Incentive and moral hazard in quality assurance, procurement management, and hierarchical control

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Incentive and moral hazard in quality assurance, procurement management, and hierarchical control an agency theoretical perspective
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Thesis (Ph. D.)--University of Florida, 1990.
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Includes bibliographical references (leaves 114-123).
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by Yeong-Ling Yang.
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Typescript.
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INCENTIVE AND MORAL HAZARD IN QUALITY ASSURANCE, PROCUREMENT MANAGEMENT,
AND HIERARCHICAL CONTROL: AN AGENCY THEORETICAL PERSPECTIVE















BY

YEONG-LING YANG


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1990






























THIS DISSERTATION IS DEDICATED TO MY PARENTS FOR THEIR EVERLASTING LOVE

MR. TZONG-GUANG YANG AND MRS FU-GUEY WON YANG













ACKNOWLEDGMENTS


I would like to express gratitude to Professor Ira Horowitz who

has been the source of guidance and stimulation. I am also indebted to

the members of my committee: Professor Edward Zabel who has helped me

with his expertise and has been very encouraging; Professor Antal

Majthay who has been very supportive. Thanks are also extended to

Professor S. Selcuk Erenguc for his assistance and kindness in my

studying years.

Special thanks go to my dear mother and father and two brothers

for their everlasting love. Sincere appreciation goes to Fa-Chung

(Fred) Chen, for without his over-decade patience, understanding and

support, this would not have been possible. I am also thankful to Jenny

Chou, Yasemin Aksoy, Chandra Chegireddy and all brothers and sisters in

the Gainesville Chinese Christian Fellowship for their friendship.

The last and the most is to give all gratefulness to the Lord for

His grace and blessing.














TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS ..................................................... iii

ABSTRACT ........................................................ vi

CHAPTERS


1. INTRODUCTION ......................................... 1

An Overview of Agency Theory ....................... 3
Objective and Organization of the Dissertation ..... 5

2. A LITERATURE REVIEW OF AGENCY THEORY ................. 8

The Basic Agency Model ................... ........... 9
The Moral Hazard Problem ..................... 9
The Self-Selection Problem ................... 15
The Extensions ..................................... 18
Summary ............................................ 30

3. QUALITY ASSURANCE AND JOB ENLARGEMENT
IN PRODUCTION MANAGEMENT ........................... 31

The Model .......................................... 33
Characterization of Optimal Contracts .............. 37
The Comparative Statics ...... ..................... 45
Summary ............................................ 59

4. PROFIT SHARING AND TARGET SETTING
IN PROCUREMENT MANAGEMENT .......................... 61

The Model ......................................... 65
Characterization of Optimal Compensation ........... 70
The First-Best Solution ...................... 70
The Second-Best Solution ..................... 74
The Comparative Statics .......... ................. 79
Summary ............................................ 84







5. DEGREE OF SUPERVISION AND MORAL HAZARD
IN HIERARCHICAL CONTROL ............................ 87

The Models ....................................... .. 91
The Comparison of the Models ....................... 105
Summary ............................................ 108

6. CONCLUSION AND FURTHER RESEARCH ...................... 109

REFERENCES ........................................................... 114

BIOGRAPHICAL SKETCH ...................... ...................... 124













Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


INCENTIVE AND MORAL HAZARD IN QUALITY ASSURANCE, PROCUREMENT MANAGEMENT,
AND HIERARCHICAL CONTROL: AN AGENCY THEORETICAL PERSPECTIVE

By

YEONG-LING YANG

December 1990


Chairman: Ira Horowitz
Major Department: Decision and Information Sciences


In this dissertation, hierarchical agency models are proposed to

discuss the cooperative nature of departmental interdependence and the

interest conflict between the principal and the agents in production

environments. There are three functional areas discussed in this study:

quality assurance, procurement management and production supervision.

These models are established according to the expertise required in the

hierarchy. It is the principal's objective to provide an appropriate

incentive to reduce the moral hazard problem.

The first model considers the recent trend of the integration of

quality and production responsibilities. Therefore, both quality and

quantity are contracting attributes. In general, the optimally designed

contracts are strictly increasing in quantity and quality produced. The

worker is better off with this arrangement, not only through the






increased compensation, but also through the enlarged job

responsibilities. The exact incentive scheme depends on the agents'

risk attitudes and the quality-enhancing technology.

The second model deals with the interaction between procurement

and production management. The production costs crucially depend on the

materials, components and subassembly purchased by the procurement

department, and on the effort expended by the production department.

Linear profit-sharing and target-setting incentive schemes are adopted

in this environment. The materials' quality is used to adjust the cost

target for the production department. It is shown that in both first-

best and second-best cases, profit-sharing compensation is always

preferred to fixed salary. The agents' compensations are tied to their

positions when the principal lacks a costless monitoring mechanism.

Agency cost is then considered as the expected value of getting a

perfect monitoring mechanism.

The third model, unlike the first two models with an existing

hierarchy, considers the necessity of production supervision and the

payoff to establishing an expanded hierarchical structure. The

principal's limited span of control and the agents' moral hazard problem

explain the desirability of separating the principal from direct

production supervision. It is shown that it is indeed to the

principal's benefit to expand the hierarchy levels and organization size

in many circumstances.













CHAPTER 1
INTRODUCTION


Increasing global interaction, competitiveness, and the advent of

rapid communication in the twentieth century have provoked a revolution

in the business environment. The interdependence and corporation among

departments in any one company could well spell the company's success or

failure. The employees' motivation, costs and efficiency are among the

elements that have significant impact on the departmental relationships

and performance, and thus require close management. This study will

express the impact each has on the other.

Typically, in a profit/non-profit organization, the largest cost

faced by the owner are those for employees' wages, salaries, and

benefits. They often absorb half or more of the organization's total

revenues (Arnold and Feldman, 1986, p. 340). Employees exchange their

time, ability, skills, and effort for valued rewards and personal

satisfaction. Therefore it is very important for organizational success

that compensation systems are designed in an effective fashion to

motivate and maintain the employees' performance. Among various types

of compensation systems, Arnold and Feldman (1986, p. 345) indicate that

payment schemes have the strongest ability to fulfill the functions of

encouraging members to join or stay, rewarding attendance, and improving

performance. A recent survey (OR/MS Today, 1990, p. 14) conducted by

the Institute of Industrial Engineers to identify issues affecting









productivity and quality in the United States workplace indicates that

the primary motivation for employees to increase productivity is

financial reward, followed by personal recognition and increased

responsibilities and decision-making opportunities. Yet, almost 80

percent of the respondents said that management lacks the commitment to

implement productivity programs for employees such as profit-sharing

plans and training programs. This is borne out by several studies.

Lawler (1971, p. 158) cites six separate studies of the

relationship between payment schemes and performance in the United

States, and finds evidence that indicates that payment is not very

closely related to performance in many organizations claiming to have

merit salary systems. More recently, the Wall Street Journal (July 10,

1990) reports a survey conducted by Brooks International. The survey

results indicate that only a quarter of the respondents believe that

management does an excellent job of rewarding work groups who make

quality improvements. Some companies press employees for quality

improvements but base rewards solely on the number of units produced.

An improperly designed compensation system will also affect the

employees' performance negatively. For instance, conventional piece-

rate schemes for production workers might sacrifice quality for

quantity, while portfolio managers paid on the basis of annual

accounting profits will sacrifice long-term profitability for short-term

earnings. Therefore, a careful examination of alternative compensation

systems becomes a necessity for any organization that searches for

success.








There are numerous theoretical and empirical studies discussing

the issue of designing incentive compensation schemes. One of the most

important and popular approaches is Agency Theory. What follows is an

overview of that theory, and a statement of the objective and

organization of this dissertation.



An Overview of Agency Theory



The term "agency" has its historic origins in Roman law (Ross,

1974). An agency relationship exists between two or more parties when

one of these, designated the principal, seeks to motivate another, the

agent, to choose his actions in a way advantageous to the principal.

Agency relationships can be widely observed, not only in history but in

the contemporary world. Arrow (1985, p. 37) has stated that "the agency

relationship is a pervasive fact of economic life analogous

interactions are virtually universal in the economy, representing a

significant component of almost all transactions." Examples include

patient/doctor, shareholder/executive, insurer/insured, manager/workers,

client/accountant, and reader/writer. The agency situation or problem

is particularly relevant when ownership and control are separate, as is

frequently the case in American corporations (Dyl, 1988). Consider this

notion of an agency problem in the context of the relationship between

stockholders and an executive officer. The stockholders act as

principals, delegating the daily operations of the firm to the

executive. Stockholders are not in a position to monitor closely the

actions of the executive, nor are they as well informed as the executive









as to what are the appropriate actions to take. In an attempt to bring

the executive's interests in line with those of the shareholders, the

executive is often given some complex compensation package.

