Multifaceted aspects of agency relationships

MULTIFACETED ASPECTS OF AGENCY RELATIONSHIPS by ELLA MAE MATSUMURA A.B., The University of C a l i f o r n i a , Berkeley, 1974 M.Sc, The University of B r i t i s h Columbia, 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Faculty of Commerce and Business Administration We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1984 © E l l a Mae Matsumura, 1984 In p r e s e n t i n g this requirements f o r an of British it freely available understood that financial shall for reference and study. I for extensive be her copying of g r a n t e d by shall not the be of make further this thesis head o f representatives. copying or p u b l i c a t i o n the University Library h i s or gain the the s c h o l a r l y p u r p o s e s may by f u l f i l m e n t of I agree that permission department or for in partial advanced degree a t Columbia, agree that for thesis this my It is thesis a l l o w e d w i t h o u t my written permission. Department o f Co mme-ree. ^ The U n i v e r s i t y o f B r i t i s h 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3 E-6 (3/81) &us/'ri£SS Columbia Aden,nisir~cut/or. ii ABSTRACT Agency theory has been used to examine the problem of stewardship of an agent who makes decisions on behalf of a p r i n c i p a l who cannot observe the agent's actual e f f o r t . E f f o r t i s assumed to be personally c o s t l y to expend. Therefore, i f an agent acts i n his or her own i n t e r e s t s , there may be a "moral hazard" problem, i n which the agent exerts less e f f o r t than agreed upon. This d i s s e r t a t i o n examines t h i s agency problem when the agent's e f f o r t i s multidimensional, such as when the agent controls several product i o n processes or manages several d i v i s i o n s of a firm. The optimal compen- sation schemes derived suggest that the widely advocated salary-plus-commission scheme may not be optimal. Furthermore, the information from a l l tasks should generally be combined i n a nonlinear fashion rather than used separ a t e l y i n compensating a manager of several d i v i s i o n s , even i f the monetary outcomes are s t a t i s t i c a l l y independent. In situations where e f f o r t i s best interpreted as time, e f f o r t can be viewed as being additive. The analysis i n this s p e c i a l case shows that the nature of the outcome d i s t r i b u t i o n , including the e f f e c t of e f f o r t on the mean of the d i s t r i b u t i o n , i s c r i t i c a l i n determining whether i t i s optimal for the p r i n c i p a l to induce the agent to d i v e r s i f y e f f o r t across tasks. These new r e s u l t s and the already existing agency theory results are applied to the sales force management problem, i n which the firm wishes to motivate a salesperson to optimally allocate time spent s e l l i n g the firm's various products. The agency model i s also expanded to allow f o r the agent's observation of the f i r s t outcome (which i s influenced by the agent's f i r s t e f f o r t ) before choosing the second e f f o r t l e v e l . The optimal compensation schemes iii both i n the absence of and the presence of a moral hazard problem are derived. The behavior of the second e f f o r t strategy i s also examined. It i s shown that the behavior of the agent's second e f f o r t strategy depends on the interaction between wealth and information e f f e c t s of the f i r s t outcome. Results s i m i l a r to those i n the multidimensional e f f o r t case are obtained for the question of optimality of d i v e r s i f i c a t i o n of e f f o r t when e f f o r t i s additive. iv TABLE OF CONTENTS Page ABSTRACT i i LIST OF TABLES vi ACKNOWLEDGEMENTS CHAPTER 1. vii INTRODUCTION 1 CHAPTER 2. NOTATION AND FORMULATION 5 Chapter 2 Footnotes 8 CHAPTER 3. ALLOCATION OF EFFORT 9 3.1 F i r s t Best 10 3.2 Second Best 11 3.3 Value of Additional Information 15 3.4 Additive Separability of the Sharing Rule 22 3.5 Additive E f f o r t 25 3.6 Application to Sales Force Management 36 3.7 Summary and Discussion 48 Chapter 3 Footnotes CHAPTER 4. ONE-PERIOD SEQUENTIAL CHOICE . 54 55 4.1 F i r s t Best 60 4.2 Second Best 63 4.3 Additive Separability of the Sharing Rule 74 4.4 Additive E f f o r t 76 4.5 Summary and Discussion 84 CHAPTER 5. SUGGESTED FURTHER RESEARCH 88 5.1 Theoretical Agency Extensions 88 5.2 Application to Variance Investigation 89 BIBLIOGRAPHY 91 V APPENDIX 1. ONE-PARAMETER EXPONENTIAL FAMILY OF DISTRIBUTIONS 94 APPENDIX 2. NORMAL DISTRIBUTION CALCULATIONS 102 APPENDIX 3. CHAPTER 3 PROOFS 108 APPENDIX 4. CHAPTER 4 PROOFS 132 vi LIST OF TABLES Page I. II. Examples i n One-product Case 40 One-parameter Exponential Family Q 94 vii ACKNOWLEDGEMENTS I would l i k e to thank Professor Jerry Feltham amd Professor John Butterworth for the many hours they spent discussing ideas and results with me. Their comments often pointed me i n new directions or helped me to look at problems from new perspectives. I am grateful to Professor Tracy Lewis for serving on my d i s s e r t a t i o n committee. I would also l i k e to thank my husband, Kam-Wah Tsui, for h i s technical assistance while I was writing this d i s s e r t a t i o n , and for h i s u n f a i l i n g encouragement the entire time I was i n the Ph.D. program. and moral support during F i n a l l y , I gratefuly acknowl- edge the f i n a n c i a l support of the University of B r i t i s h Columbia and of the Social Sciences and Humanities Research Council of Canada. 1 CHAPTER 1 INTRODUCTION Managerial accounting has t r a d i t i o n a l l y been associated with the valuation of inventories for external reporting and with information provision for internal decision making and control. Broadly speaking, the internal decision making relates to the planning of operations and the control of decentralized organizations. Variance analysis, budgeting, cost-volume- p r o f i t analysis, and the development of performance evaluation measures are t y p i c a l components of the planning and control processes. There are a number of different approaches to gaining a better understanding of the role of the accounting system i n the control of decentralized operations. Since an accounting system i s an information system, any research on the value of information, the demand for information, or the roles or uses of information has potential implications for accounting research. The body of research which examines such information issues has come to be known as information economics. Information economics uses f o r - mal economic models i n order to study the demand for information for decision making and performance evaluation purposes. In p a r t i c u l a r , information economics attempts to find economic explanations for why certain phenomena are observed (e.g., Demski and Feltham, 1978), and to uncover insights about behavior thought to be nonoptimal (e.g., Zimmerman, 1979) or behavior thought to be optimal (e.g., Baiman and Demski, 1980b). Much of the early information economics l i t e r a t u r e focused on essent i a l l y single-person decision situations (e.g., Demski and Feltham, 1976, and Feltham, 1977a), where information serves only a d e c i s i o n - f a c i l i t a t i n g purpose. That i s , the decision maker uses information about the uncertain state of nature to revise his or her b e l i e f s about the decision environment. Thus, the demand for this type of information might be called decision-mak- 2 ing demand. The recent information economics l i t e r a t u r e has incorporated agency theory i n e x p l i c i t l y modeling the multiperson nature of accounting problems (e.g., Baiman and Demski, 1980a, 1980b, Gjesdal, 1981, and Holmstrom, 1977). In multiperson situations, information can play a deci- sion-influencing r o l e . For example, i f a manager's actions a f f e c t actual production costs, and the manager i s evaluated and possibly compensated on the basis of the costs, then the manager's actions w i l l be influenced by the existence of the information system which reports the costs. The demand for this type of information might be c a l l e d performance evaluation demand, or stewardship demand. This d i s s e r t a t i o n uses the agency framework to examine some of the issues i n the development of performance evaluation measures for motivat i o n a l purposes. The basic agency model provides a means of studying s i t u a - tions i n which one individual (the principal) delegates the selection of actions to another individual (the agent). Within the context of the firm, the p r i n c i p a l might be the employer or superior and the agent might be the employee or subordinate. The agency theory l i t e r a t u r e (e.g., Harris and Raviv, 1979, Holmstrom, 1979) uses the expected u t i l i t y model to represent the preferences of the p r i n c i p a l and the agent, and generally assumes that the agent's action ( e f f o r t ) and a random state of nature determine the monetary outcome. The sharing rule (contract or compensation scheme) offered by the p r i n c i p a l to the agent specifies how much i s paid to the agent for each possible value of some performance measure or measures. The performance measure i s often taken to be the monetary outcome, or the monetary outcome and an imperfect signal about the agent's e f f o r t . The compensation can be based only on what i s j o i n t l y observable to the p r i n c i p a l and the agent, and the compensation must be adequate enough to induce the agent to work for the 3 principal. Alternative employment opportunities for the agent are thus e x p l i c i t l y considered. The p r i n c i p a l w i l l generally find i t p r o h i b i t i v e l y costly to continuously monitor the agent to determine what action ( e f f o r t ) the agent chooses. Therefore, i f the agent has d i s u t i l i t y for e f f o r t and acts i n his or her own s e l f - i n t e r e s t , the potential for a moral hazard (incentive) problem exists because of the principal's i n a b i l i t y to observe the agent's actions. If the p r i n c i p a l pays the agent a fixed wage, the agent has no economic incentive to perform the agreed l e v e l of e f f o r t , since a low outcome can be blamed on a bad state of nature rather than on shirking by the agent. At the other extreme, i f the p r i n c i p a l rents c a p i t a l or rents the firm to the agent for a fixed fee so that the agent gets the outcome less a fixed fee, the shirking problem can be avoided e n t i r e l y . The shortcoming of this type of contract i s that i t imposes a nonoptimal amount of r i s k on the agent. That i s , the p r i n c i p a l and the agent could be made better o f f i n an expected u t i l i t y sense by using some other contract. Agency theory provides a framework i n which i t i s possible to find compensation schemes which e f f i c i e n t l y motivate the agent to choose the desired actions. The idea i s to create incentives through an employment contract which imposes some r i s k on the agent i n order to provide incentives for the agent to expend some agreed l e v e l of e f f o r t . The consequences of the exis- tence of nonmonetary returns or costs, such as e f f o r t , can thus be analyzed. This i s important for the analysis of performance evaluation and managerial control systems, where incentive effects play a c r i t i c a l r o l e . The choice of variables on which compensation i s to be based can be formally derived, with implications for the design of information systems. Furthermore, the analysis c l e a r l y demonstrates how the information obtained can be incorporated for motivational purposes. 4 Most of the existing agency theory research (see Baiman (1982) for a comprehensive survey) has a rather narrow d e f i n i t i o n of e f f o r t , i n that e f f o r t i s assumed to be single-dimensional. However, people are often faced with several similar tasks which must be performed within one time period. Examples include a salesperson s e l l i n g several products for a firm, an auditor a l l o c a t i n g time to different tasks i n an audit assignment, a manager controlling several production processes, or a manager overseeing several divisions of a company. The problem of motivating the optimal a l l o c a t i o n of effort within one period i s not only interesting i n i t s own right, but also has possible implications for multiperiod problems, where e f f o r t i s a l l o cated across periods. Multiperiod problems are of interest because the eventual goal i s to be able to analyze and understand when there are current and long-term the issues involved consequences of decisions, as there are i n many accounting settings. Chapter 2 of this dissertation contains the notation used i n the remainder of the paper and a formulation of the agency problem with a l l o c a t i o n of e f f o r t . Chapter 3 describes theoretical results and an application in the a l l o c a t i o n setting, and Chapter 4 describes results i n the one-period sequential choice setting. In this scenario, after each effort l e v e l i s exerted, an associated outcome i s observed by the agent before the next e f f o r t l e v e l i s exerted. sequence of outcomes. The agent i s compensated only at the end of the The one-period sequential choice case i s an interme- diate step between the a l l o c a t i o n of e f f o r t case, i n which both the efforts are exerted before the outcomes are known, and the multiperiod case, i n which the f i r s t outcome i s observed and the f i r s t compensation i s paid before the second e f f o r t i s exerted. Chapter 5 concludes the d i s s e r t a t i o n with an outline of proposed future research. proofs appear i n the appendices. A l l technical calculations and 5 CHAPTER 2 NOTATION AND FORMULATION In order to state the agency problem with a l l o c a t i o n of e f f o r t , the following notation w i l l be used: R = the set of a l l real numbers, R_)_ = the set of a l l nonnegative real numbers, X = the set of possible monetary outcomes, x e X CZ. R i s the monetary outcome, x = (x^,...,x ) i s a disaggregation of the monetary outcome x, i . e . , n k n x= Ex. , w e R i=l i s a k-dimensional vector-valued performance measure, e.g., w = x_with k=n, s ( . ) , a real-valued function, i s a sharing rule over the arguments indicated, with s(.) e [SQ,S],*" a^ = effort expended on task i , i=l,...,d, a_ = (a^ ,... ,a^) e A CZ R^_, f(x,w|a) i s the joint density of x and w conditional on a, and i s understood to be f(x|a) i f w i s a function of x; g(«), h ( . ) , and <(>(.) w i l l also be used to denote probability d i s t r i b u t i o n s ; U(.): R + R i s the agent's u t i l i t y function over money, where U' > 0 and U " <_ 0, V(.): R^ •*• R i s the agent's d i s u t i l i t y function over e f f o r t , where 9V/ 3a > 0 and 3 V/ 3a 2 ± > 0, u = the agent's minimum acceptable u t i l i t y W(.): level, R + R i s the principal's u t i l i t y function over money, where W > 0 and W • <_ 0, argmax {. } = the set of arguments maximizing^the expression i n braces. In order to avoid side-betting issues, i t w i l l be assumed that the p r i n c i p a l and the agent have i d e n t i c a l b e l i e f s about the conditional proba- 6 b i l i t y d i s t r i b u t i o n over the outcome and performance measure, given e f f o r t a_. As i n much of the agency l i t e r a t u r e , the agent's u t i l i t y function i s assumed to be of the form U(s) - V(a). In most of the agency l i t e r a t u r e , n=d=l. The principal's problem i s Maximize // W(x-s(w))f(x,w|a) dw dx (2.1) s( .) ,a_ subject to //[U(s(w))-V(a)]f(x,w|a) dw dx > u a e argmax (2.2) {//[U(s(w))-V(a)]f(x,w|a) dw dx }. (2.3) It w i l l be assumed that (2.3) can be replaced with the conditions -JL //[U(s(w))-V(a)]f(x,w|a) dw dx = 0, i = l , . . . , d . (2.4) Furthermore, s u f f i c i e n t regularity to allow d i f f e r e n t i a t i o n inside the i n t e gral i s assumed. This permits the replacement of (2.4) with //U(s(w))f (x,w|a) dw dx = V (a), i=l,...,d, (2.5) i i with subscripts a^ denoting p a r t i a l d i f f e r e n t i a t i o n with respect to a^. a a a The principal's problem Is solved by means of a generalized Lagrangian technique. A Hamiltonian (Lagrangian) i s formed by attaching a m u l t i p l i e r X to (2.2) and m u l t i p l i e r s to each constraint i n (2.5). It w i l l be assumed that the supports of x and w do not vary as a varies, and that the p a r t i a l derivatives of f with respect to each a^ exist and are nondegenerate. The dimension d i s often taken to be equal to n, and the marginal cumulative d i s t r i b u t i o n functions are assumed to s a t i s f y f i r s t order stochastic dominance. That i s , i f F ^ x ^ a ^ ) i s the marginal cumulative d i s t r i b u t i o n func- tion of X j _ , then SF^x^\a^)/Sa^ < 0, i=l,...,n. In a framework where x.^ = h^(a£,9), where 9 represents state uncertainty, i f x^ i s increasing i n a^ (i.e., aa > 0 1 f o r a l l 9), then 3 F ( x | a ) / 3 a 1 1 i i < 0. F i n a l l y , the shar- ing rule i s assumed to be measurable and bounded. For the most part, i n t e r l o r solutions w i l l be examined. 3 7 Some of the results w i l l make use of two special classes of functions. The f i r s t i s the HARA (hyperbolic absolute r i s k aversion) class of u t i l i t y functions, whose r i s k aversion functions are of the form - U (x)/U'(x) = ,, The C = 1 case corresponds l/(Cx + D). (2.6) to U(x) = ln(x+D), the C = 0 case corresponds to U(x) = - exp[-x/D], and the other cases correspond to power u t i l i t y functions . The other class of interest i s the one-parameter exponential family of distributions. This class includes the exponential, gamma (with the shape parameter fixed), normal (with constant variance), and Poisson d i s t r i b u tions. The following representation d i f f e r s s l i g h t l y from the usual one f o r a one-parameter exponential family (see, e.g., DeGroot, 1970). Definition: A probability density function f(x|a) with respect to the measure r ( . ) w i l l be said to belong to the one-parameter exponential family Q i f i t can be written as f(x|a) = exp[z(a)x - B(z(a))]h(x), (2.7) where r ( . ) i s the Lebesgue measure when the random variable x i s absolutely continuous, and r ( . ) i s some counting measure when x i s d i s c r e t e . The representation i n (2.7) has the advantage that closed-form sions can be obtained for E(x|a) and Var(x|a). B'(z(a)) and Var(x|a) = B " ( z ( a ) ) (Peng, 1975). expres- In p a r t i c u l a r , E(x|a) = Table II i n Appendix 1 d e t a i l s the representations of some familiar d i s t r i b u t i o n s . The remainder of Appendix 1 consists of calculations which are useful i n the proofs of the results i n Chapters 3 and 4. 8 CHAPTER 2 FOOTNOTES If the sharing rule i s unbounded, an optimal solution may not exist (Mirrlees, 1974; Holmstrom, 1977, 1979). Furthermore, the agent's wealth places bounds on the possible sharing rules. Gjesdal (1981) has shown that such a u t i l i t y function for the agent ensures that nonrandomized payment schedules are Pareto optimal. His result refers to ex post (after e f f o r t selection by the agent) randomization only. Fellingham, Kwon, and Newman (1983) have shown that ex ante randomization of payment schedules i s optimal under certain conditions. It w i l l be assumed i n what follows that these conditions are not s a t i s f i e d , and hence the focus i s on pure (nonrandomized) payment schedules. That i s , the focus w i l l be on the f i r s t - o r d e r conditions, which apply to i n t e r i o r solutions. 9 CHAPTER 3 ALLOCATION OF EFFORT As stated e a r l i e r , the agency theory framework e x p l i c i t l y recognizes alternative employment opportunities for the agent, d i s u t i l i t y for e f f o r t , r i s k aversion of the agent, and the p o s s i b i l i t y of the p r i n c i p a l obtaining information about the agent's e f f o r t , a l l for situations i n which the agent has one task to perform. However, i n many situations, job e f f o r t i s multi- dimensional; the agent must allocate e f f o r t to several d i f f e r e n t , but possibly related tasks. mensional In spite of the variety of situations i n which m u l t i d i - job e f f o r t occurs, l i t t l e attention has been devoted i z i n g optimal compensation schemes for these situations. to character- S t i g l i t z (1975) considered multidimensional job e f f o r t under linear incentive schemes, and Weinberg (1975) sought an incentive compatible scheme for the problem of sales force management i n multiproduct firms. Radner and Rothschild (1975) examined the properties of three h e u r i s t i c strategies an agent might employ when faced with the problem of a l l o c a t i n g e f f o r t . More recently, Gjesdal (1982) allowed for multidimensional e f f o r t and focused on the value of information. The focus of this chapter i s the characterization of optimal incentive schemes for the agency problem with a l l o c a t i o n of e f f o r t across several tasks. The issues of separability of the optimal sharing rule across tasks and the value of additional information are examined, and the results suggest that certain compensation schemes that are widely advocated may not be optimal. In p a r t i c u l a r , commission schemes and l i n e a r sharing rules are shown not to be optimal, i n general. The special case of additive e f f o r t i s discussed, and the results are applied to the problem of sales force management . 10 3.1 FIRST BEST Suppose that i n addition to observing the aggregated or disaggregated outcome ( i . e . , w = x or w = x), the p r i n c i p a l can observe the agent's effort. These cases may be called complete contractual information cases, since the p r i n c i p a l can observe the agent's choice of e f f o r t . These " f i r s t best" situations are interesting as benchmarks for comparison with "second best" situations, those i n which there i s less than complete contractual information. The characterizations of the optimal sharing rule for these f i r s t best cases are obtained by solving the problem given by (2.1) and (2.2). As i n the single-dimensional e f f o r t case, i f one individual i s r i s k neutral and the other i s r i s k averse, then the r i s k neutral individual bears a l l the r i s k . pal s(x) Thus, i f the agent i s r i s k neutral (U(s) = s) and the p r i n c i - i s r i s k averse, then Pareto optimal sharing rules are s(x) = x - k and = x - k, where k i s a fixed fee paid to the p r i n c i p a l . Conversely, i f the p r i n c i p a l i s r i s k neutral and the agent i s not, the p r i n c i p a l bears the r i s k , receiving a share of x - c, while the agent receives a constant wage c. In the event that both the p r i n c i p a l and the agent are r i s k neutral, the a^'s are chosen so that the agent's marginal d i s u t i l i t y for e f f o r t equals the marginal increases i n the expected outcome ( i . e . , so that 3E(x|a)/3a^ = 3V/3a^, i = l , . . . , d ) , and the sharing rule can be taken as s(.) = u + V(a*), with the p r i n c i p a l receiving E(x|a*) - u - V(a*). If both the agent and the p r i n c i p a l are r i s k averse, then they each bear part of the r i s k , as indicated i n Proposition 3.1.1 Proposition 3.1.1. below. If both the agent and the p r i n c i p a l are r i s k averse and they have homogeneous b e l i e f s , then s(x) varies only with x i n the f i r s t best case. 1 1 Because the optimal sharing rule depends only on x, s(x) Is the same for a l l jc_ that provide the same t o t a l x. The sharing rule S(JC_) therefore varies with x only for risk-sharing purposes - the makeup of x i s unimportant. Moreover, i t i s e a s i l y seen that s(x) i s increasing i n each x^, regardless of the properties of the conditional d i s t r i b u t i o n function on x. This i s i n contrast to the second best solution. 3.2 SECOND BEST Suppose now that the p r i n c i p a l cannot observe the agent's e f f o r t , and hence must present the agent with a sharing rule which induces the desired choice of e f f o r t . Since the focus i n most of what follows i s on motiva- t i o n a l , rather than risk-sharing issues, i t w i l l be assumed that otherwise stated the p r i n c i p a l i s r i s k neutral and unless the agent i s r i s k averse. As remarked above, i f there were no moral hazard problem, the p r i n c i p a l would then bear a l l the r i s k . Whatever risk i s imposed on the agent i n the second best case i s thus imposed not for risk-sharing purposes, but for motivational purposes. Letting f denote 9f/3a. , the optimal sharing rule, given that only x a I 1 is observed, i s characterized d E ji f t U'(s(x)) = X by (x|a*) a f(x|a*) + ' for almost every x such that s(x) the l e f t hand side of = s i f the opposite (3.2.1) (3.2.1) £ [SQ,S]. For a l l other x, s(x) = ~ N ( a , o ) , and x = s. 1 The 1 if s(x) is true. 2 1 SQ i s greater than the right hand side, and For example, suppose that n = 2 , U(s) = l n s, x^ and x rather 2 2 ~N(a ,a ). 2 2 Then x ~ ^(.a l + a , 2 X 2 are independent, 2 2 + a ) and 1/U'(s) 2 i n t e r i o r portion of the optimal sharing rule i s thus^ (See Table II i n Appendix 1 for the normal density) 12 x - a* 2 2— v^) s(x) = X + ( ^ + °l °2 + ( + u )(a* + a*) h ^ + 2 2 ^ 2 °1 2 + + a a 2 l + J a _ 2 2 ^ X » which can be interpreted as a compensation scheme consisting of a fixed portion plus a commission. If the agent's u t i l i t y function i s U(s) = l - e , - s then the i n t e r i o r portion of the optimal sharing rule i s x - a| - a^ s(x) = l n [ X + (^ + v^) j ^—1* °L + °2 In general, i f the p r i n c i p a l i s r i s k neutral and the agent's u t i l i t y function i s i n the HARA c l a s s , with r i s k aversion function given by -U"(s)/U'(s) = l/(Cs+D), then the i n t e r i o r portion of the optimal sharing rule i s + (x|a) Wri)—>-°^ zl c-t<x ^ £ c if <*° s(x) = (3.2.2) d T. u.f (x|a) i=l D l n ( X+ i f(xla) i f C=0. >• (C=l corresponds to U(s) = ln(s+D), C=0 corresponds to U(s) = - e ~ ^ , and s the other cases correspond to power u t i l i t y D functions.) As i n the single-task setting, the f i r s t best solution i s achievable with a fixed fee going to the p r i n c i p a l when the agent i s r i s k neutral. This can be deduced by interpreting e f f o r t to be a vector rather than a scalar i n the single-task setting proofs (e.g., Shavell, 1979). though less than complete contractual information pal Thus, even Is available, the p r i n c i - and the agent can obtain the same expected u t i l i t i e s as they could i n 13 the complete contractual information case. This i s because i n e f f e c t , the risk-neutral agent rents the firm from the p r i n c i p a l for a fixed fee. When U(.) i s s t r i c t l y concave, equation (3.2.1) implies that a neces2 sary and s u f f i c i e n t condition for s(x) to be nondecreasing i n x i s d f a < ls*> "l-Rt ^ x f(x| «) * >- ° a for a l l x corresponding < ' ' 3 2 3 ) to i n t e r i o r solutions, where the y^'s are the Lagrangian m u l t i p l i e r s associated with the optimal solution (a ,s(x)). When a i s one-dimensional, (3.2.3) reduces to f (x|a*) a a -5* forr^n i >- °> < - - > 3 since y i s positive (Holmstrom, 1979). 2 4 If (3.2.4) i s true for a l l a* e A, then f(x|a) has the monotone l i k e l i h o o d ratio property i n x (Lehmann, 1959, p. 111). Many d i s t r i b u t i o n s , including a l l those i n the one-parameter expo- nential family, have the monotone l i k e l i h o o d r a t i o property; this property i s a stronger ordering on distributions than i s f i r s t - o r d e r stochastic dominance (Lehmann, 1959, pp. 73-74). Lagrangian for multipliers If a_ i s multidimensional and the y^ are a l l nonnegative, then a s u f f i c i e n t condition s(x) to be nondecreasing i n x i s f IT 1 a (xja*) f(x| *) a 1 ^ °' f o r a 1 1 x « i = 1 .