job challenge as a motivator in a principal–agent setting

Theory and Methodology

Job challenge as a motivator in a principal±agent setting

Suresh Radhakrishnan a,*,1, Joshua Ronen b

a Leonard N. Stern School of Business, New York University, 429 Tisch Hall, 40 West 4th Street, New York, NY 10012, USAb Leonard N. Stern School of Business, New York University, 300 Tisch Hall, 40 West 4th Street, New York, NY 10012, USA

Received 1 July 1997; accepted 1 April 1998

Abstract

We analyze the tradeo� between monetary and non-monetary incentives in a principal±agent framework where the

principal chooses job-challenge (job-design) to provide non-monetary incentives. We capture the construct of job-

challenge as a motivator based on Atkinson (1958). In: Atkinson, J.W. (Ed.), Motires in Fantasy, Action and Society,

Van Nostrand, Princeton, NJ. Speci®cally, more challenging jobs reduce the probability of success but increase the

marginal productivity of the agent's e�ort. This provides the agent with a sense of accomplishment from which the

agent derives utility. We show that there are three e�ects of job-challenge when we consider information asymmetries.

The substitution e�ect arises because non-monetary incentives can be used to substitute for monetary incentives. The

risk e�ect arises because job challenge can be used to impose some of the risk needed to mitigate the impact of hidden-

action, rather than having to impose risk through monetary incentives which could be costlier to the principal. The

informativeness e�ect arises because the ®nal outcome is more informative on the hidden-action of the agent and thus

less risk needs to be imposed through the monetary incentives than in a model that does not feature job-challenge. The

cost to the principal of designing challenging jobs arises from the decreased expected output. Ó 1999 Published by

Elsevier Science B.V. All rights reserved.

Keywords: Gaming; Agency theory; Task Design; Information asymmetry

1. Introduction

Studies in hidden action models have focused on a principal±agent relationship wherein the agent iswork averse (see Holmstrom, 1979; Shavell, 1979). These studies show that incentives need to be providedto induce the agent to exert unobservable productive action. Furthermore, the induced productive actionprecludes the ®rst-best action choice (and hence, ®rst-best risk sharing).

European Journal of Operational Research 115 (1999) 138±157

* Corresponding author. E-mail: [email protected] We are grateful to J. Glover, S. Jackson and two anonymous reviewers for their insightful comments.

0377-2217/99/$ ± see front matter Ó 1999 Published by Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 2 2 1 7 ( 9 8 ) 0 0 1 9 8 - 2

The industrial psychology and organizational behavior literature have focused on job enrichment andjob enlargement as motivating factors. Atkinson (1958) proposed that motivation is the product of theprobability of success (PS), and the incentive value of success (I), where (I) is an inverse function of theprobability of success, i.e., I� 1 ) PS. Thus the product, (PS)(I) achieves a maximum when PS� 0.50. Inessence, Atkinson's (Atkinson, 1958) theory of motivation revolves around job-challenge as a motivatingfactor. In particular, job-challenge would be maximized when the probability of success equals the prob-ability of failure; and since the agent has an intrinsic desire to be successful he would be motivated to exerte�ort. Janz (1982) found support for the theoretical relationship between job-challenge and performanceproposed by Atkinson (1958).

On the other hand, the industrial engineering literature on job design recommends simpli®cation andspecialization. This approach hypothesizes that bene®ts emanate from e�cient operations, which in turnshould lead to higher probability of success. Up until a decade ago, the American organization was in-¯uenced largely by Fredrick Taylor (the father of scienti®c management). The Taylor system of shopmanagement espoused specialization of jobs for individual employees (agents). Taylor's system was largelysuccessful in increasing productivity from the 1930s through the 1960s (see Skrovan, 1983).

Comparing the industrial engineering approach with the organizational behavior approach, what standsout is that (a) to increase motivation jobs need to be challenging in the sense that the probability of successshould not be high, but (b) to increase productivity (which would occur if the probability of success is high)jobs need to be simple, mechanistic and repetitive. This points to a tradeo� between the motivational e�ectand the mechanistic e�ect that needs to be considered in job-design. This tradeo� is the focus of our study.

During recent years ®rms have adopted precepts such as ¯exible assignments, 2 job enlargement and jobenrichment. 3 These practices are adopted for generally improving the productivity of the organization(Broadwell, 1985). In particular, the concept of moving away from specialization for improving produc-tivity is primarily based on the success of such techniques in Japan. For instance, Takayanagi (1985) states,``Hitachi Shipbuilding Company changed its employees' jobs from single tasks to multiple tasks. Employeeshad to perform several tasks, each requiring complicated and sophisticated operations. Handling multiplejobs requires more ¯exibility. But as the employee performs complex jobs, his pride in his work increases.By doing so Hitachi increased productivity by 20%''. Takayanagi reports a similar experience for Mit-subishi. The Japanese experience suggests that the bene®ts created from the sense of accomplishment(through higher job challenges) outweigh the costs caused by the concomitant loss of specialization andsimpli®cation. In particular, these anecdotes suggest that the motivational aspect of job design is an im-portant factor to be explicitly considered in the incentive literature.

The psychology-based models of job enlargement and design (see Hackman and Oldham, 1980; Herz-berg, 1966) focus on the motivational aspects as bene®ts. Recently, Campion and McClelland (1991, 1993)using an inter-disciplinary approach have considered the costs and the bene®ts of job enlargement. Inparticular, they measure the incremental training requirements, incremental error possibilities and thedegradation of customer service as the costs of providing motivational bene®ts in the form of employeesatisfaction. Their main ®nding suggests that job enlargement not only increases motivational bene®ts butalso increases costs.

2 Job sharing is a form of ¯exible assignment where the traditional one-to-one relationship of agents to jobs is changed. A group of

say three employees are given a set of jobs. The group members can assign the individual tasks to any member (by covering up, ®lling in

etc.). The important aspect here that the set of jobs are accomplished (see Broadwell, 1985; Juran, 1978).3 Here again the important concept is that an individual employee is provided a variety of tasks to do, as against a specialized task in

the scienti®c management era (see Broadwell, 1985).

S. Radhakrishnan, J. Ronen / European Journal of Operational Research 115 (1999) 138±157 139

The incentive-based agency literature in economics suggests that incentives are required to induce workaverse agents to work `hard'. 4 The motivational literature, on the other hand, suggests that non-com-pensatory means such as ®ne-tuning of the level of challenge o�ered by di�erent designs of jobs could beused to induce agents to work `hard'. In this paper we incorporate the motivational aspect of job-challengeas a primitive in the agent's utility function. We then use the principal±agent framework with unobservableagent's e�ort. As in the agency models (see Holmstrom, 1979; Shavell, 1979) the agent derives utility fromwealth and is e�ort averse. In addition, the agent is modeled as deriving utility from job-challenge (di�-culty). The principal (employer) designs the job and the monetary incentives based on which the agent thenchooses the optimal e�ort. A job design with increased job-challenge reduces the probability of success, butprovides the agent with motivational utility. If the agent shirks, he derives no utility from job-challenge.However, when the agent works hard increased job-challenge increases the agent's utility. But, increasedjob-challenge is costly for the principal in the sense of increasing the probability of the lower outcome. 5

Here, the tradeo� between the non-monetary incentive (such as job enlargement) and monetary incentivesto induce the agent to work hard is examined.

When the e�ort of the agent is observed by the principal (i.e., a ®rst-best solution obtains), the optimaljob design requires some job-challenge, if and only if, the marginal productivity of e�ort is increasing injob-challenge. Conversely, the agency is better-o� with specialized jobs (i.e., no job-challenge) if job-challenge decreases the marginal productivity of e�ort. The bene®t to the principal arises from substitutingnon-monetary incentives for non-monetary incentives ± a substitution e�ect. If the agent does not derive anyutility from job-challenge then the substitution e�ect is absent, and, hence, the lowest level of job-challengeis optimal.