This problem is non-trivial because generally the agent's choice

of effort level and ability are known only to himself, and there is no

immediate incentive for him to truthfully reveal that information to the

principal. In addition, because of environmental risk and the agent's

typically risk-averse attitude, the agent's behavior involving his

assignment is usually not in the principal's best interest. The

principal, on the one hand, tries to have the agent share uncertain

environmental risk and, on the other hand, must provide enough incentive

to attract the agent to the job, or to motivate him to work harder.

Thus, agency models incorporate two basic phenomena of organizations:

incomplete information and a goals conflict between members of the

organization. The trade-off between providing motivation and

encouraging risk sharing becomes a general feature of agency problems.

Agency theory views this situation through the design of contracts

that maximize the principal's utility and take several factors into

account: (a) the relationship between output and the incentive scheme

offered; (b) the alternative job opportunities from outside markets; (c)

the allocation of risk between the agents and the principal; and (d) the

preferences of the principal and the agents with respect to income and

nonpecuniary outcomes. The various factors involved and the unavoidable

need to deal with stochastic payoffs and costs often make the problem

difficult to formulate and to solve. Restrictive results from some

basic models do offer rich understanding about incentives and conflicts







5

within organizations, and provide useful insights in the construction of

contracts to guide and influence agency relations in the real world.

In the agency literature, many models come from popular phenomena

in the economy. Plentiful examples include insurance policy

(Zeckhauser, 1970; Pauly, 1968; Shavell, 1979; Harris and Raviv, 1978;

Rothschild and Stiglitz, 1976; Wilson, 1977), auditing and

responsibility accounting (Ng and Stoeckenius, 1979; Atkinson, 1978;

Holt, 1980; Antle, 1982, 1984; Suh, 1987, 1988; Demski and Sappington,

1989), portfolio selection (Jensen, 1976, 1983; Eaton and Rosen, 1983;

Narayanan, 1985; Cohen and Starks, 1988), salesforce compensation (Basu

et al., 1985; Nalebuff and Stiglitz, 1984; Coughlan and Sen, 1985; John

and Weitz, 1985; Lal, 1986; Lal and Staelin, 1986), government

contracting (McAfee and McMillan, 1986, 1987), organization behavior

(Williamson et al., 1975; Stiglitz, 1975; Fama, 1980; Fama and Jensen,

1983; Tirole, 1986), national defense contracts (Berhold, 1971; Cummins,

1977), resource allocation (Harris et al., 1982; Harris and Townsend,

1981), and public organization (Hansmann, 1981; Becker and Stigler,

1974; Ross, 1979).



Objective and Organization of the Dissertation



In his book on "The New Science of Management Decision," Herbert

Simon stated a general picture of an organization (1960, p. 40):

An organization can be pictured as a three-layered cake. In
the bottom layer, we have the basic work process. In the
middle layer, we have the programmed decision-making
processes, the processes that govern the day-to-day
operation of the manufacturing and distribution system. In
the top layer, we have the nonprogrammed decision-making









processes, the processes that are required to design and
redesign the entire system, to provide it with its basic
goals and objectives.

Hierarchy stands not only for degrees of highness or lowness, for this

tends to hide its significance. Each level is an inclusive clustering

or combination of interdependent groups, to handle those aspects of

coordination that are beyond the scope of any of its components

(Thompson, 1967, p. 59). In this dissertation, hierarchical agency

models are utilized to discuss the cooperative nature of departmental

interdependence and the interest conflict between the principal and the

agents in production environments. A perfectly competitive market is

assumed for the final products. Three hierarchical agency models are

proposed to discuss the agency relationships in quality assurance, in

procurement management, and in production supervision, respectively.

The ideas developed in this study are partly inspired by the significant

improvement in industries through implementing Just-in-Time philosophy

in various areas such as the consolidation of production and quality,

the vertical integration of procurement and production, and the highly

autonomic spirit among employees.

This study provides several interesting insights into how to

resolve conflicts of interest within organizations by carefully designed

compensation schemes. Unlike conventional contracts, the pattern and

design of incentive systems depend on factors such as the agents'

attitudes of sharing risk and expending effort, as well as on their

production contributions. This study also shows how the organization

can be more profitable and efficient by improving several exogenous

variables or by adjusting its structure.







7

A literature review of agency theory is given in chapter 2. An

agency model of quality assurance and job enlargement in production

management is presented in chapter 3. An agency model considering

linear profit-sharing and target-setting compensations in procurement

management is presented in chapter 4. An agency model regarding

production supervision and moral hazard in hierarchical control is

presented in chapter 5. This dissertation concludes with a summary and

further extensions of each model in chapter 6.













CHAPTER 2
A LITERATURE REVIEW OF AGENCY THEORY


Since the original papers by Wilson (1968), Spence and Zeckhauser

(1971), and Ross (1973, 1974), substantial attention has been given to

the development of agency theory. Agency theory has been viewed as the

neoclassical response to the questions raised by March and Simon (1958)

regarding the behavior of an organization of self-interested agents with

conflicting goals in a world of incomplete information (Levinthal,

1988). There are basically two types of incomplete information under

discussion. The first, referred to as the moral hazard or incentive

problem, reflects the inability of the principal to costlessly observe

the agent's decision. The second, referred to as the self-selection or

adverse-selection problem, reflects the unwillingness of the agent to

reveal his private information about the state of nature, his abilities

or productivity. Most agency literature has focused on the moral hazard

issue, but the more recent work has introduced elements of self-

selection (Myerson, 1982; Baron and Myerson, 1982; Demski and

Sappington, 1984, 1987, 1988). This chapter focuses on formal,

mathematical statements of the agency relationship and largely ignores

the less formal stream of positive research by Jensen and Meckling

(1976), Jensen (1983), and Fama and Jensen (1983).









In what follows, the basic models and their mathematical

justifications are presented. Then, various extensions and related

literature are discussed. The chapter concludes with a summary.



The Basic Agency Model



The basic models dealing with the moral hazard problem and the

self-selection problem are presented as follows.



The Moral Hazard Problem

The basic agency model focuses on a two-player (the principal and

the agent), one-period situation. The agent's effort (or action), a,

together with a random state of nature, 0, are assumed to generate

outcome x according to x=x(a,0). The principal, on the one hand, wants

the agent to share risk and, on the other hand, must provide enough

incentive to attract the agent to the job, or to motivate him to work

harder. The difficulty comes from the fact that the principal can only

observe the agent's outcome x, but not the agent's effort a, or the

random variable 0. Given the observed outcome, the principal can only

make probability judgments about the agent's effort. It is assumed that

both parties are rational and use von Neumann-Morgenstern utility

functions in their decision-making processes, where the utility

functions are known to both parties. The principal is typically assumed

to be less risk averse than the agent (Mirrlees, 1976). Let the

principal be risk neutral, and the agent be risk averse with utility

function H(w,a) defined over income (w) and effort (a). The model









further assumes the utility function is additively separable into a

utility for income, U(s), and a disutility of effort, V(.), i.e.

H(w,a)-U(w)-V(a) with U'>0 and U"<0. The interpretation is that a is a

productive input with direct disutility for the agent and this creates

an inherent difference in objective between the principal and the agent

(Holmstrom, 1979); Harris and Raviv (1979) have proved that the moral

hazard problem can be avoided when the agent is risk neutral. The agent

is effort averse, and his disutility of effort increases at a

nondecreasing rate, i.e. V'>0 and V">0. Mirrlees (1974, 1976) suggests

that 0 be suppressed and views x as a random variable with a

distribution F(x;a), parameterized by the agent's action. Given a

distribution of 0, F(x;a) is the distribution induced on a via the

relationship x x(a,0). The principal's problem is to design a sharing

rule or contract that elicits an appropriate effort level from the

agent. The sharing rule or contract, S(x), which is constructed on the

basis of the principal's observation of the agent's performance, can be

generated by the following program:

(M) Max f[x-S(x)]f(x;a)dx (2.1)

Subject to

E{H(S(x),a)) JfU(S(x))f(x;a)dx V(a) H* (2.2)

a e Argmax [fU(S(x))f(x;a')dx V(a')] (2.3)
a'eA

where E({) denotes the expectation operator, the notation "argmax"

denotes the set of arguments that maximize the indicated objective

function, H* is the agent's market reservation value or opportunity

cost, and A is the set of all possible actions. The first constraint is







11

the so-called "individual rationality constraint" that reflects the fact

that the contract must offer the agent an expected utility that is at

least as great as his opportunity cost (utility). The second constraint

is the so-called "incentive compatibility constraint" that indicates

that the agent chooses the action that maximizes his own utility.