---» d When a i s single-dimensional, the f i r s t - o r d e r stochastic dominance property means that as a increases, the d i s t r i b u t i o n f(x|a) s h i f t s to the right. It i s this property that accounts for the monotonicity mal sharing r u l e . of the o p t i - When a_ i s multidimensional, the problem of determining an ordering over the e f f o r t vectors a r i s e s . Condition (3.2.3) states that the d i r e c t i o n a l derivative of log f(x[a) i n the d i r e c t i o n of y = (y^,...,u^) at 14 a* be nonnegatlve i n order Thus, u provides f o r the o p t i m a l s h a r i n g r u l e to be monotonic. a d i r e c t i o n i n which to measure the s h i f t i n g of f(x|a_). Because of the c r i t i c a l r o l e that the m u l t i p l i e r s ing r u l e , i t i s of i n t e r e s t to t r y to determine whether they are positive. A p a r t i a l answer i s provided s i t u a t i o n i n which the v e c t o r x_ = P r o p o s i t i o n 3.2.1. F p l a y i n the ( il i) a x a ^» < i= i n P r o p o s i t i o n 3.2.1 It V(a) of with s t r i c t and can be i n e q u a l i t y f o r some x^ v a l u e s . strictly convex i n a, other a2» concave i n a. then both and and g( •) and concave i n a_. h( •) are e x p o n e n t i a l Then at 2^s, utility In t h i s case, i f the p r i n c i p a l i s r i s k n e u t r a l , are p o s i t i v e . P r o p o s i t i o n 3.5.9 and i n s e c t i o n 3.5 sharing Suppose f(xjj0 i s of the form g i v e n i II g (x i=l provides are p o s i t i v e . remark on the c h a r a c t e r i s t i c s of the o p t i m a l be made at t h i s p o i n t . = distribu- r e s p e c t i v e l y , then the agent's expected c o n d i t i o n s under which both A final and Suppose shown that i f the agent's u t i l i t y f o r wealth i s U(s) is strictly the must be p o s i t i v e . t i o n s with means a^ and is below, f o r i s observed. (x^,X2) f u r t h e r that the agent's expected u t i l i t y i s s t r i c t l y l e a s t one strictly Suppose x^ = (x^,X2), f(xj.a) = g(x^|a^)h(x2|a2), l»2> shar- rule can in Proposition d 3.2.1, i . e . , f ( x | a ) = merely i n d i c e s . The with x replaced by _x. 1 to u t ( s ( x optimal |a ), where the s u p e r s c r i p t s on g( •) are 1 sharing r u l e s i s c h a r a c t e r i z e d as i n (3.2.1), In the s p e c i a l case under c o n s i d e r a t i o n , d ) ) = x + \Z 2 a ( x |a*)/g(x ± ± |a*) belongs to the one-parameter e x p o n e n t i a l ( »)/g*( *) each g* a t h i s reduces i s a constant . I f , f u r t h e r , each g i ( •) family Q described m u l t i p l i e d by by (2.7), then (x^ - H j ^ a ^ ) ) , where M j ^ a ^ is i the mean of x^ g i v e n a^. The norms, so t h a t the o p t i m a l standards ( c f . C h r i s t e n s e n , means M^(a^) can be sharing 1982). thought of as standards or r u l e i s a f u n c t i o n of d e v i a t i o n s from T h i s i s c o n s i s t e n t with managerial 15 accounting's focus on variances (deviations from standards) as an aid i n performance evaluation. 3.3 VALUE OF ADDITIONAL INFORMATION A question which naturally arises at this point i s : Would the p r i n c i - pal be better off knowing each rather than only x? That i s , w i l l the p r i n c i p a l always be s t r i c t l y better off with disaggregated tion? or f i n e r informa- More generally, under what conditions w i l l the p r i n c i p a l or the agent be made s t r i c t l y better off by information i n addition to the aggregate outcome, x? I n t u i t i v e l y , the more (imperfect) information the p r i n c i p a l has about the agent's e f f o r t , the more e f f i c i e n t l y the agent can be motivated effort. to exert Consequently, the p r i n c i p a l ' s expected u t i l i t y should increase i n most situations where additional information i s a v a i l a b l e . A number of people have addressed this problem. Holmstrom (1979), for example, showed that i f the additional information i s of value (that i s , i f i t s optimal use w i l l lead to a Pareto superior pair of expected u t i l i t i e s for the p r i n c i p a l and the agent), then the additional information must be informative i n the sense that i t contains information about the agent's e f f o r t that i s not in the output. 1982) The converse was also shown to be true. contained Gjesdal (1981, examined the relationship between Blackwell (1953) informativeness and the value of information. In order to define Blackwell informativeness, l e t ft be a set of possible performance measures to. output, x. space Y. In this section, u> i s assumed to include the An information system n i s a function from ft to some s i g n a l Let y denote an arbitrary element in Y and l e t A, the set of a l l possible actions, be f i n i t e . mative than another system y : Information system n i s Blackwell more i n f o r -»• Z i f and only i f P (z|a) = 16 / P(z|y)dP (y|a) for each action a i n A and each signal z i n Z. Y It should 11 be noted that although may n i s said to be Blackwell more informative than y , n actually be only equally informative as y i s . Amershi (1982, Appendix 1) has generalized the d e f i n i t i o n for the case where A i s i n f i n i t e . Amershi (1982) re-examined the value of additional Information problem, and corrected and generalized the results of Holmstrom (1979, 1982) Gjesdal (1981, 1982). and Amershi (1982) showed that a risk neutral p r i n c i p a l weakly prefers an information system that i s Blackwell more informative than another. That i s , the principal's and the agent's expected u t i l i t i e s are at least as high with the Blackwell more informative system than with the other. A r i s k averse p r i n c i p a l requires that the Blackwell more informative system also provide a s p e c i f i c form of information about the output Proposition 3.3.1 below). (see These results d i f f e r from the single-person deci- sion maker case, where r i s k attitudes are immaterial. Intuitively, a risk neutral p r i n c i p a l i s concerned only with the incentive properties of contracts, whereas a r i s k averse p r i n c i p a l i s concerned with both the incentive and risk-sharing properties of contracts. accounts for the conditions on the output More s p e c i f i c a l l y , Proposition 3.3.1 The risk-sharing aspect i n Proposition 3.3.1 below. says that information system n i s at least as preferred as information system y i f n i s Blackwell more i n f o r mative than y with respect to the e f f o r t , a, and ( i ) there i s no risk-sharing involved, or ( i i ) y says nothing more about the output x than n does, or ( i i i ) the signal provided by n i s enough to determine the Proposition 3.3.1 (Amershi (1982, Theorem 3.1)). output. Let an information system n : ft -»• Y be more informative i n the Blackwell sense than the system y : ft -*• Z with respect to the family of measures P^ = {p(o>|a) : aeA}. pose also, at least one of the following conditions hold: i s r i s k neutral, Sup- ( i ) The p r i n c i p a l ( i i ) The output variable and the information system y are 17 conditionally independent given n. ( I i i ) The output can be expressed as x = h(n(u))) f o r some measurable function h : Y * R. Then the p r i n c i p a l weakly prefers n over y. In this proposition and i n the other propositions i n this section, to i s a vector of performance measures that includes the output, x. e f f o r t variable, a, i s taken to be single dimensional, Although the the proof holds for finite-dimensional e f f o r t vectors as w e l l . Proposition 3.3.1 i d e n t i f i e s conditions under which information system n i s at least as preferred to information system y. It i s of interest to i d e n t i f y conditions under which n i s s t r i c t l y preferred to y. Amershi's (1982) s t r i c t preference ficient statistics. results r e l y on the concept of suf- Using the notation above, a s t a t i s t i c T : ft + K i s suf- f i c i e n t for the family of measures P^ = {P(to|a) : aeAJ i f and only If there exists a nonnegative function h : ft + R + and functions g( *|a) : K •*• R such that f(to|a) = h( to)g(T( to) |a) for a l l weft and aeA, where f( •) i s a density i f the random variable i s continuous, or a mass function i f the random variable i s d i s c r e t e . viewed as an information system. A s u f f i c i e n t s t a t i s t i c may be A minimal s u f f i c i e n t s t a t i s t i c i s a s u f f i - cient s t a t i s t i c T : ft -*• L that i s a function of every other s u f f i c i e n t statistic. An, agency s u f f i c i e n t s t a t i s t i c (Amershi, 1982) "F i s equal to a suf- f i c i e n t s t a t i s t i c T on ft i f the p r i n c i p a l i s risk neutral, or (X,T) i f the p r i n c i p a l i s risk averse. *P i s called a minimal agency s u f f i c i e n t i f the s u f f i c i e n t s t a t i s t i c T i s minimal. statistic F i n a l l y , a contract (s*,a*) i s called a best agency contract i f there i s no other contract based on any information system on ft that i s s t r i c t l y preferred to i t . The proposition below uses the concept of agency s u f f i c i e n t to characterize s t r i c t preferences statistics f o r information systems. E s s e n t i a l l y , 18 the p r i n c i p a l w i l l s t r i c t l y prefer an agency s u f f i c i e n t s t a t i s t i c n over another system y which does not generate a best contract. This i s because a best contract must be a function of the minimal agency s u f f i c i e n t statistic, which extracts a l l relevant information from cu about a (Amershi (1982, Coro l l a r y 3.3)). Proposition 3.3.2 below provides a s i t u a t i o n i n which the information system y cannot generate a best contract. (Amershi (1982, Proposition 3.4)). Proposition 3.3.2 Suppose a best con- tract exists and at each best agency contract, W'(x-s*(S(w))) U'(s*(f3(u>>> where 3 i s an information system which leads to a best agency contract. The p r i n c i p a l s t r i c t l y prefers an agency s u f f i c i e n t s t a t i s t i c n over a system y If -sg- log f(oj|a ^) i s not a function of Y i f the p r i n c i p a l i s r i s k neutral 1 (or not a function of (x, Y) i f the p r i n c i p a l i s r i s k averse). Here (s*,a*) is the optimal contract based on Proposition 3.3.2 Y' holds for the multidimensional e f f o r t case, with cations of Amershi's (1982) c o r o l l a r i e s to his Proposition 3.4. are immediate. Corollary 3.3.3 Their proofs deals with the s i t u a t i o n i n which an addi- tional signal z would be of positive value given an information system which reports the outcome x and another signal y. As i n Proposition 3.3.2, a con- d i t i o n i s provided which implies that Y(x,y,z) = (x,y) cannot generate a best contract. Since t i c , Corollary 3.3.3 3.3.4 n(x,y,z) = (x,y,z) i s t r i v i a l l y a s u f f i c i e n t follows d i r e c t l y from Proposition 3.3.2. statis- Corollary provides a s i t u a t i o n i n which a s u f f i c i e n t s t a t i s t i c i s s t r i c t l y pre- ferred to a nonsufficient s t a t i s t i c . 19 Corollary 3.3.3 (Gjesdal (1982, Proposition 1)). x,y,z are from some spaces }. Letft= {u> = (x,y,z) : Let n be the information system that reports (x,y,z), and l e t Y be the information system that reports (x,y). Assume that f o r 8 = n and 8 = Y, W'(x-s*(B((D))) U'(s*(g(u))) n = X + i = \ \ TT ^ H a * ) f ± (3.3.1) g holds at the contracts (n, s*, a*p and (Y, s*, a*). Then the signal z has marginal value given (x,y) (that i s , the p r i n c i p a l s t r i c t l y prefers n over Y) i f 3 n Z i=l p 1 ^ i log f(o)|a*) ~ (3.3.2) i s not a function of (x,y). Corollary 3.3.4 (Holmstrom (1982, Theorem 6)). Suppose the p r i n c i p a l i s risk neutral, and suppose that for some system y : ft * Z, the expression i n (3.3.2) i s not a function of y at each a eA. prefers any s u f f i c i e n t s t a t i s t i c Then the p r i n c i p a l s t r i c t l y n over y i f equation (3.3.1) holds at any best agency contract generated by information system 8. As Amershi (1982) remarks, these s t r i c t preference results do not establish that an agency s u f f i c i e n t s t a t i s t i c i s always s t r i c t l y preferred to a nonsufficient s t a t i s t i c . In order for a s u f f i c i e n t s t a t i s t i c to be s t r i c t l y preferred to a nonsufficient s t a t i s t i c , the p r i n c i p a l must use Information which i s provided by the s u f f i c i e n t s t a t i s t i c but not provided by the nonsufficient s t a t i s t i c . In addition, the p r i n c i p a l ' s r i s k attitude i s a factor, as shown i n the proposition below. Part (2) of Proposition 3.3.5 says that s u f f i c i e n c y alone cannot determine s t r i c t preference ing of information systems i f the p r i n c i p a l i s r i s k averse. order- 20 Proposition 3 . 3 . 5 (Amershi ( 1 9 8 2 , Proposition 3 . 5 ) ) . Let n be the minimal s u f f i c i e n t s t a t i s t i c and x be the output. (1) A r i s k neutral p r i n c i p a l s t r i c t l y prefers n over any system y which i s not a s u f f i c i e n t s t a t i s t i c i f and only i f every best agency contract (s*,a*) i s such that s* i s a s u f f i c i e n t (2) statistic. y A r i s k averse p r i n c i p a l s t r i c t l y prefers (x,n) over any system such that (x, y) i s not an agency s u f f i c i e n t s t a t i s t i c i f and only i f every best agency contract (s*,a*) Is such that (x,s*) i s an agency sufficient statistic. Again, although the e f f o r t variable i s single-dimensional, the result holds even i f e f f o r t i s multidimensional. Amershi (1982) next developed the following r e s u l t . 3 minimal s u f f i c i e n t s t a t i s t i c . Suppose n(w) Is a 3 If -sg- log f(co|a*) = log k( n( ui) | a*) i s an i n v e r t i b l e function of n(w), then a r i s k neutral p r i n c i p a l s t r i c t l y prefers n over any system y that i s not a s u f f i c i e n t s t a t i s t i c , and a r i s k averse p r i n c i p a l s t r i c t l y prefers (x,n) over any system (x, y ) which i s not an agency s u f f i c i e n t statistic. Unlike Amershi's (1982) previous results, which were easily extended to the multidimensional e f f o r t case, the i n v e r t i b i l i t y result above does not lend i t s e l f to the multidimensional e f f o r t case. I n t u i t i v e l y , the dimension of a s u f f i c i e n t s t a t i s t i c cannot be less than the dimension of the vector of parameters to be estimated. For example, suppose that x^,...,x ( n 2 ) are n observations from a normal d i s t r i b u t i o n with unknown mean 9 and unknown var2 iance o . - 2 2 Then a s u f f i c i e n t s t a t i s t i c for the vector of parameters (9,0" ) - 2 i s (x,s ), where x i s the sample mean and s i s the sample variance. More- over, i t i s obvious that more than one observation i s needed i n order to 2 make inferences about ( 9, a ). Thus, a s u f f i c i e n t s t a t i s t i c i n the m u l t i d i - mensional e f f o r t case w i l l generally be multidimensional. The impossibility 21 of inverting a one-dimensional value to obtain a multidimensional s t a t i s t i c precludes the use of Amershi's i n v e r t i b i l i t y result i n the a l l o c a t i o n of e f f o r t problem. For example, i n the a l l o c a t i o n of e f f o r t problem, x_= (x^,...,x ) i s n p o t e n t i a l l y observable, with the d i s t r i b u t i o n of x parameterized by a = n ( a ^ , . . . , a j ) . The s t a t i s t i c x = E x . can only be s u f f i c i e n t for (x,x) i f a 1-1 i s not r e a l l y multidimensional, i . e . , i f there i s some known functional relationship among the a^s so that knowledge of one a^ Is s u f f i c i e n t to perf e c t l y i n f e r the others. A special case of this type of relationship occurs when i t i s known that the agent w i l l always choose the a^s to be equal. the a l l o c a t i o n problem, i t i s very u n l i k e l y that a_ i s not r e a l l y In multidimen- sional, and therefore i n general, x i s not s u f f i c i e n t for (x,3c_), i . e . , the minimal s u f f i c i e n t s t a t i s t i c i s multidimensional. Continuing with the focus on the value of additional disaggregated information, the principal's weak preference for the additional information i s e a s i l y established. A multidimensional-effort version of Proposition 3.3.1 shows that the information system reporting x_= (x^,...,x ) i s weakly n preferred to the information system reporting only E x., no matter what i=l n the p r i n c i p a l ' s or the agent's r i s k attitudes are. If the p r i n c i p a l can observe _x, the i n t e r i o r portion of the optimal sharing rule i s characterized by . U'(s(x)) d = A + E u, ^ / i 8 / ls) x a 1 g(x|a) ' To i l l u s t r a t e , suppose again that n=2, (x-^.x^ and U(s) = l n s, but l e t jc_ = - N( a_, E), where E i s the covariance matrix . 2 / \ po^Og 2 22 Then the i n t e r i o r portion of the optimal sharing rule i s (see Appendix 2 for b i v a r i a t e normal calculations) 2 8 tel§) a s( ) = x + £ p - J r . 1 X x r = x + F T i l" l Hi z 1 5 P(x2-a ) a n x -a 2 2 2, ~—r-rri ) ^(^(l-p ) 0^(1-p "2 [ „ 2 1 or(l-p ) 0 O (l-p ) 2 ]L 1 a (l-p ) x 2 U jp" " 1 2,'—7—27^ PV^ Uj^ = X + (x -a )[-^ 1 + p(x>a) 2 pi^ 2 77— - + (x -a )[-^ 2 oj^Cl-p ) 2 CT 2 ]. ^ (1-P ^^(l-p ) This compensation scheme may be interpreted as a commission scheme with d i f ferent commission rates for each task. sion rates w i l l be the same for both tasks. e r a l , even i f x^ and x 2 = ov, and If = y , the commis2 It should be noted that i n gen- are independent, the optimal commission rates need not be equal across tasks. This i s because when x^ and x 2 are independent ( P=0), s(x) = X - a ^ / o ^ - \ + x i , J i / ^ + x^/a . c 2 In this case, the commission rate for task i depends only on the variance of x^ and the m u l t i p l i e r u^. Since the sharing rule depends on each Xj>, the 2 s i g n a l jc, obtained i n addition to x, i s valuable (unless M^/o^ 2 = l^/^)* This can be deduced formally from Proposition 3.3.2. 3.4 ADDITIVE SEPARABILITY OF THE SHARING RULE Once the p o s s i b i l i t y of observing each x^ i s introduced, the question of whether or not to reward the agent for each outcome separately a r i s e s . For example, should a manager of two divisions that are geographically d i s persed be rewarded for the performance of each separately? Analytically, the question i s whether the optimal sharing rule i s a d d i t i v e l y separable i n 23 the x^'s. This question w i l l be addressed for the HARA class of u t i l i t y functions. V(x) = \ + E ]i.g (x|a)/g(x|a). As before, i f the agent's Let l a utility ± function i s i n the HARA class, with -U"(s)/U*(s) = l/(Cs+D), then the i n t e r i o r portion of the optimal sharing rule i s given by i((V(x)) s(x) = C - D), if C *0 < (3.4.1) D ln(V(x)), i f C = 0, for almost every jc_ such that s(x) £ . If the p r i n c i p a l i s r i s k [SQ,S] averse, with u t i l i t y function i n the HARA class and with i d e n t i c a l cautiousness C (see (2.6)), then the i n t e r i o r portion of the optimal sharing rule i s ( V(x)) (Cx+D ) - D C 1 2 if C * 0 C(l + ( V ( x ) ) ) C s(x) = < (3.4.2) D D l n V(x) + D x 1 2 2 D where 1 + D i f C = 0, 2 corresponds to the p r i n c i p a l , and T>2 corresponds to the agent. Equation (3.4.1) implies that i f the p r i n c i p a l Is r i s k neutral and the agent's u t i l i t y function i s i n the HARA class, then a necessary condition for the optimal sharing rule to be additively separable i s that C=l, i . e . , that the agent have a log u t i l i t y function. strong form of independence Given that U(s) = l n s, a of the outcomes, x^,...,x , i s a s u f f i c i e n t conn d i t i o n for the optimal sharing rule to be additively separable. cifically, l e t g^(x^|a£) be the density of outcome x^ given effort a^, and l e t g(x|a_) be the joint density of jc_ given a_. g(x ,...,xja ,...,a ) = 1 More spe- 1 n n^g ( x | a ) i i Then (3.4.3) 24 Is a s u f f i c i e n t condition for additive separability of the optimal sharing n rule, given that U(s) = In s. i i s (x ) = u — 1 In this case, s(x) = \ + Is i=l i (x ), where 1 . The example i n Section 3.3 shows that given U(s) = VUja.) l n s, independence i s a s u f f i c i e n t but not a necessary condition for separa b i l i t y of the sharing r u l e . One might conjecture that there are other common d i s t r i b u t i o n s of dependent random variables which, when U(s) = l n s, y i e l d a separable sharing r u l e . However, no other common joint d i s t r i b u - tions which seem appropriate (see, e.g., Johnson and Kotz, 1972), seem to lead to such a r e s u l t . In general, then, the optimal sharing rule w i l l not be additively separable. It i s interesting to note that (3.4.3) i s not s u f f i c i e n t to y i e l d a separable sharing rule i f U(s) l n s. Furthermore, (3.4.3) i s not s u f f i - cient to y i e l d a separable sharing rule i f both the p r i n c i p a l and the agent are risk averse, with HARA-class u t i l i t y functions and i d e n t i c a l cautiousness C. This i s easily seen from equations (3.4.2). Hence, even i f the p r i n c i p a l and agent have i d e n t i c a l log u t i l i t y functions, a separable sharing rule i s not optimal. These results d i f f e r from those i n the cooperative setting, i n which a weighted sum of the p r i n c i p a l ' s and the agent's expected u t i l i t i e s i s maximized (no Nash constraint i s necessary). In the cooperative case, i f b e l i e f s are i d e n t i c a l and the p r i n c i p a l and agent are s t r i c t l y r i s k averse, then the optimal sharing rule i s linear for a l l weights i f and only i f the individuals have HARA-class u t i l i t i e s with i d e n t i c a l cautiousness (Amershi and Butterworth, 1981). accounts Thus, the moral hazard problem p a r t i a l l y for the generally nonlinear form of the optimal sharing r u l e s . One additively separable compensation scheme which i s commonly used i s the commission scheme. This scheme has the further r e s t r i c t i o n that 25 s ^ ( x ^ ) = c^x^ + b^ , a l i n e a r f u n c t i o n of x^. t i o n f o r a commission scheme ( l i n e a r s h a r i n g As above, a necessary r u l e ) to be optimal p r i n c i p a l be r i s k n e u t r a l and that the agent have a l o g u t i l i t y Given U(s) = I n s, whether or not the optimal sharing on the c o n d i t i o n a l d i s t r i b u t i o n of the outcomes g i v e n 3.5 condi- i s that the function. r u l e i s l i n e a r depends effort. ADDITIVE EFFORT T h i s s e c t i o n examines the s p e c i a l case where e f f o r t when e f f o r t represents intrinsic disutility time spent on d i f f e r e n t t a s k s , and where there f o r any p a r t i c u l a r t a s k . f u n c t i o n f o r e f f o r t expended on d tasks T h i s case n e c e s s i t a t e s i s a d d i t i v e , as In t h i s case, the d i s u t i l i t y can be w r i t t e n as V(a^+..,+a^). only minor changes i n the a n a l y s i s ; p a r t i a l t i v e s of V ( . ) with r e s p e c t i s no t o a± a r e r e p l a c e d by V ' ( . ) . deriva- The assumption t h a t the p r i n c i p a l i s r i s k n e u t r a l and the agent i s r i s k averse w i l l be maintained i n t h i s section. Suppose that there i s one outcome x^ a s s o c i a t e d with each a^, and that d the mean of each x^ i s k m ( a ) , so that E(x) i and m|( •) > 0. i In the f i r s t i best t h a t the agent r e c e i v e a constant 8E(x|a)/9 The m i( i) a cates = sk^jlaj) ai simplest a i» ^ order conditions require wage and that = XV'(.), f o r a l l i . case i s that of constant (3.5.1) marginal p r o d u c t i v i t y , where f u r t h e r , k^ = k f o r a l l i , then (3.5.1) i n d i - o r that (3.5.2) i hence any mix of e f f o r t s s a t i s f y i n g (3.5.2) i s e q u a l l y a c c e p t a b l e to both the p r i n c i p a l and the agent. e f f i c i e n c y of e f f o r t The k^'s may be thought of as measures o f ( S h a v e l l , 1979). boundary s o l u t i o n r e s u l t s . The Z k^m^a.^), where k^^ > 0 case, the f i r s t k/ X = V ( Z a ) , and = I f a l l the k^'s are unequal, then a In p a r t i c u l a r , a l l but one of the a^'s a r e z e r o . problem i s thus e s s e n t i a l l y one of choosing on which task of many to 26 expend e f f o r t . Suppose there are two tasks, with k^ > k . 2 In this s i t u a - tion, the optimal solution i s to devote e f f o r t exclusively to task one. These results are summarized as Lemma 3A.2 i n Appendix 3, where the proofs can also be found. Comparison of two one-dimensional e f f o r t situations with k^ > k shows 2 why the p r i n c i p a l Is better off with a^ > 0 and at. = 0 than with a^ = 0 and ASj > 0. Since k^ > k , there i s a higher return per unit of e f f o r t for task 2 one than from task two. Furthermore, i t i s worthwhile for the p r i n c i p a l to induce more e f f o r t f o r task one than for task two (see Proposition 3A.3 and i t s proof i n Appendix 3). The combined productivity gains ( r e c a l l that E(x^) = k^a^) outweigh the required increased fixed wage compensation to the agent, who would receive the same expected u t i l i t y for either task. The principal's s i t u a t i o n can be depicted graphically as follows: ^i J _+ s* k a l * " a i ^^-f^k a 2 ^ ^ & - T * k 2 2 • a s* 1 1 at, a£ For general m^a^), (3.5.1) implies that k^m^(a^) = k^ml(aj), i,j=l d. The marginal impacts of the a^'s on the expected outcomes are balanced, and hence the solution w i l l generally be i n t e r i o r . If the mean functions are i d e n t i c a l , then the optimal e f f o r t s w i l l be equal. Although the agent's u t i l i t y for wealth i s not important i n determining the principal's choice of the a^'s i n the f i r s t best case, i t i s important 27 in the second best case. Assuming an Interior solution, the f i r s t order conditions i n the second best case require that g U ( s Q Q ) _ 3EU(s(x)) I /o c o\ . . , » - i> J _ l > • • • »Q ^ J . J . J ; j Since the agent's effort i s not observable i n this case, the p r i n c i p a l must induce the agent to exert the optimal amount of e f f o r t at one or more tasks. The p r i n c i p a l may at find i t optimal to devote resources to preventing shirking only one task even i f multiple tasks are a v a i l a b l e . the p r i n c i p a l could, by imposing less r i s k , motivate It i s possible that the agent more e f f i - c i e n t l y i f the agent were induced to devote effort to only one task. Since the risk-averse agent must be compensated for bearing r i s k , the p r i n c i p a l may be better off imposing r i s k related to just one task. The propositions i n the remainder of this section describe situations in which a boundary solution or an i n t e r i o r solution w i l l be optimal, and characterize i n t e r i o r solutions. Before stating the propositions, a simple example w i l l be used to introduce the issues. Suppose there are two independent and i d e n t i c a l tasks, whose outcomes are represented by X^ and X 2 is Suppose further that the agent's action space {(2a*,0),(0,2a*),(a*,a*),(a*,0),(0,a*),(0,0) }, where an e f f o r t l e v e l of 0 represents the minimal e f f o r t the agent w i l l exert. b i l i t i e s of X^ given a are: P r o b a b i l i t i e s given that a=0 a=2a* a=a* $1 .10 11/12 1/2 1/12 - .10 1/12 1/2 11/12 x i E(X la) 1.00 0.50 0.00 VarCXi la) 0.11 0.36 0.11 1 Suppose that the proba- 28 The joint outcomes occur with the following p r o b a b i l i t i e s : P r o b a b i l i t i e s given that a^=2a* Reward (X ,X ) 1 a^=a* a =0 a =0 1 1 a =0 a =2a* a =a* 2 a =0 a =a* a =0 2 2 a^=a* 2 2 2 2 si (1.1,1.1) 11/144 11/144 1/4 1/24 1/24 1/144 s 2 (1.1,-.1) 121/144 1/144 1/4 11/24 1/24 11/144 s 3 (-.1,1.1) 1/144 121/144 1/4 1/24 11/24 11/144 s 4 (-.l.