In a setting with moral hazard (i.e., the e�ort of the agent is not observed by the principal), additionalrisk needs to be imposed on the agent to mitigate the hidden-action e�ect. We ®nd that all of the risk neededto induce the high e�ort can be imposed through the job design (i.e., the ®rst-best solution obtains) if andonly if the marginal utility from job-challenge is greater than the marginal disutility from e�ort for theagent. We term this the risk e�ect. In other words, the ®rst-best solution obtains whenever the ®rst-best jobdesign provides the agent with more utility from job-challenge than the disutility he su�ers from exertinge�ort. In such a case, the job-challenge can be set as if the e�ort of the agent is observable. When the agentderives no utility from job-challenge, the principal nonetheless ®nds it bene®cial to design challenging tasksbecause the challenging tasks make the outcome more informative on the unobservable e�ort ± an infor-mativeness e�ect. When the agent derives utility from job-challenge and the ®rst-best solution does notobtain, the substitution e�ect, the risk e�ect and the informativeness e�ect interact simultaneously. Becauseall three e�ects yield bene®ts, the optimal job-challenge is set at a level higher than the ®rst-best level of job-challenge. In general, the monetary incentives could either increase or decrease with increases in job-challenge utility, depending on the interactions among the three e�ects.

When in addition to moral hazard, the agent alone knows his utility from job-challenge (i.e., the adverseselection case), we ®nd that the ®rst-best solution does not obtain. In general, an additional amount of riskneeds to be imposed on the agent for inducing the truth. We show that the additional risk is imposedthrough job-challenge. Speci®cally, the agent who derives more utility from job-challenge is provided with

4 A recent article in Fortune, 24 December, 1994, argues that there is surely more than mere monetary incentives that induce people

to work. Our study looks at the motivational impact of job-challenge as a factor potentially providing people with a sense of

accomplishment.5 The job challenge concept operationalized here makes the outcome more informative on the hidden-action. That is, as the job-

challenge increases the higher outcome implies a higher likelihood that the higher e�ort was exerted by the agent. In e�ect, job

challenge makes the performance easier to measure. This view is di�erent from Milgrom and Roberts' view (Milgrom and Roberts,

1992), who claim that increased variety of tasks makes performance evaluation increasingly di�cult.

140 S. Radhakrishnan, J. Ronen / European Journal of Operational Research 115 (1999) 138±157

more challenging jobs but a lower level of monetary incentives. On the other hand, the agent who derivesless utility from job-challenge is provided with a more risky monetary incentive to induce the higher e�ort.When the agent's utility from job-challenge is su�ciently high such that under moral hazard alone the ®rst-best solution obtains for both types of agents, the job-challenge is set at a level higher (lower) than the ®rst-best level for the agent who derives higher (lower) utility from job-challenge. This highlights the risk e�ectof job-challenge in the adverse selection setting.

The rest of the paper is organized as follows: Section 2 describes the model; Section 3 contains thebenchmark results when the e�ort that the agent exerts is observable to the Principal; Section 4 considersthe case of moral hazard; Section 5 examines the case where the agent alone knows his utility from job-challenge; and Section 6 contains some concluding remarks.

2. The model

Consider the problem of a risk-neutral principal who contracts with a risk averse agent. The principal'sobjective is to maximize the expected value of the random outcome, x, less the payments made to the agent.The random outcome can take on two values x1 and x2, with x2 > x1. The agent's e�ort, e, in¯uences theoutcome stochastically. The productive e�ort can take on two values with eH > eL. The outcome is alsoin¯uenced stochastically by a job-challenge parameter h 2 �h; h�. The technology is given by p�xi j ej; h�. 6

The e�ort of the agent is not observed by the principal (i.e., moral hazard exists) who prefers that the agentexert the higher e�ort, eH .

We assume that the marginal product increases in e�ort, i.e., p�eH ; h� > p�eL; h�. We also assume thatph�:� < 0 and phh�:� < 0, i.e., the probability of the higher outcome decreases at an decreasing rate in h. 7

This implies that as job-challenge (h) increases the probability of the higher outcome decreases. Contrastthis with repetitive, mechanistic jobs (i.e., specialization) that promise a high probability of the higheroutcome but o�er little challenge. This captures Atkinson's, and Campion and McClelland's concept(Atkinson, 1958; Campion and McClelland, 1991, 1993) that designing jobs that are more challengingimposes the cost of reduced probability of higher outcomes. The decrease in the probability of the highoutcome occurs at a decreasing rate, i.e., by designing more challenging jobs the cost increases but at amuted rate. Were this not to be the case, i.e., were the cost to the principal of increasing job-challengeincrease at an increasing rate the least challenging job would always be optimal. We assume thatph�eH ; h� > ph�eL; h�, i.e., the marginal productivity of e�ort increases with more challenging jobs.

The principal provides the risk and e�ort averse agent with a monetary incentive denoted by si for eachxi. In addition to deriving utility from wealth and su�ering disutility from e�ort, the agent also derivesutility from job-challenge. Speci®cally, the agent's expected utility is given by

RA�:� � U�s� � a�e��p�x2 j eH ; h� ÿ p�x2 j eL; h�� ÿ V �e� with a�eL� � 0 and a�eH � � a: �1�

The ®rst term of the agent's expected utility denotes the utility that the agent derives from wealth. Thesecond term in the agent's expected utility function captures the utility that the agent derives from job-challenge in the following sense. Suppose the agent chooses to shirk; then, for any level of job-challenge, theagent derives no utility from job-challenge (i.e., a�eL� � 0). In contrast. the utility that the agent deriveswhen he exerts the high e�ort is proportional (a) to the marginal productivity of e�ort [p�eH ; h� ÿ p�eL; h�].This captures the notion that the agent derives more utility when he knows that his e�ort is not spent in vain

6 Henceforth we denote p�x2 j ej; h� as p�ej; h�; and p�x1 j ej; h� as �1ÿ p�ej; h��.7 Subscripts denote partial derivatives.


(as suggested by Atkinson (1958)). The third term in the agent's utility is the disutility from e�ort. Theagent's reservation utility is given by �U .

To reiterate, the job-challenge is given by h and the agent's e�ort is given by e. The motivational impactof the job design is captured by �p�eH ; h� ÿ p�eL; h��, the marginal productivity of e�ort. Productivity of theagency is captured by �RP=E�s�� where RP is the expected utility for the principal and is given byRP � x1p�eH ; h� � x2�1ÿ p�eH ; h�� ÿ E�s� with E�s� � s1p�eH ; h� � s2�1ÿ p�eH ; h��. The events unfold asfollows:· the principal designs the incentive scheme and the job-challenge;· the agent then chooses productive e�ort so as to maximize his expected utility;· the outcome is realized and the contract is settled.

The problem for the principal when moral hazard exists is given in MH.Moral hazard (MH)

maxsi ;h

RP �:jeH � � �x1 ÿ s1��1ÿ p�eH ; h�� x2 ÿ s2�p�eH ; h�s:t: RA�:jeH � � U�s1��1ÿ p�eH ; h�� U�s2�p�eH ; h� � a�p�eH ; h� ÿ p�eL; h�� ÿ V �eH �P �U ; �2�

RA�:jeH � � U�s1��1ÿ p�eH ; h�� U�s2�p�eH ; h� � a�p�eH ; h� ÿ p�eL; h�� ÿ V �eH �P U�s1��1ÿ p�eL; h�� U�s2�p�eL; h� ÿ V �eL� � RA�:jeL�: �3�

Eq. (2) is the participation constraint that guarantees the reservation utility to the agent. Eq. (3), theincentive compatibility constraint, recognizes that the agent chooses e�ort to maximize his expected utility.

Our model di�ers from the basic principal±agent model (for example see Holmstrom, 1979; Shavell,1979) in the sense that we explicitly recognize the utility that the agent derives from job-challenge (i.e., themotivational e�ects of job design). The principal designs the incentive scheme by choosing the levels of job-challenge and providing monetary incentives.

Before proceeding to analyze the model with moral hazard, we establish some benchmark results inSection 3.

