor designof the management incentive compensation plans. These plans compel the managers “to concentrateon short-run results and adopt policies that may discourage growth and acceptance of reasonable risk”(p. 370). Stock options and long-term contracts have been used to help correct the myopic tendency inagents. However, as Rich and Larson (1987) point out, since these plans pay out at the end of a four- tofive-year period, while annual bonuses offer opportunities for substantial rewards in the near future, theagents are still motivated to direct more attention to annual performance goals as opposed to long-termgoals. They prefer to take the cash and let the credit go.Rappaport suggests three possible approaches that firms could take to better integrate managementincentives and strategic planning. One of these approaches is termed a strategic factors approach. “Thisinvolves identification of the strategic factors governing future profitability, periodic measurement of theprogress achieved in accomplishing each goal, and incorporation of the results in incentive packages” (p.372). Some examples of such strategic factors are:Chapter 5. Effort Allocation and Job Design 115• target market share,• productivity levels,• product quality measures,• product development measures, and• personnel development measures.A similar suggestion was made by Ira Kay (1991) in the article “Beyond Stock Options: EmergingPractices in Executive Incentive Programs”. He suggests that shorter-term strategic circumstances orachievements are important since they serve as critical performance markers. Thus, they should bedesigned into the annual incentive plans. Kay gives the following examples of strategic mileposts:• Progress or achievements in the research and development of new products;• The development of proprietary/unique production methods;• Improved employee productivity not attained through capital substitution;• An improvement in the capability and potential of employees, particularly managers and middle-level people;• Improved marketing methods resulting in greater market share; and• The successful development of a plan, such as a strategic business plan.An example of a firm that integrates management incentives with strategic planning is McDonald’s.McDonald’s identifies the following six areas as key success factors that affect long-run profits:• Product quality• ServiceChapter 5. Effort Allocation and Job Design 116• Cleanliness• Sales volume• Personnel training• Cost controlAccordingly, McDonald’s assesses its store managers based on their performance in these areas. “Focusing on these key success factors, rather than short-run profits, identifies these factors as the key influenceson long-run profitability” (Kaplan and Atkinson, 1989). The effectiveness of these incentive plans maybe a contributing factor to the immense success that McDonald’s enjoys worldwide. Similarly, General Electric uses multiple measures of divisional performance like market position, product leadershipand personnel development. To be able to use these success factors as performance measures requiresthat the management accountants design and maintain appropriate information systems to reflect thenecessary information. As Kim and Suh (1991) point out, different information systems may result indifferent optimal incentive plans and different optimal effort levels. They analyze the effect of differentinformation systems on the corresponding expected minimum compensation costs in inducing a giveneffort level when the agent is responsible for only one task. They provide a ranking of the informationsystem in inducing a given effort level when the agent has a square root utility function.5.3 Objectives of the ModelIn this part of the dissertation, we consider a two task situation, in a one-period setting. Both tasksshould be undertaken for the well-being of the firm. The agent’s attitude towards performing the setof tasks is captured in the agent’s personal cost of effort function. We call this attitude the interactiveeffect on the agent’s personal cost of effort function. It can either be negative, zero or positive. Anegative value implies that the two effort levels are complementary in the agent’s cost function, i.e.,Chapter 5. Effort Allocation and Job Design 117the marginal disutility of achieving one task decreases as the effort level in the other task increases. Azero value implies that the two effort levels are independent in the agent’s cost function. A positivevalue implies that the two effort levels are substitutable in the agent’s cost function, i.e., the marginaldisutility of achieving one task increases as the effort level in the other task increases. The groupingof tasks determines the value of the interactive effect, and we examine how this effect, together withthe incentive contracts, affect the cost to the principal of extracting high effort levels from the agent.We vary the precision of the performance measure in the second task and explore how this affects thetask assignment and optimal effort levels. In the extreme case, we consider what happens if there isno costlessly available performance measure on this second task. The principal explores the option ofinvesting in a costly monitoring technology to extract a signal that can be used in the compensationcontract. The more precise the information to be extracted, the higher the monitoring cost, and weexplore what factors determine the optimal monitoring level.The analysis yields the following results.1. The analysis emphasizes that when an agent is responsible for more than one task, incentive issuesshould not be addressed task by task. It is necessary that the principal studies the incentiveproblems for all the tasks together. Since the grouping of tasks affects the agent’s personal costof effort function, the principal chooses the grouping that affects the agent most favorably. Theprincipal is then able to more efficiently motivate higher effort levels and achieve higher profitability.2. A good job design and a well-designed incentive plan are both necessary to motivate the agent toexert the optimal effort levels at the minimum cost. A good incentive plan with poor job designeither limits the ability of the principal to extract optimal effort levels from the agent or if theeffort levels are achieveable, raises the compensation cost substantially. On the other hand, a goodjob design with a poorly-designed compensation plan result in suboptimal effort allocation.3. The relative precision of the performance measures determines the cost to the principal of extractingChapter 5. Effort Allocation and Job Design 118high effort levels from the multiple tasks. If the agent is responsible for both task 1 and task 2,the performance measure of task 2 is relatively very noisy compared to that of task 1, and theinteractive effect on the agent’s personal cost function is positive, there can arise cases when theprincipal is better off removing task 2 from the agent so that he concentrates on only task 1.4. The principal should not just settle for costlessly available but highly noisy information since theuse of such measures increases the cost to the principal of extracting high effort levels. Rather, heshould investigate the potential benefits from investing in a costly monitoring technology to obtainmore precise information before deciding on the basis for the incentive contracts.5.4 Agency Literature ReviewThe following three classes of literature are relevant for this part of the dissertation:• Multitask setting;• Job design; and• Investment in Monitoring Technology.5.4.1 Multitask LiteratureThere are a number of articles in the literature examining different aspects of the multitask setting.Our approach is similar to the multitask model examined by Holmstrom and Milgrom (1991). Theyfocus on the case where the agent’s personal costs depend only on the total effort the agent devotes toall his tasks, i.e., all activities are equal substitutes in the agent’s cost function. They conclude thatthe desirability of providing incentives for any one activity decreases with the difficulty of measuringperformance in any other activities that make competing demands on the agent’s attention. Incentivesfor a task can be rewarded in two ways: either the task itself is rewarded or the marginal opportunityChapter 5. Effort Allocation and Job Design 119cost for the task can be lowered by removing or reducing the incentives on competing tasks. Our analysisdiffers from theirs in that we allow for a range of interactive effects on the agent’s cost of effort, i.e., thetasks may not be equal substitutes in the agent’s cost function and they could even be complementaryin the agent’s cost function. This has an important effect on how jobs should be designed.Feltham and Xie (1994) explore the economic impact of variations in performance measure congruenceand the use of multiple measures to deal with both problems of goal congruence and the impact ofuncontrollable events on performance measures. They assume that the effort levels are independent inthe agent’s cost function, i.e., the interactive effect on the agent’s personal cost of effort function iszero. Their analysis shows that a contract based on a noncongruent measure induces suboptimal effortallocation across tasks, whereas noise in a performance measure results in suboptimal effort intensity.Our study differs from theirs in that we assume congruent performance measures and we focus on theimpact of changes in the relative precision of the performance measures on the effort levels. We alsoexamine the impact on the optimal effort levels of variations in the interactive effect on the agent’s costof effort function.Bushman and Indjejikian (1993) study the use of both accounting earnings and stock price in compensation contracts for executives involved in two tasks. Their analysis focuses on the role of accountingearnings as the information content of earnings varies. In Paul (1991), the agent performs two taskswhich he interprets as pertaining to short and long run cash flows. The agent’s contract is a function ofthe stock price. The analysis shows that depending upon which type of information has the more pronounced effect on price, overemphasis on either short run or long run actions can occur. In our analysis,if the agent is responsible for two tasks pertaining to short and long run cash flows, we suggest that forthe task affecting long run cash flow, a strategic milepost instead of stock price is used to motivate theagent to allocate effort to long-term projects. The strategic milepost may be costly to obtain and weexamine the principal’s decision to invest in obtaining the information.Chapter 5. Effort Allocation and Job Design 1205.4.2 Job DesignItoh (1991) examines the factors that lead a principal to choose to induce workers to work separatelyon their tasks rather than to induce them to spend some effort helping others. The two determiningfactors are strategic interaction between agents and their attitudes towards performing multiple tasks.The latter factor is similar to our consideration of the interactive effect on the agent’s personal cost ofeffort function. Itoh (1991) only allows for the interactive effect to be positive or zero. He obtains theresult that the principal wants either an unambiguous division of labour or substantial teamwork.Holrnstrom and Milgrom (1991) also examine the issue of job design. They obtain the result thateach task should be made the responsibility of just one agent, i.e., an unambiguous division of labour.This is because they assume that the agent’s effort types are perfect substitutes in the agent’s costfunction so that the interactive effect is positive. In our analysis, we allow for the interactive effect torange from negative to positive and we examine the impact on job design and optimal effort levels.While behavioral scientists assert that job enlargement and enrichment can motivate workers to workhard, our analysis attempts to indicate how the benefit is achieved. Job enlargement and enrichmentmay lead to a decrease in the value of the interactive effect on the agent’s cost of effort function. Thismakes it less costly for the principal to motivate higher effort levels.5.4.3 Investment in Monitoring TechnologyMost of the studies dealing with the choice of monitoring systems assume a costless monitoring technologywith an exogenously specified quality in a single-task setting. Shavell (1979) and Holmstrom (1979) showthat an information system which reports both the output and an imperfect monitor of the agent’s effortis more valuable than one which reports only the outcome if the monitor conveys information abouteffort not already conveyed by the output. This arises because a contract with improved motivationaleffects is achieved. In the studies that examine the choice of acquiring costly signals, most assumeChapter 5. Effort Allocation and Job Design 121that the principal has in place a costless information system that reveals the production output. Theprincipal’s problem is then when to acquire the costly additional signal. Singh (1985) derives the amountof monitoring of the agent’s effort endogenously in such a setting, and shows that if the marginal costs ofgathering information are always positive, there is a minimal optimal level ofmonitoring by the principal.Baiman and Rajan (1994) study the design of costly post-decision monitoring systems when there isno alternative costlessly available signal. The system reports either success or failure. The principal’smonitoring decision is to choose the probability that the desired action generates the ‘failure’ signal.Our model assumes a very different monitoring technology. The principal’s monitoring decision is toselect the precision level of the signal, thus the signal is not dichotomous but is continuous. Also, weexamine the principal’s decision to invest in the monitoring technology in a multitask setting when effortallocation becomes an issue.5.5 ConclusionThis part of the dissertation focusses on a two-task setting. We examine the role of an incentive planfor effort allocation, and consider the implications of variations in the interactive effect on the agent’spersonal cost of effort function. Our study also explores how changes in the relative precision of theperformance measures affect the optimal combination of effort levels in the multiple tasks. Finally, weexamine the principal’s decision to invest in a costly monitoring technology when performance measuresare not costlessly available or they are relatively very noisy.In the next chapter, we look at a single-task principal-agent model. Costly monitoring is engaged andwe examine the principal’s selection of the optimal level of monitoring. This provides us with valuableinsights when we examine the two-task model in chapter 7. When there is more than one task, effortallocation and job design become critical and we analyze the principal’s monitoring decision in such asetting.Chapter 6Single-Task Principal-Agent Model with Costly Monitoring Technology6.1 IntroductionIn this chapter, we consider a single period principal-agent model, in which the agent is responsiblefor one task. The outcome of the task is not observable when the agent is paid for his effort. Thissituation arises, for example, if the agent is responsible for research and development projects or long-term investment projects, in which the outcome of these activities are not realized till some periods later.Thus, the principal and the agent cannot contract on the outcome of the task, since it is not observablewhen the agent is to be paid. To motivate an effort level higher than the minimal level, the principalemploys a costly monitoring technology which provides information on the agent’s effort. We examinethe principal’s monitoring decision. This provides us with useful insights when, in the next chapter, weanalyze a two-task situation in which effort allocation becomes critical.The next section presents the general model. In section 6.3, we use specific linear profit and quadraticcost functions to facilitate the derivation of closed form expressions for the incentive rate, the activitylevel and the monitoring level. We also examine how these optimal levels vary as the exogenous variableschange. Proofs of the lemma and the propositions are provided in appendix 6A.6.2 The ModelLet t be a measure of the activity or accomplishment level that the agent can choose with certainty forthe task. The agent supplies t at a personal cost of V(t), which we assume to be convex and increasing122Chapter 6. Single-Task Principal-Agent Model with Costly Monitoring Technology 123with t. The incremental profit (before any wage payment to the agent) is given by 11(t), which accruesdirectly to the principal. 