In most of the literature (except Grossman and Hart's (1983)

convex programming approach) it is assumed that the second constraint in

(P) can be replaced by the first-order condition derived from the

agent's expected-utility-maximizing efforts, i.e.

JU(S(x))f,(x;a)dx V'(a) 0, (2.4)

where f(*.) is the derivative of f(x;a) with respect to a. Mirrlees

(1974) points out that the first-order condition may not in general be

necessary for the incentive compatibility constraint, unless the

contract elicits a unique action on the part of the agent at the

optimum. If there is not a one-to-one correspondence between contract

and action, then the first-order condition is not even a necessary

condition for the optimality of the contract. Rogerson (1985) suggests

the correct treatment of the conditions sufficient for necessity of the

first-order condition: notably, if the probability density function of

the outcome, f(x;a), has the monotone likelihood ratio (MLR) property

and the cumulative distribution function (CDF), F(x;a), is convex, then

the first-order approach is valid. The MLR condition states that more

effort (a) yields more outcome (x), and implies that increases in effort

will shift the distribution of x to the right in the sense of first-

order stochastic dominance (SDC) (Milgrom, 1981). By the MLR, F(x;a)

decreases in a, i.e. the probability of an outcome less than or equal to







12

some x" decreases as the agent works harder. The CDF convexity

condition requires that this function decreases at a decreasing rate,

i.e. the CDF is a form of stochastic diminishing returns to scale

(Rogerson, 1985).

Jewitt (1987, 1988) argues that the CDF condition is not fulfilled

by most of the distributions commonly occurring in statistics, and

Rogerson's conditions cannot be directly adapted to the multidimensional

case. Jewitt further suggests an over-sufficient but more easily

verified total positive (TP) density function, along with a specified

utility function as the justification. A function f(x;a) is said to be

total positive of degree n (TPn) if for each x1
we have

f(x1,a1) *** f(xl,ak)
I *I
S0 for k-1,2,...,n.
I I
f(xk,aC) *** f(Xk,ak)


If f(x;a) is TP for all n, then it is said to be totally positive (TP).

Jewitt indicates that any exponential family (including normal,

exponential, Poisson, binomial, etc.) is totally positive in an

appropriate parameterization. The utility condition can be satisfied by

any constant absolute risk-averse utility function and any nondecreasing

relative risk-averse utility function with an Arrow-Pratt coefficient of

relative risk aversion bounded above one half. In order to avoid

problems of existence of an optimal sharing rule, bounds are placed on

the allowable payments to the agent, and the set of possible sharing

rules is assumed to be compact. The bounds on possible payments may be








justified on the basis of institutional features such as limited

liability and bankruptcy.

Given that the first-order condition holds, determination of the

contract S(x) can be treated as an isoperimetric optimal control problem

(Intriligator, 1983, p. 318). Letting A and p be multipliers on (2.2)

and (2.4), respectively, the Hamiltonian is

L J -[x-S(x)] + A[U(S(x))-V(a)-H]

+ p[U(S(x))f,(x;a)/f(x;a) V'(a)] f(x;a)dx. (2.5)

Since S'(x) does not appear in L explicitly, when the optimal S(x) is in

the interior of the set of feasible contracts, it can be characterized

as the solution to the necessary Euler condition, i.e. aH/BS(x) 0.

Thus, the optimal sharing rule can be characterized by

1 fa(x;a)
= A + p (2.6)
U'(S(x)) f(x;a)


The above approach is called "pointwise optimization" in the agency

literature.

There are two solution concepts regarding program (M). The

solution of (2.1) subject to (2.2) alone is referred to as the first-

best solution, which entails Pareto-optimal risk sharing. The solution

of (2.1) subject to (2.2) and (2.4) is referred to as the second-best

solution, which entails suboptimal risk sharing.

The first-best solution. From Borch's (1962) work, S(x) is

Pareto-optimal from a risk-sharing point of view only if the right-hand

side in (2.6) is a constant so that the agent is paid a fixed wage

independent of the outcome and the risk-neutral principal bears all









risk. The first-best contract is analogous to a wage contract. The

second term of the right-hand side in (2.6) is not a constant by the

fact that p>0 (Holmstrom, 1979; Jewitt, 1988). Therefore, optimal risk

sharing can be achieved by dropping the (2.4) constraint. Since A>0,

constraint (2.2) is binding, which implies that the principal can infer

the agent's action.

The first-best solution can also be achieved if the agent is

risk neutral (Harris and Raviv, 1979). In this case, the principal

receives a fixed payment and the agent receives all the residual outcome

(Shavell, 1979). Alternatively, if the principal can discover the

state of nature or monitor the agent's effort directly, the risk

associated with an agency relationship can also be eliminated (Wilson,

1968; Ross, 1973, 1974; Mandlker and Raviv, 1977). Finally, there are

asymptotic results regarding the attainment of the first-best solution

as the "efficiency" of the agent's effort tends either towards zero or

infinity (Shavell, 1979; Grossman and Hart, 1983).

Let the index of the productivity of the agent's effort be denoted

as 6, with the probability density of outcomes becoming f(xl6a). If

6-0, the distribution of outcomes is not affected by the agent's effort

and the problem is merely one of risk sharing. If 6-*, the difference

between the second-best solution and the first-best solution tends to be

zero (Levinthal, 1988). But, these conditions are quite extreme.

The second-best solution. The shadow price p in (2.6) can be

proved to be positive by first-order stochastic dominance (Proposition

1, Holmstrom, 1979) or by the fact that the covariance of U(S(x)) and

l/U'(S(x)) is nonnegative (Lemma 1, Jewitt, 1988). Thus, (2.6) shows









that the second-best contract is larger (less) than the first-best

contract when the marginal return from effort is positive (negative) to

the agent. The second-best contract imposes risk on the agent. The

agent bears responsibility over which he has no control. It is "second

best" because it sacrifices some risk-sharing benefits in order to

trigger a proper effort level. From the principal's viewpoint, the

second-best solution is strictly inferior to a first-best solution. The

ratio of Ifa/f may be interpreted as a benefit-cost ratio for deviation

from optimal risk sharing (Holmstrom, 1979). In general, the solution

to the agency model is one that is second best.



The Self-Selection Problem

The self-selection problem arises when before entering into a

relationship both parties have different beliefs concerning the

exogenous uncertainty. In most circumstances, the agent's information

regarding the uncertain state of nature (such as the agent's ability or

knowledge of the state of nature) is superior to the principal's. The

principal must design some mechanisms (i.e. set of rules) in order to

evaluate prospective agents. Salop and Salop (1976) indicate two

commonly-used methods: a screening device and a self-selection device.

A screening device takes some set of observable characteristics (e.g.

past work experience, educational records, etc.), and ranks an

applicant's prospective job performance on the basis of his endowment of

these characteristics. A self-selection device is a contracting scheme

that causes the applicant to reveal truthful information about himself









by his market behavior. In the agency literature, it is the self-

selection device that is under discussion.

The following example considers the case in which the agent has

private information about his productivity states, while the principal

is unable to enforce the full-information solution. It is modified from

Demski and Sappington's (1984) and Demski, Sappington, and Spiller's

(1988) models. Consider a risk-neutral principal who owns one

productive technology that requires as an input the effort, a, of an

agent. The agent's effort, together with the realization of a random

productivity factor 0, determines the output, x, produced according to

the known relationship x=x(a,8). The random productivity factor is

assumed to be binary, with O1
agent in a more productive setting, i.e. x(a,0h)>x(a,01) for any a>0.

(x(0,0)-O for all 0.) The agent is risk averse with a von Neumann-

Morgenstern utility function that is additively separable in monetary

reward, R, and effort, a. Specifically, H(R,a)=U(R)-V(a) where U'>0,

U"<0, V'>0 and V"20. If the agent acquires perfect productivity

information before a contract is signed, then his effort choice is

equivalent to a choice of output level. Thus, the agent's utility

function can be defined over 8 and x as H(R,x;8)=U(R)-D(x,0) where

D(x,8) is the disutility incurred by the agent when he produces x in

state 8. It is assumed that D(x,0h)0, and D(x,0)>0 and

D,(x,8)20 Vx>0 and VB, where subscripts denote partial derivatives.

Finally, it is also assumed that the marginal disutility of effort for

the agent is smaller in a more productive state, i.e. Dx(x,0h)
Vx> .








