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INCENTIVE AND MORAL HAZARD IN QUALITY ASSURANCE, PROCUREMENT MANAGEMENT, AND HIERARCHICAL CONTROL: AN AGENCY THEORETICAL PERSPECTIVE BY YEONGLING YANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1990 THIS DISSERTATION IS DEDICATED TO MY PARENTS FOR THEIR EVERLASTING LOVE MR. TZONGGUANG YANG AND MRS FUGUEY WON YANG ACKNOWLEDGMENTS I would like to express gratitude to Professor Ira Horowitz who has been the source of guidance and stimulation. I am also indebted to the members of my committee: Professor Edward Zabel who has helped me with his expertise and has been very encouraging; Professor Antal Majthay who has been very supportive. Thanks are also extended to Professor S. Selcuk Erenguc for his assistance and kindness in my studying years. Special thanks go to my dear mother and father and two brothers for their everlasting love. Sincere appreciation goes to FaChung (Fred) Chen, for without his overdecade patience, understanding and support, this would not have been possible. I am also thankful to Jenny Chou, Yasemin Aksoy, Chandra Chegireddy and all brothers and sisters in the Gainesville Chinese Christian Fellowship for their friendship. The last and the most is to give all gratefulness to the Lord for His grace and blessing. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ..................................................... iii ABSTRACT ........................................................ vi CHAPTERS 1. INTRODUCTION ......................................... 1 An Overview of Agency Theory ....................... 3 Objective and Organization of the Dissertation ..... 5 2. A LITERATURE REVIEW OF AGENCY THEORY ................. 8 The Basic Agency Model ................... ........... 9 The Moral Hazard Problem ..................... 9 The SelfSelection Problem ................... 15 The Extensions ..................................... 18 Summary ............................................ 30 3. QUALITY ASSURANCE AND JOB ENLARGEMENT IN PRODUCTION MANAGEMENT ........................... 31 The Model .......................................... 33 Characterization of Optimal Contracts .............. 37 The Comparative Statics ...... ..................... 45 Summary ............................................ 59 4. PROFIT SHARING AND TARGET SETTING IN PROCUREMENT MANAGEMENT .......................... 61 The Model ......................................... 65 Characterization of Optimal Compensation ........... 70 The FirstBest Solution ...................... 70 The SecondBest Solution ..................... 74 The Comparative Statics .......... ................. 79 Summary ............................................ 84 5. DEGREE OF SUPERVISION AND MORAL HAZARD IN HIERARCHICAL CONTROL ............................ 87 The Models ....................................... .. 91 The Comparison of the Models ....................... 105 Summary ............................................ 108 6. CONCLUSION AND FURTHER RESEARCH ...................... 109 REFERENCES ........................................................... 114 BIOGRAPHICAL SKETCH ...................... ...................... 124 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy INCENTIVE AND MORAL HAZARD IN QUALITY ASSURANCE, PROCUREMENT MANAGEMENT, AND HIERARCHICAL CONTROL: AN AGENCY THEORETICAL PERSPECTIVE By YEONGLING YANG December 1990 Chairman: Ira Horowitz Major Department: Decision and Information Sciences In this dissertation, hierarchical agency models are proposed to discuss the cooperative nature of departmental interdependence and the interest conflict between the principal and the agents in production environments. There are three functional areas discussed in this study: quality assurance, procurement management and production supervision. These models are established according to the expertise required in the hierarchy. It is the principal's objective to provide an appropriate incentive to reduce the moral hazard problem. The first model considers the recent trend of the integration of quality and production responsibilities. Therefore, both quality and quantity are contracting attributes. In general, the optimally designed contracts are strictly increasing in quantity and quality produced. The worker is better off with this arrangement, not only through the increased compensation, but also through the enlarged job responsibilities. The exact incentive scheme depends on the agents' risk attitudes and the qualityenhancing technology. The second model deals with the interaction between procurement and production management. The production costs crucially depend on the materials, components and subassembly purchased by the procurement department, and on the effort expended by the production department. Linear profitsharing and targetsetting incentive schemes are adopted in this environment. The materials' quality is used to adjust the cost target for the production department. It is shown that in both first best and secondbest cases, profitsharing compensation is always preferred to fixed salary. The agents' compensations are tied to their positions when the principal lacks a costless monitoring mechanism. Agency cost is then considered as the expected value of getting a perfect monitoring mechanism. The third model, unlike the first two models with an existing hierarchy, considers the necessity of production supervision and the payoff to establishing an expanded hierarchical structure. The principal's limited span of control and the agents' moral hazard problem explain the desirability of separating the principal from direct production supervision. It is shown that it is indeed to the principal's benefit to expand the hierarchy levels and organization size in many circumstances. CHAPTER 1 INTRODUCTION Increasing global interaction, competitiveness, and the advent of rapid communication in the twentieth century have provoked a revolution in the business environment. The interdependence and corporation among departments in any one company could well spell the company's success or failure. The employees' motivation, costs and efficiency are among the elements that have significant impact on the departmental relationships and performance, and thus require close management. This study will express the impact each has on the other. Typically, in a profit/nonprofit organization, the largest cost faced by the owner are those for employees' wages, salaries, and benefits. They often absorb half or more of the organization's total revenues (Arnold and Feldman, 1986, p. 340). Employees exchange their time, ability, skills, and effort for valued rewards and personal satisfaction. Therefore it is very important for organizational success that compensation systems are designed in an effective fashion to motivate and maintain the employees' performance. Among various types of compensation systems, Arnold and Feldman (1986, p. 345) indicate that payment schemes have the strongest ability to fulfill the functions of encouraging members to join or stay, rewarding attendance, and improving performance. A recent survey (OR/MS Today, 1990, p. 14) conducted by the Institute of Industrial Engineers to identify issues affecting productivity and quality in the United States workplace indicates that the primary motivation for employees to increase productivity is financial reward, followed by personal recognition and increased responsibilities and decisionmaking opportunities. Yet, almost 80 percent of the respondents said that management lacks the commitment to implement productivity programs for employees such as profitsharing plans and training programs. This is borne out by several studies. Lawler (1971, p. 158) cites six separate studies of the relationship between payment schemes and performance in the United States, and finds evidence that indicates that payment is not very closely related to performance in many organizations claiming to have merit salary systems. More recently, the Wall Street Journal (July 10, 1990) reports a survey conducted by Brooks International. The survey results indicate that only a quarter of the respondents believe that management does an excellent job of rewarding work groups who make quality improvements. Some companies press employees for quality improvements but base rewards solely on the number of units produced. An improperly designed compensation system will also affect the employees' performance negatively. For instance, conventional piece rate schemes for production workers might sacrifice quality for quantity, while portfolio managers paid on the basis of annual accounting profits will sacrifice longterm profitability for shortterm earnings. Therefore, a careful examination of alternative compensation systems becomes a