11(t) is assumed to be weakly concave. No discounting is considered in themodel.The principal is risk neutral. The agent is risk averse, and has exponential utility Um(Z) — exp(—rz),where r is the agent’s coefficient of absolute risk aversion and z is the agent’s net income, which consistsof his wage w minus the personal cost of effort VQ), i.e. z = w — V(t). Thus, we assume that V(t) isexpressed in dollar terms.In the first-best setting in which the effort of the agent is observable and the agent could be severelypenalized if he does not provide the required level of t, the principal uses a fiat wage contract. Weassume that the certainty equivalent of the agent’s reservation net income level is zero. The interiorsolution is characterized by the following:11’(t) = V’(t)w = V(t).In the second-best setting, without any signal about the agent’s effort, if the principal uses a fiatwage contract, the agent puts in the minimal level of effort, which we assume to be zero. The expectedprofit would also be zero.Now, we assume that a costly monitoring technology is available and it provides a noisy signal of theagent’s effort. This signal could then be used for compensation purposes. Our model is a special caseof the model used by Huddart (1993). We work with a risk neutral principal who holds all the shares ofthe firm. We assume that the signal obtained from the monitoring technology is related to the agent’seffort and the level of monitoring in the following manner:y=t+6, 9N(O,1).The cost of the signal depends on h, C(h), where h equals the precision of the monitoring technology.Chapter 6. Single-Task Principal-Agent Model with Costly Monitoring Technology 124Assume that C(.) is continuous and monotonically increasing, and thatlim 0(h) = oo and urn 0(h) = 0.h—÷oo h—*OThe higher the level of h, the more precise the signal and the more costly the monitoring. The cost of aperfect signal is infinite, while there is no cost if the principal decides not to use the costly monitor. Wealso assume that the level of h is observable and verifiable, thus it is contractible. For example, h maybe related to the number of auditor or computer hours devoted to retrieving the data.With the signal from the monitor, the principal can now base the wage contract on the signal, Werestrict our analysis to the use of linear wage contracts, which take the following form, w(y) = ay + b.The manager’s problem ismax Eum[w(y) — V(t)].Since his utility function is exponential and all random variables are normally distributed, maximizingthe expected utility is equivalent to maximizing the certainty equivalent, which is given by:1 a2CEm(a,b,t) = at+b— V(t) —That is, the agent’s certainty equivalent consists of the expected wage less the personal cost of effortand less a risk premium.The principal’s objective is to maximize his expected profit, subject to the agent’s participation andincentive compatibility constraints.max 11(t) — 0(h) — E{w(y)]st. CEm(a,b,t) 0and t E argmaxt’ CEm(a,b,t’).Under the linear wage scheme, the certainty equivalent of the principal is given by:CEp(a, b, d, t) = 11(t) — C(h) — (at + b).Chapter 6. Single-Task Principal-Agent Model with Costly Monitoring Technology 125The total certainty equivalent of the principal and the agent is then given by:1 a2CE + CErn = 11(t) — C(h) — V(t) — -r.Note that the total certainty equivalent is independent of the intercept component of the wage contract,b. It serves only to allocate the total certainty equivalent between the two parties, such that the agent’sreservation utility level is reached. As explained in Holmstrom and Milgrom (1991), this implies thatthe incentive-efficient linear contracts are those that maximize the total certainty equivalent subject tothe incentive compatibility constraints. Therefore the problem reduces to the following:max 11(t) — C(h) — V(t) —s.t. I argmaxt’ at’ — V(t’).From the constraint, we know that a = V’(t), which can be substituted into the objective function toobtain the following:max 11(t) — C(h) — V(t) —[V’(t)]2Using the first-order conditions on Ii and t, we characterize the optimal level of effort t and the optimallevel of monitoring h at the optimal t.V’t —____________6 1“I— h+rV”(t)C’1h —11 (t) 12 6 2“ ‘— 21h+rV”(t)6.3 Quadratic Cost SettingIn this section, we analyze in greater depth settings in which the cost function is quadratic and theprofit function is linear. Let 11(1) = 1, V(t) = 6t2 and C(h) = ch, c > 0. Then 11’(t) = 1, V’(t) = 61,V”(t) = 6 and C’(h) = c.Chapter 6. Single-Task Principal-Agent Model with Costly Monitoring Technology 1266.3.1 First-best SettingIn the first-best setting, the optimal effort level and the optimal wage contract are as follows:—8* 1w =The total certainty equivalent is given by:CE(FB) = 11(1*) - V(t*)1286.3.2 Use of Monitoring TechnologyUsing (6.1) and (6.2), we characterize the optimal levels of effort t and monitoring h as follows:V’(t) ==(6.3)C(h) = c= 2(h+r6)(6.4)The principal monitors the agent only when the benefit exceeds the cost of monitoring. Lemma 6.1establishes when monitoring is worthwhile for the principal.Lemma 6.1: Necessary and sufficient condition for the principal to engage in monitoring is that c <As the risk aversion of the agent increases, a higher risk premium is required when imperfect information is used. As the agent’s private cost of effort increases, the agent requires more compensation fortaking effort. Thus, the bound on the cost of monitoring tightens as r increases or as 6 increases.Chapter 6. Single-Task Principal-Agent Model with Costly Monitoring Technology 127Proposition 6.1: When c < the principal engages in monitoring and the optimal level of monitoring, h, the optimal commision rate, a and the optimal effort level, I are:h == /(f){i — 6(2rc)}, (6.5)a = 1—&/(2rc), (6.6)1t = .—Q(2rc). (6.7)The deviation from first-best effort level is ,/(2rc). First-best effort intensity is achieved if the agentis risk neutral or the monitoring technology is costless, i.e., if rc = 0. Costless monitoring technologyimplies that the intensity of monitoring is infinite, so that perfect information is obtained.6.3.3 Comparative StaticsNext, we examine how the the optimal monitoring level, incentive rate and the effort level vary with thevarious parameters.Proposition 6.2:1. An increase in the cost of monitoring (c) results in reduced monitoring (Ii), reduced incentive rate(a), and reduced effort level (1).2. (a) An increase in the agent’s risk aversion (r) results in reduced incentive rate (a), and reducedeffort level (1).(b) Monitoring level is concave in r and is most intense at r =Chapter 6. Single-Task Principal-Agent Model with Costly Monitoring Technology 128To provide incentive for the agent to work at a level higher than the minimal level, the principal imposesrisk on the agent who is then compensated for bearing this risk in the form of a risk premium. When ris low, the risk premium required by the agent for a given effort level is not high, the principal settlesfor noisy information and he uses a less intense level of monitoring. The expression for h shows that asr approaches zero, the level of h approaches zero. For high levels of r, the principal uses weak incentivesto reduce the risk imposed on the agent, he settles for a low effort level, and we expect the intensityof monitoring to be reduced. Huddart (1993) states that “monitoring is valuable only when coupledwith an incentive scheme responsive to the signals generated”. For very risk averse agents, the principalsettles for low monitoring, low incentives, and low output.Chapter 6. Single-Task Principal-Agent Model with Costly Monitoring Technology 129Appendix 6A(1) Derivation of (6.1) and (6.2)Substituting a = V’(t) into the objective function, we obtain the following:maxll(t) — C(h) — V(t) —We have the following first-order conditions on t and h:ll’(t) — V’(t) — V’(t)V”(t) = 0—C’(h) + [Vf(t)j2 =Thus, the optimal level of t is characterized by:v’ — ll’(t)hh+rV”(t)The optimal level of h is characterized by:C’(h) =At the optimal level oft, we obtain the following characterization of the optimal h:C’ h— r1 ll’Q) 12( 2h+rV”(t)(2) Proof of Lemma 6.1When no monitoring is undertaken, ii = 0, a = 0 and t = 0 and the total certainty equivalent = 0. Withpositive level of monitoring, the total certainty equivalent is:CE = 11(1)— VQ)1 {V’(t)]2— C(h)=2hh+r81—ch.From (6.3), t = (hr6)• By substituting for t and simplifying, we obtain the following:CE= -cli.26(r6+h)Chapter 6. Single-Task Principal-Agent Model with Costly Monitoring Technology 130Monitoring is strictly worthwhile if, and oniy if, CE> 0 for some h E (0, oo), which implies thath—ch > 026(r6+h)1= h[ —c] > 0.28(rö+h)Hence, if C < 2r6+26h’ the certainty equivalent is srictly greater than 0 if monitoring is undertaken.Therefore, h > 0 and c < 2rb4h imply that c < From (6.5), the optimal level of h ish = /(f)[l — 6/(2rc)]. Thus, h > 0 implies that c <(3) Proof of Proposition 6.1To obtain the optimal level of h, we use (6.4).C= 2(h+r6)Hence, we obtain h = /(-) — r6.To obtain the optimal level oft, we use (6.3).hWe substitute for the optimal level of h derived above as follows:-________— 6(/(f)-r6-fr6)= -(2rc).The optimal level of a is characterized by a = V’(t). Hence, we obtain:a = 6t1== 1 — ö/(2rc).Chapter 6. Single-Task Principal-Agent Model with Costly Monitoring Technology 131(4) Proof of Proposition 6.2By differentiating (6.5), (6.6) and (6.7) of Proposition 6.1, the results in parts (1), (2) and (3a) areobvious.To prove (3b):dh = v’(f)—rS.dr - 2.,/(2cr) -.d2h< 0.drThus, ii is at its maximum when12J(2cr) =1=, r =8c62Chapter 7Multitask Principal-Agent Model with Costly Monitoring Technology7.1 IntroductionIn this chapter, we consider a single period principal-agent model. The principal owns the firm andthus owns the outcome of all tasks undertaken for the firm. There are two tasks which the agent isemployed to perform. In such a setting, the principal is not only concerned with motivating hard work,he is also concerned with directing the agent’s attention between the two tasks. We assume that thepost-action value of the firm is not observable prior to the termination of the agent’s contract. Incentivecontracts are based on imperfect performance measures associated with each task undertaken by theagent. For one of the tasks, we let the precision of the performance measure vary. At the extreme, weassume that the performance measure is not costlessly available and the principal considers investing ina costly monitoring technology. We examine the principal’s monitoring decision in such a setting wheneffort allocation is critical, and compare the results with those in a one-task setting (which we analyzein Chapter 6) when effort allocation is not an issue.In the next two sections, we present and analyze the general model. In section 7.4, we use specificlinear profit and quadratic cost functions to facilitate the derivation of closed form expressions forthe incentive rates, the activity levels, and monitoring level. We focus on the use and the value ofthe costly monitoring technology when the performance measure of one of the tasks is not costlesslyavailable. We also examine how the optimal incentive rates, activity and monitoring levels vary as theexogenous variables change. In section 7.5, we consider some implications of the results for job design132Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 133and organization structure. Proofs of the lemmas and the propositions are provided in appendix 7A.7.2 The Model7.2.1 General CharacteristicsLet t1, I = 1, 2 be a measure of the activity or accomplishment level that the agent can choose withcertainty for task j1 The agent makes a one-time choice of a vector of activity levels I = (ti, 12) at apersonal cost of V(t), which we assume to be increasing with t and convex, that is, 14(t) > 0, 14(t) > 0and VnV22 — V?2 > 0. The subscripts on V(t) refer to the first and second partial derivatives withrespect to t. For both tasks, a public signal on the agent’s activity level y is observed at the end of theperiod, and is related to the agent’s activity level in the following manner:y =t+6, 9 - N(0,h1) 1=1,2, (7.1)where h is the precision (i.e., the inverse of variance).2 We assume that the agent’s activity level 11does not influence the precision of the signal, h1, and that 61 is independent of 62. The principal couldcontract with the agent based on Yi and Y2, since they are observable and contractible.3The principal is risk neutral. The agent is risk averse, and has exponential utility um(Z) = — exp(—rz),where r is the agent’s coefficient of absolute risk aversion and z is the agent’s final income, which consistsof the realized wage w minus the personal cost of effort incurred to achieve the activity level, V(t), i.e.z = w — V(t). Thus, we assume that V(t) can be expressed in dollar terms. Assume that the principaluses a compensation contract linear in y, I = 1,2.The timing of the game is as follows:‘This activity level may be a transformed measure. For example, we may transform the number of hours worked bythe agent into t which may represent the degree of completion of the task or the expected outcome. Thus, t may beinterpreted as an output measure.2See appendix 7B for the model’s application to more general expressions of signals.may be a financial measure which provides noisy information about the agent’s choice of activity level. For example,1/i may represent reported accounting income. As a result of uncontrollable events influencing sales demand and pricesas well as input prices, accounting income provides a noisy measure of the activity level of the agent. Also, accountingadjustments and provisions may cause reported income to be a noisy measure of the agent’s effort.