Since the principal cannot force the agent to provide truthful

productivity information, he must design the choice mechanism that does

not give an incentive for dishonesty. It is proved by the Revelation

Principle (Dasgupta et al., 1979; Myerson, 1979) that for any

equivalence class of any choice mechanism, with no loss of generality,

there is an equivalent mechanism that induces truthful revelation of 0.

Therefore, the principal need only consider the mechanisms in which each

agent is constrainted to truthfully reveal his productivity parameter as

a Nash response. The principal's optimal strategy can be described by

the solution to the following program:

(SS)

Max pilxi-RiI (2.7)
Ri, xi

Subject to

U(Ri)-D(xi,0 i)>H, i=l,h, (2.8)

U(Ri) -D(xi, i)>U(Rj) -D(xj, i), i,j=l,h, (2.9)

where Pi probability that 0=8-, i-l,h; xi the output that the agent

will produce when 0-01, in return for reward Ri, i-l,h; and H* is the

market opportunity cost (utility).

The individual rationality constraints (2.8) guarantee that the

agent will receive at least his reservation utility level if he

truthfully reveals his private productivity information. The self-

selection or truth-telling constraints (2.9) ensure that the agent will

prefer to tell the truth rather than lie about his actual productivity.

Demski et al. (1988) have shown that the agent receives his reservation

level of expected utility when he observes the low productivity state.









The agent earns strictly positive rents if the high productivity is

realized, and the agent is induced to produce the most output in the

high productivity state.



The Extensions



Numerous extensions of the basic agency model have been developed.

I classify them into the following categories and discuss them in order:

the monitoring model, the multi-agent model, the multi-level model, the

dynamic model and the self-selection model.



The Monitoring Model

Incorporating monitoring arrangements is a natural extension of

the basic agency model. Since agency costs come from the

unobservability of the agent's effort, any extra information about the

agent's effort will lead to a Pareto improvement toward the first-best

solution (Harris and Raviv, 1979). This is true even if the monitoring

information is imperfect. Holmstrom (1979) indicates a necessary and

sufficient condition for imperfect monitoring information to be of

value. Let y be the monitoring signal. Holmstrom proves that y is of

value if and only if [f,(x,y;a)/f(x,y;a)]-h(x;a) is false. When the

contract is contingent on an additional informative measure, the agent

will bear less risk associated with the state of nature. Shavell

(1979), however, shows that the value of information goes to zero when

the efficiency of the agent's effort goes to zero or infinity. Harris

and Raviv (1979) have characterized the form of the optimal monitoring









contract. The latter adheres to the following rule: if the agent's

effort is judged acceptable on the basis of the monitoring information,

the agent is paid according to a prespecified fee schedule; otherwise,

he receives a less-preferred fixed payment. Demski and Feltham (1978)

derive the same type of contract (called a budget-based contract),

except that they divide performance into favorable or unfavorable on the

basis of the outcome.

Baiman and Demski (1980) turn their attention to "investigation

strategies" as to when to undertake costly monitoring. Their results

include the Pareto optimality of "one-tailed" investigations, wherein

either very large or very small variances of costs (but not both) should

be investigated by the principal. One of the sufficient conditions for

their result is that the agent has a utility function that belongs to

the hyperbolic absolute risk-aversion (HARA) class. Young (1986)

derives conditions under which "two-tailed" investigations are optimal,

assuming non-HARA utility functions for the agent. In a different

setting, Dye (1986) proposes a "lower-tailed" investigation strategy

that specifies a cutoff value such that an outcome below that cutoff

value will trigger an investigation.

Dyl (1988) conducts a regression analysis to examine the effect of

monitoring activities on managerial compensation. The results support

his hypothesis that in closely-held companies major shareholders engage

in monitoring activities that reduce the residual loss portion (due to

excessive levels of compensation) of agency costs. The natural

logarithm of the percentage of the firm owned by the five largest

stockholders is employed as the measure of the degree to which the firm







20

is closely held. Data regarding the total remuneration of the CEOs of

271 major industrial corporations listed in the Fortune 500 shows that

an increase of one unit in the natural logarithm of corporate control is

associated with a 9.75 percent reduction in the level of top-management

compensation.

Other works on investigation policies include Townsend (1979),

Evans (1980), Kanodia (1985) and Lambert (1985).



The Multi-Agent Model

If all agents face similar states of nature, then comparing

performance across agents can mitigate the common risk associated with

the outcomes (Holmstrom, 1982; Lazear and Rosen, 1981; Green and Stokey,

1983; Nalebuff and Stiglitz, 1983, 1984). Research examining the risk-

sharing benefits of multi-agent contracts has focused on the comparison

of the efficiency of individual contracts, which reward each agent for

his own outcomes, with tournaments under which reward is a function of

the rank order of performance relative to other agents (Levinthal,

1988).

Holmstrom (1982) argues that the rank order of agents is not a

sufficient statistic for individual output except in special

circumstances as discussed in Lazear and Rosen (1981). Rank-order

tournaments may be informationally quite wasteful if outcomes can be

measured cardinally rather than ordinally. Holmstrom further suggests

that the agent should be evaluated relative to the average performance

of all agents. As the number of agents increases, the effect of the

common disturbance term can be decreased. His conclusion, however, is







21

restricted to the additive or multiplicative technology assumption, i.e.

xi(ai,i)-=a+,i or xi(ai,1)-a(01) i-1,2...,n, with 01-((,Ei) where q is a

common uncertainty parameter and ei's are idiosyncratic risks.

Green and Stokey (1983) consider a model with a risk-neutral

principal and a group of homogenous, risk-averse agents. They conclude

that the tournament dominates an optimal individual contract if the

common environmental risk is large. Nalebull and Stiglitz (1984)

consider the case where the outcome is assumed to be linear in effort.

They find that the use of a tournament as an incentive device can induce

agents to abandon their natural risk aversion and supply more effort.

They also find that the first-best optimum can be approached when there

is a large number of contestants, and a penalty to the lowest-ranked

individual will be superior to a prize to the highest-ranked individual

in motivating effort.



The Multi-Level Model

Another topic to which agency ideas can be usefully extended is

the theory of hierarchy. The study of hierarchical organization was

first mentioned by Simon (1957), Williamson (1967) and Lydall (1968),

but was not precise enough to generate testable hypotheses.

Mirrlees (1976) is the first to extend the basic agency model to a

multi-level structure. In his model, other than the output, the agent's

observed performance is also assumed to be uncertain. The accuracy of

the observation depends upon the time devoted by the principal to make

the observation. Thus, the principal has to decide on the time spent in

observation, as well as on the payment function. Production is carried









out by the bottom-line workers only. Each level of the rest of the

hierarchy decides the monitoring time and the payment function for its

next-level subordinates. Unfortunately, Mirrlees does not provide any

well-founded conclusion.

Antle (1982, 1984) considers the owner-manager-auditor

relationship and examines the game-theoretic foundations of such an

expanded agency model. The auditor's primary role is to produce

stewardship information, i.e. information used by an owner and a manager

for contracting purposes. Modeling the auditor as a strategic player

introduces two complexities. First, the naive mathematical extension of

the basic model may yield unreasonable solutions. Second, the nontrivial

nature of the subgames implies that randomized strategies by the auditor

and the manager may be of crucial importance. These difficulties are

illustrated by many examples (Antle, 1982). If the auditor and manager

are assumed to play pure strategies, however, inducing truthful

reporting by the manager and auditor is optimal. Later, Antle (1984)

restricts attention to pure strategies and provides a definition of

auditor independence which is a controversial topic in the accounting

literature. Basically, Antle assumes that an independent auditor, in

choosing among several subgame equilibria in which he receives the same

expected utility, will select the one that the owner most prefers,

regardless of the effects on the manager. The implications of auditor

independence for the optimal compensation scheme for the auditor are

discussed via a series of examples.

Baiman et al. (1987) also analyze the effects of adding an auditor

to the original principal-agent problem. When the original optimal









contract is second best, they establish conditions that are sufficient

to ensure the value of hiring a utility-maximizing auditor. Their

model differs with Antle's model in three respects. First, there is no

moral hazard problem for the manager. Second, the auditor observes the

report submitted by the manager before conducting his audit. Third, the

principal cannot observe the outcome and only receives the transferred

amount of payment, which is independent of the outcome, from the

manager. The results of Baiman et al. (1987) are restricted by the

uniform probability distribution assumption over outcomes.