3. Benchmark results

In this section, we assume the absence of moral hazard. The question is how the principal should choosethe level of job-challenge in this case when the e�ort is observable? The principal can design a job that ismechanistic (i.e., h� � h), promising a higher probability of the high outcome, and hence, a higher expectedoutput. However, since the marginal productivity of e�ort is increasing with job-challenge, the agent wouldexperience a lower sense of accomplishment and thus, his satisfaction/motivational score will be low. Thelow motivational score of the agent would have to be o�set by the principal with higher monetary in-centives. Will there be residual motivational concerns at optimum, where the principal can force the agentto exert the optimal e�ort? This question is answered in the following Proposition for (a) the case where theagent derives no utility from job-challenge (a � 0), and (b) the case where the agent derives utility from job-challenge (a > 0). 8

Proposition 1. (1) When the e�ort exerted by the agent is observable and the agent does not derive utility fromjob-challenge (i.e., a � 0):

(a) the agent is paid a constant, i.e., s1 � s2 � s;

8 Proofs for all the propositions are in Appendix A.


(b) the lowest level of job-challenge is optimal, i.e., h� � h:(2) When the e�ort exerted by the agent is observable and the agent derives utility from job-challenge (i.e.,

a > 0):(a) the agent is paid a constant, i.e., s1 � s2 � s;(b) a job-challenge higher than the lowest level is optimal if and only if the marginal productivity of e�ortincreases in job-challenge, i.e., h� > h i� ph�eH ; h� > ph�eL; h�;(c) if h� > h the monetary incentives decreases monotonically with increases in a, i.e., ds=da < 0; and(d) if h� > h; then dh�=da > �<�0 for a�p�eH ; h

�� ÿ p�eL; h�� < �>��H 0�Y �=H 00�Y �� where H�:� � Uÿ1�:�,

h� indicates the optimum job-challenge in the ®rst-best solution, and

Y �h� � U � V �eH � ÿ a�p�eH ; h� ÿ p�eL; h��: �4�In the ®rst-best case, the cost for the principal of choosing challenging tasks (i.e., h > h) arises from the

decrease in expected output. When the agent does not derive any utility from job-challenge (i.e., a � 0) theprincipal does not obtain any bene®t from substituting non-monetary incentives for monetary incentives.Hence, the lowest level of job-challenge is optimal. This is shown in Proposition 1.1.

Proposition 1.2(b) shows that when the agent derives utility from job-challenge (i.e., a > 0) the costs ofchoosing challenging tasks are lower than the bene®ts in terms of substituting non-monetary incentives formonetary incentives if and only if �ph�eH ; h� ÿ ph�eL; h�� > 0. The condition implies that the marginalproductivity of e�ort should be increasing in the job-challenge, and thus, increasing job-challenge wouldincrease the agent's utility. The bene®t to the principal is derived from the substitution e�ect, i.e., thesubstitution between the monetary and non-monetary incentives. 9

Proposition 1.2(c) shows that as a increases less monetary incentives are required because non-monetaryincentives are substituted for monetary incentives. This demonstrates the substitution e�ect. But the e�ectof an increase in a on the level of job-challenge is not unambiguous. The condition in Proposition 1.2(d)indicates that the e�ect of increases in a on job-challenge depends on the agent's risk aversion and the utilityfrom job-challenge. Considering U � w1=2, it can be shown that the job-challenge increases up to a pointand then decreases. 10

4. Moral hazard

In this scenario, the e�ort exerted by the agent is not jointly observable. The principal chooses acombination of job-challenge and monetary incentives �h; si� to induce the high e�ort (eH ). The problem forthe principal is given by the program in MH. We consider the following three cases (a) the case where the®rst-best solution obtains, (b) the case where a � 0, and (c) the case where a > 0 and the ®rst-best solutiondoes not obtain. These results are then illustrated with a numerical example.

4.1. The case of the ®rst-best solution

In general, when moral hazard exists, the ®rst-best solution does not obtain (see Holmstrom, 1979;Shavell, 1979). However, under MH the principal can, in addition to monetary incentives, also use non-

9 Note that the condition states that the job-challenge should make the output more informative on e�ort, even though the e�ort is

observable.10 We present a numerical example in the next section to illustrate this phenomenon.


monetary incentives. Given the two instruments at her disposal, the following proposition examines con-ditions where the principal can design jobs and obtain the ®rst-best solution.

Proposition 2. Under MH, the ®rst-best solution obtains if and only if the marginal utility from job-challenge isgreater than the marginal disutility from e�ort, i.e., the agent is paid a constant �s1 � s2 � s� and the ®rst-bestjob-challenge is chosen �h � h�� i�

a�p�eH ; h�� ÿ p�eL; h

��P V �eH � ÿ V �eL�: �5�

Proposition 2 provides the condition under which the ®rst-best solution obtains even in the presence ofmoral hazard. In such a case, the ®rst-best level of job-challenge automatically induces the agent to choosethe high e�ort. Consider a situation when the agent shirks. By shirking the agent might not even earn hisreservation utility since a smaller monetary payment is o�ered in light of the utility that the agent derivesfrom job-challenge. Of course, the agent will choose not to exert the higher e�ort, if the increased cost ofexerting the higher e�ort is much larger than the bene®t that the agent derives in terms of utility from job-challenge. Hence, the marginal utility from job-challenge needs to be greater than or equal to the marginaldisutility from e�ort for the ®rst-best solution to obtain. In other words, the agent su�ers an opportunityloss by choosing the lower e�ort in terms of not obtaining the utility from job-challenge. Even in theabsence of monetary risk, the agent faces the risk of incurring the opportunity loss. The opportunity loss ofnot getting any utility from job-challenge by choosing the lower e�ort substitutes for the monetary risk thatearlier studies have shown as being necessary to induce the agent to exert the higher e�ort. This is the riske�ect.

4.2. The case with no utility from job-challenge

In the scenario where the e�ort is observable it was shown that the lowest level of job-challenge isoptimal when a � 0. However, if the marginal productivity of e�ort increases with increases in job-chal-lenge, i.e., ph�eH ; h� > ph�eL; h� more challenging jobs would increase the informativeness of the output. Thequestion of whether the additional informativeness of job-challenge is used by the principal when the e�ortis not observable and the agent does not get any utility from job-challenge is examined in Proposition 3.

Proposition 3. Let the agent derive no utility from job-challenge and U�w� � w1=2. Then under MH the ®rst-best solution does not obtain.1. The agent is paid a non-constant compensation. Speci®cally,

s2�NJ � � H�Y �h� � D�NJ �p�eH ; h��; �6�s1�NJ � � H�Y �h� ÿ D�NJ ��1ÿ p�eH ; h��; �7�

D�NJ � � V �eH � ÿ V �eL�p�eH ; h� ÿ p�eL; h��

; �8�

where Y �h� is given in Eq. (4) and the NJ in parenthesis denotes the optimum when a � 0:2. Job-challenge set greater than the lowest level is optimal if ph�eH ; h� ! 0 and ph�eL; h� < 0:

Proposition 3 provides su�cient conditions for job-challenge greater than the lowest level being optimaleven when the agent does not derive any direct utility from job-challenge. The condition speci®es that at thelowest level of job-challenge the informativeness of output with respect to the unobservable e�ort increasesat a su�ciently high rate with job-challenge. Suppose that the marginal increase in job-challenge from the


lowest level does not a�ect the probability of outcomes when the high e�ort is exerted, while the job-challenge decreases the probability of outcome when the low e�ort is exerted. Then, the principal will gainfrom increasing job-challenge at least marginally, because the cost of job-challenge is almost zero while thebene®t in terms of informativeness of output increases. By increasing job-challenge the principal canimprove the informativeness of outcome. That is, the higher outcome would imply a higher likelihood ofthe high e�ort, the informativeness e�ect. The informativeness e�ect decreases the cost of risk imposedthrough monetary incentives under MH.