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 1341. The principal offers the agent a wage contract;2. The agent selects his activity levels, t = (tj,t2);3. yi and y2 are observed publicly;4. The agent is paid.For subsequent analyses in this section, we consider two different cases:• Case 1: Interior solution with (t1,2)>> 0.• Case 2: Corner solution with either t1 or t2 = 0.7.2.2 Interdependency in either V(t) or 11(t)We assume that there is some form of interdependency of the two activity levels in either V(t) or11(t). This is essential, otherwise the problem reduces to two single-task principal-agent problems. Theinterdependency in V(t) is characterized by i, j = 1, 2, and i j. T4j could be either negative,positive or 0. A negative T../j means that the two activity levels are complementary in the agent’s privatecost function, i.e. the marginal disutility of achieving task i decreases as the activity level in task jincreases. This means that achieving a higher level on one task makes achieving a higher level on theother task less costly (painful). Perhaps this could be due to “learning” or due to “variety” in the sensethat “a break is as good as a rest”. If is positive, the two activity levels are substitutable in theagent’s cost function, i.e. the marginal disutility of achieving task i jncreases as the activity level intask j increases. Here, the tasks are rather similar and they make competing demands on the agent,thus, the agent is only concerned about the total activity level. When Vj = 0, the two activity levelsare independent and the marginal disutility of achieving task i is not affected by the activity levels intask j. In this case, the tasks may be of very different types and the agent has task specific disutilityfor each task.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 135To illustrate, we consider a two-product firm. For each product, both the marketing and the after-sales/customer service aspects are critical for success. The firm employs two agents and considers thefollowing options:• The firm may be organized by product line and each agent is put in charge of a product. He isresponsible for both the marketing and the after-sales service of the product.• The firm could be organized by functions, i.e., each agent is put in charge of a particular function.In the first option, the agent is responsible for both the marketing and the after-sales service of a productto a particular set of customers. Such organization is generally termed divisionalization. There is taskvariety which may add to the job satisfaction of the agent so that his effort levels are complementary inthe agent’s cost of effort function, i.e., Vi,j is negative. In the second option, each agent is responsiblefor either the marketing or the after-sales service of both products. This option is generally termed afunctional structure. Since the tasks are similar, they make competing demands on each of the agent.This is likely to result in each agent’s effort level being substitutes in the agent’s cost of effort functionand l4 is positive.Interdependency in II is characterized by ll, i,j = 1,2 and i j. llq could also be negative,positive or zero. A negative implies that the marginal profit from task i decreases as the activitylevel in task j increases.4 A positive ll implies that the marginal profit from task i increases as theactivity level in task j increases. This could be due to the presence of an indivisible input which isshared between the two activities,5 A zero means that the marginal profit of task i does not dependon the activity level of task j. Here, the production technology for the two activities and the marketsfor the resulting products may be very different, thus the two activities are independent in their impacton the expected profit.41f so, there is no reason to produce the two products/services together, unless it is legally required.5For example, II = R1 + )?2 — I, where I is the cost of the input, If Iq <0, then Ilj > 0.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 1367.2.3 First-Best SettingWe first characterize the first-best situation: the activity levels of the agent in the two tasks are observable. We assume that the agent could be severely penalized if he does not provide the required levelof t1 and t2. The wage contract is a constant, w, since there is no incentive problem. The principal’sobjective is to maximize his profit subject to the agent’s participation constraint. We assume that thecertainty equivalent of the agent’s reservation net income level is zero. The first-best problem can bestated as follows:max ll(t)—ws.t. m[w — V(t)]> um(O).Case 1Assuming an interior solution, it is characterized by the following:w = V(t)ll = 1’(t), i= 1,2,where the subscripts on II and V denote the partial derivatives with respect to t. Notice that anycomplementarities in the principal’s profit function and the agent’s cost function do not enter directlyinto the determination of the optimal effort level.6Case 2When a corner solution applies, the principal sets the appropriate t to be zero, and solves the problemfor a single-task situation.6The second order conditions for a maximum require that the following conditions are satisfied.1.I1—V,<O, irl,2.2.(H,, — V)(H3—Vj,)—(H—Vj)2 0, i,j = 1,2.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 1377.3 Analysis of the Second-Best Setting7.3.1 General Solution for Costless Performance MeasuresIn the second-best situation, the agent’s activity levels are noncontractible. The principal utilizes twocostless signals, which are observable in this current period, to provide the necessary incentives for theagent. The wage contract is assumed to be linear in Yl and Y2 and is given by w(yi, y2) = alyl +a2y+b.The manager’s problem ismax Eum[w(yi,y2)— V(t)].Since his utility function is exponential and all random variables are normally distributed, maximizingthe expected utility is equivalent to maximizing the certainty equivalent, which is given by:CEm(ai, a2, b, t) = a1t +a2t + b — V(t) — +That is, the agent’s certainty equivalent consists of the expected wage less the personal cost of effortand less a risk premium.The principal’s objective is to maximize his expected profit, subject to the agent’s participation andincentive compatibility constraints.max 11(t) — E[w(yi,y2)]s.t. CEm(ai,a2,b, t) 0and t é argmaxi CEm(ai,a2,b,t’).Under the linear wage scheme, the certainty equivalent of the principal is given by:CE(ai, a2, b, t) = 11(t) — (aiti + a2t2 + b).The total certainty equivalent of the principal and the agent is then given by:CE + CErn 11(t) - V(t) — +Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 138Note that the total certainty equivalent is independent of the intercept component of the wage contract,b. It serves only to allocate the total certainty equivalent between the two parties, such that the agent’sreservation utility level is reached. As explained in Holrnstrom and Milgrom (1991), this implies thatthe incentive-efficient linear contracts are those that maximize the total certainty equivalent subject tothe incentive compatibility constraints. Therefore the problem reduces to the following:[P7.1]ala2,t11(t) — VQ) — r[ +s.t. t E argmaxtl aitc + a2t — V(t’).Case 1If both ti and t2 are strictly positive, the incentive compatibility constraint is:a=V(t), i=1,2. (7.2)The solutions of a1 and a2 are then given by:7a1= 12 + r11 V21 0-1ii(7.3)a2 V12 V22 0 112Note that 11, and Vj are functions of t, but to simplify the notation, (t) is dropped from 11:(t) andV(t). Expression (7.3) can be rewritten as follows:h[U1(rT4j+ h) — 11rV] . .a:=(Vh)(Vh)2V2 z,,=1,2andzj. (7.4)Notice that if the activities are technologically independent, i.e. V,j = 0, i,j = 1,2 and i j, thena = 11h[rV + 1i], i = 1,2. In the model, the error terms have been assumed to be stochasticallyindependent, i.e. 9 is assumed to be independent of 92. The rates a1 and a2 are set independently ofeach other, except for the fact that H may depend on the optimal level of the other task. Observe thatthe smaller is h, i.e., the greater the noise in the performance measure, the smaller is the rate a and7See appendix 7A for derivation of (7.3).Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 139the lower is the effort exerted on task i.Case 2If the optimal solution entails that either t1 or t2 is set to zero, then the principal sets the appropriatet2 to be zero, and solves the problem for a single-task situation.7.3.2 Costless Performance Measure Available for Only Task 1We now consider the situation where the precision of the signal on the agent’s activity level for the secondtask, h2, is assumed to be zero. This implies that a costless, independent, noisy performance measureis available for only the first task. If an incentive contract based on the signal for the agent’s activitylevel in the first task is used, the agent may be motivated to work only on the first task and neglect thesecond task. If a fiat wage contract is used, the agent chooses the minimal activity levels for both tasks.Without loss of generality, we assume that this minimal activity level is zero. The incremental expectedprofit from the two tasks (before any wage payment to the agent) is given by ll(t1,t2)= 11(t), whichaccrues directly to the principal. 11(t) is assumed to be weakly concave. No discounting is consideredin this model. We assume that the principal and the agent cannot contract based on 11(t) because it isnot observable and verifiable prior to the termination of the agent’s contract.Case 1Assuming an interior solution, we apply (7.4) which gives the formula for the incentive rates when noisyperformance measures are available for both tasks. By setting h2 = 0 and simplifying, we obtain:8h1(11 —112)a1=hi+r(Vjj —)a2 = 0. (7.5)The agent’s choice of the optimal effort level is characterized by a, = (t), i = 1,2. Notice that thedenominator of a1 is strictly greater than zero.9 To ensure that a1 > 0, some restrictions apply to8See appendix 7A for the details.9Convexity of the cost of effort function implies that V11V22 — Vf2 > 0.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 140the ratio of the marginal profits, namely,‘ > It is then logical to ask why the agent should bemotivated to work on task 2, when he is not explicitly rewarded for his effort.If the two activity levels are complementary in the agent’s private cost function, that is, V12 < 0, thena1 is strictly greater than 0, and the more negative is V12, the higher is a1. A negative V12 implies thatthe marginal disutility of achieving task 1 decreases as the activity level in task j increases. Althoughthe agent is paid on only the outcome of task 1, he can decrease the marginal disutility of achievingtask 1 by working on task 2. Thus, there is incentive for the agent to work on task 2, so as to incur less“cost” on task 1.Case 2If the two activity levels are substitutes in the agent’s cost function, i.e. V12 > 0, there is no incentive forthe agent to work on task 2 when the compensation contract takes the form of w(yi) = aiyi + b. Givensuch a contract, and assuming an interior solution, the agent’s reduced incentive constraints becomeT/1(t) = a1 and V2(t) = 0. However, for a1 > 0, this implies that t1 and t2 must have opposite signs tosatisfy the two reduced incentive constraints, i.e. if t > 0, then t2 < 0, and vice versa. This solution isnot feasible and the principal should reconsider his problem by setting t2 = 0 and solving the problemas in a single-task situation.[P7.2] max ll(t1,0) — V(t1,0)— 2 h1s.t. ai=Vi(ti,0).The solution of a1 is given byHi h1a1h1 + rV11(7.6)In section 7.4, we analyze in more depth a quadratic cost setting which further clarifies the explanationsgiven above.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 1417.3.3 A Costly Monitoring TechnologyIn this section, we investigate the principal’s decision to invest in a costly monitoring technology, giventhat no costless signal on the agent’s activity level in the second task is available. The monitor providesa noisy signal of the agent’s activity level for the second task. This signal could then be used for compensation purposes. We could interpret this costly signal to include nonfinancial measures, for example,on-time delivery performance, response time to customers’ requests, and defect rates detected on shippedproducts and during manufacturing. These measures are not readily available in the accounting records,thus extra cost must be incurred to extract this information. The signal may also include a consultant’sreport. The principal chooses the monitoring intensity to obtain the desired precision on this costlysignal. The higher the desired precision, the greater the monitoring intensity and the higher the cost.Assume that the costly signal obtained is related to the agent’s activity level and the monitoring intensityin the following way:1112 = t2 + 702, 02 N(O, 1).V”2Assume that 0 arid 02 are independent of one another, that is, the error terms are stochasticallyindependent. The cost of the signal depends on the level of h2 and we denote the cost by C(h2). Assumethat C(.) is continuous and monotonically increasing, and thatlim C(h2) = cc and lim C(h2) = 0.h2—.oo h2—OThe higher the level of h2, the more precise the signal and the more costly the monitoring. The cost ofa perfect signal is infinite, while there is no cost if the principal decides not to use the costly monitor.We also assume that the level of h2 is observable and verifiable, thus it is contractible. For example, h2may be related to the number of auditor or computer hours devoted to retrieving the data.We assume an interior solution for the optimal level of monitoring. With the signal from the monitor,the principal can now base the wage contract on the signal. The wage contract takes the following form,Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 142w(yl, y2) = alyl + a2y2 + b. The total certainty equivalent of the principal and the agent is now givenby:CE + CErn 11(t) — C(h2)— V(t) — + j.The principal determines the optimal wage contract, monitoring and effort levels by solving the followingreduced problem:[P7.3J max 11(t) — C(h2)— V(t) — +a1,a2hs.t. t E argmaxi aitç + a2t — V(t’).Case 1Assuming an interior solution, the solutions of a1 and a2 are given as follows:’0—1a1 V11 V2, 0 11= 12+r . (7.7)a2 V12 V22 0 112Simplified further, we obtain the following expressions:— hi[11i(rV22+h2) —112rVi]a1— (rVii + hi)(rV22 +h2) —— h2[11(rVii + hi) — llirVi2]7 8a2— (rVii + hi)(rV22 +h2) —r2VThe expressions for a, i = 1,2 are similar to those obtained in the second-best setting and both outcomesare observable. The difference is that in the present situation, the precision of the signal on the agent’seffort in task 2, h2, is endogenously determined. The principal chooses the optimal level of monitoringh2, determined by the following first-order necessary conditions on h2, given by2ra2U’ I,II)= 2The level of h2 is determined by substituting for a2.11 The commission rates a1 and a2 are then setAppendix TA for the derivation.that the second order conditions are also satisfied. C” (h2)—Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 143accordingly, and the agent chooses his optimal action, given byl’(t)=a, i=1,2.Case 2If a corner solution in which t2 = 0 applies, then the principal does not undertake any monitoring andhe sets a2 = h2 = 0 and solves the problem for a single-task situation. Such a corner solution occurs ifthe cost of obtaining and using the signal on the agent’s activity level in task 2, y2, is too high.7.4 Quadratic Cost Setting7.4.1 IntroductionIn this section, we analyze in more depth a setting in which the agent’s personal cost function is quadraticand the expected payoff function is linear. We assume that the profit function, 11(i) is additively separablein the two effort types. By appropriately re-expressing the problem, this assumption allows us, withoutloss of generality, to concentrate on a linear profit function. A concave profit function can be transformedinto a linear function, This transformation changes the activity level measure and the agent’s personalcost function becomes more convex. Appendix 7C gives an example of the tranformation. Therefore,for subsequent analysis, we use a linear profit function, denoted by 11(t) = t1 + t2. We consider ageneral symmetric quadratic cost of effort function, given by V(t) = 6(t? + + vtit2). Then ll 1,= 6(t + vt), = S and l’j = vS for i,j = 1,2 and i j. These functions allow us to derive closedform expressions for the optimal activity levels and the incentive components. S influences the overallcost of attaining the two profit levels and ji influences the complementarity of these costs. To satisfy theconvexity requirements, i’2 < 1, i.e., —1 < ii < 1. If 0 < v < 1, the two effort types are substitutable inthe agent’s cost function, and if —1 < v < 0, the two effort types are complementary in the agent’s costfunction. ii = 0 implies that the two activities are technologically independent.’2 Figure 7.13 illustrates‘2Synetry is a key eernent of this example.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 144Figure 7.13: Behavior of V(11,t2)with changes in t2the behavior of the cost of effort function for a fixed level of effort in task 1, 4.Note that at t2 = 0,<0, ifv<0V2(t) =0, ifv=0>0, ifv>0.From (7.2), the agent’s incentive constraint is given by= a, i = 1,2. (7.10)Differentiating (7.10), we obtain the following:.‘- -‘ 6 zi6ôti —v66By the inverse function theorem, we obtain the following:1_8a1 812 = 1’v2)(7.11)-p 1ôa, 8a2 6(1—i’2) 6(1—i’2)Equation (7.11) characterises how changes in the incentive rates a affect the activity level that will besupplied.V(f11t) v, 0aChapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 145When the two effort types are complementary in the agent’s personal cost function, an increase ina: affects the activity levels in both tasks positively. The principal uses both a1 and a2 to motivate theagent’s effort in both tasks. On the other hand, when the effort types are substitutable in the agent’spersonal cost function, an increase in a: results in an increase in activity level t and a decrease in t,i j. The principal can only use incentive component ai to motivate effort in task 1 and a2 to motivateeffort in task 2. In this latter case, relatively high a1 implies that the opportunity cost of working intask 2 is high, thus the agent’s attention is partially directed from task 2. Similarly, relatively high a2discourages effort in task 1. We observe that the higher is IvI, the more responsive is the activity level tto a change in a1. From (7.11), we note that 6 also determines how responsive the agent is to incentives.High 6 lowers the responsiveness to incentives.7.4.2 First-best SolutionIn the first-best situation, a flat wage contract is paid. Optimal effort levels are t ==. Thetotal certainty equivalent is given by:CE = ll(t*) — V(t*)= 6(1+v)(7.12)As 11 increases, t, t and thus, CE decrease. This implies that the optimal effort levels are lower if thetwo effort types are substitutes in the agent’s cost function, as compared to when they are complements.Similarly, as 6 increases, t, t and thus, CE decrease. A high S indicates that it is more costly toemploy the agent, thus the optimal effort levels are lower.appendix 7A for details of the derivation.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 1467.4.3 Second-best Solution - Costless Noisy Performance MeasuresIn the second-best situation, if independent, costless, noisy performance measures are available for bothtasks, then substituting ll = 1, Vj = 6 and V,j = vS, i j into (7.4), we obtain the following:—.— h{h1 + r6(1 — v)]7 13a:—(r6+h)(r6+h)—v2The agent’s choice of activity level is characterized by (t) = a, i = 1, 2. Since V = 6(t +vt3), i 1, 2,we have two equations for the two tasks which we solve simultaneously to obtain the following:’4i,j = 1, 2, and i j. (7.14)Substituting (7.13) for ã and a3 gives-— hh3 + r6(h — uh1)— 1 2 d 7 156(1+v)[(r6+h)(r6+h)—v2]’z,j— , ,an zj.The total certainty equivalent if both tasks are undertaken is:CE’r = llQ)_VQ)_r[1+?j— r6(1 — v)(hi + h2) +2h1h (7 16— 26(1 + v)[fr6 + hi)(r6 + h2) —v2r8]We make the following observations on the behavior of the total certainty equivalent:’5• CET is increasing and concave in h1 and h2. The principal is better off with more precise performance measures.• The levels of ij, i = 1,2 and CEp are higher when v is negative than when v is positive, holdingall other parameters constant. In fact, < 0, i.e., the total certainty equivalent decreases as vincreases. Recall that a negative z’ implies that the marginal disutility of achieving task i decreasesas the activity level in task j increases, thus it is not surprising that the principal is better off the‘4See appendix 7A for details.‘5See appendix 7A for details.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 147lower the value of ii. This implies that if the principal has perfect freedom in the grouping andassignment of tasks to the agents, he is better off if such grouping and assignment achieve as lowa value of v as possible. The principal is able to motivate higher effort levels more efficiently andearn higher profit levels with a more negative value of 11.• We denote the loss to the principal from being unable to observe the agent’s effort by L. ThenL = CE — CEp, where CE is the total certainty equivalent in the first-best setting. WhileCE is independent of the risk aversion of the agent r, GET is decreasing in r, thus, the level ofL increases as r increases. Also, observe that lim,.0L(r) = 0, i.e., when the agent is nearly riskneutral, the loss to the principal approaches zero. This result is analogous to Grossman and Hart(1983) (Propositions 15 and 16) in a single-task pure moral hazard setting with binary outcomes.Arya, Fellingham and Young (1993) obtain a similar result in a setting in which the agent hasprivate productive information, They express the efficiency loss to the principal as the sum ofexpected lost production and a risk premium, and they show that the loss in expected productionincreases as the agent becomes more risk averse.Next, we examine if the principal is always better off using both performance measures in the incentivecontract and motivating the agent to work in both tasks. First, consider the case when v < 0. If theprincipal chooses to use only the costless performance measure for task 1, then the total certaintyequivalent is as follows (see equation (7.21)):- hi(1—v)26(1 + v)[hi + r6(1 — v2)]Figure 7.14 shows that the principal always achieves a higher certainty equivalent using both pieces ofinformation, regardless of how noisy the performance measure of task 2 is.Chapter 7. Multit ask Principal-Agent Model with Costly Monitoring Tech nology 148Figure 7.14: Behavior of total certainty equivalent with h2 (v < 0)On the other hand, if ii> 0, depending on the relative levels of h1 and h2, there can arise situationswhere the principal is better off motivating the agent in just one of the tasks. Assume that h1 > h2.The total certainty equivalent if only task 1 is undertaken is given by:= 2(h,+ r6)(7.17)Then the principal motivates the agent to work on both tasks if, and only if,’6- iirôhi[r6(l — i’)(2 + v) + 2h1]h2 > h2= (1 — u)[h, + r6]2(7.18)Note that h, increases as h1 increases. This implies that the higher is h,, the higher h2 must be beforetask 2 is undertaken. The intuition for this is very straightforward. The higher is h,, the more attractiveis task 1 and a higher incentive rate for task 1 results. This implies that to motivate task 2, a relativelyhigh incentive rate is required. However, this is costly for a low h2, and the principal does not motivatethe agent to work on task 2 if h2 < h2.(7.18) holds if, and only if, the total certainty equrvalent when both tasks are undertaken exceeds that whenonly task 1 is underaken. See appendix 7A fo derivation of h2.GETCE7,Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 149Figure 7.15: Behavior of total certainty equivalent with h2 (i’> 0)There are two cost elements in motivating the agent to work on task 2, namely, the required riskpremium as a result of incentive rate a2, and the increased cost of motivating the agent in task 1. Thelatter cost element arises because with a2 > 0, the opportunity cost of working on task I increases, sothat it is now more expensive to motivate any particular level of effort in task 1. Figure 7.15 depicts therelationship between the respective total certainty equivalent and the level of h2.If h2 is less than h2, the information is too noisy to be of any value to the principal.t7 The optimalactivity level in task 2 is low and the principal is better off motivating the agent to work on task 1 alone.This is consistent with Itoh (1991) in a multi-agent setting. He investigates whether it is optimal forthe principal to induce teamwork or unambigious division of labour. He concludes that in a situationsimilar to z.’> 0, the principal wants either a specialized structure or substantial teamwork. A low levelof help is suboptjmal.’8‘7The cost of using the noisy information outweighs the benefit.18Itoh (1991) uses a binary outcome structure in his analysis, i.e., outcome is denoted by either success or failure.CETCEi1%1%Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 150Lemma 7.1 applies for both positive and negative values of i’. We assume that when 11 > 0, it isoptimal for the principal to motivate the agent to work on both tasks.Lemma 7.1:• If hi = h2, then a1 = a2 and i =•Ifh1>,thenãáand.Lemma 7.1 states that if there is relatively more precise information available on the activity level ofone task, then the principal pays a higher incentive rate and induces a higher activity level in that task.Lal and Srinivasan (1993) demonstrate a similar result in a multiproduct salesforce setting for the casewhen the agent’s effort types are perfect substitutes in the agent’s cost function. Products with loweruncertainty should be given higher commission rates to generate maximum profits to the firm.7.4.4 Comparative Statics - Costless Noisy Performance MeasuresWe examine how the optimal incentive components and the optimal effort levels vary as h2, v and rvary. Proposition 7.1 examines the effect of changes in h2, the precision of the performance measure intask 2. For v > 0, we assume that as the various parameters vary, it is still optimal for the principal tomotivate the agent to undertake both tasks.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 151Proposition 7.1:An increase in the precision of the performance measure for task 2 (h2) results in:1. a decrease (increase) in the incentive rate for the performance measure of task 1 if the activities- 1<0 for—1<v<0are complements (substitutes), i.e., ji-> 0 for0<v<1;2. an increase in the incentive rate for the performance measure of task 2, i.e., fr > 0;3. an increase (decrease) in the level of activity of task 1 if the activities are complements (substitutes),- I > 0 for —1<v<0at’‘< 0 for0<v<1;4. an increase in the level of activity in task 2, i.e., fr- > 0.As the precision of the signal of agent’s effort in task 2 increases, the risk imposed on the agent for usingthe signal to motivate a given effort level decreases. It is now less costly to motivate the agent to workon task 2, thus the principal increases a2 and 12. For —1 < ii < 0, the principal reduces the incentivecomponent a1 since the marginal cost of motivating the agent through a2 is now relatively lower. 1 alsoincreases since the overall marginal cost of motivating the agent is lower. For 0 < v < 1, as a2 increases,the opportunity cost of working in task 1 increases, thus, a1 increases to balance the agent’s motivationto work on task 1. ii decreases, since the relative cost of motivating effort in task 2 decreases and theprincipal partially redirects the agent’s attention from task 1 to task 2. Note that a change in h1 affectsa1, a2, 1 and 12 the same way that a change in h2 affects a2, a1, 12 and ij, respectively.The next proposition examines how changes in v and r affect the optimal incentive rates and activitylevels. When —1 < ii < 0, an increase in the value of v implies that the complementary effect of theeffort types in the agent’s cost function decreases. When 0 < ii < 1, an increase in the value of v impliesthat the substitution effect of the effort types in the agent’s cost function increases.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 152Proposition 7.21. An increase in the value of the interactive effect of the two effort types in the agent’s cost function(v) results in:(a) a decrease in the incentive rate for both measures, i.e., j < 0, i = 1, 2;(b) a decrease in the level of activity for both tasks if the activities are complements, while it isunclear what happens if the activities are substitutes,19 i.e.,-. 