-.l) 11/144 11/144 1/4 11/24 11/24 121/144 E(X +X |a) 1.00 1.00 1.00 0.50 0.50 0.0 Var(X +X |a) 0.22 0.22 0.72 0.47 0.47 0.22 1 1 2 2 Let s_= (s^,s ,S3,S4). Suppose the p r i n c i p a l ' s problem i s : 2 Maximize E(250X! + 250X ) - E(s) 2 subject to EU(_s) - V(ai + a ) >^ u 2 (a^,a ) maximizes |EU(J3) - V(a^+a )} • 2 2 Let IKSJ^) = Ss^ and u = 10, and a* = 1. The optimal solution i s for the p r i n c i p a l to induce the agent to exert 2a* at one task, with the reward for the one task as follows: s = 148.84 and s 0 = 96.04, where s i s paid i f the outcome i s 1.1, and SQ i s paid otherwise. If the p r i n c i p a l desired to induce the agent to exert a* at both of the tasks, the following sharing rule would be optimal: B[ = 207.40, s 2 - s 3 = 144, s£ = 92.16. Looking at the variance as a measure of r i s k , we note that the outcome is r i s k i e r when a_ = (a<£,a*,) than when a_ = (2a*,0). However, this r i s k i s not d i r e c t l y of concern to either the p r i n c i p a l or the agent, because the p r i n c i p a l i s r i s k neutral and the agent i s not concerned about the riskiness of the outcomes per se, but rather about the effects on the compensation 29 received. In the example above, Var(_s) = 213.0 while Var(j_') = 1667.2, and E(j_) = 144.44, which i s less than E(s_') = 146.88. The p r i n c i p a l can thus motivate the agent more e f f i c i e n t l y with a boundary solution rather than with an i n t e r i o r solution. In this case, the principal's expected payments to the agent are lower for the sharing rule which imposes less r i s k (as measured by the variance) on the agent. Although the variances of the outcomes are not d i r e c t l y of concern to either the p r i n c i p a l or the agent, they are i n d i r e c t l y of concern. X 2 and are not only outcomes, but also signals about the agent's e f f o r t s ; as such, they provide information about the e f f o r t s . The r e l a t i v e magnitudes of the variances of the outcomes are potential surrogates for measures of informativeness, since the variances indicate how the signals about the e f f o r t s w i l l vary as the e f f o r t s vary. (information) In the example above, a t o t a l e f f o r t l e v e l of 2a* w i l l provide the same t o t a l expected outcome, regardless of whether a* i s devoted to each of two tasks, or 2a* i s devoted to a single task. However, the variance of the outcome i s smaller when 2a* i s devoted to a single task than when the e f f o r t i s allocated Since the expected to two tasks. outcomes are the same, the risk-neutral p r i n c i p a l desires to allocate e f f o r t i n the way that provides the most information about shirking. That i s , information issues become dominant i n the p r i n c i p a l ' s choice of the a l l o c a t i o n of e f f o r t . A s i t u a t i o n similar to the discrete outcome example above occurs when the X^'s are independent and i d e n t i c a l l y distributed with a normal d i s t r i b u tion with mean ka and variance . If e f f o r t a* i s devoted to each of two independent tasks, then E(X +X |a^=a*,a =a*) = 2ka* and 1 Var(X +X 1ai=a*,a =a*) = 2<?. 1 2 2 2 2 If e f f o r t 2a* i s devoted to just one task, say the f i r s t task, then the expected outcome i s 2ka*, which i s equal to E(X-^+X 1a^=a*,a =a*). 2 2 However, i f the agent i s compensated only on the 30 2 the corresponding variance i s cr , which i s s t r i c t l y less than basis of X^, Var(X^+X |a^=a*,a =a*). 2 In this situation, then, we might conjecture that a 2 boundary solution i s optimal. The two examples above had 2E(X |a*) = E ^ ^ a * ) . Clearly, 1 this can hold for a l l e f f o r t levels only when the means are linear i n e f f o r t . The examples also had Var(X +X |a =a*,a =a*) > V a r ( X | = 2 a * ) . 1 2 1 2 x (3.5.4) a] Thus, one might conjecture that a boundary solution i s optimal i n cases where (3.5.4) holds and there are independent and i d e n t i c a l l y distributed outcomes, with the means proportional to e f f o r t . It should be pointed out, however, that the a d d i t i v i t y of e f f o r t would also be c r i t i c a l for this result. If the X ^ s have Poisson d i s t r i b u t i o n with E(X |a =a) = ka = i i Var(X^|a^=a), then the variances change as the e f f o r t s change. If a* i s exerted at each of two independent tasks, then E(Xj+X |a^=a*,a =a*) = 2ka* = 2 Var(Xj+X |a^=a*,a =a*). 2 2 If 2a* i s exerted at one task, say task one, 2 E(X |a =2a*) = 2ka* = E(X +X |a =a*,a =a*) = Var(X |a =2a*). 1 1 X 2 x 2 x then Therefore, we L might expect that the p r i n c i p a l would be indifferent between a boundary solution and an i n t e r i o r one. F i n a l l y , consider the exponential d i s t r i b u t i o n , where E(X^|a^=a) = ka and Var(X^|a^=a) = k a . If a* i s exerted at each of two independent tasks, then E(X +X |a =a*,a =a*) = 2ka* and Var(X +X | =a*,a =a*) = 2 k a * . 2 1 2 1 2 1 2 a] If 2 2 2a* i s exerted at one task, then E(X |a =2a*) = 2ka* = E(X +X |a =a*,a =a*) 1 but Var(X |a =2a*) = 4 k a * 2 1 2 1 1 1 > Var(X +X |a =a*,a =a*). ] 2 x 2 2 1 2 Thus, i n this situa- tion, we might conjecture that an i n t e r i o r solution, rather than a boundary solution, would be optimal. The propositions below substantiate the i n t u i t i v e arguments above concerning when an i n t e r i o r solution or a boundary solution i s optimal, given 31 that the expected outcomes of independent and i d e n t i c a l tasks a r e p r o p o r t i o n a l to e f f o r t expended. effort, then the s i t u a t i o n s become more Initially, complicated. the normal d i s t r i b u t i o n with constant a f u n c t i o n of e f f o r t w i l l est, I f the expected outcomes are n o n l i n e a r i n be c o n s i d e r e d . variance but w i t h mean This case i s o f p a r t i c u l a r i n t e r - s i n c e i t i s the only d i s t r i b u t i o n i n Q (see (2.7)) whose v a r i a n c e i s independent o f the agent's e f f o r t . The f o l l o w i n g p r o p o s i t i o n s t a t e s condi- t i o n s under which a boundary s o l u t i o n i s o p t i m a l . Proposition x^ and x 2 3.5.1. Suppose the p r i n c i p a l i s r i s k n e u t r a l , U(s) = 2v^T, and a r e c o n d i t i o n a l l y independent and i d e n t i c a l l y d i s t r i b u t e d n o r m a l l y w i t h mean ka and constant variance. Suppose f u r t h e r that V(a) = V ( Z a ^ ) . Then a boundary s o l u t i o n i s o p t i m a l . ^ The p r o p o s i t i o n below c h a r a c t e r i z e s o p t i m a l Proposition averse, 3.5.2. and g O ^ a ) = f ( x j j a ^ ) f ( x | a ) , i . e . , x^ and x V( Ea^). 2 I f a unique i n t e r i o r 2 and u 2 2 are c o n d i t i o n a l l y Suppose f u r t h e r that V(a) = s o l u t i o n i s optimal, a^ = a*, and u* = u^, where described solutions. Suppose the p r i n c i p a l i s r i s k n e u t r a l , the agent i s r i s k independent and i d e n t i c a l l y d i s t r i b u t e d . has unique i n t e r i o r then the o p t i m a l solution a r e the L a g r a n g i a n m u l t i p l i e r s earlier. T h i s r e s u l t i s independent of the u t i l i t y agent or the d i s t r i b u t i o n of x^ g i v e n f u n c t i o n of the r i s k - a v e r s e a^; the c r i t i c a l element i s that the outcomes a r e c o n d i t i o n a l l y independent and i d e n t i c a l l y d i s t r i b u t e d . This r e s u l t does n o t say that a l l agency problems such t h a t the p r i n c i p a l i s r i s k neutral, the agent i s r i s k averse, and the outcomes a r e c o n d i t i o n a l l y i n d e - pendent and i d e n t i c a l l y d i s t r i b u t e d have s o l u t i o n s o f a^ = a | and u£ = u|; t h i s i s evident the o p t i m a l from P r o p o s i t i o n 3.5.1. P r o p o s i t i o n 3.5.2 i n d i c a t e s that i f s o l u t i o n has the agent a l l o c a t i n g nonzero e f f o r t then the e f f o r t s should be equal at each task to each i f the t a s k s present task, indepen- 32 dent and i d e n t i c a l expected returns to the p r i n c i p a l . The following propo- s i t i o n , which applies to the one-parameter exponential family (see (2.7)), describes conditions under which an i n t e r i o r solution i s optimal. These conditions are s u f f i c i e n t but not necessary. Proposition 3.5.3. 8 (51 §L) = Suppose the p r i n c i p a l i s r i s k neutral, U(s) = f ( i ! ^ ) f ( 2 l 2^» x a x a w n e r where M(0) _> 0 and M'(a) > 0. e and f('|a) belongs to Q and has mean M(a), Suppose further that V(a_) = V ^ E a ^ . Let a* be the optimal e f f o r t i n the one-task problem, (i) If M(a) i s concave and z (a*)M'(a*)/[z (a*/2)M'(a*/2)] , < 1/2, (3.5.5) < 1/2, (3.5.6) , then a boundary solution i s not optimal. (ii) If M(a) i s s t r i c t l y concave and z'(a*)M'(a*)/[z (a*/2)M (a*/2)] , , then a boundary solution i s not optimal. In both cases, i f a unique inter- i o r solution i s optimal, then a^ = a*, and y£ = u^. As shown i n below i n Corollary 3.5.4, z'(a)/z'(a/2) i s often independent of a, and hence one need not actually solve for the optimal one-task effort. Corollary 3.5.4. Under the conditions i n Proposition 3.5.3 i f M(a) = ka and z'(a*)/z (a*/2) < 1/2, then an i n t e r i o r solution i s optimal. f In p a r t i c u l a r , ( i ) For the exponential d i s t r i b u t i o n with parameter l/(ka), an i n t e r i o r solution i s optimal (z'(a)/z'(a/2) = 1/4). ( i i ) For the gamma d i s t r i b u t i o n with parameters n/(ka) and n, an i n t e r i o r solution i s optimal (z (a)/z'(a/2) = 1/4). f The following cases do not s a t i s f y (3.5.5) but are included for purposes of comparison: ( i i i ) The Poisson d i s t r i b u t i o n with mean ka has z'(a)/z'(a/2) = 1/2. 33 (iv) The normal d i s t r i b u t i o n with mean ka and constant variance has z'(a)/z'(a/2) = 1. The normal d i s t r i b u t i o n should not, of course, s a t i s f y (3.5.5) i n view of Proposition 3.5.1. In each of the cases in Corollary 3.5.4 l i n e a r l y with the agent's e f f o r t s . the expected outcomes increase In case ( i i i ) , the variances of the out- comes also increase l i n e a r l y with the agent's e f f o r t s . variances of the outcomes are unaffected by the e f f o r t s . In case ( i v ) , the In cases ( i ) and ( i i ) , the variances of the outcomes increase quadratically with the e f f o r t s . A boundary solution i s optimal i n case ( i v ) , where the rate of increase i n the variance i s s t r i c t l y less than the rate of increase i n the mean. i n t e r i o r solution i s optimal i n cases ( i ) and An ( i i ) , where the rates of increase in the variances are s t r i c t l y greater than the rates of increases in the means. The following two propositions characterize optimal i n t e r i o r solutions when the means of the outcomes are l i n e a r i n e f f o r t . Proposition 3.5.5. g( l§.) x = Suppose the p r i n c i p a l i s r i s k neutral, U(s) = 2/s, f( lI i)f( 2I 2^» x a x a w n e r belongs to Q and has mean M(a). 2 e Suppose further that V(a_) = V(Ta ). ± If M(a) = ka and z " ( a ) / z ' (a) i s s t r i c t l y monotonic, then an optimal i n t e r i o r solution i s unique and a l = a 2 a n d Hi = ^* ^ e s t r * c t and has monotonicity i s s a t i s f i e d by the exponen- t i a l and gamma d i s t r i b u t i o n s (given that M(a) = ka), but not by the normal or Poisson d i s t r i b u t i o n s . Proposition 3.5.6. Suppose the p r i n c i p a l i s r i s k neutral, U(s) = 2/s g(x|a) = f(x^|a^)f(x2|a2), where f(.|a) has mean M(a). 2 V(a_) = V ^ E a ^ . If M(a) , and Suppose further that = ka and I'(a)/I (a) i s s t r i c t l y monotonic, where 1(a) = /fg/f' dx, then an optimal i n t e r i o r solution i s unique and has a^ = a^ and y* = p*. 34 1(a) i s called Fisher's information about a contained i n x, and i s a useful concept i n mathematical s t a t i s t i c s (see, e.g., Cox and Hinkley, 1974). The next corollary demonstrates that i n part, the shape of the expected outcome function determines whether the optimal solution w i l l be i n t e r i o r . Corollary 3.5.7. Under the conditions i n Proposition 3.5.3 i f M(a) = a", then an i n t e r i o r solution i s optimal i f f(.|a) i s ( i ) Normal (M(a),cr ) and 0 < ct<_ 1/2 or 2 ( i i ) Exponential (1/M(a)) and 0 < a < 1 or ( i i i ) Poisson (M(a)) and 0 < o< 1. It i s well known that knowing f / f i s equivalent to knowing the l i k e l i a hood of a given the observations. For the exponential family Q, f / f i s given by z'(a)(x-M(a)), where M(a) i s the mean of x conditional on a. a It i s z'(a) and M(a) which play an important role i n determining whether a boundary solution or an i n t e r i o r solution i s optimal. for This might be expected, the x^'s are not only outcomes, but also signals about the efforts that have been expended. x, z'(a) and M(a) Since f / f i s s u f f i c i e n t for the likelihood of a given a together measure, to a certain degree, the informativeness of x about a. It i s interesting to compare the results for the second best case with those for the f i r s t best case. In the f i r s t best case, (3.5.1) indicates that i f M(a) = ka, then whatever the d i s t r i b u t i o n of x given a, the p r i n c i pal w i l l be i n d i f f e r e n t between an i n t e r i o r solution or a boundary one, as long as the t o t a l amount of e f f o r t expended i s the same i n both cases. the second best case, however, Proposition 3.5.1 In says that i f the d i s t r i b u - tion i s normal with mean ka and constant variance, then a boundary solution is optimal. On the other hand, i f the d i s t r i b u t i o n i s exponential or gamma with mean ka, then an i n t e r i o r solution i s optimal (Corollary 3.5.4). 35 If, i n the f i r s t best case, the means are concave i n e f f o r t , then (3.5.1) indicates that an i n t e r i o r solution i s optimal, and the optimal efforts are equal i f the mean functions are i d e n t i c a l (M'(a^) = M'(a^) implies that a* = alSf). Corollary 3.5.7 indicates that for a s p e c i f i c second best case with concave means, a similar result concerning the optimality of an i n t e r i o r solution holds. The ally results up to this point have assumed that independent and i d e n t i c a l l y d i s t r i b u t e d . and x The next two 2 are condition- propositions deal with the case of conditionally independent but nonidentically d i s t r i buted x^'s. Proposition 3.5.8. Suppose the p r i n c i p a l i s r i s k neutral, U(s) = 2/s, and g(x|a) = f ( x | a ) h ( x | a ) , where f ( . | a ) and h(.|a ) belong to Q and 1 1 2 E(x |a=a ) = k a±. i i 2 1 Suppose further that V(a) ± 2 = V( Ea.^) . ( i ) If x^ has an exponential d i s t r i b u t i o n with mean k^a^, implies that a^ > a^ and u£ > then k^ > k > k 2 2 implies 2 u£ > uJ. ( i i i ) If x^ has a normal d i s t r i b u t i o n with mean k^a^ and constant then k u^. ( i i ) If x^ has a gamma d i s t r i b u t i o n with mean k^a^, that a^ > ai£ and then k^ > variance, implies that the optimal solution i s a boundary solution, with a* > 0 and a* = 0. (iv) If x^ has a Poisson d i s t r i b u t i o n with mean k^a^, then k^ > k 2 implies that the optimal solution i s a boundary solution with a^ > 0 and a^ =0. The following proposition states that at least for a s p e c i f i c second best case, the optimal Lagrangian multipliers are p o s i t i v e . Signing the multipliers i s of importance because of their c r i t i c a l role i n the determination of the optimal sharing rule. For example, i f the density of x^ given a^ s a t i s f i e s the monotone l i k e l i h o o d ratio property i n x^ for a l l i , then the p o s i t i v i t y of the u!s guarantees that the optimal sharing rule i s 36 increasing i n each x^. It can be shown that under c e r t a i n conditions i n the second best case, not a l l the yjs can be zero, and hence in the situations above where the optimal m u l t i p l i e r s y* are equal, they must be p o s i t i v e . Proposition 3.5.9. g(l§.) x = Suppose the p r i n c i p a l i s r i s k neutra 1, U(s) = l/s, f ( i | i ) h ( x 1 a ) , where f(.|a^) and h(.|a ) belong to Q. x a 2 further that V(a) = V(Ea^). and 2 2 and Suppose If an i n t e r i o r solution i s optimal, then y£ > 0 y* > 0. Note that i n Proposition 3.5.9, x^ and x 2 need not be i d e n t i c a l l y d i s - tributed, although they are conditionally independent. Furthermore, the result holds for general V(a), as long as 9V/3a^ > 0, i=l,2. 3.6 APPLICATION TO SALES FORCE MANAGEMENT In this section, the previous analysis of multidimensional tions i s applied to the problem of sales force management. effort situa- Steinbrink (1978) depicts the c r i t i c a l role of compensation of a sales force as follows: Any discussion with sales executives would bring forth a consensus that compensation i s the most important element i n a program for the management and motivation of a f i e l d sales force. It can also be the most complex. Consider the job of salespeople i n the f i e l d . They face direct and aggressive competition d a i l y . Rejection by customers and prospects is a constant negative force. Success i n s e l l i n g demands a high degree of s e l f - d i s c i p l i n e , persistence, and enthusiasm. As a result, salespeople need extraordinary encouragement, incentive and motivation i n order to function e f f e c t i v e l y . . . .A properly designed and implemented compensation plan must be geared to the needs of the company and to the products or services the company s e l l s . At the same time, i t must attract good salesmen i n the f i r s t place . . . Management of the sales force has been the focus of a great deal of research, much of i t empirical. Steinbrink (1978), i n a survey of 380 com- panies across 34 industries, found that most companies favored a combination of salary, commission, and bonus schemes. Typical commissions used were 1) Fixed commissions on a l l sales 2) Different rates by product category 37 3) On sales above a determined goal 4) On product gross margin. These commission schemes are a l l examples of linear sharing rules. Farley (1964), Berger (1975), and Weinberg (1975, 1978) problem of " j o i n t l y optimal" compensation schemes. studied the They assumed a given compensation system (a commission scheme based on gross margin) and sought to determine i f that system i s incentive compatible, meaning that the salesperson w i l l be induced to choose levels of e f f o r t which the company desires. In these analyses, the measure of e f f o r t i s taken to be time spent s e l l i n g . The t o t a l time available i s assumed to be fixed and the decisions are how to allocate the t o t a l time across several products. Farley (1964) demonstrated that i f a commission system based on gross margin i s used, the commission rates should be the same for a l l products i n the case where both the firm and the salespeople are income maximizers. Weinberg (1975) extended Farley's result to include the choice of discounts on each product as well as the choice of time spent s e l l i n g each product. Both papers assume that the time spent s e l l i n g one product does not affect the sales of any other product. Furthermore, sales are considered to be a deterministic function of time, although the conclusions are unaffected by uncertainty because of the assumed r i s k n e u t r a l i t y of both the firm and the salespeople. Weinberg (1978) maintained the assumption of r i s k neutrality of the salespeople, and further extended his and Farley's analyses by allowing for interdependence of product sales and relaxing the assumption that salespeople maximize income. Even i n these situations, an equal gross mar- gin commission system i s incentive compatible i f the firm's objective i s to maximize expected gross p r o f i t s . Berger (1975) examined the combined effects of uncertainty and non-neut r a l risk attitudes on the part of the salespeople. He retained Weinberg's 38 and Farley's assumption of constant marginal cost per product, but treated sales of each product as a random variable parameterized by the time spent s e l l i n g that product. Berger demonstrated that i f a commission scheme i s used i n this situation, i t may be undesirable for the firm to set equal com- mission rates for a l l products. The agency model allows for many of the important factors i n the sales force managment problem and provides a more complete analysis of the problem by determining an incentive scheme which motivates the salesperson to make decisions that are Pareto optimal, rather than taking the compensation scheme as given. framework.^ The problem w i l l therefore be examined below i n an agency Interdependence of products, for the salesperson provision of enough net benefits to join and stay with the firm, and also the salesper- son's tradeoff between money and time spent s e l l i n g are incorporated. connection In with t h i s , the t o t a l time spent s e l l i n g i n a given time period w i l l be a choice v a r i a b l e . In order to focus on the motivational rather than r i s k sharing aspects of the problem, i t w i l l be assumed that the salesperson i s risk averse, and the firm i s risk neutral and therefore desires to maximize expected p r o f i t . The agency theory analysis isolates conditions under which some sort of commission scheme i s Pareto optimal, and shows that even when a commission scheme i s Pareto optimal, the commissions are genera l l y unequal. In this analysis, e f f o r t w i l l be interpreted as "time spent s e l l i n g , " and n w i l l represent the number of products available to be sold. It w i l l be assumed that the salesperson has no i n t r i n s i c d i s u t i l i t y for s e l l i n g particular product, so that the d i s u t i l i t y function may V(Ea^), with V increasing and convex. The any be taken to be remaining notation w i l l be as defined previously, with x^ denoting the difference between sales revenue and variable noncompensation costs for product i.^ 39 Suppose that the p r i n c i p a l (firm) and the agent (salesperson) have identical beliefs. hired. This might be the case when the salesperson i s f i r s t Suppose also that the firm and the salesperson are i n a f i r s t best s i t u a t i o n , either because they are acting cooperatively or because the firm can perfectly observe the times spent s e l l i n g each product. Recall that i n the f i r s t best situation, i t makes no difference whether the p r i n c i p a l observes only the t o t a l outcome, x, or the vector of outcomes, x_. Since the firm i s r i s k neutral and the salesperson i s r i s k averse, the optimal sharing rules i s a constant salary c = U * ~ ^ ( l / X) for the salesperson, with the firm receiving the remainder, x-c. The firm requires that the salesperson choose sales e f f o r t so that 3E(x|a)/3a = 9E(x|a)/9a^, i i,j=l,...,n. (3.6.1) The interpretation i n the cooperative setting i s that the salesperson happ i l y supplies e f f o r t levels a^ s a t i s f y i n g (3.6.1) i n return for the salary c, since i n doing so, he or she receives the market u t i l i t y , u. In the per- fect observability setting, the firm pays the salesperson a salary c i f e f f o r t levels a^ s a t i s f y i n g (3.6.1) are exerted, and pays nothing otherwise. Observe that i n the f i r s t best case, the salesperson chooses e f f o r t levels according to their effect on mean outcome. Suppose E(x^) = M^c^a^, where represents the contribution margin (sales revenues minus variable noncompensation costs) per unit of product i , and c^a^ represents the expected quantity of product i that w i l l be sold i f e f f o r t a^ i s exerted. ^2 2» i C , e , » If t n e The analysis i n Section 3.5 then applies. If M^c^ = contributions per unit of time spent s e l l i n g are equal, the t o t a l e f f o r t expended i s the only concern. If M^c^ > M2C2, then the Pareto optimal strategy i s for the agent to devote e f f o r t only to the f i r s t product. Under a more general return structure, the efforts a^ and a2 w i l l 40 be nonzero and unequal. If the mean functions are i d e n t i c a l and nonlinear and monotone i n a^, then the optimal e f f o r t s are such that a^ = a « 2 In practice, a straight salary i s seldom used for salespeople because of imperfect observability and imperfect cooperation (moral hazard). such situations, a second best analysis i s appropriate. In Consider f i r s t the one-product case, i n which the i n t e r i o r portion of the sharing rule i s characterized by I g (*|a) _ U'(s(x)) " a X + g(x|a) » P where x and a are univariate and the subscript a denotes d i f f e r e n t i a t i o n with respect to a. Examples of s p e c i f i c sharing rules are provided i n Table I for two members of the HARA class of u t i l i t y functions and two members of the one-parameter exponential family Q. Sharing Rule Given g(x|a) = U(s) l s n (M(a)) exp[-x/M(a)], _1 s l/b X + > b > 1 M'(a) > 0 (2Tra) exp[-(x-M(a)) /2a ], M'(a)>0 ^ i^l(x-M(a*)) M (a*) 2 (^/(b-l) _1 X _ 2 V*«a*)M'(a*) <T + 2 £ M X </ ^ / ( b - l ) Table I. Examples i n One-Product Case Observe that when U(s) = l n s, the sharing rules shown (and others corresponding to d i f f e r e n t members of Q, the one-parameter exponential family) can be interpreted as a salary plus commission on the outcome x, a scheme commonly found i n practice. If the agent's u t i l i t y function i s a concave power function, then the resulting sharing rule i s a convex power function of a l i n e a r form. The compensation schemes which pay a salary plus bonus commis- sions (e.g., s(x) = m + m^x i f x < XQ, S ( X ) = m + m^x + m (x-xo) i f x > X Q ) 2 can be considered as approximations to these sharing r u l e s . 41 The case where n > 1 i s more complicated i f the agent's u t i l i t y t i o n i s a power function, since cross terms i n the x^'s appear. examine conditions under which i t i s optimal to use a func- In order to salary-plus-commission scheme, the agent's u t i l i t y function w i l l be taken to be U(s) = l n s, since this i s the only s i t u a t i o n i n which a linear scheme can be optimal (see Section 3.4). The examples below employ the normal d i s t r i b u t i o n because of i t s convenient representation for dependent random v a r i a b l e s . For purposes of i l l u s t r a t i o n , i t suffices to take n = 2. Suppose then that U(s) = l n s, n = 2, and that x ~ N(9(a),E(a)), where 9(a) = ( 9^(a), 9 (a)) and £(a) i s the covariance matrix I^2( a ) p(a) ^ ( a ) a (a) 2 \ 2 2 o^a) p(a) o^a) a (a) 2 \ / At this l e v e l of generality, the optimal sharing rule i s quite complicated (see Appendix 2). It i s not separable i n x^ and x , salary plus commission scheme. (independent 2 and therefore i s not a A c o r r e l a t i o n c o e f f i c i e n t which i s constant of a) i s not s u f f i c i e n t for the sharing rule to be a salary plus commission scheme, although p = 0 does lead to a sharing rule which i s additively separable i n x^ and x « 2 S u f f i c i e n t conditions for a salary plus commission scheme to be optimal are that both the c o r r e l a t i o n c o e f f i c i e n t 2 and the variances be constant, with P * 1. by The commissions are determined 2 p, marginal increases i n the means 9^(a) at a*, the variances and the multipliers u^. Three especially interesting results of the example above are: (1) In the case of the normal d i s t r i b u t i o n with log u t i l i t y , independence of the products i s enough to guarantee additive separability ( i n x^ and x) 2 of the optimal sharing rule, but i s not enough to guarantee that the optimal 42 sharing rule w i l l be a linear sharing r u l e . That i s , the optimal sharing rule i s not a (salary plus) commission scheme, l e t alone an equal commission rate scheme. (2) It i s not necessary for the products to be independent i n order for a salary plus commission scheme to be optimal, or for a separable sharing rule to be optimal. (3) The optimal commission rates are generally not equal across products. The agency analysis applied to the sales force management problem i n d i cates that only under very special circumstances Pareto optimal. i s a commission scheme In practice, of course, commission schemes are favored because of their simplicity and ease of application, as well as their recognized incentive e f f e c t s . salespeople who If commission rates are used with r i s k averse face uncertainty i n sales, the rates should most l i k e l y not be equal across products, according to the analysis above. The results i n Section 3.5 on the a l l o c a t i o n of additive e f f o r t with independent outcomes provide some further insights about optimal compensation schemes for salespeople. It should be recalled that most of the results i n Section 3.5 were proved only for U(s) = 2v^\ Thus, the remarks that follow are r e s t r i c t e d by the assumption of that p a r t i c u l a r utility function for wealth for the agent. A p r i n c i p l e commonly taught i n managerial accounting texts i s that under certainty, i n order to maximize p r o f i t s given one scarce factor of production, a firm should manufacture the product which returns the highest contribution margin per unit of the scarce factor (see, e.g., Horngren (1982, p. 373)). setting. This principle does not necessarily hold i n the agency In the f i r s t best case, i f the means are linear i n e f f o r t , then the p r i n c i p l e holds. In addition, Proposition 3.5.8 indicates that in the second best case, i f expected returns are linear In e f f o r t , then a l l the 43 agent's e f f o r t should be put into s e l l i n g the product with the highest expected return per unit of e f f o r t i f the underlying d i s t r i b u t i o n i s normal with constant variance, or Poisson. However, i f the underlying d i s t r i b u t i o n i s exponential, then more e f f o r t should be put into s e l l i n g the product with the higher expected return per unit of e f f o r t , but both e f f o r t s w i l l be positive unless the expected returns per unit of e f f o r t are very d i f f e r e n t . (See the discussion after Proposition 3.5.8.) For the exponential d i s t r i b u t i o n with E(x^|a^) = k^a^, the optimal sharing rule i s given by s ( x ,x ) = [ X + t j ( x - k a*) + 2 k If k^ > k2» then L l l a > ^ ( 1 k " 2 2*^ ^ " k x 2 a 2 2 a and a| > a^. Equation [3] i n the proof of Proposi- tion 3.5.8 shows that y*/a* 2 2 2 2 = u|/a^ . Therefore, M*/(k a* ) < p*/(k a* ). x This implies that when k^ i s greater than k 2 2 (the expected return per unit of e f f o r t i s greater for product one than for product two), the agent's compensation per unit of x^ (the return on product one) i s less than the compensation per unit of x . 