Demski and Sappington (1987) consider a regulatory control model

with three individuals: a consumer, a regulator, and a firm. The

consumer (acting as the principal) wants to motivate the regulated firm

to reduce cost and also motivate the regulator to acquire information

that is valuable in directing the firm's activities. Only the firm can

expend effort designed to reduce production cost. The regulator's role

is to gather information about the firm's cost-reducing ability. Demski

and Sappington conclude that the regulator is rewarded if the firm's

cost result is consistent with his claim, and is penalized the maximum

amount otherwise.

Tirole (1986) considers the coalitions in a three-tier

principal/supervisor/agent model. The agent's unobservable production

effort, together with an exogenous productivity shock, affects the

principal's profit. The supervisor's role is to obtain more information

about the agent's activity, and his supervisory effort is assumed

exogenous. Thus, there is no moral hazard problem for the supervisor.

Under the supervisor/agent coalition, Tirole shows that the constraint







24

that induces the supervisor to reveal that the state of productivity is

low is not binding, i.e. the supervisor acts as an advocate for the

agent. Tirole also shows that even if the agent can produce verifiable

information himself, most likely there is still scope for a supervisory

function.

Suh (1987, 1988) considers a sequential department setting that

consists of an intermediate-product division and a final-product

division. Suh shows that the optimal allocation of the noncontrollable

cost is inconsistent with the concept of responsibility accounting which

distinguishes and rewards components of the outcomes for which each

division is responsible. One of the reasons is the existence of

collusion between divisions. One division could collude with another to

affect an apparently noncontrollable cost indirectly. Other reasons are

the technological dependence and asymmetric information of the

production environment. Thus, basing the final-product division

manager's evaluation on noncontrollable intermediate-product costs can

serve as an alternative device to costly monitoring or communication.



The Dynamic Model

Radner (1981) and Becker and Stigler (1974) have shown that

dynamic treatments of the agency model induce more efficient results

that may achieve the first-order solution asymptotically. Holmstrom

(1979) suggests that when the agency relationship repeats itself over

time, the effects of uncertainty tend to be reduced, and dysfunctional

behavior is more accurately revealed, thus alleviating the problem of

moral hazard. As the number of periods increases, the variance in the









average outcomes decreases and more accurately reflects the agent's

effort.

Radner (1980) and Rubinstein and Yaari (1980) examine infinite-

horizon models in which the same single-period situation is repeated

over time, and there is no utility discount over time. Radner (1981)

once again considered the same model but repeated a finite number of

times. He introduces the notion of an epsilon equilibrium as a means of

achieving the first-best outcome in a finite repeated relationship. An

epsilon equilibrium is a set of strategies such that each player's

average expected utility is within epsilon of being a best response to

the other players' strategies.

Lambert (1983) allows the principal and the agent to discount

their utilities and assumes the production functions to be separable

over time. He derived the optimal long-term contract and found that the

agent's compensation depends both on his current and past performance,

whether he precommits to the long-term contract or not. Murphy (1986)

extends Lambert's results to an incentive-based theory of executive

earnings dynamics, both theoretically and empirically. He compares two

not mutually exclusive hypotheses: incentives and learning. The

incentive model implies that the earning-performance relation and the

variance of individual earnings increase with experience, while the

learning model implies that the earning-performance relation is

strongest during early periods and the variance of individual earnings

declines with experience. Both hypotheses predict upward-sloping

experience-earnings profiles and a positive relation between

compensation and performance.









Harris and Holmstrom (1982) provide new predictions for the

behavior of aggregate wages over time. Both the principal and the

agents are assumed to be uncertain about agents' abilities, but by a

normal learning process, they gradually find out agents' abilities.

Harris and Holmstrom conclude that the optimal long-term contract

entails a downward rigid wage (i.e. under this contract one's wage never

falls over time). The wage is not fully rigid, because the threat of

quitting will force the wage to be bid up whenever the market wage is

higher than the current wage. The market wage is found to be a function

of the worker's current mean perceived productivity minus a term that

depends on his age and the precision of beliefs about his productivity.

Thus, senior workers earn more on average because they have had more

time to have their wages bid up by the market.

Narayanan (1985) borrows Harris and Holmstrom's (1982) model to

analyze the portfolio manager's decision in financial policies under

asymmetric information. The manager may have incentives to make

decisions that result in short-term gain at the expense of the

shareholders' welfare, even if long-term contracts are offered.

Narayanan shows that this manager's decision is related to the manager's

experience, length of his contract, and the risk of the projects. The

more experienced the manager is or the longer the duration of the

manager's contract, the lower is the probability that the manager would

opt for short-term profits. When the projects are riskier, only a

smaller proportion of the deviations from the expected cash flows will

be attributed to the manager's ability, and this reduces his incentives









for sacrificing the long-term interests of the shareholders for short-

term profits.

Holmstrom (1983), based on Azariadis' (1975) work, has explored

the dynamics of a contractual labor market in an equilibrium setting.

The model assumes homogeneous risk-averse agents, and quits and layoffs

are allowed at the beginning of second period. He shows that long-term

contracts emerge both when labor is specified and cannot move in the

second period and when mobility is costless. Wages are downward rigid

but not fully rigid. Holmstrom also points out the seniority rules:

different generations of newcomers may have different incomes;

especially, younger generations generally earn less, not because of

productivity differences but because they enter later. Therefore, in

Holmstrom's model, contracts create biases against young members.



The Self-Selection Model

Salop and Salop (1976) assume that agents differ exogenously only

in their probabilities of quitting. The principal's objective is to

minimize the turnover costs in a perfectly competitive labor market.

This is equivalent to identifying the slow quitters among the applicants

and hiring them. Salop and Salop suggest a two-part wage (TPW) scheme

as a self-selection device. The TPW would operate in the following

manner. The new employee pays the firm an entrance fee of DI, in return

for which he receives the market average wage, W*, and some fixed

amount, D2. The principal sets D2 and DI such that slow quitters prefer

this structure while fast quitters prefer the flat wage structure of

other firms. Finally, in equilibrium the agents pay their own training









costs and receive the full value of their marginal revenue product as

wages.

Rothchild and Stiglitz (1976), and Stiglitz (1977) examine the

equilibrium under asymmetric information in the competitive insurance

market and the monopolistic insurance market, respectively. Rothchild

and Stiglitz (1976) show that even in a competitive market, a small

amount of asymmetric information will exclude a single-price

equilibrium. High-risk individuals purchase complete insurance, while

the low-risk group purchases partial insurance. Market equilibrium,

when it exists, consists of contracts that specify both prices and

quantities. But under many conditions, equilibrium may not exist. In a

monopoly with imperfect information and nonlinear pricing (which charges

customers an amount proportional to the quantity consumed), Stiglitz

(1977) proves that the same contract will never be purchased by both

high- and low-risk individuals. But, the low-risk individual may not

purchase any insurance at all. The high-risk individual, however,

always purchases complete insurance.

Harris and Townsend (1981) and Harris, Kriebel, and Raviv (1982)

consider intrafirm resource allocation under asymmetric information.

Harris and Townsend (1981) prove that the equilibrium parameter-

contingent (P.C.) allocations of any mechanism must satisfy certain

self-selection properties and such P.C. allocation can be achieved under

some mechanism (i.e. some set of rules). Harris, Kriebel, and Raviv

(1982) further point out that one of the efficient mechanisms is the

transfer pricing scheme. Under certain constraints, headquarters

announce a schedule of transfer prices, and each division is asked to









choose a transfer price among them. By the choice of a transfer price,

all of the information required for an optimal allocation will be

revealed.

Lal (1986) addresses the issue of delegating pricing

responsibility to the salesperson. He shows that delegating pricing

responsibility to the salesperson is as profitable as centralization

when both the salesperson and the sales manager have identical

information about the selling environment. But delegation may be more

profitable when the salesperson's information is superior to that of the

sales manager. Lal and Staelin (1986) follow the same framework but

relax the assumption of salesforce homogeneity. They present the

conditions under which it may be advantageous for a profit-maximizing

sales manager to offer his salespersons the opportunity to choose from a

menu of compensation plans. Nevertheless, they are not able to provide

the specific design of optimal contracts.

McAfee and McMillan (1986, 1987) introduce a principal-agent

framework to the service market. Potential agents, risk neutral and

heterogeneous in ability, compete with each other for the contract with

the principal. The principal, without knowing the agents' types,

designs a contract that exploits the competition among the potential

agents and induces them to reveal their types. In this setting, the

contract trades off adverse selection against moral hazard.

Surprisingly, McAfee and McMillan proved that the optimal contract is

linear in the observed outcome in a broad range of circumstances.









Summary



This chapter introduced the basic models of agency theory and its

numerous extensions. Agency theory appears to be a promising research

area and there are various avenues of enrichment that will result in

valuable propositions (MacDonald, 1984). It has been developed markedly

over last 20 years and the impact of this research has been significant

(Levinthal, 1988). This study intends to apply agency theory to a

hierarchical production environment. This literature review constitutes

the basis for the suggested models in this study.