1 <0 for —1<v<0isopambiguous for 0< v <1, i= 1,2.2. An increase in the agent’s coefficient of absolute risk aversion, (r) results in:(a) a decrease in the incentive rate for both measures if the activities are substitutes, while it is am--. I ambiguous for —1< v< 0biguous if the activities are complements, i.e.,<0 for0<v<1, i=1,2;(b) a decrease in the level of activity for both tasks, i.e., < 0, i = 1,2.When v is negative, motivating a higher level of 11 has a double benefit, since the direct cost of motivatingt2 becomes less costly. As xi becomes less negative, this benefit decreases. Generally, as xi increases, itbecomes relatively more expensive to motivate any particular level of effort. We expect ã and t, i = 1,2to decrease. However, for xi> 0, a decrease in ã implies that the opportunity cost of working in task j,i j, is decreased, and the indirect effect is an increase in tj. Thus, we see that the effect of an increasein xi for xi >0 on f, i = 1,2 is ambigious.2°19While it is generally possible to characterize the sign of for alternative sets of parameter values, we do not becausethe expressions are complicated and yield little economic insight. Tbis note applies to subsequent cases of ambiguityas well. However, if the characterizations are useful, we place them in the appendix and provide the discussion on theeconomic insight in the section subsequent to the respective proposition in this main paper.20 direct effect on i (due to a change in aj) is relatively larger than the indirect effect (due to an equal change incii). However, the rate of change in ej due to changes in z’ may be different from that of ej depending on the precision ofthe signal.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 153As r increases, the required risk premium to motivate a given effort level increases and the cost ofmotivating the agent increases. We expect ii, i = 1,2 to decrease. For v > 0, j, i = 1,2 decreases asthe principal settles for a lower activity level. For v < 0, the behavior of a appears to depend on therelative level of h to h3. If h is comparable to h1, both ã and ãj decrease. If h is relatively very largecompared to h, then it is possible for ã to increase and a3 to decrease as r increases. There appears tobe a substitution effect taking place. The principal now emphasizes the use of the signal of task 1 dueto its relatively high precision and focuses less on the use of the signal of task 2.7.4.5 Costless Performance Measure Available for Only Task 1From Proposition 7.1, we learn that a change in the precision of the signal on task 2, h2, affects theincentive rates and activity levels differently depending on whether the effort types are complementaryor substitutable in the agent’s cost function. In this section, we consider the extreme case in which h2equals zero, i.e., costless performance measure is available for only task 1.Case 1If the effort types are complementary in the agent’s cost function, i.e. v < 0, then substituting ll = 1,= 6 and V,j = v6 into (7.5), we obtain the following:- hi(1—v)= hi+r6(1—v2)a2 = 0. (7.19)Without any information about the second task, the principal relies on information about the first taskto motivate agent’s effort in both task 1 and task 2. The combination of effort levels in the two tasksthat can be induced are thus restricted. The agent’s choice of activity level is characterized by l’(t) = a,i = 1,2. Substituting = 6(t1 + vt) into V = a, i = 1,2 and solving simultaneously, we obtain theChapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 154following:2111= 6(1—i’2)6(1+zi)[hi+r6(1—v)]- —va1t2= 6(1—zi)—vh17 20— 6(1 + v)[hi + r6(1 —Since v < 0, we obtain a1 > 0, 1 > 0 and t2 > 0. Recall that the incentive component, a1 is usedto motivate both 11 and t2. Thus, for any changes in the exogenous parameters, a1, l and t2 move inthe same direction. As h1 increases, the cost of using a1. to motivate effort decreases, resulting in anincrease in a1, 11 and 12. As v or r increases, the cost of motivating the agent increases, thus, a1, 11 and12 decrease. The total certainty equivalent is given by:CET(V<0) = ll(1)_V(1)_i-— h1(1—v)‘72126(1+i’)[hi+r6(1—v)]Case 2If the effort types are substitutable in the agent’s cost function, i.e. i’ > 0, then from (7.6),- h1a1 =h1 + r6a2 = 0. (7.22)Without any information about the second task, it is impossible to induce effort in that task. The agentsets t2 = 0 and his choice of activity level in t1 is characterized by V1(t1, 0) = a1. Since Vi (t1, 0) = 6t1,we obtain=21See appendix 7A for details.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 1555(h1 + rö)t2 = 0. (7.23)Notice that the optimal values of a1 and 11 are independent of both the nature of task 2 and anyinteraction between the two tasks. The result is consistent with Holmstrom & Milgrom (1991). Theyshow that when effort types are perfectly substitutable in the agent’s cost function, and if costlessperformance measure is not available for say, task 2, then the use of an incentive contract on task 1implies that the agent allocates no effort to task 2. Our analysis indicates that this result holds for any0 < v < 1, and not only when the effort types are perfectly substitutable in the agent’s cost function.Observe that à and t1 increase with h1 and decrease with r. The total certainty equivalent is givenby:ET(V> 0)= 26(h1+(7.24)The moral hazard problem created by the unobservable nature of the agent’s effort and the additionalincentive problem caused by having an observable performance measure for one task only, result in anefficiency loss and the principal is worse off than in the first-best setting in which the agent’s effort inboth tasks are observable and verifiable.7.4.6 Second Best Solution - Costly Monitoring TechnologyIn this subsection, we assume that there is no costlessly observable performance measure for the secondtask. Instead, the principal uses a costly monitoring technology. Assuming that the optimal monitoringlevel is strictly positive, then by substituting for ll = 1, V,j = 6 and Vj = v6, i, j = 1, 2, i j into(7.8), we obtain the following:—h[h + r6(1—ii)]— 1 2 7 25a— (rS+h)(r6+h)—v26’z,j , , zj. ( . )Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 156The agent chooses the optimal activity levels such that 4(t) = 6(t + vt) = a. Therefore, by solvingsimultaneously, we obtain=i,j = 1,2, i j. (7.26)Notice that these expressions are identical with those in the second-best setting with both outcomescostlessly observable (equations (7.13) and (7.14)). In that situation, the precision of the signal on theagent’s effort in the second task is exogenously specified, while in this case, it is endogenously determined.We assume that the cost of the monitoring technology is C(h2) = ch2, c> 0. The principal chooses theoptimal level of monitoring h2 which is determined as follows:h2 = max{0, h2},where 1t2 is defined below. From (7.9), the level of A2 is determined as follows:2hHence, we obtain 112 =By substituting for a2 as given in (7.25) and simplifying, we obtain22— h + r6(1 — u) — 6/(2rc)[hi + rö(1 — v2)]7 272—)(hi+r8)When h2 = 0, no monitoring is undertaken and the results in the previous section in which only oneperformance measure is available applies. Also, we observe from (7.27) that limo h2 = oo, implyingthat as the cost of obtaining information approaches zero, the principal purchases perfect information.23We assume an interior solution for 112. Thus, h2 = h2. Using (7.27) for the optimal level of /12 andsubstituting into (7.25) and (7.26), the expressions for a and t,, i 1,2, we obtain the following:— h1(1 — v6/(2rc))a1— hi+r622See appendix 7A for details of the derivation.23From (7.27), we also observe that limr_o h2 = 0, i.e., as the risk aversion of the agent approaches zero, the principaldoes not purchase any information. When the agent is risk neutral, the incentive rates are set at a = 1 and the agentpays a fixed fee to the principal.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 157— h1 + r6(1 — v) — 6,./(2rc) [hi + rS(1 — v2)]a2 — (7.28)h1 + r6— h1 — vr6[1 — 6/(2rc)(1 + ii)]1— 6(h1 + r6)(1 + v)-— 1 — 6/(2rc)(1 + ii)— 6(1 + ij’)7.29)We make the following observations of the optimal level of effort in task 2 i’2 24• It is independent of h1, which is the precision of the signal of the agent’s effort in task 1. Here, theprincipal determines the precision of the signal of the agent’s effort in task 2 through the monitoringintensity, h2. Since y provides no information of the agent’s effort in task 2, the principal offsetsany variation in h1 through his choice of h2 and maintains the level of t2. On the other hand,when the precision of the signal of the agent’s effort in task 2 is exogenously specified, the agent’sactivity level in task 2 depends on h1. For —1 <ii < 0, t2 increases with h1 and for 0 < ii < 1, t2decreases with h1. See Proposition 7.1.• If ii = 0, £ = — /(2rc). This is also the optimal effort level if task 2 is the only task and costlymonitoring technology is employed.• Under the first-best setting, the optimal effort level in task 2 is given by t = The deviationof t2 from first-best is thus given by ,/(2rc). From (6.7), we note that if task 2 is the only task,which implies that effort allocation is not an issue, and costly monitoring technology is employed,then the deviation from first-best effort level is also /(2rc). This implies that in the two-tasksetting, the deviation from the first-best effort intensity depends on only the risk aversion of theagent, and the cost of monitoring, and is not affected by the effort allocation issue. This resultis consistent with Feltham and Xie (1994). They show that a contract based on a noncongruentmeasure induces suboptimal effort allocation across tasks, whereas performance measure noiseresults in suboptimal effort intensity. In our model, the performance measures used are congruent24Note that a2 > âi * t2 > ti. t2 > t holds when /(2rc) < h+r6(1+v) Thus, low c and low h1 may result in t2 > t1.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 158with the principal’s expected profit, thus suboptimal effort allocation is not an issue.• First-best effort intensity in task 2 is achieved if the agent is risk neutral or the monitoring technology is costless (i.e., if rc = 0). A costless monitoring technology implies that perfect information isobtained and the performance measure is noiseless. As the cost of monitoring or the risk aversionof the agent increases, the deviation from first-best increases. This is a result observed in mostmodels in the principal/agent literature.Next, we derive the condition on c for monitoring to be worthwhile for —1 <v < 0 and 0 < i’< 1.From (7.16), for a given level of h1 and h2, the total certainty equivalent before deducting the cost ofmonitoring if both effort levels are strictly positive isrö(1 — v)(hi + h2) + 2h1h30CEp(2)— 26(1 + v)[frS + hi)(r6 + h2) —v2r6)(7. )CET(h2)is increasing and concave in h2 which implies that the principal experiences diminishing returnsto monitoring.Case 1If —1 < ii < 0 and no monitoring is undertaken, the agent still works on both tasks and the totalcertainty equivalent is given by (see (7.21)):h1(1—CEr(h2 = 0)26(1 + v)[hi + r6(1 —When monitoring is costly, Lemma 7.2 provides a condition for monitoring to be worthwhile.Lemma 7.2: For —1 < v < 0, a necessary and sufficient condition for the principal to engage in monitoringis{h1 + rS(1 — v)]2c< 2r6[hi + r6(1 —v2)]Chapter 7. Multit ask Principal-Agent Model with Gstly Monitoring Technology 159ACET(h2 = 0)Figure 7.16: Determining the optimal level of monitoring (v <0)At the optimal level of monitoring, the marginal benefit of monitoring equals the marginal cost ofmonitoring c. The marginal benefit (MB) of monitoring is as follows:25MB—r[hi+r6(1—v)]2731— 2(r6(1 — i.’2) + r6(hi + h2) +h1]2At h2 = 0, MB(h2 = 0) =2Jj’p. Thus, if c MB(h2 = 0), the principal does not engagein monitoring and he relies on the costless signal of the agent’s effort in task 1 to motivate the agent’seffort in both tasks. He settles for a lower effort level in task 2 than if he engages in monitoring. As hincreases, the bound on the cost of monitoring tightens. Figure 7.16 shows that once the condition inLemma 7.2 is met, monitoring is always worthwhile even when the optimal level of monitoring is verylow.Case 2If 0 < v < 1 and no monitoring is undertaken, it is impossible to induce effort in task 2 and the agentworks on only task 1. The total certainty equivalent is given by (see (7.24)):CEp(h2 = 0)= 26(h1+r6)CET (h2)See appendix 7A for details.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 160A necessary condition for the principal to motivate the agent to undertake both tasks is that CET(h2>0) > 26(hr6) This implies that (see (7.18))— vr6hi[r6(1 — v)(2 + ii) + 2h1]2> 2— (1 — v)[hi + r6]2When the monitoring intensity is too low, the information obtained is too noisy to be of any value.Using the expression for optimal h2, we determine the upper bound for the cost of monitoring, c.Lemma 7.3: For 0 < ii < 1, necessary conditions for the principal to engage in monitoring and motivaté the agent to work on both tasks are:1 <(1—v)2{h+r6hi(2—v)+rv)}’ dCr6( + ){h+r6(1—v)[2hi+r6(1—v)]}, an2. h1 > vrö[1 — 6/(2rc)(1 + v)]If the cost of monitoring is too high, the principal does not engage in monitoring. He foregoes anybenefit from task 2 and concentrates on task 1 alone. Similarly, if h1 is too small, i.e., the informationon activity level in task 1 is very noisy, the principal can be better off concentrating on task 2 alone,even though the signal for activity level in task 1 is costlessly available. Note that the lower bound onh1 depends on the level of c. As c increases, the lower bound on h1 decreases. Figure 7.17 illustratesthat the principal does not engage in a low level of monitoring.Propositions 7.3 and 7.4 compare the optimal levels of ti, t2 and ai in the non-monitoring (denotedby ) and monitoring (denoted by ) environments. In the former environment, a2 o.Chapter 7. Multit ask Principal-Agent Model with Costly Monitoring Technology 161CEp(h2 = 0)Proposition 7.3: If the activities are complements, i.e., —1 <ii < 0, the following relations hold:1. i >2. 12 > 12, and3. a1 < a.The principal chooses to extract higher effort levels, and can do so more efficiently, with the monitoringtechnology. With the information about the agent’s effort in task 2, the principal uses both a1 and a2to motivate the agent in the two tasks. Without the monitoring technology, no information is availableabout the agent’s effort in task 2 and the principal uses only a1 to motivate the agent.CE7(h2)Figure 7.17: Determining the optimal level of monitoring (ii > 0)Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 162Proposition 7.4: If the activities are substitutes, i.e., 0 < v < 1, the following relations hold. Recallthat t2 = 0.1. i <ii,2.>,and3. &i < i.In this setting, incentive component a1 motivates the agent to work on task 1 only. Without themonitoring technology, no information is available about the agent’s effort in task 2, thus, the principalmotivates the agent to concentrate totally on task 1. With the monitoring technology, the principalredirects the agent’s attention partially to task 2.