necessity for any organization that searches for success. There are numerous theoretical and empirical studies discussing the issue of designing incentive compensation schemes. One of the most important and popular approaches is Agency Theory. What follows is an overview of that theory, and a statement of the objective and organization of this dissertation. An Overview of Agency Theory The term "agency" has its historic origins in Roman law (Ross, 1974). An agency relationship exists between two or more parties when one of these, designated the principal, seeks to motivate another, the agent, to choose his actions in a way advantageous to the principal. Agency relationships can be widely observed, not only in history but in the contemporary world. Arrow (1985, p. 37) has stated that "the agency relationship is a pervasive fact of economic life analogous interactions are virtually universal in the economy, representing a significant component of almost all transactions." Examples include patient/doctor, shareholder/executive, insurer/insured, manager/workers, client/accountant, and reader/writer. The agency situation or problem is particularly relevant when ownership and control are separate, as is frequently the case in American corporations (Dyl, 1988). Consider this notion of an agency problem in the context of the relationship between stockholders and an executive officer. The stockholders act as principals, delegating the daily operations of the firm to the executive. Stockholders are not in a position to monitor closely the actions of the executive, nor are they as well informed as the executive as to what are the appropriate actions to take. In an attempt to bring the executive's interests in line with those of the shareholders, the executive is often given some complex compensation package. This problem is nontrivial because generally the agent's choice of effort level and ability are known only to himself, and there is no immediate incentive for him to truthfully reveal that information to the principal. In addition, because of environmental risk and the agent's typically riskaverse attitude, the agent's behavior involving his assignment is usually not in the principal's best interest. The principal, on the one hand, tries to have the agent share uncertain environmental risk and, on the other hand, must provide enough incentive to attract the agent to the job, or to motivate him to work harder. Thus, agency models incorporate two basic phenomena of organizations: incomplete information and a goals conflict between members of the organization. The tradeoff between providing motivation and encouraging risk sharing becomes a general feature of agency problems. Agency theory views this situation through the design of contracts that maximize the principal's utility and take several factors into account: (a) the relationship between output and the incentive scheme offered; (b) the alternative job opportunities from outside markets; (c) the allocation of risk between the agents and the principal; and (d) the preferences of the principal and the agents with respect to income and nonpecuniary outcomes. The various factors involved and the unavoidable need to deal with stochastic payoffs and costs often make the problem difficult to formulate and to solve. Restrictive results from some basic models do offer rich understanding about incentives and conflicts 5 within organizations, and provide useful insights in the construction of contracts to guide and influence agency relations in the real world. In the agency literature, many models come from popular phenomena in the economy. Plentiful examples include insurance policy (Zeckhauser, 1970; Pauly, 1968; Shavell, 1979; Harris and Raviv, 1978; Rothschild and Stiglitz, 1976; Wilson, 1977), auditing and responsibility accounting (Ng and Stoeckenius, 1979; Atkinson, 1978; Holt, 1980; Antle, 1982, 1984; Suh, 1987, 1988; Demski and Sappington, 1989), portfolio selection (Jensen, 1976, 1983; Eaton and Rosen, 1983; Narayanan, 1985; Cohen and Starks, 1988), salesforce compensation (Basu et al., 1985; Nalebuff and Stiglitz, 1984; Coughlan and Sen, 1985; John and Weitz, 1985; Lal, 1986; Lal and Staelin, 1986), government contracting (McAfee and McMillan, 1986, 1987), organization behavior (Williamson et al., 1975; Stiglitz, 1975; Fama, 1980; Fama and Jensen, 1983; Tirole, 1986), national defense contracts (Berhold, 1971; Cummins, 1977), resource allocation (Harris et al., 1982; Harris and Townsend, 1981), and public organization (Hansmann, 1981; Becker and Stigler, 1974; Ross, 1979). Objective and Organization of the Dissertation In his book on "The New Science of Management Decision," Herbert Simon stated a general picture of an organization (1960, p. 40): An organization can be pictured as a threelayered cake. In the bottom layer, we have the basic work process. In the middle layer, we have the programmed decisionmaking processes, the processes that govern the daytoday operation of the manufacturing and distribution system. In the top layer, we have the nonprogrammed decisionmaking processes, the processes that are required to design and redesign the entire system, to provide it with its basic goals and objectives. Hierarchy stands not only for degrees of highness or lowness, for this tends to hide its significance. Each level is an inclusive clustering or combination of interdependent groups, to handle those aspects of coordination that are beyond the scope of any of its components (Thompson, 1967, p. 59). In this dissertation, hierarchical agency models are utilized to discuss the cooperative nature of departmental interdependence and the interest conflict between the principal and the agents in production environments. A perfectly competitive market is assumed for the final products. Three hierarchical agency models are proposed to discuss the agency relationships in quality assurance, in procurement management, and in production supervision, respectively. The ideas developed in this study are partly inspired by the significant improvement in industries through implementing JustinTime philosophy in various areas such as the consolidation of production and quality, the vertical integration of procurement and production, and the highly autonomic spirit among employees. This study provides several interesting insights into how to resolve conflicts of interest within organizations by carefully designed compensation schemes. Unlike conventional contracts, the pattern and design of incentive systems depend on factors such as the agents' attitudes of sharing risk and expending effort, as well as on their production contributions. This study also shows how the organization can be more profitable and efficient by improving several exogenous variables or by adjusting its structure. 7 A literature review of agency theory is given in chapter 2. An agency model of quality assurance and job enlargement in production management is presented in chapter 3. An agency model considering linear profitsharing and targetsetting compensations in procurement management is presented in chapter 4. An agency model regarding production supervision and moral hazard in hierarchical control is presented in chapter 5. This dissertation concludes with a summary and further extensions of each model in chapter 6. CHAPTER 2 A LITERATURE REVIEW OF AGENCY THEORY Since the original papers by Wilson (1968), Spence and Zeckhauser (1971), and Ross (1973, 1974), substantial attention has been given to the development of agency theory. Agency theory has been viewed as the neoclassical response to the questions raised by March and Simon (1958) regarding the behavior of an organization of selfinterested agents with conflicting goals in a world of incomplete information (Levinthal, 1988). There are basically two types of incomplete information under discussion. The first, referred to as the moral hazard or incentive problem, reflects the inability of the principal to costlessly observe the agent's decision. The second, referred to as the selfselection or adverseselection problem, reflects the unwillingness of the agent to reveal his private information about the state of nature, his abilities or productivity. Most agency literature has focused on the moral hazard issue, but the more recent work has introduced elements of self selection (Myerson, 1982; Baron and Myerson, 1982; Demski and Sappington, 1984, 1987, 1988). This chapter focuses on formal, mathematical statements of the agency relationship and largely ignores the less formal stream of positive research by Jensen and Meckling (1976), Jensen (1983), and Fama and Jensen (1983). In what follows, the basic models and their mathematical justifications are presented. Then, various extensions and related literature are discussed. The