4.3. The second-best case

We next consider the case where the agent derives utility from job-challenge and the condition speci®edin Eq. (5) is not satis®ed. In other words, the ®rst-best solution is precluded. Note that increased job-challenge increases the risk imposed on the agent. Also, from the earlier results in principal±agent theory(see Holmstrom, 1979), we know that risk needs to be imposed on the agent to induce him to exert thehigher e�ort. From the propositions above it is clear that in this case the substitution, risk and informa-tiveness e�ects will interact. Proposition 4 examines this interaction.

Proposition 4. Let U�w� � w1=2. Then, under MH, when the condition speci®ed in Eq. (5) is violated the ®rst-best solution does not obtain.

1. The agent is paid a non-constant compensation. Speci®cally,

s2�MH � � H�Y �h� � D�MH �p�eH ; h��; �9�s1�MH � � H�Y �h� ÿ D�MH ��1ÿ p�eH ; h��; �10�

D�MH � � V �eH� ÿ V �eL�p�eH ; h� ÿ p�eL; h� ÿ a

� �; �11�

where Y �h� is given in Eq. (4) and the MH in parentheses denotes the optimum.2. A higher than ®rst-best level of job-challenge is chosen, i.e., h�MH� > h� if

ph�eH ; h�� ÿ ph�eL; h

��p�eH ; h

�� ÿ p�eL; h��

� �> 1 and �p�eH ; h

��2 < 0:5:

3. �dh�MH�=da� > �<�0 for fa�p�eH ; h� ÿ p�eL; h�� ÿ �H 0�Y �=H 00�Y ��gH 00�Y ��ph�eH ; h� ÿ ph�eL; h��ÿH 00�Y �D�MH�Z�:��dD�MH�=da� > �<�0 where

Z � �1ÿ 2p�eH ; h��ph�eH ; h�p�eH ; h��1ÿ p�eH ; h��

Dh

D

� �p�eH ; h��1ÿ p�eH ; h��:

Proposition 4 characterizes the solution under MH when the ®rst best solution does not obtain. Spe-ci®cally, some risk needs to be imposed through the monetary incentives. However, the risk imposedthrough monetary pay-o�s is muted to the extent that the agent derives utility from job-challenge. To seethis compare Eqs. (9)±(11) with Eqs. (6)±(8). The non-monetary incentive is used to impose additional riskon the agent to induce him to exert the high e�ort. For this purpose a higher level of job-challenge than the®rst-best level is chosen if the job-challenge makes the output su�ciently informative on e�ort, i.e.,

ph�eH ; h�� ÿ ph�eL; h

��p�eH ; h

�� ÿ p�eL; h��

� �> 1:


The condition states that the decrease in the probability of the high outcome should be a lot smaller forhigh e�ort than low e�ort. In addition, the ®rst-best level of job-challenge should be such that the prob-ability of the high outcome is not extremely high, i.e., �p�eH ; h

��2 < 0:5: If the probability of the highoutcome is su�ciently high, then the cost of increasing the job-challenge in terms of decreased expectedoutcome for the principal would be so high as to preclude the choice of a higher level of job-challenge. Also,the job-challenge increases with a over the region where it increases in the ®rst-best case.

Proposition 4 demonstrates the importance of designing jobs when e�ort cannot be observed. In largedecentralized organizations where the e�ort of the agent is typically not directly observable, it may benecessary to design jobs that are more challenging than what is ®rst-best. For the organization's empiricalobserver, the implication is that the bene®t from increased job-challenge should not be restricted to thecalibration of job satisfaction (utility) alone, but should be extended to include measures of the reduced costof inducing unobservable e�ort.

4.4. Numerical example

We illustrate the results in the ®rst-best and the second-best using an example. For purposes of theexample, we set U�s� � s0:5; x1 � 10; 000; x2 � 20; 000; �U � 75; eL � 2; eH � 5; p�e; h� � 1ÿ �2h2=e��

;V �e� � 0:5e2; with h � 0:1 and h � 1. We let a range from 0 to 100. Table 1 provides the numerical results.

Table 1

Numerical example ± moral hazard

a h s1 s2 RP P

Panel A: First-best solution

0 0.100 7656.250 7656.250 12303.750 1.607

10 0.100 7645.754 7645.754 12314.246 1.611

20 0.100 7635.264 7635.264 12324.736 1.614

30 0.100 7624.782 7624.782 12335.218 1.618

40 0.417 6943.333 6943.333 12361.111 1.780

50 0.833 4446.666 4446.666 12777.778 2.874

60 0.942 3086.346 3086.346 13364.198 4.330

70 0.974 2271.057 2271.057 13934.239 6.136

80 0.977 1737.440 1737.440 14444.444 7.314

90 0.967 1369.384 1369.384 14890.260 10.874

100 0.950 1112.223 1112.223 15277.778 13.736

Panel B: Moral hazard

0 0.430 0.001 8930.250 10990.631 1.329

10 0.448 228.383 8555.763 11309.956 1.434

20 0.485 1259.559 8062.120 11637.033 1.568

30 0.535 3004.681 7379.894 11976.124 1.741

40 0.606 5211.973 6367.110 12333.629 1.990

50 0.833 4446.666 4446.666 12777.778 2.874

60 0.942 3086.346 3086.346 13364.198 4.330

70 0.974 2271.057 2271.057 13934.239 6.136

80 0.977 1737.440 1737.440 14444.444 7.314

90 0.967 1369.384 1369.384 14890.260 10.874

100 0.950 1112.223 1112.223 15277.778 13.736

Parameter values: P � �RP=E�s��; U�s� � s:5; x1 � 10; 000; x2 � 20; 000; �U � 75; eL � 2; eH � 5; p�e; h� � 1ÿ �2h2=e�� ; V �e� � 0:5e2;

with h � 0:1 and h � 1:


Panel A of Table 1 provides the optimum ®rst-best solution. For a � 0 through a � 30 the lowest levelof job-challenge is chosen. This is because the cost of decreased expected output is not su�ciently o�set bythe substitution of non-monetary incentive. 11 The level of job-challenge increases for a � 40 througha � 80 and then decreases, illustrating the tradeo� between the substitution e�ect and the cost of decreasedexpected output for the principal. As shown in Proposition 1, the interaction among these e�ects does notlead to unambiguous results. In this example, the substitution e�ect dominates the risk-aversion e�ect whenthe utility from job-challenge is small and vice-versa. That is, when the utility from job-challenge is small,the principal ®nds it pro®table to substitute more non-monetary incentives by increasing the job-challenge.

Panel B of Table 1 provides the optimum solution under MH. Firstly, note that when a � 10 through 40the condition speci®ed in Eq. (5) is not satis®ed while for a � 50 through 100 the condition speci®ed inEq. (5) is satis®ed. 12 As a increases, the spread in the monetary incentives �s2�MH� ÿ s1�MH�� decreases,due to the risk and the informativeness e�ects. The expected payment to the agent decreases due to thesubstitution e�ect. For a � 50 and above, all the risk required to induce the agent to choose the high e�ortis imposed through job-design. In e�ect, for aP 50, the risk e�ect is ``strong'' enough to completelyeliminate the impact of unobservability of e�ort, thus the ®rst-best solution is obtained. The behavior of theoptimal level of job-challenge after this point is the same as that of Panel A. That is, all the risk is imposedthrough job-challenge to induce the desired level of e�ort. Put di�erently, the motivational e�ect of job-challenge is high enough such that all the risk for inducement can be shifted to the non-monetary incentives.

Comparing Panels A and B, the optimal level of job-challenge in the ®rst-best is weakly lower than theoptimum under MH. This is so because, in the region wherein the utility from job-challenge does notcompletely mitigate the unobservability of e�ort, the principal ®nds it bene®cial to increase job-challengebecause of the risk and informativeness e�ects.

For empirical tests the results point out that the cost of increased job-challenge needs to be compared tonot only the increased motivational level of the employees but also the additional bene®t of inducingunobservable e�ort.

We next turn to the case where the agent possesses pre-contract private information on his marginalutility from job-challenge.