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 1637.4.7 Comparative Statics - Costly Monitoring TechnologyWe examine how the intensity of monitoring, the incentive components and the effort levels vary as theother variables vary. We assume interior solutions for both the monitoring level and the effort levels.Proposition 7.5:An increase in the cost of monitoring (c) results in:1. reduced monitoring, i.e., < 0;2. increased (decreased) incentive rate for task 1 if the tasks are complements (substitutes), i.e.,1> 0 for —1<v<0Sc< 0 for0<v<1;3. decreased incentive rate for task 2, i.e.,-< 0;4. decreased (increased) effort level in task 1 if the tasks are complements (substitutes), i.e.,I < 0 for —1<v<0Sc> 0 for0<v<1;5. reduced effort level in task 2, i.e., < 0.As the costliness of monitoring, c, increases, it becomesrelatively more expensive to motivate the agentusing incentive component a2. The intensity of monitoring, h2, and consequently, the incentive component on task 2, a2, decrease. For —1 < v < 0, since the effort types are complements in the agent’sprivate cost function, the principal now uses a relatively cheaper means of motivation, and 1 increases.Both i and E2 also decrease since the cost of motivation has increased and the principal could not bebetter off than before the increase in c. For 0 < xi < 1, with the decrease in a2, the principal seeks toreduce the opportunity cost of working on task 2 to maintain a proper allocation of the agent’s effortbetween the two tasks, and à decreases. The principal partially redirects the agent’s effort from task 2Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 164to task 1, since it is now relatively cheaper to motivate effort in task 1 as compared to task 2. Thus, iincreases while t2 decreases.Proposition 7.6: An increase in the precision of the performance measure for task 1 (h1) results in1. reduced (increased) monitoring if the tasks are complements (substitutes), i.e.,I < 0 for —1<v<0&h1> 0 forO<v<1;2. increased incentive rate for task 1, i.e., > 0;3. decreased (increased) incentive rate for task 2 if the tasks are complements (substitutes), i.e.,I < 0 for —1<v<0> 0 for0<v<1;4. increased level of activity for task 1, i.e., fr- > 0;5. no change in the activity level for task 2, i.e., fr- = 0.The optimal level of 2 does not depend on h1, since the signal Yl is not informative on the agent’s effortin task 2. We compare this result with the case when the precision of the signal of the agent’s effort intask 2 is exogenously specified. Proposition 7.1 tells us that as h1 increases, the optimal level of t2 isnot maintained. Rather, as h1 increases, t2 increases for —1 < v < 0, while for 0 < v < 1, t2 decreases.As h1 increases, it is now less expensive to motivate effort in task 1 using incentive component a1, sothe principal increases a1 and p For —1 < z.’ < 0, with the increased level of a1, the principal reducesthe intensity of monitoring. 112 and a2 decrease, and the level of t2 is maintained. For 0 < v < 1, sinceä increases, the opportunity cost of working on task 2 increases. The principal increases the intensityof monitoring, 112, and the incentive component on task 2, &2, otherwise the agent’s effort in task 2 ispartially redirected to task 1.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 165Proposition 7.7: An increase in the value of the interactive effect of the two effort types in the agent’scost function (v) results in:1. reduced monitoring, i.e., < 0;2. reduced incentive rates for both measures, i.e., < 0, i = 1,2;3. reduced level of activity for task 1 if the two tasks are complements, while the relationship is1 <0 for —1<v<0unclear if the two tasks are substitutes, i.e.,-( ambiguous for 0 < u < 1;4. reduced level of activity for task 2, i.e., < 0.As v varies, &i, 1i and a2 should move in the same direction since there is no gain to using one incentivecomponent over another. As ii increases, it becomes relatively more expensive to motivate any particularlevel of effort, thus, âi, h2 and a2 decrease, Also, we expect both 1 and t2 to decrease as v increases.However, we observe that for , this behavior does not always hold for v> 0. There can arise situationswhen i may increase as v increases. This is because the rate of decrease in the incentive rate a2 maybe much faster than that of ài (due, for example, to a very low h1). Thus, the attractiveness of workingin task 2 drops by more than that for task 1, and the opportunity cost of working in task 1 decreases.The agent’s attention is partially redirected from task 2 to task 1. These relationships are illustrated inFigure 7.18.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 166Figure 7.18: Behavior of h7, 1, and a1 as v variesValues of the parameters: c = 0.2, 6 = 0.5, r = 5 and h = 0.1Proposition 7.8: Increasing the agent’s coefficient of absolute risk aversion (r) results in:1. reduced incentive rate for task 1 if the tasks are substitutes, but the relationship is unclear if the- I’ ambigtous for — 1 <i’ < 0tasks are complements, i.e., 1.<0 for0<v<1;2. reduced level of activity for task 1 if the tasks are complements, but the relationship is unclear if- 1 <0 for—1<v<0the tasks are substitutes, i.e., j- çj ambiguous for 0 < ii < 1;3. reduced level of activity for task 2, i.e., < 0.We were unable to establish the behavior of the level of monitoring and the incentive rate on task 2 withchanges in r. However, numerical examples indicate that the level of monitoring is concave in r andis most intense at an intermediate level of risk aversion. This result is similar to that for a single-task0Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 167model with costly monitoring. Proposition 6.2 states that monitoring is most intense at an intermediatelevel of risk aversion.It is also interesting to contrast the results with Proposition 7.2 for the case when the precision of thesignal of the agent’s effort in task 2 is exogenously specified. As r increases, the required risk premiumfor a given effort level increases and the activity levels of the two tasks decrease. On the other hand,when the monitoring precision is endogenous, as r increases, the principal is able to partially controlfor the increased risk premium by choosing a higher precision. This benefit is offset by the direct costof monitoring. Using cost-benefit analysis, the principal determines the optimal precision and activitylevels. As Proposition 7.8 shows, as r increases, the behavior of the incentive rates and activity levels isnot clear.We first examine the case where —1 < v < 0. The incentive rate a is concave in r and reaches amaximum at a very low level of r. The incentive rate a2 may increase or decrease in r. It appears toincrease when both h1 and c are at low levels. As r increases, a higher risk premium is required fora given incentive rate. When h1 and c are low, it becomes relatively more attractive to obtain moreprecise information on task 2 so that the principal can use a higher a2. At the same time, he reducesthe use of a1 to avoid the high risk premium due to the low h1. Recall from (7.11) that an increase in a:affects the activity levels in both tasks positively. The diagrams below show some of these relationships.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring TechnologyFigure 7.19: Behavior of h2, i and a1 as r variesValues of the parameters: c = 0.01, 6 = 0.5, ‘ = —0.3 and h1 = 5Figure 7.20: Behavior of h2, , and a1 as r variesValues of the parameters: c = 0.01, 6 = 0.5, v = —0.3 and h1 = 1168a’.FIAAtirChapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 169CFigure 7.21: Behavior of h2, , and a1 as r variesValues of the parameters: c = 0.2, 6 = 0.5, v = —0.3 and h1 = 5Figure 7.22: Behavior of h2, and a1 as r variesValues of the parameters: c = 0.2, 6 = 0.5, ii = —0.3 and h1 = 1Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 170Figure 7.23: Behavior of h2, , and ã as r variesValues of the parameters: c = 0.2, 6 = 0.5, v = 0.2 and h1 = 5Next, we look at the case where 0 < v < 1. The incentive rate a2 decreases in r for v < /(), whilethe relationship is unclear for v > The relationship of the activity level of task 1 with r is notclear, but generally, it appears that ij decreases as r increases. We illustrate some of these relationshipsin Figure 7.23.7.4.8 The Value of MonitoringThe role of monitoring in this problem is performance evaluation rather than belief revision or information-verification. Without any information about task 2 and without monitoring, the combination of effortlevels in the two tasks that can be induced are restricted or it may be impossible to induce effort in task2. The availability of noisy performance measures on both tasks enables the principal to motivate highereffort levels and more satisfactory effort allocation. The issue then is the determination of the optimaleffort level in each task given the acquired information. However, this information is imperfect. In fact,t,rChapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 171the principal chooses the level of precision of the information, but since the cost of perfect informationis infinite, the principal never acquires perfect information. The use of imperfect information in the incentive contract increases uncertainty to the agent. If the principal uses only the imperfect informationabout the agent’s effort in task 1, he just needs to pay a risk premium on that piece of information.However, with an imperfect monitor of task 2, the principal also chooses to pay a risk premium on theuse of this second piece of information. The total risk premium is + ]. The cost of monitoringincludes:1. the direct cost of monitoring, C(h2) = ch2; and2. the risk premium required on the second piece of imperfect information.The availability of monitoring allows the principal to motivate higher effort levels or more satisfactoryeffort allocation. In fact, when v> 0, the second task is not undertaken if there is no monitoring. Thebenefits of monitoring are:1. additional profit arising from higher effort levels or a more satisfactory effort allocation; and2. the savings on the required risk premium on a1, since the incentive component on task 1, a1, islower in the monitoring environment.The principal has two main decisions to make about monitoring. First, he decides whether monitoring is to be undertaken. Generally, if the benefits of monitoring exceed the cost of monitoring, thenmonitoring is undertaken. This is equivalent to the total certainty equivalent in the monitoring environment being greater than that in the environment without monitoring. Next, the principal decideson the intensity of monitoring, represented by the precision of the signal obtained from monitoring, h2.Our analysis does not distinguish between the two decisions. In fact, the first is not explicitly modelled.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 172Ignoring the direct cost of monitoring, the analysis above shows that:1. For —1 <v < 0, information, no matter how noisy it is, is always valuable;2. For 0 < ii < 1, information that is too noisy is not valuable. The principal is better off not usingthe information.Here, we show that in a two-task setting, the value of a little bit of information depends on the degree ofsubstitutability of the two effort types in the agent’s cost function v. It is positive for v < 0. However,when v > 0, a little bit of information is not valuable.Next, we discuss how the intensity of monitoring varies with the agent’s risk aversion coefficient, r.When r is very low, the risk premium required by the agent for a given effort level is not high, theprincipal settles for noisy information, and he uses a less intense level of monitoring. As the expressionfor h2 (see (7.27)) shows, as approaches zero, the level of h2 approaches zero. For high levels of r,the principal chooses to use weak incentives to reduce the risk imposed on the agent, and we expect theintensity of monitoring to be reduced. Huddart (1993) states that “monitoring is valuable only whencoupled with an incentive scheme responsive to the signals generated”. For very risk averse agents,the principal settles for low monitoring, low incentives and low output. Numerical examples indicatethat the intensity of monitoring is highest at intermediate levels of risk aversion. A similar behavior isobserved in a one-task setting when a costly monitoring technology is employed.26The principal may also find it worthwhile to invest in monitoring even when there is costlesslyavailable information. This occurs when the costlessly available information is very noisy and the costof monitoring is relatively low. With the advances in the information technology, we expect that thepresent cost of data collection and information analysis is very low. With such low cost monitoring,the principal should no longer just settle for freely available information which is collected for differentpurposes altogether. Using highly noisy information as a performance measure forces the principal26See Proposition 6.2.Chapter 7. Multit ask Principal-Agent Model with Costly Monitoring Technology 173Figure 7.24: Behavior of total certainty equivalent with rValues of parameters: 6 = 0.5, h1 = 5, h2 = 0.5, ii = —0.5, and c = 0.1to settle for low effort intensity and low output. Here, we are assuming that the costlessly availableinformation is a noisy version of the monitoring information, thus it has zero value given the availabilityof monitoring information. We construct two numerical examples to illustrate that the principal may bebetter off investing in inexpensive costly monitoring technology than settling for highly noisy information.We assume that the principal seeks to motivate the agent to work on both tasks. Recall that CET isthe total certainty equivalent when costless performance measures are available for both tasks and bothtasks are undertaken. For the two examples, let h2 = 0.5. When a costless performance measure isavailable for only task 1 and the principal invests in monitoring, we let the total certainty equivalent netof monitoring cost be denoted by CE, and we let c = 0.1. The first example is for the case when theagent’s effort types are complementary in his cost function. We let v = —0.5. Figure 7.24 depicts thebehavior of the optimal monitoring level and the total certainty equivalent as r varies.The second example is for the case when the agent’s effort types are substitutable in his cost function,and we let i’ = 0.5. The behavior of the optimal monitoring level and the total certainty equivalent asAChapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 174r varies is similar to that for v = —0.5 as depicted in Figure 7.24. Note, however, that when v < 0, theprincipal is able to achieve higher certainty equivalent levels.From Figure 7.24, we observe three partitions to r. In region A, the agent is not very risk averse.Even when the information is very noisy, the required risk premium is low and the principal can still usestrong incentives to motivate high output. The principal does not invest in monitoring. As the agentbecomes more risk averse, a higher risk premium is required for the same incentive rate. We observe inregion B that (JET > CET, i.e., the principal is better off investing in monitoring to obtain more preciseinformation to motivate higher effort levels, As the agent becomes