CHAPTER 3
QUALITY ASSURANCE AND JOB ENLARGEMENT IN PRODUCTION MANAGEMENT



Prior to World War I there was very little separation between

responsibility for production and responsibility for product quality in

the United States. There was rarely a separate functional department

for quality distinct from that for production. Between World War I and

World War II the concept of "central inspection" began to form (Juran,

1974, p. 7-6). By 1940, the distinction between the functions of

quality control and production became solidified (Lubben, 1988, p. 68).

By the 1950s, the importance of preventing defects from occurring began

to generate serious consideration as a result of the increasing pressure

to meet pressing delivery schedules and to maintain higher levels of

product quality and reliability. Hence, the quality engineering and

reliability engineering functions were introduced. Since then, the

separation of responsibilities between quality and production have

become more pronounced. By the late 1970s, worldwide competition

imposed much greater pressure upon U.S. manufacturing. The quality of

Japanese semiconductors and automobiles had reached a much higher level,

for example. The term "parts per million" instead of "parts per

hundred" emerged as the accepted defect level in electronic components

(Lubben, 1988, p. 69).









The trend of the 1980s reverses the process, separating the

interrelated functions of quality and production. Almost every book on

quality mentions the importance of the production department's

responsibility for product quality. Management realizes that as long as

there is a separation between the product and the quality of that

product, there will remain a lack of commitment, motivation and

performance in a manufacturing or service system. Thus, the success of

modern industry depends on the full integration of the control of

quality and production. But not all quality-related issues are a direct

consequence of the production process. Some issues stem from design

specifications and supporting guidelines, and others from procurement

problems. Therefore, quality control is a responsibility that should be

shared by the whole company.

One of the significant concepts of Just-in-Time (JIT) is to return

the quality-enhancing function to where the products are produced.

Quality assurance is then assigned a more productive role in supporting

the manufacturing system. Quality assurance should guide, support and

monitor the production process, and develop support programs such as

reliability, failure analysis, data analysis, information feedback, and

continuing education. The real control of quality is not in the hands

of a quality assurance department; rather it is in the hands of the

bottom-line production department. From the production employees'

viewpoint, the new arrangements will enhance their interest and

involvement, and create a break from their previous task-specific jobs.

That is, this new responsibility serves as a job enlargement factor for

production employees.









Recent trends in the manufacturing concept make the introduction

of more appropriate compensation systems a necessity. The old piecework

scheme based on production quantity does not provide incentives for

quality improvement. The nature of Japanese compensation systems

attaches more weight to factors such as effort, cooperativeness,

conscientiousness, and the display of initiatives. In fact, many

experts suggest that the development of a highly-motivated workforce,

and not just a highly-skilled workforce, is the most exacting challenge

facing Western managers attempting to establish a JIT production system.

In this chapter, a hierarchical model is proposed to simulate the

integration of quality assurance and the production process. Most

multi-level agency models consider the added level simply as the source

of monitoring information. The added agent provides no productive

action and hence cannot affect the realized outcome. In this model, the

middle-level quality control manager, in contrast, does have a direct

influence on the realized outcome. In what follows, the model is

described and the characterization of optimal contracts is then

specified. Comparative statics are then analyzed for an additive

quality-enhancing technology under certain assumptions. This chapter

concludes with a summary of the findings.



The Model



Suppose a firm consists of three individuals: a risk-neutral,

profit-maximizing principal, and two risk-averse, utility-maximizing

agents. The two agents are the manager of the quality control








department, and the worker in the production department. The quality

control manager is responsible for establishing quality policies and for

monitoring the production process. The worker is responsible for

producing products and for implementing the quality policies established

by the manager. Different expertise is the reason to set up this three-

level hierarchy. Production quantity (x) is influenced by the

production effort (al) expended by the worker, and a random

environmental factor (0) such as material shortage or machine

breakdowns. The quality level (y) of the final products, measured by

the rate of detectives or some other classification method, is

influenced by the manager's guidance effort (9), the worker's quality-

enhancing effort (a2), and an uncontrollable state of nature (e) such as

insufficient information of product nature or technology. Although the

quality standard is a joint outcome of the efforts of manager and

worker, collusion between agents by having a side contract (Tirole,

1986; Suh, 1987) is not considered in the model.

Let F(x,y;a1,a2,9) be the joint distribution function that

satisfies the monotone likelihood ratio (MLR) and the convexity of the

cumulative distribution function (CDF) conditions. Some Japanese

companies have found the natural rhythm of nonpositive correlation

between production and quality existing in production systems. It is of

no value for the management to press for greater production at the

expense of quality levels. Instead, they search for the balance point

that will provide the optimum level of production and quality (Lubben,

1988, p. 123). Following the same observation, it is further assumed









that the production and quality levels (x and y) are nonpositively

correlated.

The principal can observe both the total output, x, and its

quality, y. Therefore, x and y can be utilized as the basis of a

contract. The manager receives a nonnegative payment M(x,y), and his

expected utility G(M(x,y),P) is defined over that payment and his

effort. Following conventional agency assumptions, the agents are risk

averse and the disutility of effort increases at a nondecreasing rate

(i.e. effort averse). G(M(x,y),P) is assumed to be additively

separable, i.e. G(M(x,y),#) E[V(M(x,y))-W2(f)], where V'(.)>0,

V"(.)<0, W2'(.)>0 and W2"(*)0O. The worker also receives a nonnegative

payment S(x,y), and his expected utility, H(S(x,y),a1,a2), is defined

over his income and the production and quality-enhancing effort.

H(S(x,y),a1,a2) is also assumed to be additively separable, i.e.

H(S(x,y),a1,02) E[U(S(x,y))-W1(k1at+k2C2)], where U'(.)>O, U"(-)<0,

Wi'(.)>0, Wi"(-)>0, and k1, k2 are job enlargement factors with 0
and 0
S(x,y) are restricted to lie in certain intervals (Holmstrom, 1979).

Given the above assumptions and specifications, the three-level

agency model can be set up as the following program:

The principal seeks to

(OP) Max E[w] E[p(y)x-c(x)-S(x,y)-M(x,y)] (3.1)
S,M

Subject to

H[S,al,02] =E[U(S)] W1(k1a1+k2za) > HO (3.2a)

1,aez e Argmax (E[U(S)] W1(k1a1'+k2I2')) (3.2b)
l', 2'







36

G[M,3] E[V(M)] W2(f) 2 G* (3.3a)

P e Argmax (E[V(M)] W2(f')) (3.3b)


where E[*] denotes the expectation operator, the notation "argmax"

denotes the set of arguments that maximize the indicated objective

function, and G" and H* are the known market-reservation utility levels

of the manager and the worker, respectively.

The principal's expected profit is represented in (3.1). The

individual rationality constraints, (3.2a) and (3.3a), guarantee that

the worker's and the manager's expected utilities exceed their minimum

market levels so that they will be attracted to and stay in the job. The

incentive compatibility constraints, (3.2b) and (3.3b), indicate that

each agent will select his effort level from his decision set to

maximize his own expected utility.

In order to replace constraints (3.2b) and (3.3b) by their

corresponding first-order conditions, a supplementary assumption on

utility functions must be made. That is, U(U'-1(1/S)) and V(V'-1(1/M))

are concave in S and M, where U''1 and V''- denote the inverse function

of U' and V'(Theorem 2, Jewitt, 1988).

Given that the first-order conditions hold, determination of the

contracts S(x,y) and M(x,y) can be replaced by the following optimal

control program (FP):






37

(FP) Max ff[p(y)x-c(x)-S(x,y)-M(x,y)]f(x,y;a1,a12,3)dxdy (3.4)
S,M

Subject to ffU(S(x,y))f(x,y;al,a2,P)dxdy -Wl(k1a1+k2a2)> Ho (3.5)

ffV(M(x,y))f(x,y;al,a2,8)dxdy W2(P)> GO (3.6)

ffU(S(x,y))fal(x,y;a1,a2,p)dxdy -W1'kl(klac+k2a2)- 0 (3.7)

ffU(S(x,y))f (x,y;al1a2,P)dxdy -Wl'k2(k1a1+k2a2)- 0 (3.8)

ffV(M(x,y))fP(x,y;al,a2,6)dxdy W2'(f)- 0 (3.9)
where fi(x,y;a1,a2,P) is the partial derivative of f(x,y;al,a2,3) with

respect to i, i-= ,a2,0; and Wj'(-) is the first derivative of Wj('),

j=1,2.