5. Private information

In this section we consider the case where the agent alone knows a which can either be ``good'' or ``bad'',i.e., aG > aB. Prior to contracting, the principal holds a belief that the agent is of type ai with probability /i.The principal wishes to induce the higher e�ort, i.e., eH . Events unfold as follows:· the agent observes his type to be either aG or aB;· the principal designs a set of monetary and non-monetary incentive schemes;· the agent communicates her type to the principal, and chooses from the menu of incentive schemes;· the agent chooses productive e�ort;

11 This implies that the optimum is below h � 0:1: We have implicitly assumed interior solutions to the ®rst-best case while

establishing the propositions. The condition compares the cost �x2 ÿ x1�ph�eH ; h� with the bene®ts from substitution H 0�Y �h��Yh�h� > 0:

Speci®cally, �x2 ÿ x1�ph�eH ; h� ÿ H 0�Y �h��Yh�h� > 0:12 Note that the agency literature provides a rationale for moral hazards that exist in insurance relationships. The e�ect of utility

from motivation is not directly apparent in these insurance relationships. Hence, the result that some risk needs to be shifted on to

agents in terms of deductibles etc. is quite valid for the contexts in which agency theory was developed. The main point here is that even

in organizations where e�ort is unobservable, the principal can design and de®ne jobs which provide an additional mechanism for

motivating/inducing employees. We have shown that these non-monetary mechanisms could eliminate the moral hazard concerns

completely in some cases.


· the ®nal outcome is realized and the contract is settled.This setting captures the notion that individuals could derive di�erent levels of utility from job-chal-

lenge. Some individuals could be motivated more by a sense of accomplishment than by monetary in-centives, while others might exhibit the opposite preferences.

The problem for the principal (invoking the revelation principle) is given in PI below: Private Infor-mation (PI)

maxsi;hj

RP �:jeH ;m � j; n � j�s:t:

RA�:jeH ;m � j; n � j�P �U for j � G;B; �12�RA�:jeH ;m � j; n � j�P RA�:jeL;m � j; n � j� for j � G;B; �13�RA�:jeH ;m � j; n � j�P RA�:jeH ;m � k; n � j� for j; k � G;B; j 6� k; �14�

where

RP �:jeH ;m � j; n � j� �X

j�G;B

��x1 ÿ s1j�f1ÿ p�eH ; hj�g � �x2 ÿ s2j�p�eH ; hj��/j;

RA�:jeH ;m � j; n � j� � U�s1j��1ÿ p�eH ; hj�� U�s2j�p�eH ; hj� � aj�p�eH ; hj� ÿ p�eL; hj�� ÿ V �eH�;RA�:jeH ;m � k; n � j� � U�s1k��1ÿ p�eH ; hk�� U�s2k�p�eH ; hk� � aj�p�eH ; hk� ÿ p�eL; hk��;

and n � k for k � G;B denotes the private information that the agent has observed, while m � j for j � G;Bdenotes the report (or the menu) chosen by the agent.

Eqs. (12)±(14) are the participation constraints, the incentive compatibility constraints, and the truth-telling constraints, respectively. The problem in PI extends MH to a case with adverse selection. The so-lution to the program speci®ed in PI is characterized in Proposition 5.

Proposition 5. Under PI the solution exhibits the following characteristics:1. The agent of type aG gets a payo� higher than the reservation utility, i.e., RA�:jeH ;m � j; n � j� > U ;

Eq. (12) for j�G is non-binding.2. Inducing truth from agent of type aG is costly, i.e., RA�:jeH ;m � G; n � G� � RA�:jeH ;m � B; n � G�

Eq. (14) for j�G and k�B is binding.3. The job-challenge for the agent of type aG is higher than the job-challenge for the agent of type aB, i.e.,

h�aG� > h�aB�:4. The expected monetary incentive for agent of type aG is lower than the expected monetary incentive for the

agent of type aB:5. If Eq. (5) is satis®ed for faG; aBg, the job-challenge for the agent of type aG is higher than the ®rst-best job-

challenge for the agent of type aG, i.e., h�aG� > h��aG�:6. If Eq. (5) is satis®ed for faG; aBg, the job-challenge for the agent of type aB is higher than the ®rst-best job-

challenge for the agent of type aB, i.e., h�aB� < h��aB�:The results in Proposition 5 are similar to earlier results on adverse selection with moral hazard (for

example see La�ont and Tirole, 1986). The agent has to be paid an information rent and the ®rst-bestsolution is precluded. Inducing the truth from the agent who derives more utility from job-challenge iscostly. The agent who derives more utility from job-challenge would have an incentive to report that hederives less utility from job-challenge, accept the job designed for the other type and receive a highermonetary payo�. By doing so, the agent who derives more utility from job-challenge would get an expectedtotal utility that is greater than his reservation utility. Hence, the agent bene®ts by misrepresenting the hightype as the low type.


The general results of adverse selection with moral hazard, compared to the case with moral hazardalone have shown that some additional risk needs to be imposed to elicit truth-telling (see La�ont andTirole, 1986). The question that we address is whether the additional risk is imposed through the job-challenge (non-monetary incentives) or through monetary incentives. We show that the job-challenge is setat a level higher than under the ®rst-best and MH for the agent of type aG and vice-versa for the agent oftype aB: This shows that when more risk is required, imposing it through task design is pro®table for theprincipal. Imposing risk through non-monetary incentives helps the principal in decreasing the informationrent paid through the monetary incentives. The cost for the principal associated with imposing risk throughjob-challenge is the decrease in the expected outcome (or deviation from the ®rst-best or MH, as the casemay be). 13

We illustrate the insights with a numerical example. The parameter values are set to be the same as in thenumerical example in the earlier section. We illustrate two cases: (a) where under MH alone the ®rst-bestdoes not obtain for the agent of type aB, i.e., the condition speci®ed in Eq. (5) is not satis®ed given aB underMH; and, (b) where under MH alone the ®rst-best obtains for the agent of type aB, i.e., the conditionspeci®ed in Eq. (5) is satis®ed given aB under MH. Note that in the second case, where under MH alone the®rst-best obtains for aB, the ®rst-best will obtain for aG because aG > aB, by assumption. The results areprovided in Table 2.

Panel A of Table 2, provides the solution under PI when aB � 20. From Table 1, panels A and B, it isclear that the ®rst-best does not obtain when aB � 20: We let aG range from 30 through 60. The job-challenge for aB decreases with increases in aG: as the spread between aG and aB increases the elicitation oftruth becomes more costly for the principal. Hence, to impose risk the principal increases the spread in thejob-challenge. Note that for aG � 50, the upper-bound of job-challenge is achieved. Hence, for aG greaterthan 50, risk is imposed by making the job-challenge for type aB smaller.

Panel B of Table 2 provides the solution under PI when aB � 60 and aG ranges from 70 through 100. Thesolution is similar to the results obtained in Panel A. However, as aG increases the job-challenge increases

Table 2

Numerical example ± adverse selection

Adverse selection

Panel A (aB � 20) Panel B (aB � 60)

Good aG 30.000 40.000 50.000 60.000 70.000 80.000 90.000 100.000

hG 0.541 0.627 0.905 1.000 1.000 1.000 1.000 0.989

s1G 3269.426 5901.320 4444.394 3186.333 2278.194 1792.225 1384.004 1111.111

s2G 7581.883 6615.578 4444.394 3186.333 2278.194 1792.225 1384.004 1111.111

E�sG� 7077.013 6503.260 4444.394 3186.333 2278.194 1792.225 1384.004 1111.111

RP �:jaG� 11752.263 11924.224 12279.506 12813.667 13721.806 14207.775 14615.996 14976.405

P�:jaG� 1.661 1.834 20763 4.021 6.023 7.927 10.561 13.479

Bad hB 0.472 0.462 0.457 0.454 0.610 0.486 0.453 0.433

s1B 1000.978 796.784 706.157 649.748 5493.387 4391.552 3255.531 2497.463

s2B 8106.402 8141.804 8157.838 8167.976 5493.387 6451.426 6750.644 6929.239

E�sB� 7473.212 7514.704 7535.328 7548.125 5493.387 6256.812 6463.753 6596.875

RP �:jaB� 11635.652 11631.520 11629.276 11627.411 13018.213 12798.404 12715.411 12653.169

P�:jaB� 1.557 1.548 1.543 1.540 2.370 2.046 1.967 1.918

Parameter values: P � �RP=E�s��; U�s� � s:5; x1 � 10; 000; x2 � 20; 000; �U � 75; eL � 2; eH � 5; p�e; h� � 1ÿ �2h2=e�� ; V �e� � 0:5e2;

with h � 0:1 and h � 1:

13 This scheme is equivalent to a menu of job-challenges o�erred to the agent, who self-selects himself into the appropriate level of

job challenge: the aB agent selects the lower level while the aG agent selects the higher level.


and reaches the upper bound, and then starts to decrease. Note that the job-challenge is higher than the job-challenge under MH (and the ®rst-best). This illustrates the intricate tradeo� between the monetary andnon-monetary incentives and the instruments through which the principal can impose risk when faced withinformation asymmetries.