even more risk averse, the principalsettles for lower effort levels and lower incentive rates and we see in region C that the principal does notinvest in monitoring but uses the costlessly available information. When information is noisy, using lowincentive rates result in a low risk premium.7.5 Some Implications7.5.1 Job Design, Organization Structure and Incentive PlansThe above analysis shows clearly that when an agent is responsible for more than one task, incentiveissues should not be addressed task by task. It is necessary that the principal studies the incentiveproblems for all the tasks together. This allows the principal to take advantage of any interdependencyin the agent’s cost of effort function through proper job design. The presumption is that any change inthe task assignment only changes the interactive term of the agent’s cost function. The principal is thenable to efficiently motivate higher effort levels and achieve higher profitability.This benefit is derived by controlling the agent’s personal cost of effort. The principal seeks to keepthe value of the interactive effect of the agent’s effort on his cost function as low as possible. The aboveanalysis shows that the principal is clearly better off when the effort levels are complementary in theagent’s cost function, i.e., the marginal disutility of achieving task i decreases as the effort level in taskChapter 7. Multi task Principal-Agent Model with Costly Monitoring Technology 175j increases. Thus, if possible, jobs should be designed to achieve this. The lower cost of effort impliesthat it is now optimal for the principal to motivate the agent to attain a higher effort level, and a higherprofit level can be attained.27In section 7.2, we relate organization structure to the interactive effect of the agent’s effort on his costfunction. In divisionalized firms, it is likely that the agent’s effort levels are complementary in his costfunction. On the other hand, in functionally-structured firms, it is likely that the agent’s effort levels aresubstitutes in the agent’s cost function. It has been observed that the divisionalized forms have largelydisplaced the centralized functional forms as the dominant structure for such firms (Rumelt (1986), pp.63—69). Armour and Teece’s (1978) survey of the petroleum industry shows that the divisionalized firmsoutperformed the functionally-structured firms. Our analysis shows that a contributing factor to betterperformance in divisionalized-structured firms may be that the structure takes advantage of the negativeinteractive effect of the agent’s effort on his cost function. Thus, the lower cost of effort makes it efficientfor the principal to motivate a higher level of effort to achieve higher profit. However, empirical evidenceon the performance of divisionalized firms suggests that expected efficiency gains in such firms may notalways hold. In fact, a number of studies (Hill, 1985; Hoskisson and Hitt, 1988) have indicated that tightfinancial controls and incentives based on divisional performance in divisionalized organizations resultin short-run profit maximization and risk-avoidance behavior in the divisional managers. Our analysisindicates that such a result is not unexpected if proper incentives are not provided to the divisionalmanagers to allocate his effort between activities.For example, using accounting earnings as the only basis for performance evaluation is certainly notgoing to encourage agents to actively undertake and oversee risky projects that are healthy in the long-run but hurt short-term profits. In fact, if the interactive effect of the agent’s effort on his cost functionis positive, the agent concentrates on only the short-term effort. Examples of such long-term projects27h our analysis, we igrLore the interactive effect of effort on profit, i.e., we assume that the marginal profit from eachtask is not affected by the activity level in the other task.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 176are innovation and R & D work. The problem is further confounded by the accounting standards for R& D, which is very conservative. Most of the R & D costs are to be charged off as expenses of the periodin which they are incurred. Thus, if accounting earnings is used for performance evaluation, there isno incentive at all for the agent to work on R & D projects if he does not expect to be there to reapthe future benefits. Allocating effort to such a task not only reduces the effort that can be allocated tocurrent operations, but it also results in a lowering of current period’s earnings. Therefore, using thedivisionalized structure in large multiproduct firms is not sufficient for the advantages of the structure tofollow. Proper performance measures should be used and proper incentives must be provided to motivatethe agent to allocate his effort between the numerous tasks.The analysis in this paper implies that organization structure, job design and incentive plans cannotbe designed separately. The principal needs to be very clear what he wishes the agent to achieve. Properjob design and incentive plans can then be used to induce the agent to achieve these objectives.7.5.2 Investment in Monitoring TechnologyThe literature indicates extensive use of stock prices in the compensation contracts with the objectiveof motivating the agent to take a long term focus. While the stock price can be a congruent measure (inthe sense of Feltham and Xie (1994)), it is likely to lack precision. Similarly, some costlessly availableinformation, produced to meet financial reporting needs, may be too jioisy to be able to lead to highoptimal effort levels. Our analysis indicates that the principal should consider investing in a costlymonitoring technology to extract more precise and congruent performance measures. With the advancesin information technology, the cost of information extraction is unlikely to be high. The principalshould investigate the possible benefits from obtaining more precise and congruent measures and shouldcompare these benefits with the cost of monitoring. Information on the firm’s key success factors shouldnot be omitted from the firm’s information system simply because they are costly to obtain. Johnson andChapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 177Kaplan (1987) state that measuring and reporting nonfinancial indicators is important. These indicatorsshould be based on the company’s strategy and include key measures of manufacturing, marketing andB. & D success. A company should not settle for information extracted from a system designed tosatisfy external reporting and auditing requirements. Rather, a management accounting system shouldbe designed to be consistent with the technology of the organization, its product strategy, and itsorganization structure. They also warn that poor management accounting systems can contribute to thedecline of the organization. Our analysis indicates that this warning should indeed be taken seriously.When inappropriate or very noisy performance measures are used, the principal settles for low effort andlow profit.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 178Appendix 7AProofs(I) The Model(1) Derivation of incentive rates (7.3)The two constraints of problem [P7.1] areai—Vi(t) 0and a2 — V2(t) = 0.Let Ai and A2 be the respective lagrange multipliers of the two constraints. Computing the first ordernecessary conditions, we obtain:Hi — V1(t) —A1V1Q) —A2V1(t) = 0112 — V2(t) —A1V2(t) —A2V2(t) = 0 (7.32)ra1a1: ——+A=0a2 :ra2+ A2 = 0 (7.33)From (7.33), we obtain:ra1A1 — -p-—(11and A2 =ra2(7.34)From (7.2), we know that V(t) = a. Substituting a: for V(t) and (7.34) for A1 and A2 into (7.32), weobtain:ra1 Ta2— a1 — —V11 — —V21 = 0112 — a2 — V12— EV22 = 0. (7.35)Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 179Equations (7.35) can be written in matrix notation as follows:11 — a1 V11 V21 0112 a2 V12 V22 0fl a1 — V11 V21 j- 0 a1 = 0112 a2 V12 V22 0 a2 011 V11 V21 0 a1=, = 12+r112 V12 V22 0 a2a1 V11 V21 — 0 11= 12+r 1a2 V12 V22 0 112(2) Costless performance measure available for only task 1:Derivation of incentive rates [(7.5) and (7.6)]Case 1: Expression (7.5)Substituting h2 = 0 into (7.4), we obtain the following:— hi[11irV22 —112rVi]a1— (rVii + hi)rV22 —r2Vh1[11 —112]—a2 = 0.• Case 2: Expression (7.6)Let .X be the lagrange multiplier for the constraint of problem [P7.2]. The FOC are:t1 : 11 — V1(t) — )V11(t) = 0 (7.36)ra1a1: r+=O (7.37)111Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 180From (7.37), we obtainh1Substituting a1 for V1 and for into (7.36), we obtain the following:ra1a1 = Hi — -E--VllUi11h— hi+rVii(3) Using a monitor: Derivation of incentive rates (7.7)The two constraints of problem [P7.3] areai—Vi(t) = 0and a2 —V2(t) = 0.Let and )2 be the respective lagrange multipliers of the constraints. Computing the respective firstorder necessary conditions, we obtain:Hi — V1(t) —A1V1(t) —.2V1(t) = 0112 — V2(t) —)1V2Q) —)2V2(t) 0 (7.38)ra1a1:a2:ra20(7.39)—C’(h2)+ = 0 (7.40)From (7.39), we obtain:ra 1)il = —ni=(7.41)Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 181The agent chooses t so that a: = Vj(t). Substituting a for V(t) and (7.41) for )i and )2 into (7.38),we obtain:ra1 ra2llj—1—-----V--—2 = 0112112 — a — ralV — 0. (7.42)Equations (7.42) can be written in matrix notation as follows:11 — a1—rV11 V21 0 a1 = 0112 a2 V12 V22 0 — a2 0ll V11 1721 0 a1= 12+r112 V12 V22 0 - a2a1 V11 V21 - 0 11‘2+7’1a2 V12 V22 0 112Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 182(II) Quadratic Cost Setting(1) First-best setting: Derivation of optimal effort levelsThe principal’s problem is to select t to maximize 11 — V(t). Therefore, the first order condition withrespect to t is given by II, = 4(t). Thus, we obtain the following:11=1 = S(ii+vt2)112 = 1 6(t2 + vii).Hence, we obtain1The total CE is given by:CET(FB) = ll(t*) — VQ*)1—(2) Derivation of optimal effort levels: Equation (7.14)From (7.2), the optimal effort level is characterized by (t) = a, i = 1, 2. Since T’ = (t + vt), weobtain the following equations:6(ti + Vt2) = a1and 6(i2+vti) = a2.Solving the two equations simultaneously, we obtain the following:a1 — va2tl= 6(1—v2)a2 — va1t2= 6(1—v2)Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 183(3) Behavior of total certainty equivalent1— r[r26(1—v)+2r5h1—v)+ ]8h1—5—1)—r6(hj+h)> 0, since all terms are positive.2 82CE’p — _jr(r6+h2)[r6(1—v)+2r61—i’)+ ]1. [2(1 t’)+r6(hj+hh h]3< 0, since all terms in the {} bracket are positive.3 —j-.5[r52(v—1)—r.5(hj+h) hih]+v)< 0, since all terms in the {} bracket are positive, which we prove below.This is obvious for v < 0. For 0 < v < 1, the first, third and fourth terms in the {} bracket arepositive. The second term is also positive as we show below.To prove that [h? +h12(3(1 — v2) — 2v) + h] > 0:As v increases, [3(1 — v2) — 2v] decreases. As v —* 1, [3(1 — v2) — 2v] —÷ —2. Thus, at v[h? +h12(3(1 — v2) — 2zi) + h] = {h? — 2h1h+ h]= (h1—h2)> 0.Therefore, within the range of 0 < u < 1, [3(1—v2)—2v] > —2 and{h?+hih(3(1—v)—2z’)+hj>0.4 —26’(1—v)(hi+h)+4 5hih1—v)+hjôr — I- 2[r—1)—r(hi+h) hjh]< 0, since all terms in the {} bracket are positive.(4) Derivation of h2 given v> 0: Equation (7.18)The principal motivates the agent to work on both tasks if, and only if, the total certainty equivalentwhen both tasks are undertaken exceeds that when only task 1 is undertaken. Equation (7.16) givesChapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 184the total certainty equivalent when both tasks are undertaken, while (7.17) gives the total certaintyequivalent when only task 1 is undertaken. Thus, the principal motivates the agent to work on bothtasks if, and only if,GET > GEp1r6(1 — v)(hi_+_h2)_+ 2__________i.e.,26(1 + v)[(r6 + hi)(r6 + h2) —v2r6] 26(h1 + r6)vröhi[r6(1 — v)(2 + v) + 2h1J —which implies that h2 >(1 — )[h + r6]2= h2.(5) Proof of Lemma 7.1:1. We use (7.13) and (7.15). Set h1 = h2 and we obtain the following:— — —— h2[h + r6(1 — v)]a1 — a2— (r6 +h2) —v2r6- — -— h+r6h(1—v)—— 6(1 + v)[(r6 + h2) —v2r6]2. We set h1 = h2 + e, e > 0. Then, by substitution into (7.13), we obtain the following:- — (h2+e)[hr6(1—v)ja1— (r6+h+e)(r6+h)—v- — h[h+e+r6(1—v)]a2— (r6 + h2 + )(r6 + h2) —It is clear that a1 > a2. Similarly, by substituting h1 = h2 + e into (7.15), we obtain the following:-— (h2 + )h2 + r6(h2 — vh2 + )±1— 6(1 + v)[(r6 + h2 + e)(r6 + h2) —v2r6]-— f)h+rö(—11hve)— 6(1 + v)[(r6 + h2 + e)(rS + h2) —v2r6]Since —v<,Ii>I2.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 185(6) Proof of Proposition 7.1:From (7.13),-.— h[h + r6(1 — ii)]— 1 2a,—(r6+h)(r6+h)—v2ô’ZFrom (7.15),hh+r6(h,—vh)•—1 2(1+v)[(r6+h)(r6+h)—v22]’ZLet T denote positive expressions.2815i —__________________8h2 THence, the sign is the sign of ii.2 .5.2. — ro[hi+r6(1—v)j[hi+rS(1—-v)J8h2 T> 0, since all components are positive.3 .L_— vr2[h+r(1u)lTHence, the sign is opposite to the sign of v.4 — r(rô+hi)[hi+r6(1—u)]T> 0, since all components are positive.(7) Proof of Proposition 7.2:Effect on aFrom (7.13),h[h+r6(1—v)].—1 2a2— (r6 + h)(r6 + h1) —v2r6’Z ,Let T denote positive expressions.28j is generally the square of the denominator in the corresponding expression above.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 186—jT< 0, since the term in the {} bracket is positive as we prove below.To prove that h(r8 + h) + r6h(1 — 2v) > 0:From (7.15), t > 0 only if h: > For —1 < i’ < 1:v > 2v—1r6hv r6h(2v—1)>rS+h3röh(2v— 1)z4h >rö + h3= h(r6+h3) > röh,(2v—1)=‘ h(rö+h)+r6h(1—2v) > 0.2—8r TFor ii > 0, every term in the {} bracket is positive, thus the sign is negative. However, we wereunable to sign it for v < 0. We observe that if h is less than or comparable to h, then the termin the {} bracket is positive and j < 0. However, the term in the {} bracket can be negative ifh is relatively very large compared to h,. Then > 0. We show this below:[r26(1 — v)(1 — v2) + 2r6h(1 — i.’2) + zihIz3 + h)] < 0= Ii:r26(1—ii)(1—zi)+2r6h( v- --h—vh3r26(1—v)(1—v) 2r6(1—v) h=h1> +—vh, —v —iiChapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 187Effect on t1From (7.15),— h:h3 + r6(h — viz1)S(l+v)[(r6+h)(r6+h1)—2r6]1. — —4{r35(1+ v)[h(1 — 3zi) + h(2v2 — v + 1)]+r2ö{h+ hh(3(1 — v2) — 2v) + h] + 2r6hh3( +h3) + h?h}.For z’ < 0, every term in the {} bracket is positive, thus the expression is negative. However, wewere unable to sign it for v> 0.2— r6(1—v)[rö(h—vhj)+2hhj+T< 0, since the term in the {} bracket is positive as we prove below.To prove that [r8(h — viz,) + 2hh1] > 0:From (7.15), t, > 0 only if [r6(h — viz,) + hh1] > 0. If this holds, then [r5(h — viz,) + 2hh,J > 0.