Characterization of Optimal Contracts



Let AI, A2, pi1, A12, and P2 be the multipliers associated with

constraints (3.5)-(3.9), respectively. The Hamiltonian of the above

program (FP) can be written as the following function:

(H) Max L(S,M, A1, A2, I11,12 ,1 2,a12, )

Jf[p(y)x-c(x)-S(x,y)-M(x,y)]f(x,y;al,a2,f)dxdy
+ A1 (ffU(S(x,y))f(x,y;a1,a2,f)dxdy -W1(klal+k2a2) H)

+ An1(ffU(S(x,y))f I(x,y;al,a2,P)dxdy -Wl'kl(klal+k2a2))

+ A12(ffU(S(x,y))f2(x,y;1,Ca2jz,)dxdy -W1'k2(kla+k2a2))
+ A2 (ffV(M(x,y))f(x,y;al,12,)dxdy (W2() G*)

+ 2 (ffV(M(xxy))f(x,y;a,a2,)dxdy W2'()) (3.10)
with A>1O and A2O0.

If the optimal S(x,y) and M(x,y) are in the interior of the set of

feasible contracts, they can be characterized as the solutions to the








necessary Euler conditions. Thus, the optimality conditions for

contracts are


1

U'(S)


and


1

V'(M)


where the second

assumptions that

Taking the

B, the following


f ,(x,y) f2(x,y)
+ + 12 --+-
f(x,y) f(x,y)

fa,(x y) f2,(yx)
Ai + 1 n + 212 (3.11)
f(xly) f(ylx)

fp(x,y)
A2 + 2 -----
f(x,y)

fg(y x)
A2 + f2 ) (3.12)
f(ylx)

equations of (3.11) and (3.12) come from the

x=x(al,8) and y-y(a2,,fE).

first derivative of (3.10) with respect to al, a2, and

conditions are obtained:


lf[p(y)x-c(x)-S(x,y)-M(x,y)]f. 1(x,y)dxdy

+ A, (Sfu(S(x,y))f.1(x,y)dxdy -k1W1'(kla1+k2aC2))

+ pA (ffU(S(X,y))f. (x,y)dxdy -W1"k2(klal+k2a2))

+ P12 {JffU(S(x,y))f,l2(x,y)dxdy -Wl"klk2(kla1+k2a2))

+ A2 (ffV(M(x,y))f.1(x,y)dxdy)

+ A2 (ffV(M(x,y))fp.,(x,y)dxdy) = 0;


(3.13)








ff[p(y)x-c(x)-S(x,y)-M(x,y)]f~2(x,y)dxdy
+ AI (ffU(S(x,y))f 2(x,y)dxdy -Wl'k2(kC1+k2Ca2)

+ p1n(ffU(S(x,y))f,.2(x,y)dxdy -WV"k k2(kja1+k22) )

+ U12(fU(S(x,y))f,22(x,y)dxdy -W1"k (kla1+k2a2)

+ A2 (ffV(M(x,y))f z(x,y)dxdy)

+ A2 (ffV(M(x,y))f 2(x,y)dxdy) 0; (3.14)

ff[p(y)x-c(x)-S(x,y)-M(x,y)]fp(x,y)dxdy

+ AI (ffU(S(x,y))fp(x,y)dxdy)

+ Pn11ffU(S(xy))fa1#(x,y)dxdy)

+ 12{(ffU(S(x,y)) fz(x,y)dxdy)

+ >2 (ffV(M(x,y))fp(x,y)dxdy W2'(f))

+ z2 (ffV(M(x,y))fp(x,y)dxdy -Wz"(p)) = 0. (3.15)
The optimal compensation schemes and effort levels can be

determined by solving equations (3.5)-(3.9) and (3.11)-(3.15)

simultaneously, with A\?0 and A2>O. The exact solutions may be very

difficult to formulate, but some characteristics of the optimal

contracts are analyzed in the following results.

Result 3.1. Under the optimal compensation schemes, the expected

utilities for the manager and worker are exactly equal to their market-

reservation utility levels H* and G, respectively; that is, constraints

(3.5) and (3.6) hold as qualities at the optimum, and A1>0, A2>0.

Proof. By the Kuhn-Tucker condition: A1>0, A2z0. If z2=0, equation

(3.12) is rewritten as

1 fp(x,y)
) f2 (3.12')
V'(M) f(x,y)








The left-hand side of the equation (3.12') is always positive by the

risk-averse assumption. Since fffa(x,y)dxdy-0, this is a contradiction.

Hence, Az>0. Evaluating equation (3.11), X1>0 can be proved simil~p .)..

Result 3.2. Since the first-best solution cannot be reached, the

manager should share some risk for the quality-enhancing process, i.e.

12>0.

Proof. (This proof follows Jewitt (1988).)

Substituting (3.12) into (3.9) gives

ffV(M)[I/V'(M) A2]f(x,y)dxdy p2W2'(*). (3.3.2a)

Using the fact that E[fo/f]=ff(fo/f)fdxdy=fffldxdy=0, (3.12) gives

ff[l/V'(M)]f(x,y)dxdy = A2. (3.3.2b)

From Result 3.1 and equation (3.6), ffV(M(x,y))f(x,y)dxdy W2(6)+G.

Hence, (3.3.2a) states that the covariance of V(M) and 1/V'(M) is of the

sign as p2W2'(1). Since V(M) and 1/V'(M) are monotone in the same

direction, they have a nonnegative covariance. And since W2'(p) is

positive by assumption, it follows that p220. Furthermore, p-=O can be

ruled out, for then M(x,y) would be constant and this violates equation

(3.9). Q.E.D.

Result 3.3. The manager's contract is a strictly increasing

function of the quantity and quality produced.

Proof. From (3.12), M(x,y) can be written as

M(x,y) h(A2+Jz(fi/f)), where h(-)=(l/V'(-))-1.

Since V'(.) is a strictly decreasing function of M(x,y), h(-) is

strictly increasing with its argument. By the fact that A2 and p2 are

positive as proved in results 3.1 and 3.2, and that fp/f is an






41

increasing function of x and y, M(x,y) is also an increasing function of

x and y. Q.E.D.

Result 3.4. If fp(ylx)/f(ylx) is linearly increasing in y and the

conditional expected quality E(ylx) is a function of P, then

fp(ylx)/f(ylx)-[y-[y ylx)] Ep(ylx)/ Var(ylx).
Proof. Since f(x,y)-f(ylx)f(x), fp(x,y)-fp(ylx)f(x)+f(ylx)fg(x)

-fp(ylx)f(x). Recall equation (3.12)
1 fp(ylx)
= \2 + i&2
V'(M) f(ylx)


Define h(ylx)=fp(ylx)/f(ylx). Since h(ylx) is linearly increasing in y,

hy(ylx)>0 and hyy(ylx)=0. It can be shown that E[h(ylx)]

=fh(ylx)f(ylx)dy =ffg(ylx)dy=0. By Jensen's inequality and the

assumption that h is linear, E[h(ylx)]-h[E(ylx)]-0. Since hy(ylx)>0,

h(ylx)0 as y>E(ylx), and h(ylx)<0 as ysE(ylx). This suggests that

h(ylx)=K(y-E(ylx)), where E[yjx]=fyf(ylx)dy. Also, fyfp(ylx)dy=Ep(ylx).

But, fyfp(ylx)dy = fyh(ylx)f(ylx)dy Kfy(y-E(ylx))f(ylx)dy K{E[y2Ix]-

(E(ylx))2) = K Var(ylx). This implies K=Ep(ylx)/Var(ylx). Therefore,

fp(ylx)/f(ylx) = h(ylx) = [y-E(ylx)] Ep(ylx)/Var(ylx).
Q.E.D.

Remark. This result shows that both the conditional expectation and

variance are used in the evaluation. The principal may use the revised

conditional expected quality E(ylx) as a standard to give a bonus or

impose a penalty. And, the compensation is decreasing with the

variance. Thus, the conventional standard-setting and variance









computation is part of the optimal compensation scheme (Baiman and

Demski, 1980).

Result 3.5. If there is no correlation between quantity and

quality, then the optimal contract for the manager is a nontrivial

function of the quality standard alone, i.e. M(.)-M(y).

Proof. Since fp(x,y)/f(x,y)-fp(ylx)/f(ylx)-ff(y)/f(y) when x and y are

independent, the result follows from (3.12) directly. Q.E.D.