chapter concludes with a summary. The Basic Agency Model The basic models dealing with the moral hazard problem and the selfselection problem are presented as follows. The Moral Hazard Problem The basic agency model focuses on a twoplayer (the principal and the agent), oneperiod situation. The agent's effort (or action), a, together with a random state of nature, 0, are assumed to generate outcome x according to x=x(a,0). The principal, on the one hand, wants the agent to share risk and, on the other hand, must provide enough incentive to attract the agent to the job, or to motivate him to work harder. The difficulty comes from the fact that the principal can only observe the agent's outcome x, but not the agent's effort a, or the random variable 0. Given the observed outcome, the principal can only make probability judgments about the agent's effort. It is assumed that both parties are rational and use von NeumannMorgenstern utility functions in their decisionmaking processes, where the utility functions are known to both parties. The principal is typically assumed to be less risk averse than the agent (Mirrlees, 1976). Let the principal be risk neutral, and the agent be risk averse with utility function H(w,a) defined over income (w) and effort (a). The model further assumes the utility function is additively separable into a utility for income, U(s), and a disutility of effort, V(.), i.e. H(w,a)U(w)V(a) with U'>0 and U"<0. The interpretation is that a is a productive input with direct disutility for the agent and this creates an inherent difference in objective between the principal and the agent (Holmstrom, 1979); Harris and Raviv (1979) have proved that the moral hazard problem can be avoided when the agent is risk neutral. The agent is effort averse, and his disutility of effort increases at a nondecreasing rate, i.e. V'>0 and V">0. Mirrlees (1974, 1976) suggests that 0 be suppressed and views x as a random variable with a distribution F(x;a), parameterized by the agent's action. Given a distribution of 0, F(x;a) is the distribution induced on a via the relationship x x(a,0). The principal's problem is to design a sharing rule or contract that elicits an appropriate effort level from the agent. The sharing rule or contract, S(x), which is constructed on the basis of the principal's observation of the agent's performance, can be generated by the following program: (M) Max f[xS(x)]f(x;a)dx (2.1) Subject to E{H(S(x),a)) JfU(S(x))f(x;a)dx V(a) H* (2.2) a e Argmax [fU(S(x))f(x;a')dx V(a')] (2.3) a'eA where E({) denotes the expectation operator, the notation "argmax" denotes the set of arguments that maximize the indicated objective function, H* is the agent's market reservation value or opportunity cost, and A is the set of all possible actions. The first constraint is 11 the socalled "individual rationality constraint" that reflects the fact that the contract must offer the agent an expected utility that is at least as great as his opportunity cost (utility). The second constraint is the socalled "incentive compatibility constraint" that indicates that the agent chooses the action that maximizes his own utility. In most of the literature (except Grossman and Hart's (1983) convex programming approach) it is assumed that the second constraint in (P) can be replaced by the firstorder condition derived from the agent's expectedutilitymaximizing efforts, i.e. JU(S(x))f,(x;a)dx V'(a) 0, (2.4) where f(*.) is the derivative of f(x;a) with respect to a. Mirrlees (1974) points out that the firstorder condition may not in general be necessary for the incentive compatibility constraint, unless the contract elicits a unique action on the part of the agent at the optimum. If there is not a onetoone correspondence between contract and action, then the firstorder condition is not even a necessary condition for the optimality of the contract. Rogerson (1985) suggests the correct treatment of the conditions sufficient for necessity of the firstorder condition: notably, if the probability density function of the outcome, f(x;a), has the monotone likelihood ratio (MLR) property and the cumulative distribution function (CDF), F(x;a), is convex, then the firstorder approach is valid. The MLR condition states that more effort (a) yields more outcome (x), and implies that increases in effort will shift the distribution of x to the right in the sense of first order stochastic dominance (SDC) (Milgrom, 1981). By the MLR, F(x;a) decreases in a, i.e. the probability of an outcome less than or equal to 12 some x" decreases as the agent works harder. The CDF convexity condition requires that this function decreases at a decreasing rate, i.e. the CDF is a form of stochastic diminishing returns to scale (Rogerson, 1985). Jewitt (1987, 1988) argues that the CDF condition is not fulfilled by most of the distributions commonly occurring in statistics, and Rogerson's conditions cannot be directly adapted to the multidimensional case. Jewitt further suggests an oversufficient but more easily verified total positive (TP) density function, along with a specified utility function as the justification. A function f(x;a) is said to be total positive of degree n (TPn) if for each x1 we have f(x1,a1) *** f(xl,ak) I *I S0 for k1,2,...,n. I I f(xk,aC) *** f(Xk,ak) If f(x;a) is TP for all n, then it is said to be totally positive (TP). Jewitt indicates that any exponential family (including normal, exponential, Poisson, binomial, etc.) is totally positive in an appropriate parameterization. The utility condition can be satisfied by any constant absolute riskaverse utility function and any nondecreasing relative riskaverse utility function with an ArrowPratt coefficient of relative risk aversion bounded above one half. In order to avoid problems of existence of an optimal sharing rule, bounds are placed on the allowable payments to the agent, and the set of possible sharing rules is assumed to be compact. The bounds on possible payments may be justified on the basis of institutional features such as limited liability and bankruptcy. Given that the firstorder condition holds, determination of the contract S(x) can be treated as an isoperimetric optimal control problem (Intriligator, 1983, p. 318). Letting A and p be multipliers on (2.2) and (2.4), respectively, the Hamiltonian is L J [xS(x)] + A[U(S(x))V(a)H] + p[U(S(x))f,(x;a)/f(x;a) V'(a)] f(x;a)dx. (2.5) Since S'(x) does not appear in L explicitly, when the optimal S(x) is in the interior of the set of feasible contracts, it can be characterized as the solution to the necessary Euler condition, i.e. aH/BS(x) 0. Thus, the optimal sharing rule can be characterized by 1 fa(x;a) = A + p (2.6) U'(S(x)) f(x;a) The above approach is called "pointwise optimization" in the agency literature. There are two solution concepts regarding program (M). The solution of (2.1) subject to (2.2) alone is referred to as the first best solution, which entails Paretooptimal risk sharing. The solution of (2.1) subject to (2.2) and (2.4) is referred to as the secondbest solution, which entails suboptimal risk sharing. The firstbest solution. From Borch's (1962) work, S(x) is Paretooptimal from a risksharing point of view only if the righthand side in (2.6) is a constant so that the agent is paid a fixed wage independent of the outcome and the riskneutral principal bears all risk. The firstbest contract is analogous to a wage contract. The second term of the righthand side in (2.6) is not a constant by the fact that p>0 (Holmstrom, 1979; Jewitt, 1988). Therefore, optimal risk sharing can be achieved by dropping the (2.4) constraint. Since A>0, constraint (2.2) is binding, which implies that the principal can infer the agent's action. The firstbest solution can also be achieved if the agent is risk neutral (Harris and Raviv, 1979). In this case, the principal receives a fixed payment and the agent receives all the residual outcome (Shavell, 1979). Alternatively, if the principal can discover the state of nature or monitor the agent's effort directly, the risk associated with an agency relationship can also be eliminated (Wilson, 1968; Ross, 1973, 1974; Mandlker and Raviv, 1977). Finally, there are asymptotic results regarding the attainment of the firstbest solution as the "efficiency" of the agent's effort tends either towards zero or infinity (Shavell, 1979; Grossman and Hart, 1983). Let the index of the productivity of the agent's effort be denoted as 6, with the probability density of outcomes becoming f(xl6a). If 60, the distribution of outcomes is not affected by the agent's effort and the problem is merely one of risk sharing. If 6*, the difference between the secondbest solution and the firstbest solution tends to be zero (Levinthal, 1988). But, these conditions are quite extreme. The