6. Concluding remarks

In this paper, we examined cases of moral hazard and adverse selection problems, considering themotivational e�ects introduced by job-challenge (job-design). We operationalized the utility that the agentderives from job-challenge by incorporating the motivational e�ects of Atkinson (1958) explicitly in theagent's utility. Our construct of the motivational e�ect increases as the marginal productivity of the agent'se�ort increases. This captures the sense of satisfaction/accomplishment that agents gain from performingtheir jobs.

This construction is then used to analyze the tradeo� between monetary and non-monetary incentives ina principal±agent framework. The non-monetary incentives are provided by choosing the appropriate levelof job-challenge. In the ®rst-best case, a job-challenge higher than the lowest level of job-challenge is setonly if the marginal productivity of e�ort increases in job-challenge. We show that the job-challenge couldeither increase or decrease with changes in agent's utility from job-challenge. The direction depends on theextent of the agent's risk aversion to monetary incentives and the informativeness e�ect ± how the marginalproductivity of e�ort is a�ected by job-challenge.

In a setting where moral hazard alone exists, a ®rst-best solution obtains if and only if the marginalutility from job-challenge is greater than the marginal disutility from e�ort. Also, the monetary incentivesare reduced as the utility from job-challenge increases.

In cases where the ®rst-best is not achieved, in general, the job-challenge is set at a level higher than the®rst-best level. The cost to the principal of increasing job-challenge is the decrease in the expected outcome.The bene®t is derived from substituting non-monetary incentives for monetary incentives, thereby de-creasing the expected monetary payo�. That is, instead of imposing risk through monetary incentives theprincipal ®nds it bene®cial to impose risk through task design/ job-challenge. Moreover with the ®ne-tuningof task design, the observation of a high output signals higher e�ort with more accuracy. That is, theprecision of the signal is improved by the task design. This helps the principal decrease the risk imposedthrough the monetary incentives. Here again the level of job-challenge could increase or decrease withchanges in the agent's utility from job-challenge.

In a scenario where, in addition to moral hazard, the agent alone knows his utility from job-challenge,the ®rst-best solution does not obtain. The agent who derives a higher utility from job-challenge is providedwith a higher job-challenge. The expected monetary incentive for the agent who derives a higher utility fromjob-challenge is lower than the expected monetary incentive for the agent who derives lower utility fromjob-challenge. Here, also, the additional risk that is required to be imposed because of the added layer ofinformation asymmetry is mitigated partly through task design. The level of job-challenge is (weakly)higher than under MH for the agent who derives more utility from job-challenge and it is (weakly) lower forthe agent who derives less utility from job-challenge.

The total expected utility, and the productivity of, the agent who derives a high utility from job-chal-lenge is higher. This latter result highlights the importance of screening potential employees through in-terviews and other information gathering activities. Knowledge by the principal of the agent's type (hispassion for job-challenge) can help her in designing jobs more e�ectively so as to increase productivity. Theincrease in the principal's payo� resulting from knowing the agent's ``type'' is the maximum amount shewould be willing to invest in Human Resources Services for the purpose of selecting the ``challenge seekers''for employment.


Overall we show that information asymmetries (hidden-action and hidden-information) are mitigatedpartly through monetary and partly through non-monetary incentives. The role of non-monetary incentivessuch as job-challenge is threefold (a) as a substitute for monetary incentives; (b) as a mechanism for im-posing additional risk on the agent in cases of information asymmetries; (c) making the output more in-formative about the unobservable action. O�setting these bene®ts are the cost of decrease the expectedoutcome for the principal with higher levels of job-challenge.

Appendix A: Proofs of propositions

A.1. Proof of Proposition 1

The Lagrangian for the problem with no moral hazard is

L � ��x1 ÿ s1�f1ÿ p�eH ; h�g � �x2 ÿ s2�p�eH ; h�� k�U�s1�f1ÿ p�eH ; h�g � U�s2�p�eH ; h�� a�p�eH ; h� ÿ p�eL; h�� ÿ V �eH � ÿ �U �:

The conditions for an interior optimum are:

1=�U 0�s1�� 1=�U 0�s2�� k; �A:1�

��x2 ÿ s2� ÿ �x1 ÿ s1��ph�eH ; h� � k�U�s2� ÿ U�s1� � a�ph�eH ; h� ÿ kaph�eL; h� � 0: �A:2�From Eq. (A.1) since U 0�:� > 0 it follows that s1 � s2 � s and k > 0. Thus, Eq. (2) is binding, which im-plies:

U�s� � a�p�eH ; h� ÿ p�eL; h�� ÿ V �eH � � �U ; s � H� �U � V �eH � ÿ a�p�eH ; h� ÿ p�eL; h�� H�Y �:Substituting for s in RP we have,

RP �:; hjeH � � x1 � �x2 ÿ x1�p�eH ; h� ÿ H�Y �:The optimum h will satisfy the following ®rst-order condition:

RPh �:; hjeH � � �x2 ÿ x1�ph�eH ; h� ÿ H 0�Y �Yh � 0: �A:3�

Note that RPhh�:; hjeH � < 0, using H 0�:� > 0; H 00�:� > 0; phh�eH ; h� < 0; �phh�eH ; h� ÿ phh�eL; h�� < 0:

For a � 0, we have Yh � 0: Use Yh � 0 in Eq. (A.3) and noting that x2 > x1 and ph�eH ; h� < 0 by as-sumption, we have RP

h �:; hjeH � < 0: It directly follows from the Kuhn±Tucker conditions that h � h isoptimum for a � 0: This establishes Proposition 1.1.

For a > 0, we have Yh > 0 only if �ph�eH ; h� ÿ ph�eL; h�� > 0: If the condition is not satis®ed we will haveRP

h �:; hjeH � < 0 which implies h � h: This establishes Proposition 1.2(b):Di�erentiating the optimality conditions (2), (A.1) and (A.2) implicitly with respect to a we have

U 0�s� dsda

� �� a�ph�eH ; h� ÿ ph�eL; h�� dh

da

� �� p�eH ; h� ÿ p�eL; h�� 0; �A:4�

ÿ U 00�s��U 0�s��2

dsdaÿ dk

da� 0; �A:5�

Lhh�:� dhda

� �� a�ph�eH ; h� ÿ ph�eL; h�� dk

da

� �� k�ph�eH ; h� ÿ ph�eL; h�� 0: �A:6�


Note that phh�:� < 0 implies Lhh�:� < 0 [i.e., similar to the CDFC condition of Rogerson (1985)].From Eq. (A.5) since U 00�:� < 0 it follows that sign ds=da� �� sign dk=da� �. Assume that �dk=da� > 0.

Then from Eq. (A.5) �ds=da� > 0. Using this in Eq. (A.6) we have �dh=da� > 0, since ph�eH ; h� > ph�eL; h�and Lhh < 0 (the second-order su�ciency condition).