2 Continuing with the exponential d i s t r i b u t i o n case, i f kj^ = k , then 2 a^ = a^ and ii£ = u| (Proposition 3.5.5). The agent's compensation per unit of x^ i s equal to the compensation per unit of x , and the sharing rule can 2 be written as 2y* s(x ,x ) = [ X + 1 j ( i x 2 k l * a + x 2 ) " 2 ]• 1 Thus, the information (x^,x ) has no value i n addition to x^ + x . 2 lar result holds for more general situations, also. 2 A simi- Proposition 3.5.2 says that i f the p r i n c i p a l i s r i s k neutral, the agent i s r i s k averse, V(a) = V(Ea^), the x^'s given a^ are independent and i d e n t i c a l l y distributed, and a unique i n t e r i o r solution (a*^ > 0, a*, > 0) i s optimal, then a^ = aij and u* = 44 U*,. Under these conditions, the agent's compensation per unit of x^ i s equal to the compensation per unit of x . 2 It i s important to note that i f 2 x^ given a^ has a normal d i s t r i b u t i o n with mean ka^ and variance o" , and the x^'s given a^ are independent, a boundary solution (e.g., a^ > 0, a^ = 0) i s optimal. In this case, the agent would receive no compensation based on x « 2 Up to this point, the focus has been on a single agent exerting multiple e f f o r t s . A related topic i s that of multiple agents, which i s pertinent here because a firm w i l l generally have more than one salesperson. Feltham (1977b) examined the use of penalty contracts when a l l the agents are ident i c a l , and Holmstrom (1982) showed that the effectiveness of group penalties w i l l be hampered by limited endowments of the agents, especially as the number of agents becomes large. An important question in the multiple agent problem i s whether or not each agent should be rewarded independently of the others' performances. Holmstrom (1982) showed that i f the agents' outcomes are correlated with each other through the common uncertainty they face, then basing agent i ' s share on each agent's outcome helps reduce the uncontrollable randomness i n agent i ' s reward. Holmstrom (1982, p. 335) stated that . . .forcing agents to compete with each other i s valueless i f there i s no common underlying uncertainty. In this setting, the benefits from competition i t s e l f are n i l . What i s of value i s the information that may be gained from peer performance. Competition among agents i s a consequence of attempts to exploit this information. Only aggregate information about peer performance i s used i n the optimal sharing rules i f the aggregate measure captures a l l the relevant information about the common uncertainty. Of course, i f the agents' outcomes are inde- pendent of one another, then the optimal sharing rule for agent i depends only on agent i ' s outcome. One of the t r a d i t i o n a l principles i n performance evaluation within the firm Is the p r i n c i p l e that a person should be held responsible only for 45 those factors (e.g., costs or revenues) over which he or she has control. Basing the sharing rule for agent i only on agent i ' s outcome i s c l e a r l y consistent with the c o n t r o l l a b i l i t y p r i n c i p l e . Basing the sharing rule f o r agent i on the outcomes of other agents when there i s common uncertainty i s , at f i r s t glance, inconsistent with the c o n t r o l l a b i l i t y p r i n c i p l e . However, the reason that the compensation for each agent may depend on the outcomes of other agents i s that the p r i n c i p a l can gain information about the random state, and hence gain information about the e f f o r t s expended by each agent. That i s , the p r i n c i p a l can gain information about each agent's input ( e f f o r t ) , over which the agent has direct control. Thus, there i s no con- f l i c t with the c o n t r o l l a b i l i t y p r i n c i p l e i n this case. The apparent con- f l i c t occurs because the focus of the c o n t r o l l a b i l i t y p r i n c i p l e has been transferred from outputs to inputs ( c f . Baiman (1982, pp. 197-198)). The last modification to the standard agency analysis for the problem of sales force management relates to noneffort decisions. Frequently, the salesperson must not only make several e f f o r t decisions, but also make several " r i s k " decisions which do not require expenditures of e f f o r t . The choices of discounts to offer on each product are examples of such r i s k decisions. Weinberg (1975, p. 938) i d e n t i f i e s the following situations i n which an agent might have control over the price: (1) perishable a g r i c u l t u r a l products; . . . (2) sales involving trade-ins i n which the salesman has control over the evaluation of the trade-in, e.g., automobiles; (3) systems s e l l i n g i n which the salesman has a wide range of latitude i n specifying the combination of services to be provided, e.g., contractors and consultants; (4) some r e t a i l situations i n which the l o c a l store manager has control over price of at least some of the items sold i n his store; (5) l i q u i d a t i o n sales of obsolete product lines or r e t a i l e r d i s tress sales; and (6) highly competitive markets i n which customers are price bargainers . . . . One approach to the problem of incorporating both r i s k and e f f o r t decisions was taken by Itami (1979), who examined optimal linear goal-based incentive schemes under uncertainty. In his model, a r i s k decision i s made 46 by the agent before the state of nature i s observed. The agent then chooses an e f f o r t l e v e l based on the r i s k decision and the observed state of nature, resulting i n a deterministic output which i s a function of the agent's two decisions and the state of nature. For example, the d i v i s i o n a l manager of a large corporation might make investment decisions on projects before the environmental conditions are revealed. The e f f o r t expended and the known state then determine the output. The simplest agency theory approach to the problem of incorporating both r i s k and e f f o r t decisions i s to assume that both of the agent's decisions are made before the state of nature (or any other information) i s observed. This approach w i l l now be b r i e f l y discussed. optimal sharing rule i s derived rather than assumed. The form of the Furthermore, because risk-sharing aspects are important i n this setting, both the p r i n c i p a l and the agent are assumed to be r i s k averse. As Itami points out, there i s a direct and an indirect effect of the agent's e f f o r t on his or her u t i l i t y , while there i s only an indirect effect from the r i s k decisions. Up to this point, i t has been assumed that the agent's u t i l i t y i s separable i n e f f o r t and wealth. This assumption leads to a characterization of the optimal sharing rule that i s independent of the agent's d i s u t i l i t y for e f f o r t , although the indirect effects of e f f o r t expended are captured v i a the terms g /g. The more general u t i l i t y func- Si • J tion U(s(x),a) for the agent leads to a characterization of the optimal sharing rule that captures both the direct and Indirect effects of the agent's e f f o r t . for When there are no r i s k decisions, optimality requires that i n t e r i o r solutions, TW'(x-s(x)) TI/ / w U (s(x),a) j j = s X + y [ Ua..s (s(x),a) gB (xia) - '^ \ _ i _ / U (s(x),a) g(x|a) * v a + s 1 47 where the subscripts on U and g denote d i f f e r e n t i a t i o n with respect to a j , and the subscripts s denote d i f f e r e n t i a t i o n with respect to s. The major implication of nonseparability of the u t i l i t y function i s that the role of e f f o r t i s e x p l i c i t , as i s interaction between e f f o r t and compensation. It i s s t i l l true that i f the agent i s r i s k neutral, then the f i r s t best solution i s achievable by a sharing rule of the form x-k. Because the important distinguishing feature of e f f o r t decisions i s their twofold e f f e c t on the u t i l i t y function, the general form of the u t i l i t y function i s used here. Letting r^ denote the r i s k decision for task i and r = ( r ^ , . . . , r ), the p r i n c i p a l ' s problem is''' Maximize s(x),a,r / W(x-s(x))g(x|a,r) dx subject to / U(s(x),a)g(x|a,r) dx > u -Tr— / U(s(x) ,a)g(x|a,r)dx j / U(s(x) ,a)g(x|a,r)dx :A = 0, j=l,...,n :u. = 0, j=l,...,m. : 3. To the right of the constraints above are their associated m u l t i p l i e r s . The i n t e r i o r portion of the optimal sharing rule i s characterized by W'(x-s(x)) U a s ( 5 8 a ( 5 8 r. ( 5 It should be noted that there i s an Implicit assumpti on that the r.» s do not 1 s a t i s f y f i r s t - o r d e r stochastic dominance, since otherwise the p r i n c i p a l and the agent would agree on the choices of the r^'s and there would be no incentive problem with respect to the r ^ ' s . Suppose next, as Weinberg (1975) did, that the gross margin generated by sales of product I i s given by x i = P i ( 1 - r i ) Q i " i i' K Q w h e r e ^ = nominal s e l l i n g price per unit of product i , r^ = discount (decimal) on product i , 48 = quantity (units) of product i sold, = variable nonselling cost per unit of product i , and M£ = P j ^ l - r ^ ) - = gross margin on product i . Weinberg (1975) sought to determine i f an equal-commission scheme i s incen- tive compatible when there both r i s k and e f f o r t decisions. An agency theory analysis suggests that such a scheme i s not Pareto optimal. Q ± ~N( 9 ( a , r ) , a ( r ) ) . i i (P^l-r^-K pal 1 i i Then x ± ~ N(P (l-r )-K ) 8 ( a 1 i i i 1 Suppose ,r ), ± ) cr^(r ) ) . Previous analysis indicates that i f both the p r i n c i 2 and the agent are r i s k averse with u t i l i t y functions i n the HARA class, then the optimal sharing rule i s i n general not a d d i t i v e l y separable i n and x « 2 V(a), If the p r i n c i p a l i s r i s k neutral and the agent's u t i l i t y i s l n s - then previous remarks concerning the optimality of a commission scheme i n the normal d i s t r i b u t i o n example with no r i s k decisions apply. Demski and Sappington (1983) examined the s i t u a t i o n i n which i t i s desired to motivate an individual to obtain and use information which i s personally costly ( i n a pecuniary or nonpecuniary to obtain. Their analysis may sense) for the i n d i v i d u a l provide insights for the sales force manage- ment problem when the salesperson has the option or the a b i l i t y to observe private information before making r i s k decisions. 3.7 SUMMARY AND DISCUSSION This chapter derived optimal incentive schemes when the agent has sev- eral tasks over which to exert e f f o r t , and the p r i n c i p a l and the agent have homogeneous b e l i e f s about the outcome d i s t r i b u t i o n . In the f i r s t best case, where there i s no moral hazard problem, the major issue i s r i s k sharing, and the results are similar i n nature to the one-dimensional e f f o r t case. If one individual i s r i s k neutral and the other i s r i s k averse, then the r i s k neutral individual bears a l l the r i s k , receiving the uncertain outcome less a constant fee. If both individuals are r i s k averse, then the r i s k sharing 49 aspect i s prominent; even i f the disaggregated information, x = ( x ^ , . . . , x ) , n i s observed, the sharing rule depends only on the sum of the x^'s. In the analysis of the second best case, where there i s a moral hazard problem, the p r i n c i p a l was assumed to be r i s k neutral i n order to focus on motivational issues. As i n the single-task case, the f i r s t best solution i s achievable when the agent i s r i s k neutral. When the agent Is r i s k averse, the optimal sharing rule can be as simple as a salary plus commission, or can be more complicated, depending on the d i s t r i b u t i o n of the outcomes and the agent's u t i l i t y function. In general, i t i s much more d i f f i c u l t to determine when the sharing rule w i l l be increasing i n each outcome, x^, than in the single-dimensional e f f o r t and output case. the sign of each of the Lagrangian m u l t i p l i e r s There are two reasons: must be determined, the question of multivariate stochastic dominance must be addressed. and Each of these problems can be analyzed only i n special cases. The analysis of the value of additional information i s also more complicated than i n the single-dimensional e f f o r t and output case. The applic- a b i l i t y of the results of Amershi (1982) for the multidimensional e f f o r t case was discussed. The use of additional disaggregated information was demonstrated by means of examples. It was shown that i n the case where a salary plus commission scheme i s optimal, the commissions related to each task w i l l generally be unequal. The next question addressed was whether a manager should receive separate rewards for the outcomes from the different tasks. It was shown that a strong form of independence (see (3.4.1)) i s neither necessary nor s u f f i cient for an optimal sharing rule to be additively separable i n the outcomes. If the p r i n c i p a l i s r i s k neutral and the agent's u t i l i t y function i s in the HARA c l a s s , then a necessary condition for additive separability of the optimal sharing i s that the agent have a log u t i l i t y function. If the 50 p r i n c i p a l and the agent are i d e n t i c a l l y r i s k averse, with i d e n t i c a l log utility functions, then the optimal sharing rule i s not additively separ- able. The remainder of this chapter focused on situations i n which effort i s additive, as when e f f o r t represents time devoted to differen t tasks. optimal sharing rules i n the f i r s t best and second best situations were examined under various assumptions about the means and distributions outcomes. The of the In the additive e f f o r t case where there i s no i n t r i n s i c d i s u t i l - i t y for any particular task, i t i s of interest to determine whether the p r i n c i p a l can most e f f i c i e n t l y induce a r i s k averse agent to allocate a l l effort to one task, or to d i v e r s i f y by allocating e f f o r t to each task. section showed that the nature of the outcome d i s t r i b u t i o n factor i n determining whether the optimal solution i s an important w i l l be boundary ( a l l e f f o r t devoted to one task) or i n t e r i o r (effort spread across tasks). c r i t i c a l factor, however, appears to be the relationship expended and the mean of the d i s t r i b u t i o n . mal i n t e r i o r solution This The between e f f o r t Conditions under which an o p t i - i s unique were found, and i t was shown that i f an optimal i n t e r i o r solution i s unique, then the optimal efforts for both tasks are equal, as are the Lagrangian multipliers u^. The additive e f f o r t results were applied to the sales force management problem. As remarked earier, simple commission schemes are rarely Pareto optimal; even when they are optimal, the commissions are generally not equal across products. However, i f the p r i n c i p a l i s r i s k neutral, the agent i s risk averse, V(a) = V(Ea^), the x^'s given a^ are independent c a l l y distributed, and a unique i n t e r i o r solution and i d e n t i - (a'J > 0, a*, > 0) i s o p t i - mal, then the agent's compensation per unit of x^ i s equal to the compensation per unit of x . 2 The multiple salesperson firm was b r i e f l y discussed, 51 as was the addition of r i s k decisions (not involving e f f o r t ) by the s a l e s - person. It has long been recognized that dysfunctional behavior on the part of managers can be induced by their focus on short-term personal goals rather than long-term company goals. Moreover, the company may unwittingly pres- sure managers to make decisions which w i l l increase short-term the expense of long-term goals. One p r o f i t s at of the obvious aspects of a solution i s to extend the performance evaluation period from, for example, one year to several years. A brief comparison of the a l l o c a t i o n of e f f o r t problem and a multiperiod problem follows. Consider the s i t u a t i o n i n which the agent chooses one action a^ i n each of n time periods, resulting i n monetary outcomes x^ which are observed by both the p r i n c i p a l and the agent at the end of period i . The agent's action in any period and the sharing rule for each period can then depend on the outcomes from previous periods. izon w i l l be considered here. For ease of exposition, the two-period horThe p r i n c i p a l ' s u t i l i t y for the two-period horizon is now W(x^-s^(x^),x ~s (x^,x )), 2 2 U(s (x ) , s ( x x ) ,a ,a (x )) . 1 1 2 1 } 1 2 2 ,x ,a Let g ^ 1 ,a (x )) = l 2 2 1 The p r i n c i p a l ' s problem i s then h(x |a^,x^,a (x^))f(x^|a^). 2 and the agent's u t i l i t y i s 2 2 Maximize / J w ^ - s ^ x ^ , x - s ( x , x ) ) g ( x , x , a . a ^ x ^ i' i 2 2 1 2 L 2 l )dXjdx (3.7.1) 2 v subject to /]b(s (x ),a ,s (x ,x ),a (x ))g(x ,x ,a ,a (x ))dx dx 1 1 1 2 1 2 2 1 1 2 1 2 1 and a^ and a2(«) maximize the left-hand side of Let E 2 denote expectation with respect to h ( . ) . rules are characterized by 1 2 > u (3.7.2) (3.7.2). The optimal sharing 52 2 \ E < ll l> f x a i a (E2 + l^C^) < V 2 2 E D B a ) 1 1 a » -ff-jj 2 almost every f o r x, l 8 l and W s S 8 2 , . * h U S a ( 0 h l g( .) a + + . . , Wj( ]_) a a % x h ( 0 2 () > t almost every (x^, x ) , f o r 2 2 special cases are of i n t e r e s t . Two The f i r s t i s the case i n which the p r i n c i p a l ' s and agent's u t i l i t i e s are additive over time, with discount factors 3 and ot, respectively. Suppose the p r i n c i p a l and the agent agree on the contract at the beginning of the two-period horizon and each individual is committed to the contract for the entire time horizon. Then the p r i n c i - pal's expected u t i l i t y i s j W ( x - s ( x ) ) f ( x |a )dx 1 1 1 1 1 + 3/JW( 2 2^ 1 » 2 ^ ^ ( x 1 -s x x . x 1 x t&i , a ( ) ) d x d x , x 2 1 2 1 2 where g(.) = h(x 1 a^ ,x , a ( x ) ) f (x-j^ l a ^ . 2 l 2 1 The agent's expected u t i l i t y i s / b ( s ( x ) ) f ( x |a )dx 1 1 L 1 1 + cx//U(s (x , x ) ) g ( x ,x ,a , a ( x ) ) d x d x 2 1 2 1 2 L 2 1 1 2 - V^) - a/v(a (x ))f(x |a ) dxj. 2 1 1 1 The i n t e r i o r portions of the optimal sharing rules are characterized by woy'i^i)) U'Cs^x,)) f 1 = X + + "l i( il i> x . a n fCxJa,) ( 3 ' * 7 3 ) and s z a W ( x 2 ~ s (Xj_ ,x ) ) g ( x ,x ,a ,a ( x ) ) 2 2 x x 2 x 2 L = A + u, U'(s (x ,x )) 2 1 i 2 g(x ,x ,a ,a (x )) 1 2 1 2 1 g (x ,x ,a ,a ,(x )) 2 + ^ X l > g 1 2 1 2 1 (x ,x ,a ,a (x )) • 1 2 1 2 > 1 ( 3 * ' 7 where the subscripts j on the d i s t r i b u t i o n s indicate p a r t i a l derivatives with respect to a^ 4 ) 53 Equation (3.7.A) i s similar to the characterization for single-period sharing rules i n the multidimensional e f f o r t problem. Thus, the multidimen- sional e f f o r t results described e a r l i e r are useful i n extending to a certain class of f i n i t e - h o r i z o n multiperiod problems. the theory Lambert (1981) has examined the model above under the assumption that e f f o r t i n one period has no effect on the outcome i n any other period, and also examined problems which occur when the p r i n c i p a l i s committed to a two-period contract, but the agent can leave the firm a f t e r the f i r s t period. The second special case of interest i s the case i n which the p r i n c i pal's and the agent's expected u t i l i t i e s depend only on the t o t a l return over the entire time horizon. In this case, the p r i n c i p a l ' s u t i l i t y tion i s W(x^ + %2 ~ and the agent's u t i l i t y i s ( ^ ( x ^ ^ ) , a l' 2^*^* a T n ^ s s(x^,X2)), func- structure i s also appropriate for the problem of sequential a l l o c a t i o n of e f f o r t within one time period, where the time period i s said to end when the agent receives his or her compensation. The sequential a l l o c a t i o n aspect would arise because of the agent's opportunity to observe an outcome affected by the f i r s t e f f o r t choice before making any other e f f o r t choices. This s i t u a t i o n i s the focus of the next chapter. 54 CHAPTER 3 FOOTNOTES 1. Although there are technical problems connected with the use of the normal d i s t r i b u t i o n , i t i s used here for i l l u s t r a t i v e purposes because i t i s the only d i s t r i b u t i o n with a convenient representation for dependent random v a r i a b l e s . Detailed calculations and r e s u l t s for the normal d i s t r i b u t i o n appear i n Appendix 2. 2. A modified version of this result holds when the p r i n c i p a l i s r i s k averse. D i f f e r e n t i a t i n g the f i r s t - o r d e r condition characterizing the sharing rule with respect to x shows that , , , 3 a sign (s'(x)) = sign ( ^ — f ~ W"U\ 2~^ * 3 a Thus, V -K- j— > 0 implies that s'(x) > 0, but the converse does not hold. ~~ f 3. As Holmstrom (1979) points out, i f the production function x i s given by x(a, 6), where 0 represents a random state of nature, then 3x/3a > 0 implies that the d i s t r i b u t i o n of x s a t i s f i e s the f i r s t - o r d e r stochastic dominance property (provided that changes i n a have a n o n t r i v i a l effect on the d i s t r i b u t i o n ) . 4. Extending this and the other propositions which depend on the assumption of a square root u t i l i t y function to a more general class of u t i l i t y functions appears to be n o n t r i v i a l . However, i n the discrete-outcome example presented e a r l i e r , the result i s not confined to only the square root u t i l i t y function. Hence, i t appears l i k e l y that this and the other results stated for the square root u t i l i t y function hold for a more gene r a l class of u t i l i t y functions. 5. Lai (1982) also independently applied agency theory to the problem of sales force management. Much of his analysis i s for a special normal d i s t r i b u t i o n and the class of power u t i l i t y functions. He did not analyze the additive e f f o r t case. 6. Let p be the constant sales price of a product, and c be the constant noncompensation cost per unit of product. Further, l e t q be the random quantity sold as a result of the agent's e f f o r t . One question of i n t e r est i s whether the agent's compensation should be based on, for example, sales (pq) or a "contribution margin" (pq-cq). It i s easy to see that ! f (q|a*) the optimal sharing rule i s characterized by TTTT—/ w = A + u ——,—• . . . U (s( •)) f(q|a*) That i s , the optimal sharing rule does not depend e x p l i c i t l y on p or c or p-c. a 7. The formulation i s presented i n order to i l l u s t r a t e the structure of the problem. Technical problems with the properties of the functions to be maximized are not addressed. 55 CHAPTER 4 ONE-PERIOD SEQUENTIAL CHOICE In this chapter, the model i s expanded to include decisions made at different times. The extension i s to sequential decisions within one period, where a period i s defined to end at the time of payment to the agent. The one-period sequential case i s an intermediate step between the a l l o c a t i o n of e f f o r t case, i n which both the e f f o r t s are exerted before the outcomes are known, and the two-period case, i n which the f i r s t outcome i s observed and the f i r s t compensation Is paid before the second e f f o r t i s exerted. The a l l o c a t i o n and sequential situations can be depicted as follows: Allocation of e f f o r t : Agent exerts Principal chooses s ( l>*2) x a l » A P r i n c i p a l and agent observe xi,x > p r i n c i p a l 2 pays s(x^,x ) to the agent. 2 2 One period sequential choices: Principal chooses s(x ,x ) 1 2 Agent exerts a Agent observes l x Agent exerts a (0 l 2 Agent observes X 2 ; p r i n c i p a l observes x^ and X 2 and pays s ( x i , x ) to the agent 2 Two-period sequential choices: Principal chooses s s l( l) x a n d 2^ l» 2 x x Agent exerts a l P r i n c i p a l and agent observe x^; p r i n c i p a l pays s ( x ) the agent. 1 ) 1 Agent exerts a (0 2 P r i n c i p a l and agent observe X 2 ; p r i n c i p a l pays S 2 ( x ^ , X 2 ) to the agent. In each of the cases above, i f the p r i n c i p a l and the agent observe addit i o n a l valuable post-decision information about the agent's e f f o r t s , then the sharing rules w i l l depend on t h i s additional information. 56 A number of situations might be modeled i n the one-period sequential framework. In the sales force management example, the agent might spend a certain amount of time s e l l i n g products i n one t e r r i t o r y and observe the amount of the resultant sales there before beginning work i n another t e r r i tory. If there i s c o r r e l a t i o n between x^ and x , 2 information from x^ which may then the agent obtains be useful i n the decision about a2« t i o n a l post-decision information that the p r i n c i p a l may agent's e f f o r t s might be comments obtained The addi- obtain about the from personally Interviewing the agent's customers. Another one-period sequential decision setting might involve production decisions by an agent, where a^ i s the number of hours of production some sales information i s obtained. of hours of production The agent would then choose the number for the remainder of the period. the additional post-decision information obtained the number of work hours recorded until In t h i s s i t u a t i o n , by the p r i n c i p a l might be on the agent's time cards. More gener- a l l y , a manager i n a decentralized organization w i l l not be monitored d a i l y , but rather w i l l make many decisions during a given time period and w i l l be evaluated only p e r i o d i c a l l y . The one-period sequential model can be thought of as the special case of the f u l l y general two-period model In which the periods are very short, so that the p r i n c i p a l ' s and the agent's expected u t i l i t i e s depend only on their t o t a l return for the entire horizon. The one-period model can incor- porate some of the elements of the f u l l y general two-period model while providing a somewhat s i m p l i f i e d structure for analysis. For example, i n both models, the f i r s t outcome, which i s f i r s t - s t a g e post-decision information, can be used as pre-decision information for the second e f f o r t choice. The agent's precommitment to stay for the entire time horizon i s not a major 57 problem In the one-period model, since the agent i s not paid u n t i l a l l the required e f f o r t s have been exerted. In the f i r s t part of this chapter, the s i m p l i f i e d structure i n the oneperiod sequential model i s used to explore the impact of c o r r e l a t i o n of outcomes i n f i r s t best and second best s i t u a t i o n s . to the a l l o c a t i o n of e f f o r t r e s u l t s . Some comparisons are made The analysis w i l l focus on aspects which were not addressed i n the pre-decision information l i t e r a t u r e or i n Lambert's (1983) analysis of a f i n i t e - h o r i z o n multiperiod agency problem with independent outcomes. for The second part of t h i s chapter develops results the one-period sequential problem that p a r a l l e l two sets of r e s u l t s i n the a l l o c a t i o n of e f f o r t problem, namely additive s e p a r a b i l i t y of the sharing rule and d i v e r s i f i c a t i o n of e f f o r t across tasks when e f f o r t i s additive. The s i m i l a r i t i e s to and differences from the a l l o c a t i o n r e s u l t s are d i s cussed. Before proceeding to the analysis, a b r i e f review of the existing results on pre-decision information w i l l be given and Lambert's (1983) r e s u l t s w i l l be summarized. Unless otherwise stated, the "sequential e f f o r t problem" w i l l refer to the one-period sequential e f f o r t problem. A limited amount of research has been devoted to one-period agency problems with pre-decision information. Baiman (1982, p. 192) comments as follows on the increased complexity with pre-decision information: The role and value of a pre-decision information system i s more complex than that of a post-decision information system. Expanding a post-decision Information system to report an addit i o n a l piece of information w i l l always r e s u l t i n at least a weak Pareto improvement, since the p r i n c i p a l and agent can always agree to a payment schedule that ignores the additional informat i o n . However, expanding a pre-decision information system to report an additional piece of information may not r e s u l t i n even a weak Pareto improvement. The agent generally cannot commit himself to ignore the additional information, and therefore the optimal employment contract without the additional pre-decision information i s no longer necessarily self-enforcing given the additional information. This i s true whether the additional predecision information i s p r i v a t e l y reported or p u b l i c l y reported. 