secondbest solution. The shadow price p in (2.6) can be proved to be positive by firstorder stochastic dominance (Proposition 1, Holmstrom, 1979) or by the fact that the covariance of U(S(x)) and l/U'(S(x)) is nonnegative (Lemma 1, Jewitt, 1988). Thus, (2.6) shows that the secondbest contract is larger (less) than the firstbest contract when the marginal return from effort is positive (negative) to the agent. The secondbest contract imposes risk on the agent. The agent bears responsibility over which he has no control. It is "second best" because it sacrifices some risksharing benefits in order to trigger a proper effort level. From the principal's viewpoint, the secondbest solution is strictly inferior to a firstbest solution. The ratio of Ifa/f may be interpreted as a benefitcost ratio for deviation from optimal risk sharing (Holmstrom, 1979). In general, the solution to the agency model is one that is second best. The SelfSelection Problem The selfselection problem arises when before entering into a relationship both parties have different beliefs concerning the exogenous uncertainty. In most circumstances, the agent's information regarding the uncertain state of nature (such as the agent's ability or knowledge of the state of nature) is superior to the principal's. The principal must design some mechanisms (i.e. set of rules) in order to evaluate prospective agents. Salop and Salop (1976) indicate two commonlyused methods: a screening device and a selfselection device. A screening device takes some set of observable characteristics (e.g. past work experience, educational records, etc.), and ranks an applicant's prospective job performance on the basis of his endowment of these characteristics. A selfselection device is a contracting scheme that causes the applicant to reveal truthful information about himself by his market behavior. In the agency literature, it is the self selection device that is under discussion. The following example considers the case in which the agent has private information about his productivity states, while the principal is unable to enforce the fullinformation solution. It is modified from Demski and Sappington's (1984) and Demski, Sappington, and Spiller's (1988) models. Consider a riskneutral principal who owns one productive technology that requires as an input the effort, a, of an agent. The agent's effort, together with the realization of a random productivity factor 0, determines the output, x, produced according to the known relationship x=x(a,8). The random productivity factor is assumed to be binary, with O1 agent in a more productive setting, i.e. x(a,0h)>x(a,01) for any a>0. (x(0,0)O for all 0.) The agent is risk averse with a von Neumann Morgenstern utility function that is additively separable in monetary reward, R, and effort, a. Specifically, H(R,a)=U(R)V(a) where U'>0, U"<0, V'>0 and V"20. If the agent acquires perfect productivity information before a contract is signed, then his effort choice is equivalent to a choice of output level. Thus, the agent's utility function can be defined over 8 and x as H(R,x;8)=U(R)D(x,0) where D(x,8) is the disutility incurred by the agent when he produces x in state 8. It is assumed that D(x,0h) D,(x,8)20 Vx>0 and VB, where subscripts denote partial derivatives. Finally, it is also assumed that the marginal disutility of effort for the agent is smaller in a more productive state, i.e. Dx(x,0h) Vx> . Since the principal cannot force the agent to provide truthful productivity information, he must design the choice mechanism that does not give an incentive for dishonesty. It is proved by the Revelation Principle (Dasgupta et al., 1979; Myerson, 1979) that for any equivalence class of any choice mechanism, with no loss of generality, there is an equivalent mechanism that induces truthful revelation of 0. Therefore, the principal need only consider the mechanisms in which each agent is constrainted to truthfully reveal his productivity parameter as a Nash response. The principal's optimal strategy can be described by the solution to the following program: (SS) Max pilxiRiI (2.7) Ri, xi Subject to U(Ri)D(xi,0 i)>H, i=l,h, (2.8) U(Ri) D(xi, i)>U(Rj) D(xj, i), i,j=l,h, (2.9) where Pi probability that 0=8, il,h; xi the output that the agent will produce when 001, in return for reward Ri, il,h; and H* is the market opportunity cost (utility). The individual rationality constraints (2.8) guarantee that the agent will receive at least his reservation utility level if he truthfully reveals his private productivity information. The self selection or truthtelling constraints (2.9) ensure that the agent will prefer to tell the truth rather than lie about his actual productivity. Demski et al. (1988) have shown that the agent receives his reservation level of expected utility when he observes the low productivity state. The agent earns strictly positive rents if the high productivity is realized, and the agent is induced to produce the most output in the high productivity state. The Extensions Numerous extensions of the basic agency model have been developed. I classify them into the following categories and discuss them in order: the monitoring model, the multiagent model, the multilevel model, the dynamic model and the selfselection model. The Monitoring Model Incorporating monitoring arrangements is a natural extension of the basic agency model. Since agency costs come from the unobservability of the agent's effort, any extra information about the agent's effort will lead to a Pareto improvement toward the firstbest solution (Harris and Raviv, 1979). This is true even if the monitoring information is imperfect. Holmstrom (1979) indicates a necessary and sufficient condition for imperfect monitoring information to be of value. Let y be the monitoring signal. Holmstrom proves that y is of value if and only if [f,(x,y;a)/f(x,y;a)]h(x;a) is false. When the contract is contingent on an additional informative measure, the agent will bear less risk associated with the state of nature. Shavell (1979), however, shows that the value of information goes to zero when the efficiency of the agent's effort goes to zero or infinity. Harris and Raviv (1979) have characterized the form of the optimal monitoring contract. The latter adheres to the following rule: if the agent's effort is judged acceptable on the basis of the monitoring information, the agent is paid according to a prespecified fee schedule; otherwise, he receives a lesspreferred fixed payment. Demski and Feltham (1978) derive the same type of contract (called a budgetbased contract), except that they divide performance into favorable or unfavorable on the basis of the outcome. Baiman and Demski (1980) turn their attention to "investigation strategies" as to when to undertake costly monitoring. Their results include the Pareto optimality of "onetailed" investigations, wherein either very large or very small variances of costs (but not both) should be investigated by the principal. One of the sufficient conditions for their result is that the agent has a utility function that belongs to the hyperbolic absolute riskaversion (HARA) class. Young (1986) derives conditions under which "twotailed" investigations are optimal, assuming nonHARA utility functions for the agent. In a different setting, Dye (1986) proposes a "lowertailed" investigation strategy that specifies a cutoff value such that an outcome below that cutoff value will trigger an investigation. Dyl (1988) conducts a regression analysis to examine the effect of monitoring activities on managerial compensation. The results support his hypothesis that in closelyheld companies major shareholders engage in monitoring activities that reduce the residual loss portion (due to excessive levels of compensation) of agency costs. The natural logarithm of the percentage of the firm owned by the five largest stockholders is employed as the measure of the degree to which the firm 20 is closely held. Data regarding the total remuneration of the CEOs of 271 major industrial corporations listed in the Fortune 500 shows that an increase of one unit in the natural logarithm of corporate control is associated with a 9.75 percent reduction in the level of topmanagement compensation. Other works on investigation policies include Townsend (1979), Evans (1980), Kanodia (1985) and Lambert (1985). The MultiAgent Model If all agents face similar states of nature, then comparing performance across agents can mitigate the common risk associated with the outcomes (Holmstrom, 1982; Lazear and Rosen, 1981; Green and Stokey, 1983; Nalebuff and Stiglitz, 1983, 1984). Research examining the risk sharing benefits of multiagent contracts has focused on the comparison of the efficiency of individual contracts, which reward each agent for his