Use ph�eH ; h� > ph�eL; h�; �dk=da� > 0; �ds=da� > 0; and �dh=da� > 0 in Eq. (A.4) to obtain a con-tradiction. Hence, �ds=da� < 0. This establishes Proposition 1.2(c).

Di�erentiate Eq. (A.3) with respect to h to get

RPhh�:; hjeH � dh

da

� �� RP

ha�:; hjeH � � 0:

RPhh�:; hjeH� < 0 (second-order condition). Therefore sign dh=da� �� sign RP

ha�:; hjeH �:

RPha�:; hjeH � � �ph�eH ; h� ÿ ph�eL; h��H 0�Y � ÿ aH 00�Y �fp�eH ; h� ÿ p�eL; h�g�:

It directly follows that if afp�eH ; h� ÿ p�eL; h�g > H 0=H 00 then RPha�:; hjeH � > 0 and vice-versa. h


Denote the ®rst-best solution by fs�; h�g: We need to show that the ®rst-best solution obtains if and onlyif a�p�eH ; h

�� ÿ p�eL; h��P V �eH� ÿ V �eL�. We will show that under the speci®ed condition the ®rst-best

solution is feasible under MH. It is direct to check that the individual rationality constraint given by Eq. (2)is satis®ed at equality. It remains to check if the incentive compatibility constraint is satis®ed by fs�; h�g.Note that Eq. (3) is

�U�s2� ÿ U�s1� � a��p�eH ; h� ÿ p�eL; h��P V �eH � ÿ V �eL�: �A:7�Using s1 � s2 � s� and h� in Eq. (A.7) we have


��P V �eH � ÿ V �eL�:This implies that the ®rst-best solution is feasible under MH, if and only if Eq. (5) is satis®ed. Because, the®rst-best is upper envelope of the Principal's expected utility, when fs�; h�g is feasible under MH, that is theoptimum under MH also. h


The Lagrangian for MH is

L � ��x1 ÿ s1�f1ÿ p�eH ; h�g � �x2 ÿ s2�p�eH ; h�� k�U�s1�f1ÿ p�eH ; h�g � U�s2�p�eH ; h� � a�p�eH ; h� ÿ p�eL; h�� ÿ V �eH� ÿ �U �� l�fU�s2� ÿ U�s1� � agfp�eH ; h� ÿ p�eL; h�g ÿ fV �eH � ÿ V �eL�g�:

The conditions for an interior optimum are:

1=�U 0�s1�� kÿ lp�eH ; h� ÿ p�eL; h�

1ÿ p�eH ; h��

; �A:8�


1=�U 0�s2�� k� lp�eH ; h� ÿ p�eL; h�

p�eH ; h��

; �A:9�

��x2 ÿ s2� ÿ �x1 ÿ s1��ph�eH ; h� � k�U�s2� ÿ U�s1� � a�ph�eH ; h� ÿ kaph�eL; h�

� l�U�s2� ÿ U�s1� � a��ph�eH ; h� ÿ ph�eL; h�� 0: �A:10�It follows from Proposition 2, that the ®rst-best solution does not obtain because since


�� < V �eH � ÿ V �eL�. Therefore, l > 0. Using l > 0, it is direct to verify that s2 > s1

from (A.8) and (A.9). This implies that both the (2) and (3) constraints are satis®ed at equality. Solving forU�s1� and U�s2� for any h we have:

U�s1� � U � V �eH � � ap�eL; h� ÿ V �eH� ÿ V �eL�p�eH ; h� ÿ p�eL; h� ;

U�s1� � Y ÿ Dp�eH ; h�; �A:11�

U�s2� � U � V �eH � ÿ a�1ÿ p�eL; h�� V �eH � ÿ V �eL�p�eH ; h� ÿ p�eL; h� ;

U�s2� � Y � D�1ÿ p�eH ; h��; �A:12�where,

Y � U � V �eH � ÿ a�p�eH ; h� ÿ p�eL; h��;

D � V �eH � ÿ V �eL�p�eH ; h� ÿ p�eL; h� ÿ a

� �:

For a � 0 it directly follows that fs1�NJ�; s2�NJ�g are given by Eq. (7) and Eq. (9); and D�NJ� is given byEq. (8). Substituting for fs1; s2g from Eqs. (7) and (6) in RP �:jeH ; h� we have

RP �:jeH ; h� � x1 � �x2 ÿ x1�p�eH ; h� ÿ H�Y � Y2�p�eH ; h� ÿ H�Y ÿ Y1��1ÿ p�eH ; h��;where Y2 � Dp�eH ; h� and Y1 � D�1ÿ p�eH ; h��:

The ®rst-order condition for h is given by

RPh �:jeH ; h� � �x2 ÿ x1�ph�eH ; h� ÿ �H�Y � Y2� ÿ H�Y ÿ Y1��ph�eH ; h�

ÿ H 0�Y � Y2��Yh � Y2h�p�eH ; h� ÿ H 0�Y ÿ Y1��Yh ÿ Y1h

��1ÿ p�eH ; h��:Substituting for H�w� � w2 and H 0�w� � 2w and rearranging we have

RPh �:jeH ; h� � �x2 ÿ x1�ph�eH ; h� ÿ YhH 0�Y � ÿ 0:5D2H 00�Y ��1ÿ 2p�eH ; h��ph�eH ; h�

ÿ DDhH 00�Y �p�eH ; h��1ÿ p�eH ; h��: �A:13�Noting that for a � 0 we have Yh � 0 and evaluating RP

h �:jeH ; h� at h � h we have

RPh �:jeH ; h� � f�x2 ÿ x1� ÿ 0:5D2H 00�Y ��1ÿ 2p�eH ; h��gph�eH ; h�

ÿ DDhH 00�Y �p�eH ; h��1ÿ p�eH ; h�� Qph�eH ; h� ÿ DDhH 00�Y �p�eH ; h��1ÿ p�eH ; h��:

Using ph�eH ; h� ! 0 the ®rst term tends to zero. Note that D > 0, H 00�:� > 0, and


Dh � ÿ �V �eH � ÿ V �eL��ph�eH ; h� ÿ ph�eL; h��p�eH ; h� ÿ p�eL; h��2

< 0;

and therefore RPh �:jeH ; h� > 0: Thus h > h is optimum. h


Following the same procedure as in the proof of Proposition 3, from Eqs. (A.11) and (A.12) we haveEqs. (9)±(11). Substituting Eqs. (9) and (10) in RP and di�erentiating with respect to h and setting it equal tozero, the ®rst-order condition satis®es Eq. (A.13). Evaluating RP

h �:jeH ; h� at h � h� where h� represents the®rst-best solution, and using Eq. (A.3) we have

RPh �:jeH ; h

�� ÿ0:5D2H 00�Y ��1ÿ 2p�eH ; h��ph�eH ; h

�� ÿ DDhH 00�Y �p�eH ; h��1ÿ p�eH ; h

��:If we show that RP

h �:jeH ; h�� > 0, then from the Kuhn±Tucker theorem it follows that the optimum h under

MH is greater than h�:Case 1: If p�eH ; h

�� < 0:5 then both the ®rst and the second terms are positive and hence,RP

h �:jeH ; h�� > 0:

Case 2: If p�eH ; h�� > 0:5:

RPh �:jeH ; h

�� ÿ0:5D2H 00�Y ��1ÿ 2p�eH ; h��ph�eH ; h

�� ÿ DDhH 00�Y �p�eH ; h��1ÿ p�eH ; h

��

� ÿH 00D2

2

�1ÿ 2p�eH ; h��ph�eH ; h

��p�eH ; h

��1ÿ p�eH ; h��

2Dh

D

� �p�eH ; h

��1ÿ p�eH ; h��

� ÿH 00D2

2p�eH ; h

��1ÿ p�eH ; h��

�1ÿ 2p�eH ; h��ph�eH ; h

��p�eH ; h

��1ÿ p�eH ; h�� ÿ

2fV �eH � ÿ V �eL�gfph�eH ; h�� ÿ ph�eL; h

��gfV �eH� ÿ V �eL� ÿ agfp�eH ; h

�� ÿ p�eL; h��g

� �� ÿH 00D2

2

p�eH ; h��fph�eH ; h

�� ÿ ph�eL; h��g�1ÿ p�eH ; h

��fp�eH ; h


� ��2p�eH ; h

�� ÿ 1��ÿph�eH ; h��fp�eH ; h


2p�eH ; h��1ÿ p�eH ; h

��fph�eH ; h�� ÿ ph�eL; h

��g ÿfV �eH � ÿ V �eL�gfV �eH � ÿ V �eL� ÿ ag

� �:

It is direct to verify that

�2p�eH ; h�� ÿ 1�

2p�eH ; h��1ÿ p�eH ; h

�� < 1

using �p�eH ; h��2 < 1;

fp�eH ; h�� ÿ p�eL; h

��gfph�eH ; h

�� ÿ ph�eL; h��g < 1

using the condition speci®ed and �ÿph�eH ; h�� < 1 because p�:� 2 �0; 1�: Also, note that

fV �eH � ÿ V �eL�gfV �eH � ÿ V �eL� ÿ ag > 1

and hence, RPh �:jeH ; h

�� > 0. This establishes Proposition 4.2.


Proposition 4.3 follows directly by noting that Dha � 0 and H 000 � 0 and di�erentiating Eq. (A.13) withrespect to a: h


The Lagrangian for PI is

L �X

j�G;B

f��x1 ÿ s1j�f1ÿ p�eH ; hj�g � �x2 ÿ s2j�p�eH ; hj��

� kj�U�s1j�f1ÿ p�eH ; hj�g � U�s2j�p�eH ; hj� � ajp�eH ; hj� ÿ V �eH � ÿ �U � ÿ kjajp�eL; hj�� lj��U�s2j� ÿ U�s1j� � aj�fp�eH ; hj� ÿ p�eL; hj�g ÿ V �eH � � V �eL��cj�fU�s1j�f1ÿ p�eH ; hj�g � U�s2j�p�eH ; hj� � aj�p�eH ; hj� ÿ p�eL; hj��gÿ fU�s1k�f1ÿ p�eH ; hk�g � U�s2k�p�eH ; hk� � aj�p�eH ; hk ÿ p�eL; hk��g�for j; k � G;B; j 6� k:

To establish Proposition 5.1, we have to show that kh � 0 and kl > 0. From the truth-telling constraintswe have

U�s1G�f1ÿ p�eH ; hG�g � U�s2G�p�eH ; hG� � aG�p�eH ; hG� ÿ p�eL; hG��P U�s1B�f1ÿ p�eH ; hB�g � U�s2B�p�eH ; hB� � aG�p�eH ; hB� ÿ p�eL; hB��> U�s1B�f1ÿ p�eH ; hB�g � U�s2B�p�eH ; hB� � aB�p�eH ; hB� ÿ p�eL; hB��P �U ;

where the ®rst inequality follows from the truth-telling constraint when j � G and k � B; the second in-equality follows since aG > aB by assumption; and the last inequality follows due to the individual ratio-nality constraint. This shows that kG � 0. To establish that kB > 0 follow the track in Grossman and Hart(1983).

Di�erentiating the Lagrangian with respect to s1G and s2G and rearranging (after using kG � 0), we have:

/G

U 0�s1G� � ÿlG�p�eH ; hG� ÿ p�eL; hG��

1ÿ p�eH ; hG� � cG ÿ cB; �A:14�/h

U 0�s2G� � lG�p�eH ; hG� ÿ p�eL; hG��

�p�eH ; hG�� cG ÿ cB: �A:15�

To establish Proposition 5.2, we have to show that cG > 0 and cB � 0. We will ®rst show that cj > 0, forj � G;B is not possible. Assume that cj > 0 for both j � G;B. Then from the Kuhn±Tucker conditions thetruth-telling constraints are satis®ed at equality. The truth-telling constraint for j � G will be

U�s1G�f1ÿ p�eH ; hG�g � U�s2G�p�eH ; hG� ÿ �U�s1B�f1ÿ p�eH ; hB�g � U�s2B�p�eH ; hB�� aG�fp�eH ; hB� ÿ p�eL; hB�g ÿ fp�eH ; hG� ÿ p�eL; hG�g�;

the truth-telling constraint when j � B will be

aB�fp�eH ; hB� ÿ p�eL; hB�g ÿ fp�eH ; hG� ÿ p�eL; hG�g�� U�s1G�f1ÿ p�eH ; hG�g � U�s2G�p�eH ; hG� ÿ �U�s1B�f1ÿ p�eH ; hB�g � U�s2B�p�eH ; hB��:

The two truth-telling equations above imply aG � aB a contradiction with the assumption that aG > aB.Hence, both cj > 0 is not possible.


We will now show that both cj � 0 for j � G;B is not possible. Assume that cj � 0 for j � G;B. The leftside of Eq. (A.14) is positive since U 0�:� > 0. Note that lj P 0 from Holmstrom (1979) and Shavell (1979).The right side of Eq. (A.14) is negative if lj > 0 and equal to zero if lj � 0; a contradiction.

Hence, at least one cj > 0 and the other cj � 0. It is directly to see that for Eq. (A.14) to hold it must bethat cG > 0 and cB � 0 since lj P 0. This establishes Proposition 5.2.

De®ne,

D � U�s1G�f1ÿ p�eH ; hG�g � U�s2G�p�eH ; hG� ÿ �U�s1B�f1ÿ p�eH ; hB�g � U�s2B�p�eH ; hB��and

Q � �fp�eH ; hB� ÿ p�eL; hB�g ÿ fp�eH ; hG� ÿ p�eL; hG�g�:Suppose, hG < hB. Then since ph�eH ; h� ÿ ph�eL; h� > 0, Q > 0. From Proposition 5.2 since cG > 0 andcB � 0 we have D � aGQ and aBQ > D. Substituting for D we have aB > aG a contradiction. Hence, hG > hB,which establishes Proposition 5.3.

From Proposition 5.3 Q < 0, hence D < 0, which establishes Proposition 5.4.If Eq. (5) is satis®ed for aG and aB then lG � lB � 0: Also, it is directly to see from the derivations above

that s1G � s2G � sG and s1B � s2B � sB: Using Eq. (12) for j � B and Eq. (14) for j � G we have

sB � H�V �eH � � U ÿ aB�p�eH ; hB� ÿ p�eL; hB�� H�YB�;sG � H�V �eH � � U ÿ aG�p�eH ; hG� ÿ p�eL; hG�� aG ÿ aB��p�eH ; hB� ÿ p�eL; hB�� H�YG�:

Note that YG > Y �:jaG� under MH and the ®rst-best solution. Substituting for sj in RP we have

RP �:jeH ; hG; hB� � x1 � ��x2 ÿ x1�p�eH ; hG� ÿ H�YG��/G � ��x2 ÿ x1�p�eH ; hB� ÿ H�YB��/B:

The optimum hG satis®es

RPhG�:jeH ; hG; hB� � �x2 ÿ x1�ph�eH ; hG� ÿ H 0�YG�aGfph�eH ; hG� ÿ ph�eL; hG�g � 0:

Noting that H 00�:� > 0 and YG > Y �MH� evaluating RPhG�:jeH ; hG; hB� at fh�G; h�Bg we have

RPhG�:jeH ; h

�G; h

�B� > 0: Therefore, it follows that hG > h�G:

Following a similar track we have

RPhB�:jeH ; hG; hB� � �x2 ÿ x1�ph�eH ; hB� ÿ H 0�YB�aGfph�eH ; hG� ÿ ph�eL; hG�g

ÿ H 0�YG�faG ÿ aBgfph�eH ; hB� ÿ ph�eL; hB�g � 0:

Noting that YB � Y �MH� and the last term above is negative, we have RPhB�:jeH ; h

�G; h

�B� < 0: Therefore, it

follows that hB < h�B: h

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job challenge as a motivator in a principal–agent setting

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