58 Some of the research concerning pre-decision Information focuses on the following question: Given that the agent has private pre-decision informa- t i o n , what i s the value of public post-decision information systems? Holmstrom (1979) showed that an informativeness c r i t e r i o n (f(x,y,z;a) * g(x,y)h(x,z;a), where z i s the pre-decision signal) i s necessary for the post-decision information system which reports a public s i g n a l , y, i n addition to x, the outcome, to provide a Pareto improvement over the information system which reports only x. Christensen (1982) expanded Holmstrom's (1979) model by allowing the agent to communicate to the p r i n c i p a l a message m about the private pre-decision s i g n a l . The agent i s assumed to select the message that maximizes h i s or her expected u t i l i t y . In Christensen's model, a generalization of Holmstrom's (1979) informativeness c r i t e r i o n i s necessary for the post-decision information system which reports y, i n addition to x and m, to provide a Pareto improvement over the information system which reports only x and m. Here, the public post-decision signal i s a s i g - nal about the agent's e f f o r t and the agent's private pre-decision information signal. Another d i r e c t i o n of the research on pre-decision information has been the value of pre-decision information systems. There are both positive and negative effects of private pre-decision information for the agent. On one hand, the agent has more information before choosing an action, and hence should make "better" decisions. On the other hand, more information may reduce the r i s k the agent faces, and hence reduce the motivation for the agent to exert e f f o r t . Christensen (1981) provided an example which shows that the p r i n c i p a l may be worse o f f when the agent has private pre-decision information (with or without communication of^a message), and also provided an example which shows that the p r i n c i p a l may be better o f f when the agent has private pre-decision information and communicates a message to the prin- 59 cipal. Christensen's examples i l l u s t r a t e the d i f f i c u l t y i n obtaining a gen- e r a l preference ordering r u l e over pre-decision information systems. A t h i r d d i r e c t i o n of research on private pre-decision information has been the value of communication of a message about the private information from the agent to the p r i n c i p a l , given the existence of the private pre-dec i s i o n information system. In the accounting context, the focus i s on the value of communication of private information i n the process of p a r t i c i p a tive budgeting. (1983), who The major result i n t h i s area i s that of Baiman and Evans provided necessary and s u f f i c i e n t conditions for communication to result In a Pareto improvement. Baiman (1982, p. 204) summarizes the result as follows: . . . If the agent's private pre-decision information i s perfect, then communication has no value. Observing the firm's output i n that case allows the p r i n c i p a l to i n f e r a l l he needs to know about the agent's private pre-decision information. However, i f the agent's private pre-decision information i s imperfect, a necessary and s u f f i c i e n t condition for communication to be s t r i c t l y valuable i s for the honest revelation of the agent's private pre-decision information to be s t r i c t l y valuable. That i s , i f any value can be achieved with the information being hone s t l y revealed to a l l , then a s t r i c t l y positive part of that value can be achieved by giving the agent sole d i r e c t access to the information and l e t t i n g him communicate i n a manner that maximizes h i s expected u t i l i t y . Lambert (1983) has examined a special case of the f i n i t e - h o r i z o n multiperiod agency problem. He assumed that both the p r i n c i p a l and the agent have u t i l i t y functions (and that the agent has a d i s u t i l i t y function) which are separable across time. independently He further assumed that the state variables are distributed across time, and that e f f o r t i n one period does not influence the monetary outcome i n any other period. Under these condi- tions, Lambert showed that the agent's compensation i n a given period w i l l depend on the outcomes i n previous periods as well as on the outcome i n the present period. He further showed that the incentive problems associated with the agent's e f f o r t choices i n each period are not eliminated. In the 60 notation of t h i s chapter, the result can be stated as (I) (ii) ^ ( x ^ > 0 for almost every x 1 > 0, and ( f i r s t - s t a g e outcome). The remainder of t h i s chapter analyzes the one-period sequential e f f o r t choice problem. The cooperative, or f i r s t best case i s f i r s t considered, and the behavior of the agent's second-stage acterized. e f f o r t choice strategy i s char- The second best case i s then analyzed. The optimal sharing r u l e i s derived and discussed, as i s the behavior of the agent's second-stage choice strategy, with and without independence of the outcomes. It i s then shown that the optimal sharing rule w i l l not be a d d i t i v e l y separable i n the outcomes, even under the conditions which were s u f f i c i e n t for such a r e s u l t in the e f f o r t a l l o c a t i o n problem. F i n a l l y , the special case of additive e f f o r t i s analyzed, and the question of the d e s i r a b i l i t y of d i v e r s i f i c a t i o n of the agent's e f f o r t s across tasks i s examined. The result i s related to the information content of the outcome about the agent's 4.1. FIRST BEST In the f i r s t best case, the p r i n c i p a l ' s problem i s : Maximize / / W(x-s(x^ ,x ) ) <f>(x^ ,x s( •) , a , a ( •) 2 1 2 ,a2( •) ) dx dx^ 2 2 subject to / / {U(s(x ,x )) - V ( a , a ( •))}•( •)dx dx 1 where effort. 2 t 2 2 1 > u, ,x 1 a^ ,a ( •) ) = f (x^^ | a^)g(x \x^ ,a^ ,a ( •) ) and a ( 0 2 2 2 2 2 indicates that e f f o r t i s i n general not a constant, but rather can the agent's second-stage depend on any information available at the time of choice. Letting X be the m u l t i p l i e r for the agent's expected u t i l i t y constraint and d i f f e r e n t i a t i n g the Hamiltonian with respect to s( •) for every (x^,x ) y i e l d s 2 W'(x-s( ,x )) Xl 2 U'(s( ,x )) X l 2 = X 61 for almost every ( x , x ) . 1 This implies that i f one person i s r i s k neutral 2 and the other i s r i s k averse, then the r i s k neutral person w i l l bear the r i s k (see Appendix 4). That i s , i f the p r i n c i p a l i s r i s k neutral and the agent i s r i s k averse, then the optimal sharing rule Is constant; i f the p r i n c i p a l i s r i s k averse and the agent i s r i s k neutral, then the principal's return i s k, a constant, and the agent receives x^+x ~k. 2 are If both individuals r i s k averse, then the r i s k i s shared; the optimal sharing rule i s a func- tion of (x^+x ). Furthermore, 2 9s/9x^ i s positive f o r i = 1,2. Finally, i f both are r i s k neutral, then the optimal sharing rule i s s = u + V(a^,a (»)). 2 These r e s u l t s are the same as those f o r the a l l o c a t i o n of e f f o r t problem. Thus, i n the f i r s t best case, the sequential nature of the e f f o r t decisions does not a f f e c t the characterization of the optimal sharing r u l e s . In this scenario, there are no signals on which the choice of a^ can be based. Whether or not a i s a function of x^ depends on the r i s k attitudes 2 of the individuals and the j o i n t d i s t r i b u t i o n <|>(x^ ,x |a^ ,a ( • ) ) . 2 2 If at l e a s t one of the individuals i s r i s k neutral and (jKxj^ ,x |a^ ,a ( •) ) = f (x^ |a^)g(x | a ( •) ) , then the optimal a ( •) i s indepen2 2 dent of x^. 2 2 2 In this case, the r i s k neutral person e s s e n t i a l l y owns the out- put of the firm, and thus bears a l l the r i s k associated with the uncertainty of x^. Furthermore, x^ conveys no information about x . 2 If both of the individuals are r i s k averse or i f <)>(•) i s the more gene r a l f ( x j j a ^ ) g ( x | x ^ , a ^ , a ( • ) ) , then the optimal a ( •) w i l l generally depend 2 on x^. ual 2 2 In the f i r s t case, the change from the s i t u a t i o n where one i n d i v i d - i s r i s k neutral occurs because each r i s k averse individual's marginal u t i l i t y depends on the f i r s t outcome, since i t determines where on his or her u t i l i t y curve the Individual i s ; a r i s k neutral individual's marginal u t i l i t y , on the other hand, would be the same no matter what the value of x^ is. This f i r s t e f f e c t of x-i can be termed the "wealth" or " r i s k aversion" 62 effect. for x 2 In the second case, i f x^ and x 2 are dependent, then may change according to the f i r s t outcome, x^. therefore wish to induce the agent to choose a («) 2 decreasing expectations The p r i n c i p a l may as an increasing or function of x^, depending on the r i s k attitudes of the p r i n c i p a l and the agent, the agent's d i s u t i l i t y for e f f o r t , and the nature of the corr e l a t i o n between x^ and x . This second effect of x^ can be termed the 2 "information" e f f e c t . The information effect of x^ i s made more precise i n the proposition below. Proposition 4.1.1. Suppose that i n the f i r s t best case, the p r i n c i p a l i s r i s k neutral, the agent i s r i s k averse, and <{>(•) = f(x^ |a^)g(x2 |x^ ,a^ ,a ( • ) ) . 2 In this case, a ( •) w i l l depend on x^. Let M^( •) denote the mean of x^ 2 given a^, and l e t M2(x^,a^,a2( •)) denote the conditional mean of x respect to g(»). Let the second and third subscripts of j on M t i a l d i f f e r e n t i a t i o n of M 2 2 2 with denote par- with respect to the j-th argument of M ( x , a , a ( •))• Then 2 1 1 2 a * « ( ) = -M /[M233 " A[ 9 V( •) / 9a ] ] • 2 Xl 2 231 For example, suppose M2(x^,a^,a2( •)) = x^^Cx^/a^ and V( •) = (a^+a2) . Then a*,'(x^) = l/(2a^A) > 0. In this case, a^(x^) increases l i n e a r l y i n x^. The e f f e c t of the nature of the correlation between x^ and x 2 i s cap- tured i n the derivatives of M ( 0 » and the effect of the d i s u t i l i t y function 2 2 2 i s captured i n the 9 V/9a 2 term. Note that a*,(x^) does not depend on the agent's u t i l i t y function for wealth. This i s because the r i s k averse agent receives a constant wage i n the f i r s t best case, and hence the agent's u t i l i t y for the wage i s constant. Note further that i f M 2 depends only on a (*)> then a*, i s constant. 2 Proposition 4.1.1 and the discussion preceding e f f o r t choice's dependence on x^, the f i r s t outcome. i t focused on the second The second e f f o r t choice, a„(»), might seem to also depend on the f i r s t e f f o r t choice, aj_. 63 However, the agent chooses the e f f o r t a^ and the e f f o r t strategy a^(. •) simultaneously at the beginning of the time horizon. The second e f f o r t choice i s therefore not viewed as a function of a^, although there i s i m p l i c i t recognition that a^ and a ( ") chosen j o i n t l y and therefore a r e 2 influence one another. However, since the f i r s t outcome i s unknown at the beginning of the time horizon, the second e f f o r t choice can p o t e n t i a l l y depend on the f i r s t outcome. 4.2 SECOND BEST In this section, the general formulation of the one-period sequential model i s f i r s t presented. Subsequently, the two extremes of independent outcomes and perfectly correlated outcomes are examined. In the f i r s t case, knowledge of x^ reveals no information about x , whereas i n the second case, 2 x^ reveals perfect information about x . The behavior of the agent's second 2 e f f o r t strategy i s i l l u s t r a t e d In the two extreme cases, and also for the intermediate case of imperfectly correlated outcomes. As before, i n order to focus on motivational issues, i t w i l l be assumed that the p r i n c i p a l i s r i s k neutral and the agent i s r i s k averse. The prin- cipal's problem i s : Maximize s( •),a ,a ( •) 1 / / (x-s(x^ ,x )) <|>(x^ ,x |a^ ,a ( «))dx dx^ 2 2 2 2 2 subject to / [ / U ( s ( x x ) ) g ( x | x , a , a ( «))dx - V ( a ,a ( •) ) ] f ( x |a )dx 1 } 2 / / u(s(-))[g { /U(s(0)g 1 2 1 f + gf ^]dx d a 2 2 X l 2 1 2 - / (V^f + vf ^)d a x Xl 1 2 a 1 1 >u = 0 (*)dx - V ^( • ) } f ( x | a ) - 0 for almost every a ;L x, x where <«x^,x 1a^,a ( •)) = f(x^ |a^)g(x |x^,a^,a ( •)) and d i f f e r e n t i a t i o n 2 respect to a 2 2 2 i s pointwise for each x^. 2 with The i n t e r i o r portion of the optimal 64 sharing rule i s characterized by (s( )) u t = ^i+a ^* X + where X, then A a l f ° ra l m o s t e v e r y (x ,x ), 1 2 and ^ ( x ^ ) are m u l t i p l i e r s for the three constraints above. Here, 6 a " ^ l ^ a ^ * + x /<(> = f l a / f + g /g and $ l l 2 a a /<\> = g 2 /g. If ai does not influence xo, a /<(. = f / f . l a The characterization of the i n t e r i o r portion of the optimal sharing rule i n the sequential e f f o r t case i s similar to that i n the a l l o c a t i o n of e f f o r t case, except that here y*, and a*, may depend on x^. depends on x^. However, i f the agent i s r i s k neutral and <}>(x^ >x |a^ ,a ( •)) 2 = f ( x | a ) g ( x |a ( •)), then a*,( •) does not depend on x^. 1 1 2 In general, a*,(«) 2 2 If the x^s are conditionally correlated, then a*,( •) w i l l depend on x-^ even i f the agent i s r i s k neutral. These r e s u l t s are d i r e c t consequences of the a c h i e v a b i l i t y of the f i r s t best solution i n the second best case i f the agent i s r i s k neutral (see Shavell (1979)). The proposition below describes aspects of the second stage problem f o r a p a r t i c u l a r u t i l i t y function for the agent, and f o r several commonly used d i s t r i b u t i o n s for the independent outcomes. Proposition 4.2.1. Suppose that i n the second best case, the p r i n c i p a l i s r i s k neutral, and the agent's u t i l i t y function for we a l t h i s U(s) = 2/s. Suppose also that <(>(•) = f (x^ | a^)g(x | a ( •)) , where f( •) and g( •) are i n Q^, 2 2 the class consisting of the exponential, gamma, and Poisson d i s t r i b u t i o n s represented i n Appendix 1. Define a^ and a a^ and the mean of g ( x | a ) i s a . 2 2 2 i f 8V/3a i s p o s i t i v e at a* then 2 (a) y (x..) i s p o s i t i v e , and 0 so that the mean of f(x^|a^) i s Then, assuming that the optimal e f f o r t s are nonzero, (i) 2 65 a sufficient (b) c o n d i t i o n f o r the agent's expected second stage net u t i l i t y to be i n c r e a s i n g i n x^ i s t h a t a*,( •) be a d e c r e a s i n g (ii) if f u n c t i o n of x^; i s p o s i t i v e , then (a) the agent's expected u t i l i t y f o r the second stage p e c u n i a r y r e t u r n , E(u(s(x))|x^}, i s an i n c r e a s i n g f u n c t i o n o f x^, (b) the c o n d i t i o n s V > 0, V 2 jointly sufficient x^. 3V/3a c o n d i t i o n that strictive. conditions 2 2 2 > 0, and 2 in (ii)(b). The to the partial > 0 1 2 2 f u n c t i o n of differentia- one, and i s nonre- forms o f d i s u t i l i t y f u n c t i o n s s a t i s f y f o l l o w i n g , f o r example, s a t i s f y the V ( a ^ , a ) = h(a^) + a , where m m 2 are j-th effort variable. be p o s i t i v e i s a standard A number of g e n e r a l V f o r a*,(*) to be a d e c r e a s i n g Here, s u b s c r i p t s j on V r e p r e s e n t t i o n with respect The > 0, V 2 2 and > 1 and a 2 > the conditions: 0, 2 2 V(a^,a ) = 2 a 1 a where a± 2 V(a ,a ) = h(c a 1 2 1 1 a and 2 1 2 c^ and c 2 > 0, h " ' > 0, and (see Appendix 4 ) . T h i s i s c o n s i s t e n t w i t h the results with a r i s k neutral p r i n c i p a l , o f the outcomes. In g e n e r a l , first a r i s k averse agent, and There i s n e i t h e r an i n c e n t i v e problem nor an to induce the dependence o f a though, a 2 2 the positive. are i s z e r o , so t h a t t h e r e i s no i n c e n t i v e problem, then a depend on x^ effect > 0, 2 + c a ) , where h' > 0, h * constants If > 0 and on 2 does not best independence information x^. w i l l depend on x^. t h a t i n some p a r t i c u l a r s e t t i n g s , the o p t i m a l P r o p o s i t i o n 4.2.1 states second stage e f f o r t will decrease as the f i r s t outcome i n c r e a s e s . R e c a l l t h a t x^ determines a p o i n t on the u t i l i t y curve f o r the agent b e f o r e the second stage e f f o r t i s chosen. Because the agent's m a r g i n a l u t i l i t y f o r w e a l t h i s a d e c r e a s i n g function and 66 the agent's marginal d i s u t i l i t y f o r e f f o r t i s an increasing function, i t i s more costly for the p r i n c i p a l to induce a given l e v e l of a2, the higher x^ is. The r e s u l t that a 2 i s decreasing i n x^ should thus hold for other con- cave u t i l i t y functions for wealth, coupled with convex d i s u t i l i t y functions. Proposition A.2.1 also provides conditions under which the agent's second stage expected u t i l i t y w i l l increase as the outcome increases. Under the given conditions, E[U(s(x))|x^] i s increasing i n x^, and -VCa^,a (x^)) 2 i s increasing i n x^ because a 2 i s decreasing i n x^. Thus, the agent's expected second stage net u t i l i t y i s increasing i n x^. The independence of x^ and X2 i n Proposition A.2.1 means that there i s no information e f f e c t of x j . If x^ and X2 are correlated, then the behavior of a*,( •) would depend additionally on the nature of the c o r r e l a t i o n . In order to examine the information effect of xj_, the extreme case of perfect c o r r e l a t i o n of the outcomes w i l l next be analyzed. When the outcomes are perfectly correlated, then a j o i n t density f o r x^ and X2 does not e x i s t . Since the lack of a j o i n t density precludes using the previous analysis d i r e c t l y , a modified approach must be taken i n order to examine the nature of the sharing rule and the agent's second-stage e f f o r t choice when the out- comes are p e r f e c t l y correlated. Let x^ = x^(0,a^), where 0 i s an uncertain state that influences both the outcomes. It w i l l be assumed that for any fixed a^, x^ can be Inverted to obtain 0 = 0(x^,a^), The p r i n c i p a l ' s and the agent's common b e l i e f s about the outcomes w i l l be expressed as <|>(x^ ,x 1 a^ ,a ( •) ) = f(x^|a^) i f 2 X 2 = x ( » 2^ l^ 9 2 a x a n d 8 = ^ x ^ 3 2 ^ otherwise, <{>(») = 0. In order to describe the sharing r u l e , l e t a^ be the agent's f i r s t stage e f f o r t choice that i s induced by the sharing r u l e , and l e t a^(x^) be the agent's second-stage e f f o r t strategy that i s induced i f x^ i s observed and i t i s assumed that a^ = a*. Because of the perfect c o r r e l a t i o n between 67 x^ and x , 2 form: the sharing rule s(x^,x ) can be viewed as being of the following 2 s(x^,x ) = s(x^) i f x 2 = x (9,a*(x^)) and 9 = 9(x^,a*); otherwise, 2 2 s( •) i s a penalty wage which i s possibly negative. The sharing rule can be viewed as being dichotomous with respect to and varying continuously only with x^. 2 2 and the inferred value of 9. 2 A l t e r n a t i v e l y , the sharing rule can be viewed as being a function of the t o t a l output, x^ + x , condition that the observed x x subject to the i s i n agreement with the observed value of x^ In either view of the sharing rule, lack of agreement between the observed values of x and x^ i s taken as evidence of 2 shirking; accordingly, a penalty i s imposed i n such s i t u a t i o n s . If the pen- a l t y i s s u f f i c i e n t l y severe, the penalty need never be imposed, since the agent w i l l choose to avoid the penalty by choosing a*(x^). Determination of the optimal sharing rule can hence be confined to determination of the o p t i mal function s(x^); furthermore, no f i r s t order condition i s required i n order to induce a*,(x^), as long as a^ i s properly induced. The p r i n c i p a l ' s problem can therefore be written as follows: Maximize s(x ),a ,a (x ) 1 1 2 subject to + x ( 9 ^ , 3 ^ . a ^ x ^ ) - sCx^ )f ( x J a ^ d X j ^ / 2 1 /[U(s( )) - V(a ,a (x ))]f(x |a )dx Xl 1 /[U(s( )) - V ( Xl - /V a a i 2 ,a ( 2 1 X l 1 ))]f 1 (x |a )dx 1 a (a ,a (x ))f(x |a )dx 1 2 1 1 > u 1 1 1 1 1 = 0. In order to determine the f i r s t order conditions, l e t X and u be the Lagrangian m u l t i p l i e r s for the f i r s t and second constraints, respectively, and form the Hamiltonian H i n the usual way. D i f f e r e n t i a t i n g H with respect to s(») for every x^ y i e l d s W'( Xl ^ ^ l ' V + x (8(x ,a ),a (x )) - s ^ ) ) 2 1 1 2 U'(s( )) X l 1 = X + U f( X l | a i ) ' 68 which i s o f t h e usual Jw' ( 0 f ( •)dx - 2V a Finally, form. (Of l a differentiating 3x w (•) H with respect (OdXj a + u/{[U(0 - V ( 0 ] f (.)f(0}dx. l a t o a-^ y i e l d s (0 = 0. 1 H with respect to a f o revery 2 yields x ^ ? •) f( -~ - xv ( . ) f ( •) - u [ v ( O f ( 0 + v ( O f ( O] = 0 . a <«a a 2 If + /W( - ) f 1 " V l (0 l a Differentiating thep r i n c i p a l a 2 i s risk focus on m o t i v a t i o n a l rather neutral, a 2 a^ 1 2 a s i s commonly a s s u m e d i n o r d e r t o than r i s k - s h a r i n g i s s u e s , then t h e f i r s t order c o n d i t i o n s above reduce t o f U ' ^ ) ) ~ / 1PT Ia7 f ( A x l l i/{tU(s(x )) + l - 2V ( a i ,a 2 ( X a i ) l ) d l x / + W - ( <'-> 42 1 ) f a i ( x l l - V(a ,a (x ))]f 1 a a L f'xjap ' v + (x | a 2 1 l ))f 1 d l x 1 1 - V 1 ) (x |a )dx 1 (x |a ) a l a (a g & 1 .a^x^ ) f (xj_ | a^ jd^ = x 0, (4.2.2) and ^ f - ( x ^ v Dividing a. 3x l l a 2 l > a ( a i » (4.2.3) a 2 ( a2 x by f 2 W (a ,a (x ))f(x |a ) " i ) > f 1 a 1 ( 1 2 x l l f(x^|a^) a l ) 1 + V a i a 1 2 ( a l ' a 2 and r e a r r a n g i n g ( x l ) ) f ( x l l a l ) ] = °* ( - - ) 4 2 3 yields (x,|a.) a^ l 1' 1 *2 Substituting = 3a 2 (4.2.1) T7T7-F—vT U'(s(x^)) into V a (4.2.4) yields ( a . , a , ( x ) ) + uV 1 2 1 l 2 a (a 2 1 a (x ) ) . 2 1 (4.2.5) 69 It i s e a s i l y seen that i f (4.2.4) i s to hold for almost every X p a ( •) must i n general vary with x-^. then The "wealth" and "information" e f f e c t s 2 of x-j^ described i n the f i r s t best analysis can be seen i n (4.2.4). The wealth e f f e c t of x^ results from the interaction of the agent's marginal utility for wealth and marginal d i s u t i l i t y for e f f o r t . The information e f f e c t of x^ refers to the information that x^ provides about x « 2 In the perfect c o r r e l a t i o n case, the state 9 i s Inferred from x^ and a^, and x hence a deterministic function of a agent's perspectives. the 9 x / 9 a 2 2 is from both the p r i n c i p a l ' s and the 2 The information e f f e c t of x^ i s therefore captured i n term i n (4.2.5). 2 The behavior of a*, as x^ varies can be determined by d i f f e r e n t i a t i n g (4.2.4) with respect to x^ to obtain P 2 v V 9 3 a- 3x. / l> f _3_ ^ 2 39 ^ 3 a / a Q ; f 2 — 3a 2 (4.2.6) (X + u - ^ ) V 2 2 a 1 2 When x 2 J59 3x. a a a a 2 i s l i n e a r i n a , the 3 x / 3 a 2 - uV l 2 2 2 2 term i n the denominator i s zero. Two special cases of interest are ( i ) x^ = 9 + a^, where 9 Is purely noise, and ( i i ) x^ = 9a^, where 9 reveals information about the production technology. In case ( i ) , the marginal output per unit of e f f o r t i s one, regardless of the value of 9. In case ( i i ) , however, the marginal output per unit of e f f o r t i s 9. 2 2 To i l l u s t r a t e the r e s u l t s , suppose that V(«) = a^a,,. For case (i) , 2 2 2 assume that 9 " N(0,cr). Then x^ " N(a^,a ) and f / f = ( X j - a ^ ) a . There& 2 2 fore, the numerator of (4.2.6) i s u(2a^a )/a and the denominator i s 2 2 l ~ l -(X + u — 2 — ) ( 2 a ^ ) - 4a^u. x a a l~ l The term (X + u — — ) x a 2 * s positive by the a f i r s t order condition (4.2.1), and the e f f o r t levels are assumed to be posi- 70 tive. Therefore, a*/(x^) < 0, provided that y > 0. This can also be seen by solving f o r a (x^) d i r e c t l y from (4.2.4) to obtain 2 X "3. a ( 2 ) = [2a (X + y J^i) + 4ua ]" . 2 X l 1 1 a In t h i s case, the sign of a*,' i s the same as i n the independent outcome s i t uation described i n Proposition 4.2.1, where there was no information about x 2 to be gained from x^. The case ( i ) result here can thus be interpreted as indicating that the wealth e f f e c t of x^ dominates any information e f f e c t that exists through perfect f ci or sr tr e lthat a t i o n0 of For case ( i i ) , assume ~ e xthe p ( l outcomes ) . Then .x^ ~ exp(a^) and f fl 2 /f = (Xj-a^)/a^. Equation (4.2.4) becomes l~ l 9 = (X + y — j — ^ x a a 2 a 2 l 2 a ^ + t j a i a 2 * l Sub s t i t u t i n g 0 = x^/a^ and rearranging results i n l *2< l> " — x -a 2 a ^ [ ( X + y - 4 - ^ ) + 2y] x a X 1 1 ai a l Therefore, r x a^X i a 2 , ( x i> = . t 7T 2 a + y l a i 2—) + 2y - x ( y / a ) 1 a 1 i {a (X + y ^ - ^ ) L a + 2y} 2 l The numerator of a*,'(x^) reduces to (a^X - y + 2y), which i s positive x l" l a (assuming y > 0) because a^(X + y — j — ) > 0 for x^ J> 0, and for x^ = 0 i n «! particular. for Thus, a*/(x^) > 0 i n t h i s case. A similar analysis can be done the normal d i s t r i b u t i o n example used i n case ( i ) , with the result that 71 a*,'(x^) > 0. The sign of a*'(x^) would remain the same i n cases ( i ) and ( i i ) for a wide variety of reasonable d i s u t i l i t y functions. For the normal d i s t r i b u t i o n example, the only difference i n the expres9 ^ 2 96 sion for a i ' ( x ) i s the — (——) — — term. 2 z L 90 X 1 In case ( i ) , i t i s zero, and i n a case ( i i ) , i t i s 1/a-^. Although 9 i s purely noise i n case ( i ) , r i s k i s imposed on the agent for motivational purposes i n order to induce a Pareto optimal choice of a^. The e f f o r t strategy a (x^) i s primarily determined by 2 the wealth e f f e c t of x-^, leading to a decreasing function of x^ just as i n the case when the outcomes were assumed to be independent (see Proposition 4.2.1). In case ( i i ) , where the marginal output per unit of e f f o r t i s 9, the agent receives perfect information about the production technology that was not relevant i n case ( i ) . The information effect of x^ overrides the wealth effect i n the case ( i i ) examples above, so that a*, i s now an increasing function of x^. To i l l u s t r a t e the second best r e s u l t s , suppose that the p r i n c i p a l i s r i s k neutral and the agent's u t i l i t y for wealth i s 2/s. Suppose further that x^|a^ and x | a 2 2 are Independent, f(») i s exponential with mean a^, and g(») i s exponential with mean a ( x ^ ) . 2 Then the i n t e r i o r portion of the 2 optimal sharing rule i s characterized by s(x^,x ) = P (x), where 2 x -a*(x ) " { * ). a* ( x ) 2 P(x) = \ + M ^ - ^ ) a* + P (x )( 2 1 1 i (4.2.7) x P(x) must be s t r i c t l y p o s i t i v e i n order to s a t i s f y the f i r s t order condition 1/U' = P(x). In the proof of Proposition 4.2.1, i t i s shown that P ( ) = (9V(a*)/9a )a* ( )/2 , 2 2 X l 2 Xl (4.2.8) which i s p o s i t i v e under the usual assumption that the agent's d i s u t i l i t y function i s increasing In the second e f f o r t . Furthermore, i t i s e a s i l y seen 72 that u '(x^) < 0 under the assumptions i n Proposition 4.2.1, part ( i i ) . 