own outcomes, with tournaments under which reward is a function of the rank order of performance relative to other agents (Levinthal, 1988). Holmstrom (1982) argues that the rank order of agents is not a sufficient statistic for individual output except in special circumstances as discussed in Lazear and Rosen (1981). Rankorder tournaments may be informationally quite wasteful if outcomes can be measured cardinally rather than ordinally. Holmstrom further suggests that the agent should be evaluated relative to the average performance of all agents. As the number of agents increases, the effect of the common disturbance term can be decreased. His conclusion, however, is 21 restricted to the additive or multiplicative technology assumption, i.e. xi(ai,i)=a+,i or xi(ai,1)a(01) i1,2...,n, with 01((,Ei) where q is a common uncertainty parameter and ei's are idiosyncratic risks. Green and Stokey (1983) consider a model with a riskneutral principal and a group of homogenous, riskaverse agents. They conclude that the tournament dominates an optimal individual contract if the common environmental risk is large. Nalebull and Stiglitz (1984) consider the case where the outcome is assumed to be linear in effort. They find that the use of a tournament as an incentive device can induce agents to abandon their natural risk aversion and supply more effort. They also find that the firstbest optimum can be approached when there is a large number of contestants, and a penalty to the lowestranked individual will be superior to a prize to the highestranked individual in motivating effort. The MultiLevel Model Another topic to which agency ideas can be usefully extended is the theory of hierarchy. The study of hierarchical organization was first mentioned by Simon (1957), Williamson (1967) and Lydall (1968), but was not precise enough to generate testable hypotheses. Mirrlees (1976) is the first to extend the basic agency model to a multilevel structure. In his model, other than the output, the agent's observed performance is also assumed to be uncertain. The accuracy of the observation depends upon the time devoted by the principal to make the observation. Thus, the principal has to decide on the time spent in observation, as well as on the payment function. Production is carried out by the bottomline workers only. Each level of the rest of the hierarchy decides the monitoring time and the payment function for its nextlevel subordinates. Unfortunately, Mirrlees does not provide any wellfounded conclusion. Antle (1982, 1984) considers the ownermanagerauditor relationship and examines the gametheoretic foundations of such an expanded agency model. The auditor's primary role is to produce stewardship information, i.e. information used by an owner and a manager for contracting purposes. Modeling the auditor as a strategic player introduces two complexities. First, the naive mathematical extension of the basic model may yield unreasonable solutions. Second, the nontrivial nature of the subgames implies that randomized strategies by the auditor and the manager may be of crucial importance. These difficulties are illustrated by many examples (Antle, 1982). If the auditor and manager are assumed to play pure strategies, however, inducing truthful reporting by the manager and auditor is optimal. Later, Antle (1984) restricts attention to pure strategies and provides a definition of auditor independence which is a controversial topic in the accounting literature. Basically, Antle assumes that an independent auditor, in choosing among several subgame equilibria in which he receives the same expected utility, will select the one that the owner most prefers, regardless of the effects on the manager. The implications of auditor independence for the optimal compensation scheme for the auditor are discussed via a series of examples. Baiman et al. (1987) also analyze the effects of adding an auditor to the original principalagent problem. When the original optimal contract is second best, they establish conditions that are sufficient to ensure the value of hiring a utilitymaximizing auditor. Their model differs with Antle's model in three respects. First, there is no moral hazard problem for the manager. Second, the auditor observes the report submitted by the manager before conducting his audit. Third, the principal cannot observe the outcome and only receives the transferred amount of payment, which is independent of the outcome, from the manager. The results of Baiman et al. (1987) are restricted by the uniform probability distribution assumption over outcomes. Demski and Sappington (1987) consider a regulatory control model with three individuals: a consumer, a regulator, and a firm. The consumer (acting as the principal) wants to motivate the regulated firm to reduce cost and also motivate the regulator to acquire information that is valuable in directing the firm's activities. Only the firm can expend effort designed to reduce production cost. The regulator's role is to gather information about the firm's costreducing ability. Demski and Sappington conclude that the regulator is rewarded if the firm's cost result is consistent with his claim, and is penalized the maximum amount otherwise. Tirole (1986) considers the coalitions in a threetier principal/supervisor/agent model. The agent's unobservable production effort, together with an exogenous productivity shock, affects the principal's profit. The supervisor's role is to obtain more information about the agent's activity, and his supervisory effort is assumed exogenous. Thus, there is no moral hazard problem for the supervisor. Under the supervisor/agent coalition, Tirole shows that the constraint 24 that induces the supervisor to reveal that the state of productivity is low is not binding, i.e. the supervisor acts as an advocate for the agent. Tirole also shows that even if the agent can produce verifiable information himself, most likely there is still scope for a supervisory function. Suh (1987, 1988) considers a sequential department setting that consists of an intermediateproduct division and a finalproduct division. Suh shows that the optimal allocation of the noncontrollable cost is inconsistent with the concept of responsibility accounting which distinguishes and rewards components of the outcomes for which each division is responsible. One of the reasons is the existence of collusion between divisions. One division could collude with another to affect an apparently noncontrollable cost indirectly. Other reasons are the technological dependence and asymmetric information of the production environment. Thus, basing the finalproduct division manager's evaluation on noncontrollable intermediateproduct costs can serve as an alternative device to costly monitoring or communication. The Dynamic Model Radner (1981) and Becker and Stigler (1974) have shown that dynamic treatments of the agency model induce more efficient results that may achieve the firstorder solution asymptotically. Holmstrom (1979) suggests that when the agency relationship repeats itself over time, the effects of uncertainty tend to be reduced, and dysfunctional behavior is more accurately revealed, thus alleviating the problem of moral hazard. As the number of periods increases, the variance in the average outcomes decreases and more accurately reflects the agent's effort. Radner (1980) and Rubinstein and Yaari (1980) examine infinite horizon models in which the same singleperiod situation is repeated over time, and there is no utility discount over time. Radner (1981) once again considered the same model but repeated a finite number of times. He introduces the notion of an epsilon equilibrium as a means of achieving the firstbest outcome in a finite repeated relationship. An epsilon equilibrium is a set of strategies such that each player's average expected utility is within epsilon of being a best response to the other players' strategies. Lambert (1983) allows the principal and the agent to discount their utilities and assumes the production functions to be separable over time. He derived the optimal longterm contract and found that the agent's compensation depends both on his current and past performance, whether he precommits to the longterm contract or not. Murphy (1986) extends Lambert's results to an incentivebased theory of executive earnings dynamics, both theoretically and empirically. He compares two not mutually exclusive hypotheses: incentives and learning. The incentive model implies that the earningperformance relation and the variance of individual earnings increase with experience, while the learning model implies that the earningperformance relation is strongest during early periods and the variance of individual earnings declines with experience. Both hypotheses