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 188Costless performance measure for only task 1(8) Derivation of optimal effort levels, (7.20)When i’ < 0, we obtain an interior solution for t2. The agent chooses the optimal effort levels so that= a, i = 1, 2. Since V = 6(t + vt2) and from (7.19), a2 = 0, we obtain the following equations:6(t1 + vt2) = a1and 6(t2+vui) = 0.Solving the two equations simutaneously, we obtain the following:= 6(1—va)—va1= 8(1—v2)(9) Behavior of incentive rates and effort levelsThis proposition applies for —1 <ii < 0 only.Effect on a1From (7.19),- — hi(1—v)a1— hi+r6(1—v2)Let T denote positive expressions.291—_____________9h1 T> 0, since all components are positive.2 -- — —hi[r5(l—v)2+hjJT< 0, since all components in the {} bracket are positive.29j is generally the square of the denominator in the corresponding expression above.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 1893 j — —f5h1(1—v)(1—v)T< 0, since all components in the {} bracket are postive.Effect on 1From (7.20),- h1= 6(1+v){hi+r6(1—v2)]Let T denote positive expressions.3°1r(1-v)h1 T> 0, since all components are positive.2— hj[r5(3v—1)(1+’)—hiJT< 0, since the term in the [] bracket is negative.3 j— fhl(1_V)8, — I T< 0, since all the terms in the {} bracket are positive.Effect on t2From (7.20),- —vh16(1 + v)[hi + r6(1 — v2)jLet T denote positive expressions.311 —— ur(1—v)h1 T> 0, since —v is positive.301t is generally the square of the denominator in the corresponding expression above.‘ It is generally the square of the denominator in the corresponding expression above.Chapter 7. Multitask Principal—Agent Model with Costly Monitoring Technology 1902 — hj[r6(v2(1+2v)+1)+hjl8v - T< 0, since all the terms in the {} bracket are positive.To prove that v2(1 + 2v) + 1 > 0: For —1 <z.’ < 0,1+2v > —1v2(1+2v) >= 1+v2(1 2 ) >3 hjv(1—v)8r T< 0, since v is negative.Use of monitoring technology(10) Derivation of h2, (7.27)From (7.9), the optimal level of h2 is determined as follows:rac== a2/(-).By substituting for a2 from (7.25), we obtain the following:h2[r6+hi—iir8] rh2= (r+hi)(r6+h)_z,2r2622c)—[r6 + h1 — vr6]\/() +v2r6rS + h1h— h1 +r6(1—v)+v2/()—./()r6(r6+hi)2— 1J()(r6+hi)— h1 + rö(1—ii) — 6\/(2rc)[hl + r6(1 — i.’2)]— J()(hi+r6)Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 191(11) Proof of Lemma 7.2:From (7.27),— h +r6(1 —ii) —6./(2rc)[hi +r6(1 —v2)]2— /()(h1+ r6)1. h2 > 0 when the following holds:h1 + r6(1 — 11) > 6s./(2rc)[hi + rS(1 — i.’2)].< [hi+r6(1—v)]2r6[hi + r6(1 — v2)]2. From (7.31), the marginal benefit of monitoring (before deducting the cost of monitoring) isMB— r[hj+rö(1—v)]22[r8(1 — i.’2) + r6(hi + h2) +h12]At h2 = 0,MB— [h1 + r6(1 — v)]2—2r6[h.+r6(1—v)]When c <2r[hj+r6(i—v)]’the marginal cost of monitoring is less than the marginal benefit ofmonitoring at h2 = 0. At the optimal level of monitoring, the marginal cost of monitoring equalsthe marginal benefit of monitoring. Thus, when c < 2r the optimal level of monitoring is strictly greater than zero.(12) Derivation of marginal benefit MB of monitoring, (7.31)From (7.30), for any given level of h2, the total certainty equivalent before deducting the cost of monitoring isrö(1 — v)(hi + h2) +2h1hCET(h2)= 26(1 + v)[(r6 + hi)(rS + h2) —v2r6]The MB of monitoring is given by the partial differentiation of CET(h2)with respect to h2:ÔCET — r[hi + r6(1 — v)]2— 2[r6(1— i.’2) + rö(hi + h2) +h12]Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 192The bound on the cost of monitoring is given by MB(h2 = 0) = )]2. ThenoMB(h2 = 0) v(1 — v)[hi + r6(l — v)]8h1 — 6[h1 + n5(1 —v2)]3< 0, sincev<0.(13) Proof of Lemma 7.3:1. If the signal on the agent’s effort in the second task is costlessly available and h1 > h2, the principalmotivates the agent to work on both tasks if, and only if, (see (7.18))-— vr6hi[rS(1 — v)(2 + v) + 2h1]h2 > h —(1 — v)[hi + rö]2If the signal on the agent’s effort in the second task is costly, then a necessary condition formonitoring to be undertaken is that the level of monitoring (7.27) is greater than h2, i.e.h + rS(1 — v) — S/(2rc)[hi + rö(1 — i.’2)] vröhi[rS(1 — v)(2 + v) + 2h1]()(h1+ r) > (1 — v)[hi + r6]2For the expression to hold,(1 — v)2{h? + rShi(2—v) +r26(1 —C2r6(1 +v)2{h? + r6(1 — v)[2h1 + rô(1 —2. At the optimal level of monitoring intensity, from (7.29),— h1 — vr6[1 — S./(2rc)(1 + z.’)]1— 6(h1 + rö)(1 + v)For i > 0, h1 > vr6[1 — i5/(2rc)(1 + ii)].Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 193(14) Proof of Proposition 7.3:In the monitoring environment, (7.28) and (7.29) give the optimal levels of ã and ij. In the non-monitoring environment, for v < 0, (7.19) and (7.20) give the optimal levels of ?i and i’j. Then,—— —vr[h1 + rö(1 — ii) — S/(2rc) [hi + r6(1 —1 1— (hi+r6(1—v2))(r --hi)> 0.-— hi+r6(1—z’)—6/(2rc)[hi+r5(1—vt2 — t2— 6(h1 + r6(1 — v2))> 0.— —— vhi[hi + r6(1 — v) — 6/(2rc) [h1 + rS(1 —a1 a2— (h1 + rö(1 —v2))(rS + h1)< 0.(15) Proof of Proposition 7.4:In the monitoring environment, (7.28) and (7.29) give the optimal levels of â and I. In the non-monitoring environment, for ii > 0, (7.22) and (7.23) give the optimal levels of ä and j. Note that2=0andt0. Then,- - — v[hi + r6(1 — 6/(2rc)(1 + ii))]ti — tl ——6(r6 +h1)(1 + v)< 0.— v5hi/(2rc)rS+hi< 0.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 194(16) Proof of Proposition 7.5:We assume interior solutions for optimal monitoring level and optimal effort levels. Therefore, h2 = 112.From (7.27),— h1 + rö(1 — ii) — 6/(2rc)[hi + rS(1 — v2)jh2)(h1+r6)From (7.28) and (7.29),h(1 — v6/(2rc))a1 =h1 + rö— h1 + r6(1 — v) — 6-/(2rc) [hi + rö(1 — v2)]a2 —hi + rö— h1 — vr6[1 — 6/(2rc)(1 + v)]1— S(h1 + r6)(1 + v)— 1 — 6/(2rc)(1 + v)t2— 6(1+v)These are also used for propositions 7.6, 7.7 and 7.8.18h2 —____________•— I. 4c(r6+h1)< 0, since all terms in the {} bracket are positive.2 -— vhj.J()• — 2(hi+r6)Hence the sign is opposite to the sign of v.3—__ __ _ __ __ __• 8c 1 2(hi+r5)< 0, since all terms in the {} bracket are positive.4— VkZ•8c — 2(hi+r6)Hence the sign is the sign of v.5 1a—_ /L1• — V2<0.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 195(17) Proof of Proposition 7.6:1 h2 — .J(2)vr[1—v/(2rc)J• 8h1— 2./()(r6+hi)Hence the sign is the sign of v. Recall from (7.28) that [1 — v6/(2rc)] > 0 is necessary for a > 0.2— r(1—v6.J(2rc))• — (hi+r6)2> 0, since all terms are positive.3 — vr6(1—v6./(2rc))• 9h1 — (hj+r6)2Hence the sign is the sign of ji.4 — r(1—v6/(2rc))• 8h1— (hi+rö)2> 0, since all terms are positive.5 --—0—(18) Proof of Proposition 7.7:1— j-r5[1—2vö/(2rc)]— I<0, since all terms in the {} bracket are positive.To prove that [1— 2vSV(2rc)] > 0:It is obvious if v < 0. If v> 0, we prove by contradiction:Suppose 1 — 2v6V(2rc) < 0.Then >26V(2rc)From (7.29), 2 > 0 only if ii < — 1. Thus, for an interior solution to exist:1 12ö/(2rc) < 61/(2rc) —1=‘26/(2rc)> 1Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 19612/(2rc)For 0 < v < 1,2(2rc)<2v(2rc)’2v/(2rc)1a contradiction.26sJ(2rc)Therefore, [1 — 2v6/(2rc)] > 0.2—________8v — I h1-I-r5< 0, since all terms in the {} bracket are positive.3— fr6[1—2v5,/(rc)]a — I (r6+hj)< 0, since all terms in the {} bracket are positive.4 — jr6(1—6/(2rc)(1+v)2)+hj— 1. 6(hi+r6)(1+v)The terms in the {} bracket is positive for v < 0, as we prove below. We prove that (1—6,/(2rc)(1-f-v)2) >0 for —1< v< 0:(1-f-v)2 < (1-f-i,)= 1—8/(2rc)(1 + v)2 > 1 — S/(2rc)(1 + v).From (7.29), 2 > 0 only if [1 — 6/(2rc)(1 + v)] > 0.Therefore, we conclude that [1 — /(2rc)(1 + v)2] > 0 and . < 0.For v> 0, we were unable to determine the sign of We observe that for high v and low h1, itis possible for the term in the {} bracket to be negative, thus the sign is positive.ç La— 1•—<0.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 197(19) Proof of Proposition 7.8:1. =— 4r(,.ö+hi)2{(r6hi(3zi—2)—r(1—zi)—h?)/()+2-/(2)r6(r1 v+2röh (1—v?)].We were unable to sign the expression.2— fJ(2)6hl[\/()_vr6+Yhl)1Or —- 2./()(hi+r6)The term in the {} bracket is positive for v> 0 but we were unable to sign it for i.’ < 0.To prove that for i > o, [.,/() — vrö] > 0:Suppose not. Then,< vrS6/(2rc)For t2 > 0, ii < /(2rc) — However, since 5/(2rc) > 6-,/(2rc) — 1, therefore, ii > cannothold. Thus, we conclude that {/() — vr6] > 0.Or /(2)(hi+rö)2 [hiv/( +r28(1 — v2) — hi(r6(3v2— 2) —h1)].We were unable to sign the expression.41J(2)hi—tJ(rc)(3hi+r6)Or — ../(2)(hi+r6)The term in the {} bracket is positive for ii < 0 but we were unable to sign it for v> 0.5 La—.... /(.Or — V2r<0.Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 198Appendix 7BApplicability to General Expressions of SignalThe model applies to general expressions of signals with some modifications. The signals on the agent’seffort could be related to the agent’s effort in the following manner:Yi =f1(t)+ 01, 91 ‘— N(O, h1),Y2 =f2(t)+ 02, 62 N(O, h2),where we assume that fi and f2 are increasing and concave functions. Now, let=f1(ti),P2 = f2(2).Then, yi=pi+Oi,Y2 = P2 + 92.Since a choice oft = (t1,2) is equivalent to a choice of p = (p1,p2), we could use p as the action choicevariable of the agent. Then the agent’s personal cost function V(t) and the expected profit functionfrom the two tasks will need to be re-expressed in terms of p.V(f(pi),f(p2))ll(fj(pi), f(p2)).Since functions fi and f2 are concave and increasing, their respective inverse functions are convex andincreasing. As the agent’s cost function V(t) is convex with respect to t, this implies that it is also convexwith respect to p. The expected profit function 11(t) is assumed to be concave. As such, there is noassurance that the transformed expected profit function will be concave with respect to p. A sufficientcondition to ensure that it will be concave is that 11(t) is more concave than f1(t) and f2(t).Chapter 7. Multitask Principal-Agent Model with Costly Monitoring Technology 199Appendix 7CApplicability to General Profit FunctionsLet 11(T) = gi(ri) + g2(T2), where g and g are increasing and weakly concave. Definet1 =g1(r) T g’(ti),t2 = g2(T2) = r2 = g(t2).We can re-express both the profit functions, 11(T), and the agent’s private cost function, V(T), in termsoft1 and t2, given by:11= tj + t2,V(g1(ti), g’(t2)).We use an example to show the transformation.1/2 1/211(T) T1 +T2V(r) = T?+T+vTlT2.Define:1/2 2t1 = T1 =r1t,1/2 2= T2 4’T2t.We can then re-express the problem in terms ofti and t2.11(t) = t1 -i- t2,V(t) = t+t+vtt.Bibliography[1] Amershi A.H. and SM. Datar, 1991, “Incomplete Contracts, Production Expertise and IncentiveEffects of Modern Manufacturing Practices”, Working Paper, University of Minnesota.[2] Armour 11.0. and D.J. Teece, 1978, “Organizational Structure and Economic Performance: A Testof the Multidivisional Hypothesis”, Bell Journal of Economics 9, pp. 106—122.[3] Arya A., J.C. Fellingham and R.A. Young, 1993, “The Effects of Risk Aversion on ProductionDecisions in Decentralized Organizations”, Management Science 39, No. 7, July, pp. 794—805.[4] Baiman S. and M.V. Rajan, 1994, “On the Design of Unconditional Monitoring Systems in Agencies”, The Accounting Review, vol. 69, no. 1, January, pp. 217—229.[5] Bushman R.M. and R.J. Indjejikian, 1993, “Accounting Income, Stock Price, and Managerial Compensation”, Journal of Accounting and Economics 16, pp. 3—23.[6] Dixon J.R., A.J. Nanni and T.E. Vollmann, 1990, The New Performance Challenge, Illinois: DowJones-Irwin.[7] Feitham G.A. and J. Xie, 1994, “Performance Measure Congruity and Diversity in Multi-taskPrincipal/Agent Relations”, Forthcoming in Accounting Review, July.[8] Grossman S.J. and O.D. Hart, 1983, “An Analysis of the Principal-Agent Problem”, Econometrica51, No. 1, January, pp. 7—45.[9] Hayes R.H, and W.J. Abernathy, 1980, “Managing our way to economic decline”, Harvard BusinessReview, July-August, pp. 67—77.[10] Hill C.W.L., 1985, “Oliver Williamson and the M-Form Firm: a critical review”, Journal of Economic Issues 19, pp. 731—51.[11] Hill C.W.L., M.A. Hitt and R.E. Hoskisson, 1988, “Declining U.S. Competitiveness: Reflections ona Crisis”, The Academy of Management Executive, Vol. II, No. 1 pp. 51—60.[12] Holmstrom B., 1979, “Moral Hazard and Observability”, The Bell Journal of Economics, Spring,pp. 74—91.[13] —, 1989, “Agency Costs and Innovation”, Journal of Economic Behavior and Organization 12,pp. 305—27.[14] Holmstrom B. and P. Milgrom, 1987, “Aggregation and Linearity in the Provision of IntertemporalIncentives”, Econometrica 55, pp. 303—28.[15] — and—, 1991, “Multitask Principal-Agent Analyses: Incentive Contracts, Asset Ownership,and Job Design”, Journal of Law, Economics and Organization VII, pp. 24—52.[16] Hoskisson R.E. and M.A. Hitt, 1988, “Strategic Control Systems and Relative R & D Investmentin Large Multiproduct Firms”, Strategic Management Journal 9, pp. 605—21.200Bibliography 201[17] Hrebiniak L.G., W.F. Joyce and C.C. Snow, 1989, “Strategy, Structure, and Performance: Pastand Future Research”, in: C.C. Snow, ed., Strategy, Organization Design, and Human ResourceManagement, England: JAI Press Inc., pp. 3—54.[18] Huddart S., 1993, “The Effect of a Large Shareholder on Corporate Value”, Management Science39, No. 11, November, pp. 1407—1421.[19] Itoh H., 1991, “Incentives to Help in Multi-agent Situations”, Econometrica 59, No. 3, May, pp.6 11-636.[20] Johnson H.T. and R.S. Kaplan, 1987, Relevance Lost: The Rise and Fall ofManagement Accounting,Chapter 11, Boston: Harvard Business School Press.[21] Kaplan R. and A. Atkinson, 1989, Advanced Management Accounting, 2nd ed., Prentice Hall.[22] Kay T. Ira, 1991, “Beyond Stock Options: Emerging Practices in Executive Incentive Programs”,Compensation and Benefits Review 23, pp. 18—29.[23] Kim S.K. and Y.S. Suh, 1991, “Ranking of Accounting Information Systems for Management Control”, Journal of Accounting Research, vol 29, no. 2, Autumn, pp. 386—396.[24] Lal R. and V. Srinivasan, 1993, “Compensation Plans for Single- and Multi-product Salesforces: AnApplication of the Holmstrom-Milgrom Model”, Management Science 39, No. 7, July, pp. 777—793.[25] March J.G. and H.S. Simon, 1958, Organizations, New York: John Wiley & Sons, Inc.[26] Paul J.M., 1991, “Managerial Myopia and the Observability of Future Cash Flows”, Working paper.[27] Radner R. and J.E. Stiglitz, 1984, “A Nonconcavity in the Value of Information”, in: M. Boyerand R.E. Kihistrom, eds., Bayesian Models in Economic Theory, Amsterdam: North-Holland, pp.33—52.[28] Rappaport A., 1982, “Executive Incentives vs. Corporate Growth”, in: A. Rappaport, ed., Information for Decision Making, 3rd ed., New Jersey: Prentice Hall, pp. 367—375. Reprinted from HarvardBusiness Review, July-August 1978.[29] Rich T. Jude and John A. Larson, 1987, “Why Some Long-Term Incentives Fail”, in: H.R. Nalbantian, ed., Incentives, Cooperation, and Risk Sharing, New Jersey: Rowman & Littlefield, pp.151—162. Reprinted from Compensation Review, First Quarter, 1984.[30] Rumelt R.P., 1986, Strategy, Structure, and Performance, Boston: Harvard Business School Press.[31] Shavell S., 1979, “Risk Sharing and Incentives in the Principal and Agent Relationship”, The BellJournal of Economics, Spring, pp. 55—73.[32] Singh N., 1985, “Monitoring and Hierarchies: The Marginal Value of Information in a PrincipalAgent Model”, Journal of Political Economy, vol. 93, no. 3, pp. 599—609.[33] Stiglitz, Joseph E., 1975, “Incentives, risk, and information: notes towards a theory of hierarchy”,The Bell Journal of Economics 6(2), Autumn, pp. 552—579.

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