2 I n t u i t i v e l y , the higher the f i r s t outcome i s , the less concerned the r i s k neutral p r i n c i p a l i s about motivating a high choice of a 2 This i s because the higher the outcome x^ i s , the c o s t l i e r i t becomes to induce a given l e v e l of a2« As remarked e a r l i e r , the p r i n c i p a l induces a strategy which i s decreasing a (xi) 2 In xj_. At the time of the second e f f o r t choice, the f i r s t outcome x^ i s known. x -a*,( ) 2 ^ ' 2 As Lambert (1983) notes, P(x) can be viewed as X(x^) + u ( x ^ ) ( 2 Xl which i s as i t would appear i n a one-stage, one-period agency problem, given that x^ i s f i x e d . Thus, i t i s not t o t a l l y surprising that, as i n the one- stage, one-period problem, 3s/3x u (x^) are s t r i c t l y postive. 2 ably more complicated. 2 i s s t r i c t l y p o s i t i v e , since P(x) and The behavior of s( •) as x^ varies i s consider- Substituting (4.2.8) into (4.2.7) and d i f f e r e n t i a - ting shows that f-= l 2P(x) A, a * (fL!^ + _ a-V .)/2]. 2 ( 2 2 Under the assumptions i n Proposition 4.2.1, part ( i i ) , the f i r s t and t h i r d terms i n the brackets are p o s i t i v e . The condition that x 2 < a*,(x^) i s suf- f i c i e n t f o r the sharing rule to be increasing i n the f i r s t outcome. ever, i t i s c l e a r l y possible that x 2 How- 3s/3x^ i s increasing i n x^ even i f > a*,^). An a l t e r n a t i v e approach to an analysis of the sharing rule i s i n s i g h t - ful. Recall that the f i r s t order conditions require that f U^i)T = R ( X ) - X + h a (x |a*) g 1 fUja*) + "2< l> x a (x |a*( )) 2 X l gOc.la*^)) * 73 Taking the conditional expectation of R(x) with respect to g(x | a^x^) ) 2 f a ( , ) 1 results i n the expression X + model, i f ^ ^ . As i n the one-stage, one-period > 0 annd f( •) s a t i s f i e s the monotone likelihood r a t i o property, then (X + ^ ^ Thus, the agent faces a sharing ) i s increasing i n x^. rule with similar characterization for each stage, looking only one step ahead. That i s , at each stage, the shading rule i s characterized by the condition that 1/U' = X + uJh /h. u u a A In order to i l l u s t r a t e the behavior of a (x^) when x^Ja^ and x | a 2 2 are 2 imperfectly correlated, suppose that g(x |x^,a (x^)) i s exponential with 2 mean M (x^) = x ^ a ( x ^ ) . 2 2 Since the exponential d i s t r i b u t i o n i s a one-parame- 2 ter d i s t r i b u t i o n , we may write g(x |x^,a (x^)) = g(x |M (x^)), 2 t i o n 4.2.1 can be applied. 2 2 2 and Proposi2 2 For concreteness, suppose that V(a^,a ) = a 2 2 Then V 2 ^ a 2 ' 2 = 2a^a > 0, V 2 2 2 = 2a 1 > 0, V = 0, and V 2 2 2 1 2 2 = h& 1 > 0, so that the conditions i n Proposition 4.2.1, part ( i i ) ( b ) are s a t i s f i e d . Substitui s p o s i t i v e , then M^Cx^) i s which i s s t i l l p o s i t i v e . Therefore, i f ting a (x^) = M (x^)/x^ into the expression for V yields V = 2a M (x )/x^, decreasing i n x^. That i s , x^a^x^) i s decreasing i n xj_. If b i s positive, 2 2 2 2 2 2 1 then i t i s e a s i l y seen that a (x^) i s decreasing i n x^, as when b i s zero 2 (the "independent" case). In this s i t u a t i o n , as when there i s perfect cor- r e l a t i o n with the normal d i s t r i b u t i o n i n case ( i ) , the wealth effect of x± i s dominant. Recall that case ( i i ) of the perfect c o r r e l a t i o n analysis assumed that x t = Qa.^, so that x 2 = x a (x )/a . 1 2 1 1 This seems similar to the imperfect c o r r e l a t i o n example i n which M (x^) = x ^ a ( x ^ ) . 2 2 However, the signs of a*,'(x^) are opposite i n these perfect and imperfect correlation cases. This can be interpreted as follows: i n the presence of information related to the production technology, the wealth e f f e c t of x^ i s dominant i f 74 the c o r r e l a t i o n i s imperfect; the information effect of x^ i s dominant only i f the c o r r e l a t i o n i s perfect. If b i s negative, then the behavior of a (x^) i s p o t e n t i a l l y much more 2 complex. The condition that M (x^) i s decreasing i n x^ i s equivalent to the 2 condition that bx^ a ( x ) + x^a^x^) < 0. Since b < 0, the f i r s t term i s negative; a ( x p may It could be, for example, that 1 2 2 1 thus be of any sign. because of the interactions of the wealth and information effects of x^, a (x^) i s increasing for low values of x^ and decreasing for high values of 2 x r This concludes the analysis of the effect of the information x^ on the agent's second e f f o r t strategy. The next two sections examine two aspects which were of interest i n the a l l o c a t i o n problem, namely additive separabili t y of the sharing r u l e , and additive e f f o r t . 4.3. ADDITIVE SEPARABILITY OF THE SHARING RULE In this section, the question of whether or not to reward the agent for each outcome separately i s examined. For example, suppose a salesperson exerts e f f o r t s e l l i n g a product i n one t e r r i t o r y , observes the resultant sales, and then devotes e f f o r t to s e l l i n g the same product or a d i f f e r e n t product i n another t e r r i t o r y . Should the firm compensate the salesperson with a d i f f e r e n t reward function for each outcome, as i f he or she were two separate salespeople? That i s , should the sharing rule be a d d i t i v e l y separ- able i n the outcomes? It was shown i n Section 3.4 that i n the e f f o r t a l l o c a t i o n problem, i f the p r i n c i p a l Is r i s k neutral and the agent i s r i s k averse with a HARA-class u t i l i t y for wealth, then j o i n t l y s u f f i c i e n t conditions for the optimal sharing rule to be a d d i t i v e l y separable i n x^ and x 2 are ( i ) the agent has a log u t i l i t y function for wealth and ( i i ) the outcomes are conditionally dent (see equation (3.3.2)). indepen- In the one-period sequential e f f o r t problem, 75 the optimal sharing rule w i l l not be additively separable i n x^ and x , even 2 under conditions ( i ) and ( i i ) above. This i s e a s i l y seen from the charac- t e r i z a t i o n of the i n t e r i o r portion of the optimal sharing rule: ^•[(V(x)) - D ] if C * 0 D l n 7(x) i f C = 0, C 2 s(x ,x ) = 1 2 2 where the agent's r i s k aversion function i s -U''(s)/U'(s) = l/(Cs+D ) and 2 *a ( 0 f ( x a i = X "I* f ( + X l '"a ll l> a | a i ) + ( 0 S ( x | x , a , a ( •)) a i 2 1 1 g(x |x ,a ,a (.)) 2 and d i f f e r e n t i a t i o n with respect to a 1 2 1 g ^ 2 1 + W 2 2< l x ) •) 1T0~' i s pointwise for every x^. Thus, even i f U(s) = In s ( i . e . , C = 1) and g(x |x^,a^,a (x )) = g(x2|a (x^)), the 2 2 1 2 optimal sharing rule w i l l not be a d d i t i v e l y separable i n x^ and x of the l ( i ) 8 J x 2 a /g term unless ^ ( x ^ ) = k, a constant, and g t i v e l y separable i n x^ and x . 2 fl 2 because /g i s addi- Lambert (1981, p. 90) has shown i n a similar situation that ^ ( x ^ ) > 0 for almost every xj_. Since for almost every x^, U (x^) * 0, and i t i s unlikely that ^ ( x ^ ) = k (which would require that 2 3E(x-s( •))/3a 2 2 2 = k3 E(U(s(')) ~ V(»))/3a f o r almost every xj^), the optimal 2 sharing rule w i l l almost c e r t a i n l y not be additively separable i n x^ and x . 2 A c o r o l l a r y of this result Is that i f the p r i n c i p a l i s r i s k neutral and the agent's u t i l i t y f o r wealth i s i n the HARA c l a s s , then the optimal sharing rule w i l l not be l i n e a r . Thus, the simple commission schemes often used i n practice are not the most e f f i c i e n t way to motivate a r i s k averse agent when sequential e f f o r t decisions are involved. The presence of the additional decision information, x^, for the agent, which i s the only difference between the sequential e f f o r t problem and the 76 e f f o r t a l l o c a t i o n problem, introduces more complexity Into the sharing rule (I) the m u l t i p l i e r 1^(0 in two ways: depends on x^, the d i s t r i b u t i o n of x X j j a i and x | a 2 2 depends on xj_, and ( i i ) because a ( •) 2 given a ( •) depends on x^, even i f 2 are s t a t i s t i c a l l y independent. 2 The combination of these features precludes an additively separable sharing r u l e . depends on x^ even i f x-jja^ a n c * x |a 2 Note that a ( •) 2 are s t a t i s t i c a l l y independent. 2 two Hence, a 's dependence on x^ Is not due to information that x^ provides about the 2 l i k e l i h o o d of x . 2 Rather, the dependence i s due to a wealth effect (x^ influences the agent's position on h i s or her u t i l i t y curve before the second e f f o r t i s chosen) which the p r i n c i p a l can use to e f f i c i e n t l y motivate the agent. Recall that i n the f i r s t best case, there i s no motivational problem, and therefore the optimal a 2 does not depend on x^ i f the agent i s r i s k averse, the p r i n c i p a l i s r i s k neutral, and x^|a^ and x | a 2 2 are s t a t i s - t i c a l l y independent. I f , on the other hand, x-jja^ and x | a 2 2 are dependent, then a 2 depends on x^ f o r the additional reason that x^ provides information about the l i k e lihood of x . This i s true i n both the f i r s t best and second best cases. 2 The m u l t i p l i e r u (x^) further complicates the sharing r u l e . 2 Intui- t i v e l y , i t i s a measure of the cost to the p r i n c i p a l of the motivational problem for a . 2 The result that ^ ( x ^ ) > 0 for a l l x^ means that no matter what the f i r s t period outcome i s , the p r i n c i p a l w i l l not find i t optimal to induce as high an e f f o r t l e v e l , a , as he or she could have i f there were no 2 motivational problem. 4.4 ADDITIVE EFFORT In this section, the additive e f f o r t situation described i n Section 3.5 i s examined when sequential choice i s allowed^. The p r i n c i p a l i s assumed to be r i s k neutral and the agent i s assumed to be r i s k averse. The agent i s further assumed to have no i n t r i n s i c d i s u t i l i t y for any particular task, but 77 rather i s assumed to have d i s u t i l i t y only for the t o t a l e f f o r t expended. The agent's d i s u t i l i t y i s thus represented as V(a^+a2(•))• In f i r s t best situations, i f X j j a j and x | a 2 2 are independent, the p r i n - c i p a l i s r i s k neutral, and the agent Is r i s k averse, then the optimal a ( •) 2 does not depend on x^ i n the sequential e f f o r t case. Therefore, the f i r s t best results f o r the a l l o c a t i o n of e f f o r t problem s t i l l hold for the sequent i a l e f f o r t problem. In p a r t i c u l a r , i f the means are l i n e a r i n e f f o r t , that i s , the means are given by ka^, then only the sum of the e f f o r t s i s of importance to the p r i n c i p a l and the agent. where k^ * k j , the If the means are given by k^a^, i * j , then a l l the e f f o r t should be put into the task with largest return per unit of e f f o r t . For more general unequal mean func- tions, the optimal solution w i l l involve nonzero e f f o r t s devoted to a l l tasks. If the mean functions are i d e n t i c a l nonlinear s t r i c t l y increasing functions, then the optimal e f f o r t s are equal. The second best case i s quite d i f f e r e n t because of the dependence of a ( •) on X]_. Recall that i n the a l l o c a t i o n problem, assuming an Interior 2 solution, the constraints require that 3EU(s(x)) 3EU(s(x)) 3a^ 3a 2 ' because each of the marginal expected u t i l i t i e s must equal the marginal d i s utility the from the t o t a l e f f o r t , V'(a^+a ). 2 In the sequential e f f o r t case, constraints become 3EU(s(x)) •3a 1 3EV(a +a ( •)) 1 3a 2 (4.4.1) x and 3E U(s(x)) 2 ^ = V(a^+a^i •)) f o r almost every x^, (4.4.2) 78 where E U(s(x)) = / U( s( •) )g(x | a (x^) )dx . Equation (4.4.1) requires aver2 2 2 2 aging over a l l possible values of x^ and x , because a^ i s chosen before 2 Equation (4.4.2), on the other hand, requires either outcome i s available. averaging only over a l l possible values of x , because x 2 i s the only 2 remaining uncertainty at the time the second e f f o r t l e v e l i s selected. Corollary 4.4.1 below applies Proposition 4.2.1, which characterizes the behavior of the second stage e f f o r t strategy, to the additive e f f o r t Proposition 4.2.1 case. assumed that e f f o r t s were defined such that they were the means of the outcome d i s t r i b u t i o n s . In t h i s section, e f f o r t s are assumed to be additive; assuming that e f f o r t s are simultaneously of the outcome d i s t r i b u t i o n s i s overly r e s t r i c t i v e . 4.4.1 the means Therefore, Corollary allows for a more general s i t u a t i o n i n which the means of the outcome d i s t r i b u t i o n s are functions of the e f f o r t s . This accounts for conditions on the second stage mean, M ( •), i n order to characterize the behavior of the 2 second stage e f f o r t strategy. It should be noted that the d e f i n i t i o n of e f f o r t in turn influences the description of d i s u t i l i t y captured i n the d i s u t i l i t y function V( • ) . Thus, conditions on both V(•) and M (») are either 2 i m p l i c i t l y or e x p l i c i t l y required i n order to characterize the behavior of the second stage e f f o r t strategy. Corollary 4.4.1. Assume that the conditions i n Proposition 4.2.1 except that E(xi_|aj_) = M^a^) > 0, E ( x | a ) = M ( a ) > 0, and V ( a , a ) = 2 V(a +a ), with Mj/ > 0 and M ' 1 2 2 > 0. 2 2 2 1 Let ei = M ^ a ^ and e 2 2 > 0, V " > 0, V " ' > 0, and e*(x^) to be decreasing 2 1 1 2 1 1 1 2 = M (a ). induced d i s u t i l i t y function i s then V * ( e , e ) = V ( M ~ ( e ) + M V hold, 2 (e )). 2 i n x-^ i s that 3M " 2 - ^'''M^ be nonnegative at *2ct For example, suppose M^(a^) = a^ , M ^ a ^ , where 0 < a < 1, 0< If ' < 0, then a s u f f i c i e n t condition for a (a +a„) The 8 <V2 » and a 8 = a 2 , and V(a^+a ) = > 0 for 1=1,2. 2 Then 79 a± = = ^ e a a n d e*,(x^) i s d e c r e a s i n g V" = 2 > 0, 2 2 2 0 2 " 4 - = M 2 ^ 2^ e since V 2 0 2 & 4.4.1 = 2(a^+a ) > 0 f o r a^+a 2 2 2 ^^^^Y 2^^' = M' ' = g(3-l)a 8 (S-l)(B-2)a the agent's expected The 2 i n x^, V " » = 0, 3 3 (0-l) a a " 0 2 " 2 < 0, and > 0 f o r g <V2 4 second stage net u t i l i t y 3M ' » 2 2 • shows t h a t * ®> 2 - M ' ' »M ' 2 = 2 By P r o p o s i t i o n 4.2.1, i s t h e r e f o r e i n c r e a s i n g i n x^. remainder o f t h i s s e c t i o n makes Pareto comparisons between e f f o r t s t r a t e g i e s i n the s e q u e n t i a l e f f o r t model, i n which i n f o r m a t i o n becomes a v a i l a b l e a t a f i x e d p o i n t d u r i n g the p e r i o d . lows, the agent has e f f o r t c h o i c e s f o r two observed. at In the d i s c u s s i o n t h a t tasks b e f o r e any information i s A f t e r o b s e r v i n g the i n f o r m a t i o n ( i f a nonzero e f f o r t i s exerted a t a s k , then the a s s o c i a t e d outcome i s o b s e r v e d ) , the agent can choose to b e g i n , c o n t i n u e , o r d i s c o n t i n u e e x e r t i n g e f f o r t a t the two the f i r s t fol- s u b s c r i p t on a and on x denote time ( t h e stage) tasks. and Letting letting the second s u b s c r i p t denote the t a s k s , the s e q u e n t i a l c h o i c e s c e n a r i o under d i s c u s s i o n can be d e p i c t e d as f o l l o w s : a-Q and/or Information a^ exerted x^ 2 Point: and/or x ^ a 2 1 ^ * ) and/or a 2 2 2 x ^ and/or 2 ( •) e x e r t e d x observed 2 2 observed In the a l l o c a t i o n o f e f f o r t s i t u a t i o n d i s c u s s e d i n Chapter 3, agent's e f f o r t d e c i s i o n s are made and outcomes a r e observed. e f f o r t s are e x e r t e d The the e f f o r t s are e x e r t e d b e f o r e s i t u a t i o n may be viewed as one s e q u e n t i a l l y , or simultaneously. outcomes a r e observed In e i t h e r case, thought o f as a s p e c i a l case of the Point. Neither x ^ nor x ^ 2 Is observed the The allo- situation d e s c r i b e d above, where the n u l l i n f o r m a t i o n system i s i n e f f e c t a t Information the i n which the o n l y a f t e r both e f f o r t s have been e x e r t e d . c a t i o n of e f f o r t case can be the the u n t i l the end of the 80 period. The e f f o r t s a ^ and a 2 are thus independent of x ^ 2 2 the end of the period, a^ = a ^ + a ^ and a 2 exerted, and x^ = x ^ + x ^ and x 2 2 = x^ + x 2 = a^ 2 + a 2 2 2 and x^ . At 2 w i l l have been w i l l have been observed. 2 2 Analysis similar to that i n the proofs of the propositions i n Section 3.5 establishes the sequential e f f o r t results below. 4.4.2 Part ( i ) of Proposition deals with situations with e f f o r t devoted to only one task at a time, while parts (II) and ( i i i ) deal with situations i n which e f f o r t i s devoted to more than one task at a time. Proposition 4.4.2. Suppose the p r i n c i p a l i s r i s k neutral. Suppose further that the agent's u t i l i t y for wealth i s the square root u t i l i t y function and that the agent's d i s u t i l i t y i s a function of the t o t a l e f f o r t expended. F i n a l l y , suppose that for i , j = 1,2, the x^j's given the corresponding a^j's are independent, and E(x^j|a^j) = ka^-j, where k i s a constant. (i) If i t i s optimal for the p r i n c i p a l to induce (1) a ^ a 2 1 ^ l l ^ ^ ^» and a x 2 2 > 0, a ^ 2 = 0, ( x ^ ) = 0, then i t i s also optimal for the prin- c i p a l to induce (2) a ^ > 0, a ^ or to induce (3) a-j^ = 0, a ^ to induce (4) a ^ = 0, a ^ 2 2 2 = 0, a ^(x^^) = 0, and a ( x ^ ^ ) > 0, 2 > 0» a 2 22 ^ ( x ^ ) ^ 0» 2 a n d a ( x ^ ) = 0, or 2 2 2 > 0, a ^ ( x ^ ) = 0, and a ( x ^ ) > 0. 2 2 2 2 That 2 i s , i f one of the four combinations of e f f o r t s (1) through (4) i s optimal, then the p r i n c i p a l i s i n d i f f e r e n t among the four combinations. This result holds no matter what the r i s k averse agent's u t i l i t y for wealth i s . Moreover, means that are l i n e a r i n e f f o r t are not required. 2 (ii) If xjLj|a£j i s normally distributed with mean ka^j and variance a , then (a) the best e f f o r t strategy with a ^ and a a^ 2 2 > 0, a ^ 2 = 0, a ^(x]_^) > 0, 2 ( x ^ ) > 0 i s Pareto i n f e r i o r to some e f f o r t strategy with > 0, a ^ 2 = 0, a ^ ( x ^ ) > 0, and a 2 2 2 ( x ^ ) = 0, and 81 (b) the best e f f o r t strategy with a-^ > 0, a ^ > » u a 2 1 ^ l l ' 1 2 ^ ^ ^» x x a n d a 2 2 ^ l l » 1 2 ^ > 0 i s Pareto i n f e r i o r to x x some e f f o r t strategy with and (iii) > 0, a-^2 > 0> 2 1 ^ l l » 1 2 ^ a x x E ^» a22(xn.X]^) ^ * u If x ^ j l a ^ j i s exponentially d i s t r i b u t e d with mean ka^-j, then (a) the best e f f o r t strategy with > 0, a ^ = 0, a 2 i ( x n ) > 0, 2 and a 2 2 ( x n ) = 0 i s Pareto i n f e r i o r to some e f f o r t strategy with a (b) ll ^ ^» 12 a ®* 2 1 ^ l l ^ ^ ^» = a x a n < * 22^ ll^ ^ » a the best e f f o r t strategy with a 21^ ll' 12^ x x E ^» a n c x u a n < * > 0, a^2 > 0» * 2 2 ^ l l ' 1 2 ^ > 0 i s Pareto i n f e r i o r to a x x some e f f o r t strategy with a ^ > 0, a ^ > 0, 2 l ( l l » 1 2 ) > 0» a x x 2 and 2 2 ( l l > 1 2 ) ^ * a x x u The r e s u l t s i n Proposition 4.4.2 can be depicted as follows, where s o l i d l i n e s indicate nonzero e f f o r t , and dashed l i n e s indicate zero (no) effort. The f i r s t l i n e i n each pair of l i n e s represents the second l i n e i n each pair represents the f i r s t task, and the second task, (i) (1) (2) J I a u > 0 I I a 2l( ll) x > 0 a 1 2 = 0 I 22< ll> x ~ 0 I l l a I a I > 0 a I a 12 I 2l( ll) = 0 22( ll> > 0 x L I = 0 The p r i n c i p a l i s i n d i f f e r e n t between (1) and (2). a x Alternative (4) Is similar to a l t e r n a t i v e (2), with the tasks renumbered, and a l t e r n a t i v e (3) i s s i m i l a r to a l t e r n a t i v e (1). 82 ( i i ) (a) (A.) (B) I I &11 > a 0 I 1 2l( ll) > x a I a = 0 a 1 2 I 0 I 22^ ll^ ^ x J ll > a 0 I 2l( ll) > x I 0 a 12 0 I = a 0 I 22^ ll^ ^ x 0 The p r i n c i p a l prefers some form of (B) to the best possible form of (A). (ID (b) (C) (D) l _ I a ll > 0 a I 2l( ll> 12) x I x > 0 _l I a 1 2 > 0 a 22^ ll» 12) x x > I a 0 ll I > 0 a I a 12 I 2l( ll> 12) = x x 0 1 > 0 a ! 22^ ll» 12^ x x > 0 The p r i n c i p a l prefers some form of (D) to the best possible form of (C). The results i n ( i i ) say that whether e f f o r t i s exerted at one or two tasks i n i t i a l l y , a l l e f f o r t should be concentrated i n only one task at the second stage. Because of the assumed independence, i t does not matter which task i s chosen. In part ( i i i ) of the proposition, the results In ( i i ) ( a ) and (b) are reversed. That i s , whether e f f o r t i s exerted at one or two tasks i n i t i a l l y , e f f o r t should be s p l i t across two tasks at the second stage. It i s prefer- able to Induce the agent to d i v e r s i f y e f f o r t after receipt of the informat i o n x^ when the outcomes are exponentially distributed as described, and i t i s preferable not to induce the agent to d i v e r s i f y e f f o r t when the outcomes are normally distributed as described. In part ( i ) of the proposition, d i v e r s i f i c a t i o n of e f f o r t Is not i n question. Because the outcomes condi- t i o n a l on the e f f o r t s are independent and i d e n t i c a l l y distributed, the p r i n c i p a l i s i n d i f f e r e n t among the four alternatives (1) through (4). 83 As i n Section 3.5, the results i n parts ( i i ) and ( i i i ) are partly explainable i n terms of the variances of the t o t a l outcomes. For simplic- i t y , consider a comparison between a fixed amount of e f f o r t , a, devoted to only one task, or divided across two tasks. comes of the two tasks, and l e t ka^ and k a where a^ i s the e f f o r t devoted to task i . individual 2 Let x^ and x denote the out- 2 denote their respective means, Since the means of each of the outcomes are linear i n e f f o r t , the t o t a l e f f o r t expended i s the only quantity of relevance for the purpose of comparing the means of the t o t a l outcomes (x^ i f e f f o r t i s devoted only to one task, and x^+x i f 2 e f f o r t i s devoted to two tasks). For the normal d i s t r i b u t i o n i n part ( i i ) of Proposition 4.4.2, Var(x-jJai=a) = o , and Var(xi-hx 1a]+a =a) = 2o~. 2 2 the exponential distributions i 2 For i n part ( i i i ) , Var(xjjai=a) = k^a^, and 2 2 Var(xi+x |ai+a2 a) < k^a . = 2 For the normal d i s t r i b u t i o n , the variance of the t o t a l outcome i s smaller when a l l the e f f o r t i s devoted to one task, while for the exponential d i s t r i b u t i o n , the variance of the t o t a l outcome i s smaller when a l l the e f f o r t i s divided across two tasks. can be related This observation to the Information content of the outcomes considered as s i g - nals about the agent's e f f o r t ( s ) . The quantity 1(a) = / f (x|a)/f(x|a)dx, called Fisher's information about the parameter a contained i n the data (see, for example, Cox and Hinkley, 1974), i s used as a measure of information content about a i n x. For both the normal and exponential d i s t r i b u - tions described above, 1(a) i s the reciprocal of the variance. Thus, for the normal case, there i s "more" information about the agent's e f f o r t when a l l e f f o r t i s devoted to one task than there i s when the e f f o r t i s divided across the tasks. The opposite i s true for the exponential distribution. Proposition 4.4.2 does not state what the optimal e f f o r t strategies are i n each case. The comparisons i n parts ( i i ) and ( i i i ) are between situa- tions with the same information available at the beginning of the second 84 stage. For example, i n ( i i ) ( a ) , which only the comparison i s between two situations i n i s available at the beginning of the second stage. Compari- sons of situations with d i f f e r i n g information available at the beginning of the second stage are more d i f f i c u l t to make. 4.5 SUMMARY AND DISCUSSION This chapter examined the problem of sequential e f f o r t decisions within one period. The sequential aspect arose because the agent observed an out- come affected by the f i r s t e f f o r t choice before making the second e f f o r t choice, which affected a second outcome. The agent was paid only a f t e r both e f f o r t s were exerted and both outcomes were observed. In the f i r s t best case, the characterization of the optimal sharing rule i n the sequential e f f o r t case i s similar i n s p i r i t to that i n the a l l o cation of e f f o r t case. That i s , i f one person i s r i s k neutral and the other i s r i s k averse, then the r i s k neutral person bears the r i s k . If both the p r i n c i p a l and the agent are r i s k averse, then the r i s k i s shared, with the sharing rule a function of the sum of the outcomes. The f i r s t best characterization of the optimal e f f o r t s i s different i n the sequential e f f o r t case than i n the a l l o c a t i o n of e f f o r t case. The sec- ond e f f o r t choice may now depend on the f i r s t outcome and the f i r s t choice. effort If both of the individuals are r i s k averse, then the optimal second stage e f f o r t strategy w i l l depend on x j _ , the f i r s t outcome. The second stage e f f o r t strategy w i l l also depend on x^ i f the joint density of the two outcomes given the actions i s f ( x ^ | a ^ ) g ( x | x ^ . a ^ , a ( • ) ) • 2 2 However, i f at least one of the individuals i s r i s k neutral and the joint density of the two outcomes given the actions i s f ( x ^ | a ^ ) g ( x , a ^ , a ( • ) ) > then the o p t i 2 2 mal second stage e f f o r t strategy w i l l be independent of x^. The second stage e f f o r t strategy may depend on x^ because of a "wealth" ("risk aversion") e f f e c t , or because of an "information" e f f e c t . The wealth 85 effect occurs when both individuals are r i s k averse, because a r i s k averse individual's marginal u t i l i t y varies at different points of the u t i l i t y curve. The f i r s t outcome determines where on the u t i l i t y curve the i n d i v i d - ual i s , so the individual w i l l want the second stage e f f o r t adjusted accord- ing to the value of the f i r s t outcome. the two outcomes are dependent. The information effect occurs when Depending on the nature of the c o r r e l a t i o n between the two outcomes, the p r i n c i p a l may wish to induce the agent to choose the second stage e f f o r t strategy to be an increasing or decreasing function of the f i r s t outcome. Proposition 4.1.1 provides a precise expres- sion for the derivative of the second stage e f f o r t strategy with respect to the f i r s t outcome. The analysis i n the second best case allowed for nonindependence of the outcomes. As usual, the p r i n c i p a l was assumed to be r i s k neutral and the agent was assumed to be r i s k averse. The characterization of the optimal sharing rule i n the sequential e f f o r t case i s similar to the characterizat i o n i n the a l l o c a t i o n of e f f o r t case, except that the multipler u e f f o r t strategy a x l» 2 a D e 2 may depend on x^. Although i n general, a 2 a n d t n e 2 w i l l depend on independent of x^ i f the agent i s r i s k neutral and the j o i n t density of the outcomes i s of the form f ( x ^ | a ^ ) g ( x | a ( • ) ) • 2 Proposition 4.2.1 2 assumed a square root u t i l i t y function for the agent and c o n d i t i o n a l l y independent outcomes given the actions. It was shown that the agent's second stage e f f o r t strategy w i l l be decreasing i n x^. Intui- t i v e l y , t h i s i s because the higher x^ i s , the more c o s t l y i t i s for the p r i n c i p a l to induce any p a r t i c u l a r l e v e l of a . 2 ginal u t i l i t y The agent's decreasing mar- for wealth and increasing marginal d i s u t i l i t y for e f f o r t account for the increasing costliness of inducing a . 