predict upwardsloping experienceearnings profiles and a positive relation between compensation and performance. Harris and Holmstrom (1982) provide new predictions for the behavior of aggregate wages over time. Both the principal and the agents are assumed to be uncertain about agents' abilities, but by a normal learning process, they gradually find out agents' abilities. Harris and Holmstrom conclude that the optimal longterm contract entails a downward rigid wage (i.e. under this contract one's wage never falls over time). The wage is not fully rigid, because the threat of quitting will force the wage to be bid up whenever the market wage is higher than the current wage. The market wage is found to be a function of the worker's current mean perceived productivity minus a term that depends on his age and the precision of beliefs about his productivity. Thus, senior workers earn more on average because they have had more time to have their wages bid up by the market. Narayanan (1985) borrows Harris and Holmstrom's (1982) model to analyze the portfolio manager's decision in financial policies under asymmetric information. The manager may have incentives to make decisions that result in shortterm gain at the expense of the shareholders' welfare, even if longterm contracts are offered. Narayanan shows that this manager's decision is related to the manager's experience, length of his contract, and the risk of the projects. The more experienced the manager is or the longer the duration of the manager's contract, the lower is the probability that the manager would opt for shortterm profits. When the projects are riskier, only a smaller proportion of the deviations from the expected cash flows will be attributed to the manager's ability, and this reduces his incentives for sacrificing the longterm interests of the shareholders for short term profits. Holmstrom (1983), based on Azariadis' (1975) work, has explored the dynamics of a contractual labor market in an equilibrium setting. The model assumes homogeneous riskaverse agents, and quits and layoffs are allowed at the beginning of second period. He shows that longterm contracts emerge both when labor is specified and cannot move in the second period and when mobility is costless. Wages are downward rigid but not fully rigid. Holmstrom also points out the seniority rules: different generations of newcomers may have different incomes; especially, younger generations generally earn less, not because of productivity differences but because they enter later. Therefore, in Holmstrom's model, contracts create biases against young members. The SelfSelection Model Salop and Salop (1976) assume that agents differ exogenously only in their probabilities of quitting. The principal's objective is to minimize the turnover costs in a perfectly competitive labor market. This is equivalent to identifying the slow quitters among the applicants and hiring them. Salop and Salop suggest a twopart wage (TPW) scheme as a selfselection device. The TPW would operate in the following manner. The new employee pays the firm an entrance fee of DI, in return for which he receives the market average wage, W*, and some fixed amount, D2. The principal sets D2 and DI such that slow quitters prefer this structure while fast quitters prefer the flat wage structure of other firms. Finally, in equilibrium the agents pay their own training costs and receive the full value of their marginal revenue product as wages. Rothchild and Stiglitz (1976), and Stiglitz (1977) examine the equilibrium under asymmetric information in the competitive insurance market and the monopolistic insurance market, respectively. Rothchild and Stiglitz (1976) show that even in a competitive market, a small amount of asymmetric information will exclude a singleprice equilibrium. Highrisk individuals purchase complete insurance, while the lowrisk group purchases partial insurance. Market equilibrium, when it exists, consists of contracts that specify both prices and quantities. But under many conditions, equilibrium may not exist. In a monopoly with imperfect information and nonlinear pricing (which charges customers an amount proportional to the quantity consumed), Stiglitz (1977) proves that the same contract will never be purchased by both high and lowrisk individuals. But, the lowrisk individual may not purchase any insurance at all. The highrisk individual, however, always purchases complete insurance. Harris and Townsend (1981) and Harris, Kriebel, and Raviv (1982) consider intrafirm resource allocation under asymmetric information. Harris and Townsend (1981) prove that the equilibrium parameter contingent (P.C.) allocations of any mechanism must satisfy certain selfselection properties and such P.C. allocation can be achieved under some mechanism (i.e. some set of rules). Harris, Kriebel, and Raviv (1982) further point out that one of the efficient mechanisms is the transfer pricing scheme. Under certain constraints, headquarters announce a schedule of transfer prices, and each division is asked to choose a transfer price among them. By the choice of a transfer price, all of the information required for an optimal allocation will be revealed. Lal (1986) addresses the issue of delegating pricing responsibility to the salesperson. He shows that delegating pricing responsibility to the salesperson is as profitable as centralization when both the salesperson and the sales manager have identical information about the selling environment. But delegation may be more profitable when the salesperson's information is superior to that of the sales manager. Lal and Staelin (1986) follow the same framework but relax the assumption of salesforce homogeneity. They present the conditions under which it may be advantageous for a profitmaximizing sales manager to offer his salespersons the opportunity to choose from a menu of compensation plans. Nevertheless, they are not able to provide the specific design of optimal contracts. McAfee and McMillan (1986, 1987) introduce a principalagent framework to the service market. Potential agents, risk neutral and heterogeneous in ability, compete with each other for the contract with the principal. The principal, without knowing the agents' types, designs a contract that exploits the competition among the potential agents and induces them to reveal their types. In this setting, the contract trades off adverse selection against moral hazard. Surprisingly, McAfee and McMillan proved that the optimal contract is linear in the observed outcome in a broad range of circumstances. Summary This chapter introduced the basic models of agency theory and its numerous extensions. Agency theory appears to be a promising research area and there are various avenues of enrichment that will result in valuable propositions (MacDonald, 1984). It has been developed markedly over last 20 years and the impact of this research has been significant (Levinthal, 1988). This study intends to apply agency theory to a hierarchical production environment. This literature review constitutes the basis for the suggested models in this study. CHAPTER 3 QUALITY ASSURANCE AND JOB ENLARGEMENT IN PRODUCTION MANAGEMENT Prior to World War I there was very little separation between responsibility for production and responsibility for product quality in the United States. There was rarely a separate functional department for quality distinct from that for production. Between World War I and World War II the concept of "central inspection" began to form (Juran, 1974, p. 76). By 1940, the distinction between the functions of quality control and production became solidified (Lubben, 1988, p. 68). By the 1950s, the importance of preventing defects from occurring began to generate serious consideration as a result of the increasing pressure to meet pressing delivery schedules and to maintain higher levels of product quality and reliability. Hence, the quality engineering and reliability engineering functions were introduced. Since then, the separation of responsibilities between quality and production have become more pronounced. By the late 1970s, worldwide competition imposed much greater pressure upon U.S. manufacturing. The quality of Japanese semiconductors and automobiles had reached a much higher level, for example. The term "parts per million" instead of "parts per hundred" emerged as the accepted defect level in electronic components (Lubben, 1988, p. 69). The trend of the 1980s reverses the process, separating the interrelated functions of quality and production. Almost every book on quality mentions the importance of the production department's responsibility for product quality. Management realizes that as long as there is a separation between the product and the quality of that product, there will