2 Since these charac- t e r i s t i c s hold i n general, the results i n Proposition 4.2.1 other u t i l i t y functions. should hold for 86 The c a s e rule which generally result the strictly first perfectly incorporates outcome i f about of the correlated a penalty wage f o r i n a second e f f o r t state is outcome i n in the the perfectly behavior is It ing was rule effort 2 next that case sequential is will information, x^, The f i r s t the additively separable guarantee case. best in results Section in t i o n case because of information tions ing for the under which i n xj_, the dary versus the interior with varying of the case The c o n d i t i o n a l sequential effort the that separable the hold for the sequential the differ first choice. in effort is strategy Section 3.5 obtain degrees of diversification information investigation In the in of the effort effort in as case. the alloca- provides will effort. of be condidecreas- the to The the bounthe effort results information effort. problem i s conditional somewhat related investigation The pre-decision Finally, a measure of decision were a p p l i e d of shar- allocation P a r e t o c o m p a r i s o n s between statistic, agent's 4.4.1 additive. correlated. rule. from those Corollary the an o p t i m a l allocation outcome p l a y s the The sharing rule the case of strategy. the sharing first be override additional separable second stage the in the may imperfectly outcomes to problem. effort for sequential outcome about therefore agent's in order Fisher's strategy results solution results strategies content 3 also role can presence of o u t c o m e , when e f f o r t effort to the effort agent's sequential were r e l a t e d in in to information c o n d i t i o n s which guaranteed second e f f o r t the first the reveals sharing shown decreasing outcomes a r e an a d d i t i v e l y additive state case an a d d i t i v e l y Thus, precludes problem analyzed second best shown t h a t is A The i n f o r m a t i o n e f f e c t the more c o m p l e x when t h e not effort of examined. second e f f o r t correlated changing the behavior a the next s e c o n d s t a g e was that the outcome. wealth e f f e c t , of If then first the strategy random n o i s e . production technology, increasing o u t c o m e s was to the problem, the 87 agent exerts e f f o r t , and both the p r i n c i p a l and the agent observe the outcome x. The p r i n c i p a l then has the option of observing y, an additional signal about the agent's e f f o r t . The agent's compensation i s s(x) or t( iy)» depending on what was j o i n t l y observed. x Cost variance Investiga- tion, a f a m i l i a r problem i n accounting, has been modeled as a conditional investigation problem (see, for example, Baiman and Demski, 1980a,b) i n which x i s a cost and y i s the result of an investigation to t r y to determine the reason for the cost's deviation from a preset standard. The prob- lem i s similar to the sequential e f f o r t problem i n that decisions are based on an i n i t i a l outcome. However, a f t e r the i n i t i a l outcome, the p r i n c i p a l chooses an act i n the conditional investigation problem, and the agent chooses an act i n the sequential e f f o r t problem. The major focus i n the conditional investigation problem has been on the determination of the o p t i mal investigation strategy; such a question i s not at a l l relevant i n the sequential e f f o r t choice problem. Some additional comments about the condi- t i o n a l investigation problem w i l l be made i n the next chapter. As remarked at the end of Chapter 3, the sequential e f f o r t case can be viewed as a s p e c i a l case of the two-period agency problem i n which the p r i n cipal's and the agent's expected u t i l i t i e s depend only on the t o t a l return over the entire time horizon. Thus, the sequential e f f o r t results have potential applications i n such multiperiod s i t u a t i o n s . 88 CHAPTER 5 SUGGESTED FURTHER RESEARCH This chapter concludes the thesis with suggestions for further research. The f i r s t section discusses possible extensions to t h e o r e t i c a l agency r e s u l t s , and the second section discusses possible applications of the agency theory results to a t r a d i t i o n a l accounting topic, cost variance investigation. 5.1 THEORETICAL AGENCY EXTENSIONS A number of generalizations of the results In this thesis are d e s i r - able. For example, i n the a l l o c a t i o n of e f f o r t setting with additive e f f o r t , i t i s desirable to obtain r e s u l t s for a more general class of u t i l i t y functions and for nonindependent d i s t r i b u t i o n s of incomes. A similar remark holds for some of the results i n the sequential e f f o r t setting. The situation with multiple agents was discussed b r i e f l y i n Section 3.6, where the agents were salespeople i n a firm. The important problem of c o l l u s i o n among agents i n order to conceal shirking or the theft of assets has l a r g e l y been unexplored. Beck (1982), however, has recently taken an incentive contracting approach to the problem of c o l l u s i o n for the purpose of concealing the theft of assets. As remarked e a r l i e r , many accounting addressed i n a multiperiod s e t t i n g . and other business issues are best Lambert (1981, 1983) has analyzed special case of the multiperiod agency problem i n which u t i l i t i e s tive over time and the outcomes are independent. analyzed a are addi- Chapter 4 of this thesis a d i f f e r e n t special case of the multiperiod problem. The analysis allows for nonindependent outcomes, and assumes that the agent i s paid only at the end of the time horizon, even though the e f f o r t choices and observations for of the outcomes are sequential. short-term the The analysis i s thus suitable horizons i n which the p r i n c i p a l and the agent are concerned 89 only with t h e i r t o t a l shares at the end of the time horizon. more general multiperiod situations are desirable. Results for These situations are, of course, more d i f f i c u l t to analyze. 5.2 APPLICATION TO VARIANCE INVESTIGATION A great deal of attention has been focused on strategies for investiga- ting the underlying causes of cost variances or deviations from standards. Most of the a n a l y t i c a l research has assumed that investigations reveal the state of a mechanistic production process, and that the investigator can return an "out-of-control" state to an " i n - c o n t r o l " state (Kaplan, 1975). Thus, only the correctional purposes of investigations were examined. Cor- rectional benefits occur, for example, when costs are higher for a malfunctioning machine than for a properly functioning machine. In some s i t u a t i o n s , the primary focus i s on evaluating a manager has control over a mechanistic process. In such s i t u a t i o n s , there may who be motivational as well as correctional benefits to investigating variances. The manager's actions can be influenced by the p o s s i b i l i t y of an investigation i f a reward or penalty i s based on the results of the investigation. The motivational purposes of investigations have recently come to attention in the a n a l y t i c a l l i t e r a t u r e . explored Baiman and Demski (1980a, 1980b) have the motivational aspects of variance analysis procedures i n a period agency model, with a single-dimensional the analyses, effort variable. the agent i s responsible for a production one- In both of process which gener- ates a monetary outcome determined by the agent's e f f o r t and some exogenous randomness. The monetary outcome, owned by the p r i n c i p a l , i s assumed to be j o i n t l y observable, while the agent's e f f o r t i s not. The p r i n c i p a l can, however, conduct a costly investigation i n order to obtain a further imperfect signal which i s independent of the outcome but informative about the agent's e f f o r t . The nature of the investigation strategy was characterized, 90 and the use of the information for motivational purposes was demonstrated. Lambert (1984) extended the analysis by allowing for a nonindependent addit i o n a l signal about the agent's e f f o r t , and showed that the investigation strategy would d i f f e r from that obtained by Baiman and Demski. A number of extensions to the Baiman-Demski analysis are possible. extension i s to allow for multiple e f f o r t decisions by the agent. One Feltham and Matsumura (1979), for example, suggested three d i f f e r e n t e f f o r t decisions the agent might be responsible for: control at the beginning 1) bringing the system back into of the period a f t e r detecting that I t i s out of control; 2) keeping the process i n control during the period given that the process i s i n control at the beginning of the period; 3) influencing or con- t r o l l i n g the operating costs or the outcome during the period. Their analy- s i s did not focus e x p l i c i t l y on the tradeoffs between the e f f o r t s expended by the agent. Instead, the focus was on characterizing the optimal investi- gation strategy and sharing rule for an i n f i n i t e - h o r i z o n Markov process. Another extension to the Baiman-Demski analysis i s the extension to multiple periods. horizon model. One approach would be to extend the analysis to a f i n i t e - Another approach would be to extend the analysis to an i n f i n i t e - h o r i z o n model. It has been argued that i n f i n i t e - h o r i z o n multi- period problems involving two players, the factor that overshadows a l l others i s the players' knowledge that they have arrived at the last play. 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John Wiley & Sons, Inc.: Mirrlees, J . , "Notes on Welfare Economics, Information and Uncertainty," i n M. Balch, D. McFadden, and S. Wu (eds.), Essays on Economic Behavior Under Uncertainty. North-Holland (1974). Peng, J . , "Simultaneous Estimation of the Parameters of Independent Poisson D i s t r i b u t i o n , " Ph.D. d i s s e r t a t i o n , Technical Report #78, Department of S t a t i s t i c s , Stanford University (1975). Radner, R. and M. Rothschild, "On the Allocation of E f f o r t , " Journal of Economic Theory (1975), 358-376. Rubinstein, A., "Offenses That May Have Been Committed by Accident - an Optimal Policy of Retribution," i n S. Brams, A. Schotter, and G. Schrodiauer (eds.), Applied Game Theory. Physica-Verlag: Wurzburg (1979). Shavell, S., "Risk Sharing and Incentives i n the P r i n c i p a l and Agent Relationship," B e l l Journal of Economics (Spring 1979), 55-73. Steinbrink, J . , "How to Pay Your Sales Force," Harvard Business Review (July-August 1978), 111-12 2. S t i g l i t z , J . , "Incentives, Risk, and Information: Notes Toward a Theory of Hierarchy," Bell Journal of Economics (Autumn 1975), 552-579. Weinberg, C., "An Optimal Commission Plan for Salesmen's Control Over Price," Management Science ( A p r i l 1975), 937-943. , " J o i n t l y Optimal Sales Commissions for Nonincome Maximizing Salesforces ," Manajement_S£ience_ (August 1978), 1252-1258. Zimmerman, J . , "The Costs and Benefits of Cost Allocations," The Review (July 1979), 504-521. Accounting Appendix 1 Table I I One-parameter Exponential Family Q f(x|a) = exp[z(a)x - B(z(a))]h(x) Exponential f ( x 1 I v c t ' . -x1 . MUT^W a ) Normal 1 "^= e X-(x-M(a)) Pl_ 1. 2 E(x|a) M(a) M(a) Var(x|a) M ( ) o 2 a Gamma X x " e" n n 1 Poisson Binomial , ._(M(a)) expI-MCa)]^^- Xx x M r(n) / M(a) (= n/X) ,n M(a) ( x )<-^) M(a) M(a) 2 X w d" M(a)"" -^0 M(a) M ( a ) [1 - Hll] n z ( a ) ~ MTaJ X n " HTaT In M(a) l n M(a) - In (n-M(a)) a B(z) B*(z) -In (-z) , ( , n) n In z v 1 -z 2 oz n -z 1 -2 n 7 Note: 2 —a z 2 2 ° 7 , exp (z) x v n e nz - n ln[ i z ] z 1 +e r 1 _i_ , exp (z) v r . e * .. p<z) ne z z 1 +e ne Z ^ 7 The exponential d i s t r i b u t i o n i s a special case of the gamma d i s t r i b u t i o n but i s l i s t e d separately because of i t s wide use. 95 The following calculations for the one-parameter exponential family, Q, given i n Table I I , w i l l be useful i n the proofs of the results i n Chapters 3 and 4. _a f d ln f da = ^ - [z(a)x - B(z(a))] da z'(a)x-B'(z(a))z (a) , z'(a)[x-B'(z(a))] (Al.l) z'(a)(x-E(x|a)). f (x|a) / ( | dx = / z'(a)(x-E(x|a))f (x|a)dx f x a ) a z ' ( a ) f / xf(x|a)dx - z ' ( a ) E ( x | a ) / f (xla)dx da ' ' a 1 J z'(a)^- B'(z(a)), aa since E(x|a) =B'(z(a)) and / f (x|a)dx = 0 = (z'(a)) B"(z(a)). 2 f^(x|a) (A1.2) f ) f(x|a)dx f (x|a) da Note that (z'(a*)) 2 / (x-E(x|a*)) f(x|a)dx 2 / (x-E(x|a*)) f(x|a)dx 2 = / (x-E(x|a) + E(x|a) - E(x|a*)) f(x|a)dx 2 = / (x-E(x|a)) f(x|a)dx + (E(x|a) - E ( x | a * ) ) 2 2 - 2(E(x|a) - E(x|a*)) / (x-E(x|a))f(x|a)dx = Var(x|a) + (E(x|a) - E ( x | a * ) ) . 2 96 Therefore, f!(x|a) dx = ( z ' ( a * ) ) H ^ 2 (B"(z(a))) f (x|a) a* (A1.3) = (z'(a*)) B"'(z(a*)). 3 f (x|a) / f 3 f(x|a) aa z'(a*) - i y / (x-E(x|a*))f(x|a)dx da (xla)dx = z'(a*) - \ da (E(x|a) - E(x|a*)) = z'(a*) B'(z(a)) da = z'(a*) (B"(z(a))z»(a)) = z ' ( a * ) [ B ' " ( z ( a * ) ) ( z ( a * ) ) + B"(z(a*))z»'(a*)]. , 2 Second Best, Additive E f f o r t Example (Section 3.5) Suppose the p r i n c i p a l i s r i s k neutral, <Kx ,x |a ,a ) = f ( x | a ) f ( x | a ) , 1 2 1 2 1 1 2 V ( a , a ) = V ( a + a ) , and 1 2 1 lAs) = 2JT~ Then 2 . s * ( x x ) = R (a*), where l f 2 2 R(a*) - X + ^ ± f a ( f x il l> a i x ^ 2 (A1.4) Let K(s,a_) = / 2 R(a*)4>(x|a) dx EU(s*,a) = K(s*,a) - V ( a and the p r i n c i p a l ' s expected / (x G(s*,a) = The + x L Hamiltonian, with : . + The 2 utility is <Kx|a) dx. 2 s*(x), i s 2 H = G(s*,a) + The 1) A(EU(s*,a) - u) + 3EU(s*,a) Z u k=l — k f i r s t order c o n d i t i o n s are 3G + X»0 + That i s , which imply G a, + % + u,K u Z . , i=l G and »l< ^ + ( - 0, i 3a. 9a . i J " V") K a a K a i a ' "> V 2 + + j=l,2. ^ ( ^ - V") ^ 2 ^ " "> i. a, a, + 11 u»K Hi. a, a- = G n~2 = 0 , j-1,2. =° V • a_ "2 + U.K T. a, a„ *1~2 + P~K a„a 2°2 at a* = ( a i * , a * ) 2 I.e., / 2R(a*) f ( x | a ) f ( x | a ) d x ^ - / 2R(a*) f ( x | a ) f ( x | a ) d x ^ - V (a _ 1 and = 0 that where a l l the f u n c t i o n s are e v a l u a t e d 2) utility a ) - R (a*)) 2 agent's expected ] > 1 1 2 2 2 2 + a ) = 0 + a ) = 0 2 , ] 2 98 Thus, 3a, J [ X+ E i=l 2 a f ] f ( x | a ) f ( x | a ) dx dx 2 ] f ( x | a ) f ( x | a ) dx dx 2 1 f ( 0 1 2 2 1 '> ( 1 1 2 2 L , which implies that f 3a, ' U l ^ I ' V — 3 — since a d x i = 3 -3a7 1 : y f(x„ a„) dx„ 2 2 2 v i • ! V L - T ^(^1^) l Therefore, 1^ / - j — 2 2 f 3 r d x i = for i * j . 0 a d x l = (A1.6) dx « "2 / 2 Let J denote the quantity i n (A1.6) G can be written as a l -JL [ECxJa^ E(x |a )] + 2 * 2 / [ X + 2X^ f f a + 2X 2 "2 — ! 2 ^ + 2 } [ ^ 2 l a 2 .-r-] — ] •f(x | a ) f ( x |a )dx dx ][ 1 2 2 1 2 + 2 I * l 99 9E(x |a ) 1 1 f(x |a )dx 1 . 2 9 /• a j l 1 1 } f(x |a )dx 1 f f a, 1 1 a. fCxJa^ f(x |a)dxdx ] 2 2 1 2 The last term In the sum i s 0 when evaluated at a* because x^ and x 2 are conditionally independent and f ^ l a , * ) ^ f(x |a *) 2 2 ' f< x l 2* a 2 ) d x 2 " °' Thus, at a*, SECxJap - 2XJ - u dx^. 2 f(x |a )dx + 1 1 1 * "i 2 9a, l-r f a a 1 ( x 1 l l | a ) d x l 1 1 / f(x |a )dx ]} 2 2 2 100 and K a f(x |a )dx = K l 2 a a 2 2 2 ] = 0 2 l a Therefore, (A1.5) can be written as 3E(x |a ) 1 2 , 1 - 2 XJ l a dx + 2 2 i a l f a ( i a i x l! l a ) d l x a* 9E(x |a ) 2 2 - 2XJ - 9a U r a a f 2 ( x 2 2 2 a f 2l 2 a ) d x 2 dx„ (A1.7) 2 For the exponential family, using the results i n Table II and equations (A1.2) through (A1.4), (A1.7) can be written as B"( (a *))z'(a *) - ^[(z'Ca^)) 2 1 B ' " ^ ^ * ) ) - 2(z'( *)) 3 1 B "(z(a *)) 3 , a i 1 - 2z'(a *)z"(a *)B"(z(a *))] 1 1 1 = B"(z(a *)z'(a *) - ^ [ ( z ' ( a * ) ) 2 2 2 3 B"'(z(a *)) - 2 ( z ' ( * ) ) 2 - 2z'(a *) z " ( a * ) B " ( z ( a * ) ) ] , 2 or 2 2 a ? 3 B " ' (z(a *) ) 2 101 B"( (a *))z'(a *) + ^ [ ( z ' ( Z 1 1 a i *)) B'"(z( *)) 3 a i + 2z'(a *)z"(a *)B"(z(a *))] 1 1 1 = B"(z(a *))z'(a *) + ^ [ ( z ' ( a * ) ) 2 2 B"'(z(a *)) 3 2 2 + 2z'(a *)z"(a *) B"(z(a *))] 2 2 (A1.8) 2 Equation (A1.6) can be written as (see equation (A1.2)) W (z'(a *)) L 1 2 B " ( z ( * ) ) = P2(z'(a *)) a i 2 2 B"(z(a *)) 2 . (A1.9) 102 Appendix 2 Normal Distribution Calculations This appendix contains calculations for the bivariate normal distribution, the only d i s t r i b u t i o n with a convenient representation for dependent random variables. I 2 X o-^a) p(a) 0 ^ ( 3 ) 0 - 2 ( 3 ) Suppose x ~ N( 6 ( 3 ) , £(§)), with E = p(s) 0^(3)0-2(3) \ where s_= (.a l t 6(a) = ( e ^ a ) , 9 ( a ) ) . and a ) 2 T 2 1 f (x ,x, |a. , 3 , ) = 1 x B = l" l 2 9 X l" l 9 — — X 2" 2 9 x 2" 2 2 a l °1 a 2 =— ], where (A2.1) 9 i ) - 2p(-L_^)(^—±) + (-^-^ Z 2 Then B exp[ 2 o" (§) • °2 Let D denote the argument i n the exponentiation i n (A2.1). The following quantity plays an important role i n the determination of the optimal sharing rule. 9f / f aa. 1 9 9 - 5 — [- log 2TT - log 9a log f = 9a 1 a - log 2 o_ 6 1 x 6 s 1 - \ l o g ( l - p ) + D] 2 (1) (1) (1) \ °2 - — + — + D(1) , °L °2 1-p p p V 2 ; e 103 where the superscript ( i ) denotes d i f f e r e n t i a t i o n with respect to a^. -1 x CD [ i l l ^ B ^ B ^ (1-PV ] and 1-p* r i -^V^rV^ (i)x r e i e 1 o^ x 2 1 _ e 2 2 -9{ a -(x -9.)a5 x,-9 1) 1) 1 - 2 P [ — i — 9 1 n-^-i 2 l) 1 at x.-9 -g-^^-Cx^-g )a\ ~ 2p[4-^1[ ] 2 a. 1 -t£ > -(x -e ) a j x_-9 1 0 + n—r n— 1 ) 7 1 • jL 2 af 2 Case ( a ) : 9(a) =. ( ^ ( a ) , f cf, ov, 1 2 9 (a)) T 2 . , 2 1-p , , 2 2 (1-p ) „ ., 2. 2(l-p ) Since f i s symmetric, simply replace the ( l ) ' s with (2)'s to get f (2) (2) _!?. . _ A - . ^ f °L / 2 _ + ^ _pP\ 1-P 2 P P 0 ( N 2 ) B (1-P ) 2 2 _ 1 B (2) 2(l-p ) 2 The optimal sharing rule i s not separable i n x^ and x 2 and i s not a com- mission scheme. Case (b): p = constant * 0. The optimal sharing rule w i l l c l e a r l y not be a commission scheme. Of course, i f p H 0, the optimal sharing rule w i l l be separable i n x^ and x 2 but w i l l not be a commission (linear) scheme. 104 Case ( c ) : p = constant and a 8 - r • r 2 = constant. (x -9 ) 2 a r t 1-p p ( " -^r (^> x -e ) ( 2 e^c^-e^) + • 2 a a 2 12 + 1-p ) 2 1 2 + J . 2 °2 The optimal sharing rule w i l l be a commission scheme with the c o e f f i c i e n t of X l e q u a l t (l) 0 fl r 1 . 1 2 [ 1-p n f l / (l) 1 * M,( l v Q w 2 —") «— a. 7 2o o^ 1 2 (2) (2) A P ^ i 2 + 2U „ ( 2 o V Oj a,— o—_ ) 1 2 J . The c o e f f i c i e n t of x i s 2 P . 2 1 1-p 2 a 9< 9 ^ a. a ' 1 2 2 K 2 2) a 2 a. a 2 V p0< > 2 0 1 2 Lemma 2A.1 characterizes some properties of the optimal second best solution f o r p a r t i c u l a r u t i l i t y functions when the d i s t r i b u t i o n of the outcomes i s bivariate normal. The calculations i n the proof w i l l be useful i n the proofs of propositions i n Section 3 . 5 . k Lemma 2 A . 1 . Suppose (x^, X 2 ) ~ ^ ] l l a N C A ^ » 2 CT where E = ( 1 p o l °2 _ ) . Suppose further that the p r i n c i p a l i s risk neui o 1 l J0^ t r a l , U(s) = lis, and V(a_) = V(a^ + a.^)' assuming the i n t e r i o r characp a a T n e n 105 t e r i z a t i o n of the optimal sharing rule, s(x ,x_) = (X + Ey.f /f ) , i s 2 1 v a l i d for almost every (x^ ,^2), the following results hold: (1) a j * > 0 , &2* > 0 , k^ = k • and (2) k^*k or a * = 0 . V S — In t h i s case, V =[ W Pk (x -k a ) 1 " 2 2 2 = 2 2 ~ 1 2 2 2 = (X+ 2 £ yf /f y ^ Let ) ^ pk (^(I-P) y2k . 2 5— and x = o (l-p ) Z 2 ^1^1 2 2" a (l-p ) — • a^l-p ) EW = E(x^ + x - s(x!,x )) C 2 = 2 l / d - P) . oTcZ 0 s(xi ,x ) > ' ]/(1_p PkjC^-k^) 2 2 [ 2 i 1 k (x -k a ) 2 2 dTdZ 0^ I " 2 0. 2 Proof of Lemma 2 A . 1 . a 2 implies that the optimal solution i s a boundary solution, 2 i . e . , a^* = and = o" imply that y^ = y « 2 2 2 l 3 l + k a 2 - E( X + = k l 3 l + k a 2 1 - w E( X + 2 6 2 = k 2 "^h principal's expected return i s C L a k l 3 l " o, k l a 1 l * + c g n C A 1 + 2 k 2 2 3 " ) k 0. 2 2* a 2 106 Letting k = ± V r*V i * x C and y ± = £ i~ i i k a , g the p r i n c i p a l ' s expected i return can be written as 2 l + A ) ^ - E( i a i + k a 2 - (X + A 2 - < a, E k i i a + k C x 2 " l 2 t 2> 2 C 2 2 " ( x+ a - C k + k a 2 i a i 2 i A C + A i i + A ) -i-i L = k. SEW 3a, 2 a, > 1 Cov(y ,y ) - C 1 2 - C 2 2 - 2C C p 1 (A2.2) 2 , 1-1,2, and pk p pk V [ a (l-p') " i " 1 , i,j-l,2, i * j. j ± If x^ and x >< C 2 2 Ak 3EW 3a, 2 1 L (yi) 2 2 - 2(A + A V a r Var(y ) - 2 0 ^ k ± c 2 C )^ L o " i ) 2 - (A + Aj_ + A ) 2 3EW k 2 a g < E l l k X k are independent, then p = 0 and = k - 2Ak p /a 2 ± 1 2 i Letting a = a^ + a , the agent's expected u t i l i t y i s 2 EU = 2 E( A + £ p.f / f j 2 k a - k.a * 1- k a 2 2 2 i, . 2 1-p 2 a 2 k v v l h 2 a 2 P2k (- 2 ~ vuk p p.k. P - k a * 1 " P 3EU 3a I ) - V(a) a 2 2 ^ j V ? 2 a l 2 0 p k p -) ] " V(a) 1 1 °i °o 1 2 •) - V'(a), i,j=l,2, i * j. 107 2k C = — i V'(a), 1=1,2. (A2.3) a The f i r s t order conditions require that (A2.3) i s zero for 1=1,2. There- fore, l l k C a If k^ = k \ that C = o^, then (A2.4) implies that implies that Lemma 2A.1. 2 2 °2 and 2 k = = C , which i n turn 2 (assuming p * + 1). This establishes result (1) of Note that i f p = 0, then setting (A2.3) equal to zero shows > 0 and > 0, since V (a) i s assumed to be p o s i t i v e . The Hamiltonian i s H = EW + X(EU-u) + E \i 3EU/3a J-l jw. •—- = k - 2[ X + i fe ± j = j . J 2 k.C. k.C, E (a.-a *) ] j j c 1 CTj - V"(a) E u j-l ± i-1,2. (A2.5) J 3H Setting — = 0 for 1=1,2, and l e t t i n g P denote the quantity i n (A2.4) 1 6 3 yields k t - 2P[ X + P(a +a x = k 2 It i s impossible 2 - a *-a *) ] 1 - 2P[ X+ P(a +a 1 2 2 - *-a *) ] . a]L (A2.6) 2 to s a t i s f y equation (A2.6) unless k^ = k , which estab2 l i s h e s result (2) i n Lemma 2A.1. Q.E.D. 108 Appendix 3 Chapter 3 Proofs Proof of Proposition 3.1.1. The p r i n c i p a l ' s problem i s Maximize EW(x-s(x)) = / W(x-s(x)f(x|a)dx subject to EU(s(x)) - V(a) = u. The f i r s t order condition for s*(x) requires that - W'(x-s*(x))f(x|a) + X U'(s*(x))f(x|a) = 0, or W'(x-s*(x)) = XU'(s*(x)) . This implies that x - s*(x) = W' (X U'(s*(x))) -1 = T(s*(x)), with T'(s*) > 0 since X > 0. Therefore, x = T(s*(x)) + s*(x) =Y(s*(x)), with Y'(s*) = T'(s*) + 1 > 0 . Thus, s*(x) = Y ( x ) . - 1 Q.E.D. Lemma 3A.1 below w i l l be used i n proving Proposition 3.2.1. Lemma 3A.1: Suppose f(x|a) = expected u t i l i t y n II f (x. |a.) and that the risk-averse agent's i=l i s pseudoconcave i n a. Suppose further that F a (x.|a.) < 0, with s t r i c t i 1 3EW x^-values. Then for i = l , . ..,n, i f Proof of Lemma 3A.1: (1) Inequality for some 1 > 0 . < 0, -g^— The f i r s t order conditions are (x|a*)dx + £ u.* { jb(s|x)f _ ( «)dx - V i i j i j j=l i=l,•..,n, and /W(x-s(x))f a 2 a 3 a 3 } = 0, 109 tW'(r(x)) n/ i w fa. (xia*) -'- n U'(x-r(x)) j j f(x|a*) f ( x . a.*) ] 3 « a = A* + Z u.* — r f (x.|a.*) j-l J 1 3 J J because of the independence assumption. Here, subscripts a^ and aj on f ( •) and V( •) denote p a r t i a l d i f f e r e n t i a t i o n with respect to a^ or a j , tively; A* and n *, j=l,...,n, are the optimal values of the multipliers i n the second best problem, and r(_x) = x-s(x). Suppose some < 0. Without loss of generality, l e t j = l . Consider the following a u x i l i a r y problem: Max S J W(x-s (x))f(x|a*)dx + A* [ / U(s (x))f(x|a*)dx - V(a*)] x x A n + Z \x* [ j U(s,(x))f (x|a*)dx - V (a*) ] , J=2 3 3 where a*, A* and i ^ * , . . . , ^ * terized by (1) and ( 2 ) . For x E X = { x with X j such that f ^ x ^ a j * ) > 0 } , 1 + r-yr- U'(x-r(x)) correspond to the optimal solution charac- Let r^(x) = x - s^(x). „,/ / W'(r(x)) rrrp- respec- f (x.|a.*) a. 3 3 j n = A* + Z U.* < A* ( x . | a, . *^) j=l . . 3 f-3. J J J J W'(r (x)) A U'(x-r (x)) * A f (x.|a.*) a. 3 3 j n + 1 Z \1 * j=2 . _ j T 3- f_v (x.|a.*) J J J 110 W(r(x)) Note that 777-; . . U'(x-r(x)) N r ^(5) * s a n i s decreasing i n r(x) for every fixed x. — — Further, increasing function of x^, since 3r 3r W " ( r ( x ) ) _ J u ' ( x - r ( x ) ) + W < •) U " ( • ) ( ! - ^ ) x x 2 1 U' L ° = Z ^X W'U'' implies that ^ - = „, .y, , , , > 0. r + W D Now W'(r(x)) r r r 7777 W(r(x)) 7~~\T decreasing i n r W'(r (x)) x < U'(x-r(x)) 7~sT 7777 a n 7777 d U'(x-r^x)) U'(x-r(x)) implies that r(x) > r^(x), for a l l x e X^ . Correspondingly, + r(x) < r (x) on X j _ = { x with Xj^ such that f ^ C x ^ J a ^ ) x Therefore, /W(r(x))f (x|a*)dx - / w ( r , ( x ) ) f a. ^ }. (x|a*)dx A cl ^ = t ( <*>> - W(r (x)) ] f ^(x|a*)dx + L [ W(r(x)) - W(r (x)) ] f ( »)dx > 0. l l+ w < 0 r a x X A a (x|a*)dx > 0. It remains to show that / W(r,(x))f A 3. ^ ~~* The left-hand side of ~" the expression can be written as j [ / 1 ... / W ( r ( x ) ) f ( x | a * ) . . . f ( x | a * ) d x . . . d x 2 n 2 x x x n 2 2 n n 2 n ] f^^ 1 |a *)dx 1 ]L = L T(x )f ^(x, |a *)dx > 0, as i n the one dimensional case, because of X, l a , 1 1 1 n 1 1 Ill stochastic dominance and the fact that T'(x,) = f W 1 J 3r . « f ...f dx ...dx > 0 . 3x^ 2 n n 0 Q.E.D. Proof of Proposition 3.2.1: Let A = / W(x-s(x))f(x|a)dx and B = / U(s(x),a)f(x|a)dx. Subscripts i and j on A and B w i l l denote p a r t i a l d i f f e r e n t i a t i o n with 3H respect to a^ or a j , respectively. for n=2 are (1) A (2) A + : + 2 The f i r s t order conditions = 0 + UpB^ = 0 and P B L + 2 1 = 0, where the functions are evaluated at the optimal a* and with the optimal s*(x). In matrix notation, A + B y = 0, l " l 0 l l 12 where A = ( ), B = ( ), y = ( ), and 0 ( ). A B B y2 0 A B B 2 2 1 2 2 If B i s s t r i c t l y concave i n a, then | B | * 0 and B -1 1 B 22 B 12 A I.e., A (3) ^ = (4) U2 2 12 B A l 22 B iBl V21 " Vll rgi and l - 1 exists. Therefore, 112 If B i s s t r i c t l y concave i n a_, then | B | > 0 and B ±i Now assume < 0 and < 0. < 0, i=l,2. Then by Lemma 3A.1, A^ > 0 and A 2 > 0. From (3) and (4), we have A 2 12 ~ 1 22 < 0 A 1 21 < B B A B " 2 11 A B These imply that B a n d °* 1 2 < 0 (note: (2) cannot be s a t i s f i e d . B 1 2 = B ). 2 1 But i f B Therefore, not both 1 2 < 0, then (1) and and u can be nonpositive. 2 Q.E.D. Lemma 3A.2 below deals with the problem of a l l o c a t i n g e f f o r t to two tasks considered Lemma 3A.2. simultaneously. ( F i r s t Best, Additive E f f o r t ) Suppose E(x^) = k j ^ * i=l,-..,n. (1) If k^ = k, for a l l i , then k = X V ( Ea^) implies that any nonnegative vector a_such that Ea^ s a t i s f i e s (2) If some k^ [1] below i s Pareto a boundary solution r e s u l t s . are zero except one. optimal. That i s , a l l the a^'s In the n=2 case with k^ > k , a^* > 0 and 2 a * = 0. 2 Proof of Lemma 3A.2. The principal's problem i s Maximize / (x-s(x)) g(x|a) dx s(x), a subject to / [ U(s(x)) - V(a) ] g(x|a) dx > u. H = / (x-s(x)) g(x|a) dx + X { / [ U(s(x)) - V(a) ] g(x|a) dx - u } . 1 31 -g- = -g + XU'g = 0 implies that U'(s(x)) = ~y which implies that s(x) = U ' - 1 ( i ) = C. 113 -g-= /(x-s(x)) g (x|a) dx + X { / [ U(s(x)) g ( •) ] dx - V ( E a . ) } = 0 . 3E(x|a) - 0 + X(0-V'(Ea )) = 0 implies that This establishes result (1) of Lemma k = XV ( E a ) for a l l i . [1] 3A.2. To establish result (2), r e c a l l that a^* and a * are nonnegative by 2 assumption. Let s*(x) = C*, where s*(x) i s the optimal sharing rule corre- sponding to the optimal choices a^* and and a a 2 *« ^ > 0, be a feasible e f f o r t pair given 1 2 ( i'» e t a a 2'^» w n e r e i' * 0 a C*. The agent's expected u t i l i t y for any feasible ( a i , a ) i s 2 C* - V(a + a ) = u . 2 1 Since ( a ' , a ' ) i s f e a s i b l e , C* - V(a ' + a ') 1 2 x ( a " , a " ) = ( a ^ + a ', 0). 1 2 2 2 = u. Consider the pai r This pair i s also f e a s i b l e , since a^ ' + a " 2 = a^' + a ', and the p r i n c i p a l i s s t r i c t l y better o f f with ( a i " , a ' ' ) 2 2 since his expected return i s k ^ " + k a " - C* = ^ ( a ^ + a ') 2 Therefore, 2 2 - C* > k ^ ' + k a ' - C* i f k 2 2 L > k. 2 a ' > 0 Is not optimal, and hence the optimal e f f o r t pair i s 2 such that a^* > 0 and a * = 0 . 2 Q.E.D. Proposition 3A.3 Proposition 3A.3. below compares the solutions to two one-task problems ( F i r s t Best, Additive E f f o r t ) Suppose E(x^) = K^a^, i l > 2 , and that k^ > k . = 2 Consider the two sepa- rate problems where e f f o r t i s devoted only to task i . Then (1) a^* > a * i f V i s increasing and convex, (2) a^* > a * implies that s^* > s * ( i . e . , the agent i s paid more 2 2 2 for exerting a^* at task 1 than for exerting a * at task 2), and 2 114 (3) the p r i n c i p a l i s better off with a^* > 0 and a.^* = 0 than with = 0 and a^* a^* Proof of Proposition 0. > 3A.3. The principal's problem i f e f f o r t i s devoted only to task I i s Problem i : Maximize / (x - s^(x)) g^(x|a^) dx s ^ x ) ^ subject to J u ( ( x ) ) g ( x | a ) dx - V(a ) >u i S l 1 9H. . = 0 implies that s (x) = U'~ 1 . (_ X dS^ 3H ± „ = 0 )= C and k ^ i . 1 '< i> = A V'(a ) - -r- - — r — = - . A-i (c. X" X X i F e a s i b i l i t y requires that U ( U'" which implies that U'" 1 [1] X A^ V implies that . (i i 1 a ( 1 - ) ) - V(a ) i ± ) = U'" [ u + V(a.) 1 A = u , ] . 1 Equation [2] implies that A Result (1). i k k^ > k 2 i implies that a^* > a * 2 i f V i s increasing and convex. Proof. Suppose a^* < * « a 2 U (G _1 Then + V(a *)) < U~ (G 1 L (since U ^ - + V(a *)) and V are 2 increasing) [2] [3] -1 , ' (^ V U' _ implies that ( l * J— l a x -1 '< 2* > ) <- U' ^ i — ) by [3] and [4], , _ ^( 2 ) V a ) f k k which implies that V»( *) V(a *) — r > r 2 l ai K x (since U' i s decreasing), or k 2 '( 2* k~~ * V (a *) k _ 2 V a ) > * ( i s n c e v ' increasing and a^* < a * ) , so that i s 2 ^2 ^ ^1 * Therefore, > k 2 implies that a^* > a * . 2 Result (2). a^* > a * implies that s^* > s * . Proof. a^ > a * implies that u + V ( a ^ ) > u + V ( a * ) , which 2 2 2 2 implies that U ( u + V(a *)) > U ( u + V ( a * ) ) - 1 _ 1 L 2 (since U ^ i s Increasing), so that - U'" ( V 1 ) > ' u sj* > s * by 2 Remark : a^* > a * Result (3). 2 - 1 ik- [1] ) b y I !* 3 Therefore, . also implies that > ^ I f ki > k , the p r i n c i p a l i s better o f f with 2 a j * > 0 and a * = 0 than with a ^ = 0 and a * > 0 . 2 2 U6 Proof. It i s necessary to show that 1 2 / (x - S j * ) g ( x ^ * ) dx > / (x - s * ) g (x|a *) dx, 2 that i s , k ^ * - > k a * - C 2 2 2 . 2 [5] Note that ( a * , s * ) i s feasible for Problem 1: 2 2 / U(s *) g ( x | a * ) dx - V(a *) = U(C ) - V(a *) = u . 1 2 2 2 2 2 Therefore, k j ] * ~ j * > l 2 * ~~ 2 * because °^ f e a s i b i l i t y of ( a * , s * ) 3 s k a s 2 2 for Problem 1 and optimality of ( a ^ * , s^*) for Problem 1. Furthermore, k^a * - s * > k 2 * ~ 2 * s a 2 2 b e c a u s e l > 2 ^ ^» k k 2 a n d n e n c e t l holds. 5 Q.E.D. Proof of Proposition 3.5.1: if x ~N(ka ± and an i n t e r i o r solution (a^* > 0, a * > 0) i s optimal, l t 2 then i t must be that a* = a^* + a 2 In Lemma 2A.1 of Appendix 2, i t was shown that = Uj,- Let u = , a* = (a^*, a * ) , and 2 * ' This i n t e r i o r solution s a t i s f i e s the Nash conditions f (xjlaj) a ) f(x |a )f(x |a )dx x - V'( + a ) = 0, a i 1 2 2 i=l,2. 9 The condition for i=l i s f 2 v {"857 1 f (xja^) a kil*i*> a , 2 f + 3 -£7 I or 2 p - i - / (kx ( x a L * 2' 2* a 1 f ( 2 2 3 l l K ) d x l ) L ' f(x la *) - k x 2 K^W^V** * ) •f(x |a )dx 1 1 1 1" - V ' ^ + a )= 0 2 + 2> " °» a 117 2 pk i.e., - V'(a*) = 0, which would also result from the i=2 condition. Hence, there i s r e a l l y only one Nash condition. The principal's expected utility is f / ( + x X l 2 - ( X+ y Z = ka* - X - 2u k 2 2 (x.la.*) 3 i ^ ) ) f(x |a *)f(x |a *)dx dx 1 1 2 2 (see equation (A2.2) i n Appendix 2 1 2 2). The agent's expected u t i l i t y i f e f f o r t a* i s exerted i s f./x^a.*) 2 / ( X+ y f / | x j j 1 a *) > f(x |a *)f(x |a *)dx dx 1 1 2 2 1 2 - V(a*) = u , j which implies that 2X - V(a*) = u . suppose that a =0, the minimum e f f o r t , and that x Now 2 , f (x|a) Consider s(x) = [ X + u 2 i s ignored for a compensation purposes. fOcTa) and a* are the same as i n the i n t e r i o r solution above. is ) where X, u, The Nash condition now a f (x|a) ) f(x|a)dx - V*(a) = 0 , la" or 2 U - L - / (kx - k a * ) f ( x | a ) d x - V'(a) = 0 , that i s , 2vk - V'(a) = 0 , which i s s a t i s f i e d at a=a* The p r i n c i p a l ' s expected u t i l i t y If a=a* i s f (x|a) a / ( x - ( X+ = ka* - y 2 X 2 f(x|a) - ) ) f(x|a*)dx 2 2 y k 2 2 > ka* - X - 2y k , which i s the p r i n c i p a l ' s expected u t i l i t y with an i n t e r i o r solution. Since the agent's expected u t i l i t y i s 118 unaffected, the p r i n c i p a l i s s t r i c t l y better o f f , and the Nash condition holds, a boundary solution i s optimal. Q.E.D. Proof of Proposition 3.5.2; H = E(x + x L The Hamiltonian for the two-task problem i s - s*(x)) + X [ EU(s*(x)) - 2 + Ji,_ - j ^ - [ EU(s*(x)) - V(a x + a ) ] + ^ x + a ) ] [ EU(s*(x)) - V(a + a ) - u ] 2 2 2 The f i r s t order conditions are 3H "5T7 3H = o. [i] = 0 pointwise, [2] [ EU(s*(x)) - V ( 3 l + a ) ] - 0, 2 [3] EU(s*(x)) - V ( a * + a *) = u and L 2 As before, [2] implies that U'(s*(x)) X + Z j = 1 3H 3a, = /^ V'jIV 2 + x 2 U. -=4 1 r- J ^jl^j) - s*(x))f ( x | a ) f ( x |a )dx L + ^ -1^ [ EU(s*(x)) - V( .) ] 3 a l 1 2 2 + 0 119 = 0 . = -gi / (x + x t + u, 2 - s*(x))f(x |a )f(x |a )dx 1 1 [ EU(s*(x)) - V( •) ] 2 2 + 0 = 0 . 9a2 These imply that 9H 9a, 3H 9a, [5] Similarly, i t i s necessary that / U(s*(x))f(x |a )f(x |a )dx 1 = 1 2 = V'(a * 2 1 + a *) 2 / U(s*(x))f(x |a )f(x |a )dx 1 1 2 2 [6] It i s clear that (a^* = a2* and u^* = v^*) constitute a solution to conditions [5] and [6]. Therefore, i f a unique i n t e r i o r solution i s optimal, then i t has a^* = a2* and u^* = v^*• X are determined from conditions The p a r t i c u l a r values of a^*, u^*, and [1], [3], and [4]. Q.E.D. 120 Proof of Proposition 3.5.3: Consider f i r s t the s i t u a t i o n where a the agent's compensation i s based only on xj_. 2 = 0 and Dropping the subscript for convenience, the optimal sharing rule Is f (x|a) a t ( x ) = [ X 0 + ] *b -fUTaT 2 I * = [ XQ + u z'(a*)(x - E(x|a*)) ] , 2 0 where UQ > 0 (Holmstrom, 1979). Recall that E(x|a) = B ' ( z ( a ) ) . The princi- pal's expected return i s / (x - t(x))f(x|a*)dx = B*(z(a*)) - X, - u j ( z ' ( a * ) ) / (x-E(x| a*)) f (x | a*)dx 2 2 2 = B'(z(a*)) - X, - v g ( z ' ( a * ) ) B " ( z ( a * ) ) 2 since 2 Var(x|a*) = B " ( z ( a * ) ) . The agent's expected u t i l i t y i s f (x|a*) 2 ' ( \) % + f(x|a*) ^ ( l *) f x a d x " ( *) V a u . which implies that 2 XQ - V(a*) = u . The Nash condition i s 2 ' ( \) H> f ( x | a * ) + f f O T 2 V 0 2 f ( |a*)dx x ) - V(a*> = 0, a (x|a*) / f(xla*) d X - '<"> V = °• , [1] 121 Now consider the two-task s i t u a t i o n , where fCx-^ja) = f(x2|a) i f x^ = x « 2 Let 2 s(x ,x ) = ( ^ + u Z 1 V ^ i ' V 2 ) fCx^) i = 1 . 2 _y a* where a' = (a^', a ') and a^' = a , -. 2 2 The Nash conditions are now 2 / 2 ( + f ^ y - V'(a L a < iK'> X fCxJap + a ) = 0 1 ( + ^ a i 1 1 2 2 1 2 y i ! i ' > x " Jl - V*( ai and 2 2 2 >f U |a )f(x |a )dx dx a fUJa^) )f(x |a )f (x |a )dx dx 1 1 a 2 2 2 1 2 + a ) = 0 . 2 When evaluated at a', the Nash conditions reduce to £ a, ( x ll l a f , ) " J fCxJa,') 2 d 'l • '< *> ' T a 2 ^ 2 ^ 2 ^ a » / f(x |a ') 2 2 d x 2 The Nash conditions w i l l thus hold at a' i f ^"l'V* P / fCx^a^) fj(x|a*) d x l = Ho / f(x|a*) d x that i s , i f C = yCz'Ca^)) 2 B (z(a ')) = ^(z'Ca*)) , , 1 (see equation (Al.9) i n Appendix 1). 2 B"(z(a*)) [2] 122 [2] Equation i s true i f (z'(a*)) 2 a* (z'Cy-)) B"(z(a*)) _ z'(a*)M'(a*) a* B"(z(|-)) z (— ; M}M' f — } f 4 r ,. 4 ' The p r i n c i p a l ' s expected return i s / (x + x L - s(x ,x ))f(x |a ')f(x |a ')dx dx 2 1 2 = 2B'(z(^L)) " 1 2 2 1 2 \ j - 2 u ( z ' ( ^ ) ) B"(z(|^)) . 2 Since M(a) i s concave, M(a*) < M(|-) 1 2 2 < 2M(|^-) . because M( •) i s concave. If M(0) (Proof: V2M(0) + !/2M(a*) > 0, thenV2M(a*) < M(|-)). That i s , B'(z(a*)) < 2B'(z(-2^-)) . Suppose that Z '' ^ " ' ^ <V z ' ( f - ) M'(|-) The difference between [4] and 2 [4] [5] [6] . [1] is 2B'(z(4r-)) " B'(z(a*)) - 2pC + i^C > 0 because [5] holds and because [3] H, - 2 - H , 11 0 2 , , U z If M(a) y ) K , and y ) [6] imply that i > o. '(|_)M'<f-> i s s t r i c t l y concave, then the Inequality i n [5] inequality, and hence the s t r i c t inequality i n [6] becomes a s t r i c t can be relaxed to be a nonstrict Inequality (<). F i n a l l y , the agent's expected u t i l i t y i n the two-task situation described above i s s t i l l 2XQ - V(a*) = u. Since the p r i n c i p a l i s better o f f with an i n t e r i o r solution which s a t i s f i e s the Nash conditions, a boundary solution i s not optimal. 123 Proof of Corollary 3.5.4: z'(a*)/z'(^*-) <V2 (1) If M(a) = ka, then (3.5.5) reduces to • For the exponential d i s t r i b u t i o n with mean ka, z(a) = (see Table II i n Appendix 1), and hence z'(a)/z'(f) = - i y / _ 4 T ka ka (ii) For the gamma d i s t r i b u t i o n with mean ka, z(a) = (see Table II i n Appendix 1), and hence z (a)/z'( -) = J y ka , (iii) = V4<V2 • / _ ^ . = l/4< ka a V2 • For the normal d i s t r i b u t i o n with mean ka and unit variance, z(a) = ka (see Table II i n Appendix 1). z'(a)/z (|-) = 1 >V2 . , (iv) Therefore, For the Poisson d i s t r i b u t i o n with mean ka, z(a) = l n ka = l n k + In a z'(a)/z'(f) = (see Table II i n Appendix 1). i/|=V . 2 Q.E.D. Proof of Proposition 3.5.5: In this case, B'(z(a)) = ka, B " ( z ( a ) ) z ' ( a ) = k, and B ' ( z ( a ) ) ( z ( a ) ) ( z ' ( a ) ) 1 , , 2 + B' • ( z ( a ) ) z " (a) = 0 . Equation (A1.8) reduces to k + |£ z ' ( a * ) z " ( a * ) B " ( z ( a * ) ) = k + 1 1 z' ( a * ) z " (a *)B' ' ( z ( a * ) ) , 1 2 2 2 which implies that ^ z"( a i * ) = ^ z"(a *) 2 . [1] Equation (A1.9) reduces to U ' ( * ) = li2z'(a *) . lZ ai 2 [2] 124 [1] and [2] together imply that z"(a*) z"(a*) [3] (z'( a i *)) 2 (z'(a *)) 2 2 ' 2 Let v(a) = z ' ' ( a ) / ( z ' ( a ) ) . that a^* = a *. I f v(a) i s s t r i c t l y monotone, then T h i s i n t u r n i m p l i e s , from [1] or 2 [3] Implies [2], that Q.E.D. Examples 1) -1 z(a) = ^ Exponential: ka : l 2 i, a 4 k 1 -2 , z'(a) = — j , z''(a) = — ^ » ka ka a n a * 3 v 2) e (a) = = -2ka is strictly Gamma: z ( a ) = 7 - ° - , so v (a) i s a constant lea g 2 decreasing and = 0. R If pliers Pj and Poisson: 9 and u. * = u m u l t i p l e of v ( a ) . e z'(a) = k, z''(a) = 0, s o l u t i o n i s r e q u i r e d , [2] i n d i c a t e s that the m u l t i p 2 would have to be z ( a ) = l n ka, z'(a) = equal. , z''(a) = - and _ J_ 2 = — H-1 and case, a boundary s o l u t i o n i s o p t i m a l . a v *. • z ( a ) = ka, R e c a l l that i n t h i s an i n t e r i o r i n a, so a,* = a * p^* = u^* Normal w i t h u n i t v a r i a n c e : v (a) 4) / v (a) T h e r e f o r e , a^* = a * 3) . . T h e r e f o r e , the s o l u t i o n i s not unique and ~2 a cannot say whether p^ = P2 . we 125 Proof of Proposition 3.5.6: Equation (A1.6) can be written as ^1(8^) f (x|a) f( | ) = UjKaj*). 2f f 2 d Note that a I* (a) = S x [1] d = / ( x a f ~ 3 a ) > dx and hence equation (A1.7) can be written as I'( *) = I'(a *) . a i Equations [1] and [2] together [2] 2 imply that I'(a *) _ I'(a *) 1 2 ~2 r( Therefore, 2 3 l *) ' I (a *) 2 i f T.'(a)/I (a) i s s t r i c t l y monotonic, then a j * = a * , which Z 2 implies that u^* = l ^ * (equation [1]). Q.E.D. Proof of Corollary 3.5.7. For cases ( i ) - ( i i i ) , an i n t e r i o r solution i s optimal i f (3.5.6) holds. (i) z(a) = M(a), and hence (3.5.6) requires that ( ^ f f or 2 2 2 <V , 2 which i s s a t i s f i e d i f 0 < a <V • 2 (ii) z(a) = ^ ) a r , and hence (3.5.6) requires that M'(a) . M' (—) 2> .2 K , fM'(a) MC—) 2 ,2 ; 126 2 or • (—|— ) 2 o r ,2cr2-2a <ty> , which i s s a t i s f i e d when 0 < a< 1, since 2ct 1 . 1 4^2* (iii) z(a) = l n M(a) , and hence z'(a) = ^ ' M(a) ( a ) = a Equation (3.5.6) requires that a orl ^ 3 2a i.e.,V • 2 - <V , <V , 0 2 ,a..cc-l 1 2 2 which i s s a t i s f i e d i f 0 < a < 1 . O.E.D. Proof of Proposition 3.5.8. In this case, B*(z(a)) = ka, B " (z)z'(a)=k, and B ' " ( z ) z ( a ) + B " ( z ) z " ( a ) = 0. , y Equation (A1.9) says that 2 k z ( a l l l' l* ) = u 2 2 2 k z , ( a 2* ) » [ 1 ] and equation (A1.8) says that k (i) + l ^Vl" z(a) = - i_ Equations and l * 2 a k i + l* ) = k 2 , z'(a) = ka 4. 2 2 " + k z ( a 2* ) * [ 2 ] , and z " ( a ) = ka [1] and [2] become - 1 a ( a P i 2 * 2 "R " [3] e 2 . _- 2 = k 2 + ^ . _ 2 - ,2 127 which together imply that k. - 2R a * = k - 2R a * , 1 e l 2 e 2 2 2 0 or 2R (a * - a *) = k, - k, > 0 e l 2 1 2 2 since k^ > k . Therefore, 2 a^* > a * , and hence, u-^* > u * 2 2 (from [3]). (ii) z(a) - - £ - , z'(a) = -Ar , and z " ( a ) = ka ' * " ' . 2 ' " ,3 ka ka 2 n v v o / An analysis similar to that i n ( i ) establishes the result. (iii) z(a) = ka, z'(a) = k, and z''(a) = 0. k l Equation [2] becomes = 2 » k which contradicts the assumption that kj > k . 2 optimal solution i s a boundary solution. solution has a^' = 0 and a ' > 0' ( a ± * , a 2 the Suppose the optimal It w i l l be shown that there i s a 2 Pareto superior solution Therefore, with a^* > 0 and a * = 0. * ) , 2 optimal sharing rule i f only task two has nonzero e f f o r t i s f (x |a «) 2 ^2 f(x |a «) ^ 2 a < 2> - ( 5 X X "2 + 1 2 2 Z 2 The Nash condition i s (see equation (A2.3) i n Appendix 2) 2k U2 - V ( a ' ) - 0 . 2 2 The agent's expected u t i l i t y i s 2A - V(a ') = u , 2 and the p r i n c i p a l ' s expected return i s (see equation (A2.2) i n Appendix 2) k 2 2 2' ~ * ~ a 2w 2 2 2 2 ' k The 128 Now consider the pair ( a ^ * , a * ) , where a^* = a2' and a * 2 = u 2 i and consider the sharing rule f l> t ( x where = —jk • ( = h X + (x.la,*) f^xja^) The agent's expected u t i l i t y (with e f f o r t l exerted only at task one) i s s t i l l u, and the Nash condition i s 2 2 s a t i s f i e d , since 2 ^ = 2k and a ' = a ^ . 2 Furthermore, the 2 p r i n c i p a l i s s t r i c t l y better o f f because 2 k k^* - X - l{ k 2 = 2 a ' - X - 2U* k 2 k l > k a ' - X - 2u k 2 2 (iv) 2 = Equations [1] and [2] £• u *k 1 2 become a . ~ —« • 3. 1 2 2 z(a) = ln ka, z'(a) = —,and z''(a) U *k ( J - ) 2 2 l a 2 = ^ 2 a P and 2 k l + "L* ~ ' — 2 a l 2 k = k 2 V + k '—71 l * a 2 2 which together imply that 2 k l K R since k i > ko. P R - -E. = k k 2 K x < ^ " 2 R - _E k * 2 17 > • k i " 2 k > 0 Therefore, -r— ~ 1 7 - > 0, which implies that *2 l k k l ^ 2 (contradiction). Hence, a boundary solution i s optimal. k 129 Suppose the optimal solution has a^' = 0 and a ' > 0. It w i l l 2 be shown that there i s a Pareto superior solution ( a i * , a * ) , 2 with a j * > 0 and a * = 0. The optimal sharing rule i f only task 2 two has nonzero e f f o r t i s f s(x 2> = ( X "2 + (x |a ') 2 f(x la ') 2 » 2 ^ 2 X ' = The Nash condition, evaluated at 8 2 ' » i s E x =0 "2 1 2 f -! u x ( x 2l*2 f ) f(x la ') l 2 a " ; V ' ( a 2 , : > ° » = 2 that i s , (see Appendix 1 ) , ^(z'ta^)) "2 2 "-^2 2 B"(z(a ')) - V ( a ' ) = 0 , 2 k 2 2' ~ a 2 V , ( a 2 , : > - » 0 a "2 2 k which implies that ; a V'(a ') = 0 . 2 The agent's expected u t i l i t y , evaluated at a2', i s 2X - V(a ') = u , 2 and the principal's expected return i s . k Now , 2 2 a .2 " X "2 2 k ~ * consider the pair ( a ^ * , a * ) , where a j * = a ' and a * = 0, 2 and consider the following sharing rule, 2 2 130 where = ^— The agent's expected u t i l i t y (with • effort exerted only at task one) i s s t i l l u, and the Nash condition i s s a t i s f i e d , since k^ principal = k 2 ^2 a n d l* a = 2' * a Furthermore, the i s s t r i c t l y better o f f , because \ i 2 2 \ 1 2 V 2 _ k,a * - A - -!-4 = k,a ' - X - k *1=1 1 2 lV 2 2 "2 2 —-j-. 2 > k a ' ~ X 2 2 Q.E.D, Proof of Proposition 3.5.9. 2 ^ / 2 ( x + j = The Nash conditions require that 4> ( l a ) x a ) <«x|a)dx \ ^-wnr- V ' ( * + a *) = 0, a i 2 j=l,2. For j=l, the condition i s 8 a 2l 2*) g(x |a *) 2 2 \ I f'xja^) d x l + Zv 2 I a ( x 2 2 * f (x !a *)g(x |a *)dx a i 1 1 2 2 - V ' ( * + a *) = 0, a i 2 which reduces to a l l* $ ffrja^) f 2 \ 2 ( x | a > d x l= V'( * a i + a *) . 2 Since f( •) belongs to Q, the condition can be written as (see Appendix 1) 2u (z'(a *)) i 1 2 B"(z( *)) a i = V ' ( a * + a *) . x 2 Since V > 0 and B " ( z ^ * ) ) = V a r ^ ^ a ] * ) > 0, analysis for j =2 shows that j ^ * > 0. j^*> 0 • A similar 132 Appendix 4 F i r s t Best The p r i n c i p a l ' s problem i s to Maximize / / W(x-s(x^ .x^) ) <J>(x^ ,x 1 a^,a ( •) ) dx dx^ s( •) , a , a ( •) 2 1 2 2 2 subject to / / [U(s(x ,x ))-V(a ,a («))]'()(x ,x |a ,a (0) 1 where ^(x^ ,x |a^^ ,a ( •)) 2 2 = 2 1 2 1 2 dx^x^^ > u, 2 1 f (x^ | a ) g ( x \-x^ ,a ( •)) and a ( •) indicates that 1 2 2 2 the agent's second-stage e f f o r t i s i n general not a constant, but rather can depend on any information available at the time of choice. described, a may depend on x^. 2 / / W(x-s(-))<t'(-)dx dx 2 In the scenario The Hamiltonian Is + X / / [U(s(-))-V(-)]<l>(Odx dx . 1 2 1 D i f f e r e n t i a t i n g the Hamiltonian pointwise with respect to s y i e l d s -M'lf + X U'<|> = 0 f o r almost every ( x ^ . x ^ , which implies that W'(x-s(x ,x )) r-r— = X for almost every (x. , x „ ) . ,. , 1 u 1) Q S ( , X ^ , X 2 2 , ) } 1 (A4.1) i. Risk averse p r i n c i p a l , r i s k neutral agent ( i . e . , U' = 1). Equation (A4.1) implies that W ( x - s ( x i , x ) ) = X f o r almost every 2 ( x i , x ) , which implies that x - s ( x i , x ) i s constant for almost every ( x i , x ) , 2 2 2 which i n turn implies that s ( x , x ) = x-c, where c i s a constant. 1 be shown below that a 2 2 i s independent of x^ i f x^ and x 2 It w i l l are c o n d i t i o n a l l y independent, i n which case c = E(x|a*) - V(a*) - u. 2) Risk neutral p r i n c i p a l , r i s k averse agent (W'(x-s(«)) = 1). Equation (A4.1) implies that U ' ( s ( x i , x ) ) = constant for almost every 2 ( x i , x ) , which implies that s( •) i s a constant for almost every ( x j _ , x ) ' « 2 2 x^ and x 2 are c o n d i t i o n a l l y independent, then a s( •) = U ( u + V(a*)). - 1 2 i s independent of x^ and If 133 3) Both individuals r i s k averse. Equation (A4.1) implies that x - s ( x , x ) = W' ( AU' ( s ( x , x ) ) ) = -1 1 G(s(x)), where G' > 0. H' > 0. 4) 2 x 2 Therefore, x = G(s(x)) + s(x) = H(s(x)), where Thus, s(x) = H ( x ) , where H'" _ 1 > 0. 1 Both individuals r i s k neutral. ^ In this case, the agent's expected u t i l i t y constraint implies that s ( , x ) = u + V(a*). X l 2 The choice of the agent's e f f o r t decisions w i l l f i r s t be examined i n the simplest case, where the p r i n c i p a l i s r i s k neutral, the agent i s r i s k averse, and the outcomes are conditionally independent. ( K x - p X ^ a ^ . a ^ •)) That Is, f (x^ | a^)g(x | a ( •) ) > where we allow for the p o s s i b i l i t y = 2 2 that the second e f f o r t decision depends on the f i r s t outcome. Since the optimal sharing rule Is s ( x i , x ) = s (constant), the function to be maxi2 mized i s / / (x-s) ^(x! ,x laj^ ,a ( •))dx dx 2 2 2 1 + A[ / / (U(s) - V ( a a ( 0 ) } « x , x | a , a ( « ) ) d x d x 1 > or, 2 1 2 1 2 2 1 - il] , ignoring constants, / x f(x |a )dx 1 1 1 + / / x f(x!|a )g(x |a (•))dx dx 1 2 1 2 2 2 1 (A4.2) - A / / V(a ,a (.))f(x |a )g(x |a (.))dx dx . 1 2 1 1 2 2 2 1 (A4.2) can be rewritten as / [xj_ + { / [x - X V(a ,a (0)]g(x |a («))dx }]f(x |a )dx . 1 2 2 2 2 2 1 1 1 (A4.3) For each fixed xj_, maximizing the expression inside the braces with respect to a 2 w i l l maximize (A4.3) with respect to a . 2 Since the expression depends on x^ only through a , a ( •) i s the same for almost every x^. 2 2 That i s , a 2 134 does not depend on x^. A s i m i l a r a n a l y s i s can be done f o r the case where the p r i n c i p a l i s r i s k averse and the agent i s risk neutral. Finally, If both i n d i v i d u a l s a r e r i s k a v e r s e , then the f u n c t i o n t o be maximized i s / [ / W(x-s(x))g(x |a (-))dx ]f(x |a )dx 2 2 2 1 1 + X [ / { / U ( s ( x ) ) g ( x | a ( »))dx 2 In t h i s case, a Maximizing 2 1 - V(a 2 1 > a (0)}f(x |a )dx ]. 2 1 1 1 ( 0 w i l l g e n e r a l l y depend on x^. 2 (A4.3) w i t h r e s p e c t t o a^ r e s u l t s i n the c o n d i t i o n t h a t aE(x |a' )/aa 1 Maximizing 1 = X . (A4.3) w i t h r e s p e c t to a 2 (which i s independent of x^) r e s u l t s i n the c o n d i t i o n t h a t 3E(x |a )/aa 2 2 2 = X 3V(-)/3a . 2 Proof of P r o p o s i t i o n 4.1.1. Under the g i v e n assumptions, a ( •) w i l l 2 depend on xj . denote the mean o f x^ g i v e n a^, and l e t M^x^ t i o n a l mean of x 2 w i t h r e s p e c t to g( • ) . J J ( x j + x )<j>( • ) d x d x 2 = = 2 1 - X / / V( , a ( •)) denote the c o n d i 2 The f u n c t i o n to be maximized i s a i , a ( •))•( 2 / X j f U j l a ^ d x j + / M^ •)f(x |a )dx 1 1 1 Odxjdxj - X / V ( - ) f (Xj |aj )dxj M j C a j ) + E M ( •) - X EjVC • ) , 1 2 where E^ r e p r e s e n t s e x p e c t a t i o n with r e s p e c t to f ( • ) . d i t i o n s with respect to e f f o r t 3M (a )/3a 1 Let M ^ a j ) 1 1 + B E ^ C O ^ The f i r s t order con- a r e then = X ffi^C and 3M (»)/3a 2 2 = X 3V(0/aa 2 (A4.5) 135 for almost every and for a^ = ai£. The sign of a^'Cx^) can be determined by taking the derivative of (A4.5) with respect to x^. Let the second and t h i r d subscripts of j on M 2 denote p a r t i a l d i f f e r e n t i a t i o n of M respect to the j - t h argument of M2(xpa^,a2( •)) • 2 with Taking the derivative of (A4.5) with respect to x^ results i n M 233 * ' a 2 + M 231 = [3 V(0/3a ]a « X 2 2 2 or a*'( ) = -M Xl /[M 33 - X[3 V(0/3a2]] 2 231 2 Q.E.D. Second Best Let <(<x ,x |a ,a ) = f ( x ^ | a ) g ( x | x , a , a ( •)) . 1 2 1 1 2 2 x x 2 The agent's expected u t i l i t y i s / [ / U ( s ( x , x ) ) g ( x | x , a , a ( *))dx 1 2 2 1 1 2 ,a < •) ) ] f (x | ) d x . 3 2 2 1 1 1 The Hamiltonian i s H = / / ( x - s C x ^ x ^ H C Odx + X { / [ / U(s(x ,x ))g(x |x ,a ,a («))dx 1 2 2 1 - V(a ,a (-))]f(x |a )dx 1 2 1 + 1^1 / / U ( s ( . ) ) [ g (a) f + gf a { / U(s(0)g + / y (x ) 2 1 1 1 2 2 - u} 1 J d x ^ - / (V <Odx 2 - V f + Vf fli )dx } 1 (•)}f(x |a )dx . 1 1 1 D i f f e r e n t i a t i n g H pointwise with respect to s( •) y i e l d s - <J> + X U'<f> + UjU*^ + p (x )U'g 2 1 a f = 6 for almost every ( x , x ) . 1 2 That i s , 1 u'C(x l t •a , . x )) = X y + 2 where the subscript a each fixed x^. 2 i — 1 . + W , x ^a 2 T' represents d i f f e r e n t i a t i o n with respect to a for 2 136 A t a - a * , | - - // (b) h *rr I / / < < - » f c u + ' + (c) W (-)dx (x-sCxj.Xg))* f s "al^ ^ ' [ * U ( S ( + 8 f ai a i a ) ) 8 i d x ( 0 d x 2 d x i - / <\ + V f > il d x a i 2 " V ( - ) ] f ( x | a ) d x } = 0. a 2 2 f 1 1 f + V 2 f 1 1 At a = a*, and for every fixed x^, -g- = / (x-s(x ,x ))<j) (.)dx 1 2 a2 2 f + g f ]dx + M, { / U(s( - ) ) [ g 1 a a 2 l • a a L ^(x^f + a a 2 /U(8(0)g (Odx - V 2 - (V l 2 a a (OjfCxJa^ Clearly, the strategy a*,(») depends on x^ i n general. = )} 0. However, i f the agent i s r i s k neutral, then the f i r s t best solution can be obtained (see Shavell (1979)). Proof of Proposition 4.2.1: (1981, pp. 104-105).) (Generalization of the derivation by Lambert Since f( •) and g( •) are i n Q, they can be written as f(x |a) = exp[z (a)x 1 - B ^ z ^ a ) ) ] h ( x ) and g(x |a) = e x p [ z ( a ) x 2 - B (z (a))]h (x ). 1 1 2 2 1 2 2 1 2 2 Recall that E(x |a) - B ^ z ^ a ) ) , Var(x |a) = B^'(z ( a ) ) , and ± 1 f / f = zj^(a)(x - M ( a ) ) , where Mj_ = E ( x | a ) . 1 C i 1 1 = y z ^ ( a | ) ( x - M^(a*)) , where u (X + i c l i + c ) . 2 2 1 2 Let denotes (^(x^). Then s(x) = Some h e l p f u l quantities w i l l f i r s t be calculated. (1) / / (x + x ) f ( x | a ) g ( x | a ( x ) ) d x d x 1 2 1 1 2 = / XjfCxj l a ^ d X j 2 1 1 2 + / [ / x g ( x | a ( x ) ) d x ] f ( x |a )dx 2 2 2 1 2 1 1 1 = MjCaj) + / M ( a ( x ) ) f ( x | a ) d x . 2 (2) 2 1 1 1 1 / / sCxj,x )f(xj|a )g(x |a (x ))dx dx 2 1 2 2 1 1 2 = / / (D + F + G ) f ( x | a ) g ( x | a ( •))dx dx , 1 1 2 2 1 where D = ( X + C ^ ) , F = 2 ( X + C ^ C ^ and G = C . 2 E(D) = / / [ X 2 2 + X W J Z J U J K X J - M J U * ) ) 2 2 1 = X 2 + 2 X Xl 1 + M ^) 2 since E(x-a*) X + 2 1 2 2 1 2 - Mj(a*)) U J Z J U J X M J U J ) + ^zJ (a*)tVar(Xl lap = 2 u^zJ (a*)(x -M (a*)) ]f( |a >g(x |a <•))dx dx + E(D) 2 - a t ^ a p M j U j ) + M (a*)] , 2 2 2 2 = Var x + (Ex) - 2a*Ex + a* . uJz^CajOVarCxJa*) 2 E(F) = 2 X / / u (x )z^(a*)[x -M (a*(«))]f(x | )g(x |a ( 2 1 2 2 1 ai 2 •))dx dx 2 1 2 + 2 Uj, / / z ' ( a J ) ( x - M ( a { ) ) u ( x ) z ^ ( a § ( 0 ) ( x ^ M ( a § ( « ) ) ) f ( 0 g ( 0 d x d x 1 1 2 1 2 = 2 X / u (x )z^(a*)[M (a (x )) 2 1 2 2 2 1 2 (a*(x ))]f(x |a )dXj 1 x x y + 2u z^(a*) / z ^ ( a * ( ' ) ) P ( x ) ( x - M ( a * ) ) ( M ( a ( 0 ) " M (a$( 0 ) ) f ( OdXj, 1 2 E(F)| 1 1 1 2 2 2 = 0. Is* E(G) = / / u ( x ) z ( a * ( - ) ) ( x - M ( a * ( - ) ) ) f ( ' ) g ( * ) d x d x 2 2 1 2 2 2 2 1 = / z ( a * ( - ) ) u ( x ) [ V a r ( x | a ( - ) ) +M (a (-)) 2 2 2 2 1 2 2 2 2 2M (a*(0)M (a (0) - + M ( a * ( •)) 1 f ( O d X j . 2 2 2 2 2 /' z ^'( a * ( • ") ) ^ ( x ) V a r ( x | a * ( • ) ) f ( x | a * ) d x = E(G) 2 , Z 1 2 1 1 Therefore, - (2) - ' + u^zj 2 2 2 (a*)Var( |a*) X l * + (3) / z^ (a*( •))^(x )Var(x |a*( •))f(x | * ) d 2 1 2 1 / / 2/s(x) f ( x | a ) g ( x | a ( » ) ) d x d x 1 1 2 2 1 1 • f ( ')g( • ) d x d x 1 = 2X+ 1 - 2 1 - / V ( a j , a ( 0 ) f (xj |aj )dxj 2 2 / V(a a (»))f(x |a )dx 1 > 2y z{(a*)(M (a ) 1 2 1 - (3) - / V( 2X- (4) a i 2 2 2 ,a ( •))f(x |a )dx . / V( 2 a i 1 1 1 Z. ,a ( •))f(x |a )dx = 2 1 1 1 = 2 y z | ( a * ) M j ( a ) + 2 / ^ ( x j ) z ' ( a * ( •)) [ M ( a ( • ) ) 1 - M (a|( 0 ) ] f 2 (4) 1 la* (5) 1 - ML, (a*( •) ) ] f ( O d X j , 1 1 Mj(a*)) 1 + 2 / M (x )z (a*(0)[M (a (.)) 2 X l ( x j ) z ^ ( a * ( • ) ) ( x ^ (a*( •)) ) ] = 2 / / [X+y zj(a*)(x ^ (a*)) + 1 a dxj - / V a = 2w *(a*)M^(a*) lZ f(x |a )dx a 1 Jv 1 x - / V( - ) f & &2 ^p- = |a )dx . 1 1 (x |a*)dx 1 1 1 2H (x )z'(a*(0)M^(a (0)f(x |a ) 2 2 1 1 1 ( • ) f ( x . | a ) = 0, which i m p l i e s t h a t 1 V W Clearly, - / V(-)f ^ ( x j 1 fCxJa-pdXj & 2 1 Fix x . - V 2 1 U^Xj) > 0 if V " & (a*) 2 '(a*(x ))M^(a*(x ))* Z ( •) > 0. 1 1 This e s t a b l i s h e s r e s u l t (i)(a), 139 After i s r e a l i z e d , the agent's expected u t i l i t y g i v e n x j and a^(x^) is 2 / Tsjx) g ( x | a * ( ) ) d x 2 X l - V(a*,a*(x )) 2 1 UjzJCajXxj-MjCa*)) + j^Cxj ) z ' ( a * ( X j ))(x ^M (a*(x ))] = 2 J [\+ 2 • g(x | Odx 2 2 1 - V(a*,a*( )) 2 X l = 2 [ A + UjzJCaJXxj-MjCa*))] - V(a* . a * ^ ) ) . D i f f e r e n t i a t i n g w i t h r e s p e c t t o x^ y i e l d s 2 The Vl< l> agent's s(x)) V a a V - ' ( 2 expected ) a 2 utility * f o r the second stage p e c u n i a r y r e t u r n ( i . e . , i s an i n c r e a s i n g f u n c t i o n o f x^ (assuming ( •) > 0, a s u f f i c i e n t a , ( X l ) p^ > 0 ) . c o n d i t i o n f o r the agent's Assuming t h a t expected second stage n e t 2 utility xj. t o be i n c r e a s i n g i n x^ i s t h a t a^(x^) be a d e c r e a s i n g f u n c t i o n o f This e s t a b l i s h e s r e s u l t s Now f i x x j and l e t a f-=M'(a )f denote a ( x ^ ) , and f denote 2 2 1 2 2u zJ(a*)z^(a*)p (x )(x -M (a*))M^(a )f 1 - z 2 2 2 (a*)p^( - X l 1 1 1 2 ) [ B " ( z ( a ) ) z ^ ( a ) + 2M (a )M (a ) 2 2 2 2 2 2 2 2 2 2 P [2u (x )z (a*)M'(a )f + u (x )[2p (x )z^(a*)M^'(a )f - V 1 2 2 2M (a*)M (a )]f + 3H f(xjja^). 2 -2Xu (x )z'(a*)M'(a )f 2 - ( i ) ( b ) and ( i i ) ( a ) . 2 1 1 2 2 2 - V a i 1 ^ 2 f- V^COf^l (Of]. - 2y (x )[M (a*)Xz (a*) + - H J U * ) 2 1 2 2 u l a a l a a V f 2 1 a i a 2 / f l 140 + (Note y ^ ( x ) [ 2 z ( a * ) M ' ( a * ) - z ( a * ) B ' " ( z ( a * ) ) ] = 0. , 1 3 2 that f a 2 /f 2 - z J ( a * ) ( x - M ( a * ) ) = 0.) 1 1 l the e x p r e s s i o n f o r v^ix^) from (5) above and l e t t i n g Substituting subscripts j on V r e p r e s e n t p a r t i a l d i f f e r e n t i a t i o n w i t h r e s p e c t t o the j - t h e f f o r t variable yields V v 2 22 V V z! (x, -M. ) - y , V 112 ^ V i ^ l 1 0 0 M 2 " W 2 - 2 ^ " V V ' „ 2 + 2z^M 2 4M Z 2 Z : B : * ' f ,2 = 0. D i f f e r e n t i a t i n g w i t h r e s p e c t to x^, fi a' M 2 a XV a 2 2 2 2 2 "l 122 2 [ V a a V + V r 22 2 a 2 ( z ^ 2 z (z-M 2 (z*Mp M + z M ') 2 2 2 a' 2V V M'' + V p[*' ' 2 2 22 2 2 2 + 4 ( . ) z'M' 2 2 V zJ] 2 ) 2 Z 2 ) M 2 2 V 1 Z R e c a l l that M ( a ) = a ^ 1 2 2 2 ( x a' 2V V z'B ' f 2 , 2 22 2 2 4 2 1 2 V + z^2M M ') 2 2 2 V V 2 222 22 2 l r l V M *(z^'M 2 V z M 1 2 2 + 2 z''B' ' 2 2 2 ,2 M, 1 z H z + so t h a t i 2 2 V z'B'' ' ' 2 2 2 ,2 M, V z'B'''2M'' 2 2 2 2 -] = 0. ,3 M 2 B v = 1 and M|' = 0 . Thus, the e x p r e s s i o n above reduces t o £ ., a 2 < X . l> = " "l 2 i V Z ., / D ' W h e r e /, D = (X + U l — ) a U „ V 2 2 + . V 2 22 — + V V 2 222 .. 2 22 2 —j- 141 V V 2 + 1 122 V 2 2 z B "' 2 V 2 2 2 (z 'B " + z 2 2 2 2 B "') 2 + Recall that i t i s assumed that V 2 > 0, V 2 2 > 0, V i s e a s i l y checked that for the exponential, i n 0, z' > 0, z " < 0, B , , , 2 2 2 > 0. It 1 2 2 gamma, and Poisson d i s t r i b u t i o n s » > 0, and z'^'** + z B ' * " > 0. These facts, , 2 plus the f i r s t order condition requiring that X + y. f antee that the denominator of a (x^) i s p o s i t i v e . i s the same as the sign of y^. /f be p o s i t i v e , guar- The sign of the numerator 2 function of x^. > 0, and V Hence, i f y^ > 0, then a*,( •) i s a decreasing This establishes r e s u l t (ii)(b). Q.E.D. Proof of Corollary 4.4.1: V* 2 2 > 0, and V * 22 > 0. It i s necessary to show that V* > 0, V* > 0, 2 The derivatives of M 1 w i l l f i r s t be calculated. Dropping subscripts for convenience, 1) M (M(a)) = a implies that M '(M(a))M'(a) = 1. -1 _1 Therefore, M '(M(a)) = 1/M'(a) > 0. _1 2) M "(M(a))M'(a) = - M " ( a ) / ( M ' ( a ) ) . _1 2 M "(M(a)) = - M"(a)/(M'(a)) _1 3) -1'" M" 'M' M (M(a))M'(a) = - [- 3 1 ™. * Therefore, M \\ 3M (M(a)) = Therefore, > 0. 3 2 - M''3M' M ' ^ ]. M' 1 -M = M , , 2 , , , M' ,:> Let subscripts j on V* denote p a r t i a l d i f f e r e n t i a t i o n with respect to ej. Then V* = V'M^' = V'/M'(a ) > 0, 2 142 V* 22 V"(M -1' 2 -1'' ) + V'M^ 1 V V V* 2 2 2 -1* = i ( a 2 -1' 2 -1'' [V'"(M y +V"M ] i l 2 M ^ l } [ v 2 ,,, (M (a )) 2 V"'(M ) 2 V"M '(a ) —] (M (a )) 2 2 2 2 2 + V"2M M 2 > 0, and 2 L 2 +V"M 2 (M'(a )) 2 2 3 2 (M (a )) 2 4 2 - M^'*Mp > 0 a t a*, then V * M 2 2 M 2 3 V " M " ( a ) ^ V'(3M'' Thus, I f ( 3 M 1 ' 2 (M^(a ))- 2 1 V* 222 V'M '(a ) 1 j (M^(a ))' Z 2 2 + V'M 2 2 - M "M ) 2 2 ,5 > 0, as r e q u i r e d . Q.E.D.