remain a lack of commitment, motivation and performance in a manufacturing or service system. Thus, the success of modern industry depends on the full integration of the control of quality and production. But not all qualityrelated issues are a direct consequence of the production process. Some issues stem from design specifications and supporting guidelines, and others from procurement problems. Therefore, quality control is a responsibility that should be shared by the whole company. One of the significant concepts of JustinTime (JIT) is to return the qualityenhancing function to where the products are produced. Quality assurance is then assigned a more productive role in supporting the manufacturing system. Quality assurance should guide, support and monitor the production process, and develop support programs such as reliability, failure analysis, data analysis, information feedback, and continuing education. The real control of quality is not in the hands of a quality assurance department; rather it is in the hands of the bottomline production department. From the production employees' viewpoint, the new arrangements will enhance their interest and involvement, and create a break from their previous taskspecific jobs. That is, this new responsibility serves as a job enlargement factor for production employees. Recent trends in the manufacturing concept make the introduction of more appropriate compensation systems a necessity. The old piecework scheme based on production quantity does not provide incentives for quality improvement. The nature of Japanese compensation systems attaches more weight to factors such as effort, cooperativeness, conscientiousness, and the display of initiatives. In fact, many experts suggest that the development of a highlymotivated workforce, and not just a highlyskilled workforce, is the most exacting challenge facing Western managers attempting to establish a JIT production system. In this chapter, a hierarchical model is proposed to simulate the integration of quality assurance and the production process. Most multilevel agency models consider the added level simply as the source of monitoring information. The added agent provides no productive action and hence cannot affect the realized outcome. In this model, the middlelevel quality control manager, in contrast, does have a direct influence on the realized outcome. In what follows, the model is described and the characterization of optimal contracts is then specified. Comparative statics are then analyzed for an additive qualityenhancing technology under certain assumptions. This chapter concludes with a summary of the findings. The Model Suppose a firm consists of three individuals: a riskneutral, profitmaximizing principal, and two riskaverse, utilitymaximizing agents. The two agents are the manager of the quality control department, and the worker in the production department. The quality control manager is responsible for establishing quality policies and for monitoring the production process. The worker is responsible for producing products and for implementing the quality policies established by the manager. Different expertise is the reason to set up this three level hierarchy. Production quantity (x) is influenced by the production effort (al) expended by the worker, and a random environmental factor (0) such as material shortage or machine breakdowns. The quality level (y) of the final products, measured by the rate of detectives or some other classification method, is influenced by the manager's guidance effort (9), the worker's quality enhancing effort (a2), and an uncontrollable state of nature (e) such as insufficient information of product nature or technology. Although the quality standard is a joint outcome of the efforts of manager and worker, collusion between agents by having a side contract (Tirole, 1986; Suh, 1987) is not considered in the model. Let F(x,y;a1,a2,9) be the joint distribution function that satisfies the monotone likelihood ratio (MLR) and the convexity of the cumulative distribution function (CDF) conditions. Some Japanese companies have found the natural rhythm of nonpositive correlation between production and quality existing in production systems. It is of no value for the management to press for greater production at the expense of quality levels. Instead, they search for the balance point that will provide the optimum level of production and quality (Lubben, 1988, p. 123). Following the same observation, it is further assumed that the production and quality levels (x and y) are nonpositively correlated. The principal can observe both the total output, x, and its quality, y. Therefore, x and y can be utilized as the basis of a contract. The manager receives a nonnegative payment M(x,y), and his expected utility G(M(x,y),P) is defined over that payment and his effort. Following conventional agency assumptions, the agents are risk averse and the disutility of effort increases at a nondecreasing rate (i.e. effort averse). G(M(x,y),P) is assumed to be additively separable, i.e. G(M(x,y),#) E[V(M(x,y))W2(f)], where V'(.)>0, V"(.)<0, W2'(.)>0 and W2"(*)0O. The worker also receives a nonnegative payment S(x,y), and his expected utility, H(S(x,y),a1,a2), is defined over his income and the production and qualityenhancing effort. H(S(x,y),a1,a2) is also assumed to be additively separable, i.e. H(S(x,y),a1,02) E[U(S(x,y))W1(k1at+k2C2)], where U'(.)>O, U"()<0, Wi'(.)>0, Wi"()>0, and k1, k2 are job enlargement factors with 0 and 0 S(x,y) are restricted to lie in certain intervals (Holmstrom, 1979). Given the above assumptions and specifications, the threelevel agency model can be set up as the following program: The principal seeks to (OP) Max E[w] E[p(y)xc(x)S(x,y)M(x,y)] (3.1) S,M Subject to H[S,al,02] =E[U(S)] W1(k1a1+k2za) > HO (3.2a) 1,aez e Argmax (E[U(S)] W1(k1a1'+k2I2')) (3.2b) l', 2' 36 G[M,3] E[V(M)] W2(f) 2 G* (3.3a) P e Argmax (E[V(M)] W2(f')) (3.3b) where E[*] denotes the expectation operator, the notation "argmax" denotes the set of arguments that maximize the indicated objective function, and G" and H* are the known marketreservation utility levels of the manager and the worker, respectively. The principal's expected profit is represented in (3.1). The individual rationality constraints, (3.2a) and (3.3a), guarantee that the worker's and the manager's expected utilities exceed their minimum market levels so that they will be attracted to and stay in the job. The incentive compatibility constraints, (3.2b) and (3.3b), indicate that each agent will select his effort level from his decision set to maximize his own expected utility. In order to replace constraints (3.2b) and (3.3b) by their corresponding firstorder conditions, a supplementary assumption on utility functions must be made. That is, U(U'1(1/S)) and V(V'1(1/M)) are concave in S and M, where U''1 and V'' denote the inverse function of U' and V'(Theorem 2, Jewitt, 1988). Given that the firstorder conditions hold, determination of the contracts S(x,y) and M(x,y) can be replaced by the following optimal control program (FP): 37 (FP) Max ff[p(y)xc(x)S(x,y)M(x,y)]f(x,y;a1,a12,3)dxdy (3.4) S,M Subject to ffU(S(x,y))f(x,y;al,a2,P)dxdy Wl(k1a1+k2a2)> Ho (3.5) ffV(M(x,y))f(x,y;al,a2,8)dxdy W2(P)> GO (3.6) ffU(S(x,y))fal(x,y;a1,a2,p)dxdy W1'kl(klac+k2a2) 0 (3.7) ffU(S(x,y))f (x,y;al1a2,P)dxdy Wl'k2(k1a1+k2a2) 0 (3.8) ffV(M(x,y))fP(x,y;al,a2,6)dxdy W2'(f) 0 (3.9) where fi(x,y;a1,a2,P) is the partial derivative of f(x,y;al,a2,3) with respect to i, i= ,a2,0; and Wj'() is the first derivative of Wj('), j=1,2. Characterization of Optimal Contracts Let AI, A2, pi1, A12, and P2 be the multipliers associated with constraints (3.5)(3.9), respectively. The Hamiltonian of the above program (FP) can be written as the following function: (H) Max L(S,M, A1, A2, I11,12 ,1 2,a12, ) Jf[p(y)xc(x)S(x,y)M(x,y)]f(x,y;al,a2,f)dxdy + A1 (ffU(S(x,y))f(x,y;a1,a2,f)dxdy W1(klal+k2a2) H) + An1(ffU(S(x,y))f I(x,y;al,a2,P)dxdy Wl'kl(klal+k2a2)) + A12(ffU(S(x,y))f2(x,y;1,Ca2jz,)dxdy W1'k2(kla+k2a2)) + A2 (ffV(M(x,y))f(x,y;al,12,)dxdy (W2() G*) + 2 (ffV(M(xxy))f(x,y;a,a2,)dxdy W2'()) (3.10) with A>1O and A2O0. If the optimal S(x,y) and M(x,y) are in the interior of the set of feasible contracts, they can be characterized as the solutions to the necessary Euler conditions. Thus, the optimality conditions for contracts are 1 U'(S) and 1 V'(M) where the second assumptions that Taking the B, the following f ,(x,y) f2(x,y) + + 12 + f(x,y) f(x,y) fa,(x y) f2,(yx) Ai + 1 n + 212 (3.11) f(xly) f(ylx) fp(x,y) A2 + 2  f(x,y) fg(y x) A2 + f2 ) (3.12) f(ylx) equations of (3.11) and (3.12) come from the x=x(al,8) and yy(a2,,fE). first derivative of (3.10) with respect to al, a2, and conditions are obtained: lf[p(y)xc(x)S(x,y)M(x,y)]f. 1(x,y)dxdy + A, (Sfu(S(x,y))f.1(x,y)dxdy k1W1'(kla1+k2aC2)) + pA (ffU(S(X,y))f. (x,y)dxdy W1"k2(klal+k2a2)) + P12 {JffU(S(x,y))f,l2(x,y)dxdy Wl"klk2(kla1+k2a2)) + A2 (ffV(M(x,y))f.1(x,y)dxdy) + A2 (ffV(M(x,y))fp.,(x,y)dxdy) = 0; (3.13) ff[p(y)xc(x)S(x,y)M(x,y)]f~2(x,y)dxdy + AI (ffU(S(x,y))f 2(x,y)dxdy Wl'k2(kC1+k2Ca2) + p1n(ffU(S(x,y))f,.2(x,y)dxdy WV"k k2(kja1+k22) ) + U12(fU(S(x,y))f,22(x,y)dxdy W1"k (kla1+k2a2) + A2 (ffV(M(x,y))f z(x,y)dxdy) + A2 (ffV(M(x,y))f 2(x,y)dxdy) 0; (3.14) ff[p(y)xc(x)S(x,y)M(x,y)]fp(x,y)dxdy + AI (ffU(S(x,y))fp(x,y)dxdy) + Pn11ffU(S(xy))fa1#(x,y)dxdy) + 12{(ffU(S(x,y)) fz(x,y)dxdy) + >2 (ffV(M(x,y))fp(x,y)dxdy W2'(f)) + z2 (ffV(M(x,y))fp(x,y)dxdy Wz"(p)) = 0. (3.15) The optimal compensation schemes and effort levels can be determined by solving equations (3.5)(3.9) and (3.11)(3.15) simultaneously, with A\?0 and A2>O. The exact solutions may be very difficult to formulate, but some characteristics of the optimal contracts are analyzed in the following results. Result 3.1. Under the optimal compensation schemes, the expected utilities for the manager and worker are exactly equal to their market reservation utility levels H* and G, respectively; that is, constraints (3.5) and (3.6) hold as qualities at the optimum, and A1>0, A2>0. Proof. By the KuhnTucker condition: A1>0, A2z0. If z2=0, equation (3.12) is rewritten as 1 fp(x,y) ) f2 (3.12') V'(M) f(x,y) The lefthand side of the equation (3.12') is always positive by the riskaverse assumption. Since fffa(x,y)dxdy0, this is a contradiction. Hence, Az>0. Evaluating equation (3.11), X1>0 can be proved simil~p .).. Result 3.2. Since the firstbest solution cannot be reached, the manager should share some risk for the qualityenhancing process, i.e. 12>0. Proof. (This proof follows Jewitt (1988).) Substituting (3.12) into (3.9) gives ffV(M)[I/V'(M) A2]f(x,y)dxdy p2W2'(*). (3.3.2a) Using the fact that E[fo/f]=ff(fo/f)fdxdy=fffldxdy=0, (3.12) gives ff[l/V'(M)]f(x,y)dxdy = A2. (3.3.2b) From Result 3.1 and equation (3.6), ffV(M(x,y))f(x,y)dxdy W2(6)+G. Hence, (3.3.2a) states that the covariance of V(M) and 1/V'(M) is of the sign as p2W2'(1). Since V(M) and 1/V'(M) are monotone in the same direction, they have a nonnegative covariance. And since W2'(p) is positive by assumption, it follows that p220. Furthermore, p=O can be ruled out, for then M(x,y) would be constant and this violates equation (3.9). Q.E.D. Result 3.3. The manager's contract is a strictly increasing function of the quantity and quality produced. Proof. From (3.12), M(x,y) can be written as M(x,y) h(A2+Jz(fi/f)), where h()=(l/V'())1. Since V'(.) is a strictly decreasing function of M(x,y), h() is strictly increasing with its argument. By the fact that A2 and p2 are positive as proved in results 3.1 and 3.2, and that fp/f is an 41 increasing function of x and y, M(x,y) is also an increasing function of x and y. Q.E.D. Result 3.4. If fp(ylx)/f(ylx) is linearly increasing in y and the conditional expected quality E(ylx) is a function of P, then fp(ylx)/f(ylx)[y[y ylx)] Ep(ylx)/ Var(ylx). Proof. Since f(x,y)f(ylx)f(x), fp(x,y)fp(ylx)f(x)+f(ylx)fg(x) fp(ylx)f(x). Recall equation (3.12) 1 fp(ylx) = \2 + i&2 V'(M) f(ylx) Define h(ylx)=fp(ylx)/f(ylx). Since h(ylx) is linearly increasing in y, hy(ylx)>0 and hyy(ylx)=0. It can be shown that E[h(ylx)] =fh(ylx)f(ylx)dy =ffg(ylx)dy=0. By Jensen's inequality and the assumption that h is linear, E[h(ylx)]h[E(ylx)]0. Since hy(ylx)>0, h(ylx)0 as y>E(ylx), and h(ylx)<0 as ysE(ylx). This suggests that h(ylx)=K(yE(ylx)), where E[yjx]=fyf(ylx)dy. Also, fyfp(ylx)dy=Ep(ylx). But, fyfp(ylx)dy = fyh(ylx)f(ylx)dy Kfy(yE(ylx))f(ylx)dy K{E[y2Ix] (E(ylx))2) = K Var(ylx). This implies K=Ep(ylx)/Var(ylx). Therefore, fp(ylx)/f(ylx) = h(ylx) = [yE(ylx)] Ep(ylx)/Var(ylx). Q.E.D. Remark. This result shows that both the conditional expectation and variance are used in the evaluation. The principal may use the revised conditional expected quality E(ylx) as a standard to give a bonus or impose a penalty. And, the compensation is decreasing with the variance. Thus, the conventional standardsetting and variance computation is part of the optimal compensation scheme (Baiman and Demski, 1980). Result 3.5. If there is no correlation between quantity and quality, then the optimal contract for the manager is a nontrivial function of the quality standard alone, i.e. M(.)M(y). Proof. Since fp(x,y)/f(x,y)fp(ylx)/f(ylx)ff(y)/f(y) when x and y are independent, the result follows from (3.12) directly. Q.E.D. Remark. In this case, the product quantity is noninformative (Holmstrom, 1979; Suh, 1988) regarding the manager's action choice. For the manager's performance evaluation purposes, there is no need to consider the quantity level. That is, the quality standard (y) becomes a sufficient statistic for (x,y) regarding P. If quantity and quality are correlated, even though production is beyond the manager's control, the production quantity tells the principal something about the manager's behavior. Thus, both factors are used in the manager's final evaluation. In addition, the manager tends to act as an advocate for his subordinate (Tirole, 1986). He would like to share some production risk with the worker so that the qualityimproving policy may be implemented more effectively and efficiently. Result 3.6. If the principal can observe the agents' actions or the states of nature directly, then: (i) the manager's compensation is a constant times the compensation of his subordinate, the worker; (ii) the manager's utility, at the optimum, is a linear transformation function of the worker's; and 43 (iii) the manager's coefficient of absolute risk aversion is a constant times the worker's. Proof. When the principal can observe the agents' actions or the states of nature, the firstbest solution can be achieved. That is, 1/U'A1 and l/V'=A2 with 11>0 and X2>O. Therefore, S(x,y)U'1(1/A1) and M(x,y)V'l(l/A2). The agents are paid by a constant salary, given that the specified effort levels are provided. Otherwise, they receive nothing. Conclusion (i) follows directly. From (i) A2V'(M)=AIU'(S) and M=kS, where k is a positive constant, we have that V'(kS)(A1/A2)U'(S). (3.3.6a) Integrating (3.3.6a) over S, V()=aU()+b, where a=kAl/Az, and b is a constant of integration. Conclusion (ii) follows directly. Differentiating both sides of (3.3.6a) with respect to S, we have kV"(A0/A2)U". (3.3.6b) Define the ArrowPratt measure of absolute risk aversion as R,V"/V' and Ru=U"/U'. From (3.3.6a) and (3.3.6b), it follows that RP=(l/k)Ru. Q.E.D. Remark. Herbert Simon (1957) suggested that "an executive's salary should be b times the salary of his immediate subordinates, ..." Simon's suggestion is supported under the availability of the firstbest solution. Conclusions (ii) and (iii) are consistent with the relationship of Paretooptimal risk sharing and the similarity rule of utility functions as given in Ross (1974). 44 Result 3.7. At the optimum, the worker has to share some risk of x and y, i.e. PlI>0 and pl2>0. Proof. This proof is similar to result 3.2. Substituting (3.11) into (3.7) gives ffU( (/U') A))f(x,y)dxdy pijffUf*dxdy pAlkW1' (3.3.7a) Substituting (3.8) into (3.3.7a) gives ffU((l/U')Ai))f(x,y)dxdy = piAkW,'+ pi2k2W '. (3.3.7b) By the fact that E[f, /f]=O and E[f,2/f]0, (3.11) yields Jff(/U')f(x,y)dxdy = A,. (3.3.7c) Equation (3.3.7b) states that the covariance of U and 1/U' is of the same sign as (pklk1W1'+ pik2W21') which should be nonnegative (Jewitt, 1988). Therefore, pil and Pi2 cannot both be negative. We can rule out 117j0, Pi2<0 and pIA1<0, pi2=0 cases for the same reason. /p4 and p42 cannot both be zero because of (3.7) and (3.8). We now consider the following two cases: 1. /li0, C12>0, or pli>O, A120, and 2. AII>0, p12<0, or pi1<0, P12>0. The second terms in (3.13) and in (3.14) are equal to zero by (3.7) and (3.8). The fourth term of (3.13) and the third term of (3.14) are positive by firstorder stochastic dominance. The third term of (3.13) and the fourth term of (3.14) are negative by the secondorder condition. The remaining terms of (3.13) and (3.14) are positive by the fact that M(x,y) is increasing in x and y, and firstorder stochastic dominance. Both cases violate equations (3.13) and (3.14). Hence, p 1>0 and Ip2>0. Q.E.D. Result 3.8. The worker's compensation is a strictly increasing function of the quantity and quality of products. Proof. Proof is similar to that of result 3.6. Q.E.D. The Comparative Statics In order to obtain greater insight from the model, the following additional assumptions are made: (a) The production and qualityenhancing processes follow a linear technology, i.e. x(a1,0)=1(a1)+O, y(a2,,0,)=m(a2,A)+e. B and e are bivariate standard normally distributed with covariance aog. Since oag=a=l, aep with l 0, "(al)O0, m 2>0, m9>0, m2a2 50, 