Remark. In this case, the product quantity is noninformative

(Holmstrom, 1979; Suh, 1988) regarding the manager's action choice. For

the manager's performance evaluation purposes, there is no need to

consider the quantity level. That is, the quality standard (y) becomes

a sufficient statistic for (x,y) regarding P. If quantity and quality

are correlated, even though production is beyond the manager's control,

the production quantity tells the principal something about the

manager's behavior. Thus, both factors are used in the manager's final

evaluation. In addition, the manager tends to act as an advocate for

his subordinate (Tirole, 1986). He would like to share some production

risk with the worker so that the quality-improving policy may be

implemented more effectively and efficiently.

Result 3.6. If the principal can observe the agents' actions or

the states of nature directly, then:

(i) the manager's compensation is a constant times the compensation of

his subordinate, the worker;

(ii) the manager's utility, at the optimum, is a linear transformation

function of the worker's; and







43

(iii) the manager's coefficient of absolute risk aversion is a constant

times the worker's.

Proof. When the principal can observe the agents' actions or the states

of nature, the first-best solution can be achieved. That is, 1/U'-A1

and l/V'=A2 with 11>0 and X2>O. Therefore, S(x,y)-U'-1(1/A1) and

M(x,y)-V'-l(l/A2). The agents are paid by a constant salary, given that

the specified effort levels are provided. Otherwise, they receive

nothing. Conclusion (i) follows directly.

From (i) A2V'(M)=AIU'(S) and M=kS, where k is a positive constant,

we have that V'(kS)-(A1/A2)U'(S). (3.3.6a)

Integrating (3.3.6a) over S, V(-)=aU(-)+b, where a=kAl/Az, and b is a

constant of integration. Conclusion (ii) follows directly.

Differentiating both sides of (3.3.6a) with respect to S, we have

kV"-(A0/A2)U". (3.3.6b)

Define the Arrow-Pratt measure of absolute risk aversion as R,--V"/V'

and Ru=-U"/U'. From (3.3.6a) and (3.3.6b), it follows that RP=(l/k)Ru.

Q.E.D.

Remark. Herbert Simon (1957) suggested that "an executive's salary

should be b times the salary of his immediate subordinates, ..."

Simon's suggestion is supported under the availability of the first-best

solution. Conclusions (ii) and (iii) are consistent with the

relationship of Pareto-optimal risk sharing and the similarity rule of

utility functions as given in Ross (1974).







44

Result 3.7. At the optimum, the worker has to share some risk of

x and y, i.e. PlI>0 and pl2>0.

Proof. This proof is similar to result 3.2. Substituting (3.11) into

(3.7) gives

ffU( (/U') A))f(x,y)dxdy -pijffUf*dxdy -pAlkW1' (3.3.7a)

Substituting (3.8) into (3.3.7a) gives

ffU((l/U')-Ai))f(x,y)dxdy = piAkW,'+ pi2k2W '. (3.3.7b)

By the fact that E[f, /f]=O and E[f,2/f]-0, (3.11) yields

Jff(/U')f(x,y)dxdy = A,. (3.3.7c)

Equation (3.3.7b) states that the covariance of U and 1/U' is of the

same sign as (pklk1W1'+ pik2W21') which should be nonnegative (Jewitt,

1988). Therefore, pil and Pi2 cannot both be negative. We can rule out

117j-0, Pi2<0 and pIA1<0, pi2=0 cases for the same reason. /p4 and p42

cannot both be zero because of (3.7) and (3.8).

We now consider the following two cases:

1. /li-0, C12>0, or pli>O, A12-0, and

2. AII>0, p12<0, or pi1<0, P12>0.

The second terms in (3.13) and in (3.14) are equal to zero by (3.7) and

(3.8). The fourth term of (3.13) and the third term of (3.14) are

positive by first-order stochastic dominance. The third term of (3.13)

and the fourth term of (3.14) are negative by the second-order

condition. The remaining terms of (3.13) and (3.14) are positive by the

fact that M(x,y) is increasing in x and y, and first-order stochastic

dominance. Both cases violate equations (3.13) and (3.14). Hence,

p 1>0 and Ip2>0. Q.E.D.









Result 3.8. The worker's compensation is a strictly increasing

function of the quantity and quality of products.

Proof. Proof is similar to that of result 3.6. Q.E.D.



The Comparative Statics



In order to obtain greater insight from the model, the following

additional assumptions are made:

(a) The production and quality-enhancing processes follow a linear

technology, i.e. x(a1,0)=1(a1)+O, y(a2,,0,)=m(a2,A)+e. B and e are

bivariate standard normally distributed with covariance aog. Since

oag=a=l, ae-p with -l0, "(al)O0, m 2>0, m9>0, m2a2 50,

mO5 and ma2 )0, where the subscripts denote partial derivatives. The

joint density function is as follows:

f(x,y)-(1/2w(l-p2)1/2) exp(-1/2(l-p2)[(x- )2-2p(x-2)(y-m)+(y-m)2]).

(b) The agents have power utility functions. Moreover, these utility

functions are assumed to be V(M)=2M1/2 and U(S)=3S1/3.

(c) The product price, p(y), is a linear function of quality, i.e.

p(y)=Po+py, where po>0 and p!0. This implies that products with a

better quality level will also have higher bargaining power.

(d) The production cost, c(x), is a linear function of quantity, i.e.

c(x)-co+cx, where c0>O and c>0.

The normal distribution is totally positive (Jewitt, 1987).

Assumption (b) satisfies the utility shape specified by Jewitt (1988).

The power utility functions are constant relative risk-averse functions

with a coefficient of relative risk aversion bounded above one half.









The power utility function also has larger risk tolerance than the

exponential, logarithmic and quadratic utility functions. If the

principal has a choice, the agent who is increasingly risk averse (in

the sense of Arrow-Pratt) should be preferred by the risk-neutral

principal, so that the agent is willing to pay a higher risk premium.

Assumption (b) implies that the worker is more risk averse than the

manager.

From assumptions (a)-(d), the optimality conditions for contracts

(3.11) and (3.12) can be rewritten as follows:

1/V' A2 + p2 (m#/(l-p2))[y-m-p(x-2)]

[A2-P2(pmP/(l-p2))(m-2)] -p2(pmP/(l-p2))x + pz(mP/(l-p2))y (3.11')

a2 + b2x + c2y;

1/U' = AI + pll('/(l-p2))[x-~-p(y-m)] +/12 (m2/(l-p2))[y-m-p(x- ~)]

--[,I-(A11.2,1(l-P2) -12Pm 2 /(I-P2) ) + ('l1Pg,/(I-P2) -12Ma2 /(i-P2) )m]
-+ -((A1'p/(l1p pm-p2) /2 -11/2 -P2))m]

+ (11'/(l-p2)- 12pm /(l-2))x + (-_ /( 2) 2/(l- )y

aI + blx + cly, (3.12')

ai where ai can be treated as the base salary, bi as the commission rate

of production, and ci as the commission rate of quality (i-1 for the

worker, i-2 for the manager). From results 3.1, 3.2, 3.3, 3.7, and 3.8,

ai>O, bi>O, and ci>O. The following result can be observed immediately.

Result 3.9. The agents are compensated by a lower salary and

higher commission rates when there is a negative correlation between the

quantity and quality of products.

Proof. If p-0, the optimality conditions of contracts are

1/V' = Az + p2mp(y-m), and (3.11")

1/U' = Ai + Ap11'(x- ) + A12mz(y-m). (3.12")








Using the fact that all multipliers are positive and M(x,y) and S(x,y)

are increasing in their arguments, and comparing (3.11'), (3.12') with

(3.11") and (3.12"), respectively, yields the desired result. Q.E.D.

Remark. When quantity and quality of products are negatively

correlated, the principal needs to provide more incentive for the risk-

averse agents to overcome this additional uncertainty. A contract with

higher commissions and lower salary will induce more effort.

The discussion of comparative statics will utilize the following

preliminary results.



Preliminaries of The Comparative Statics

For each effort level, the optimal contracts for the manager and

the worker are specified by substituting the multipliers into (3.11')

and (3.12'). Then, by comparing the solutions across effort levels, the

optimal effort levels can be determined.

The manager. From (3.11), (3.11') and the power utility function

assumption, we have

M(x,y) = [A2 + A2zf/f]2


(A2 + 2 (m/(a-P2))[Y-m-px-l)]}2

Therefore, the manager's utility function can be written as

V(M(x,y)) 2[A2 + A2fl/f].

Substituting (3.17) in (3.6) with the equality holding yields

W2+G = 2ffS[A2 + 2f,/f]fdxdy

2A2 + 2p2ff[fo/f]fdxdy

= 2\2.

Therefore, X2=(W2+G*)/2.


(3.16)



(3.17)


(3.18)








From (3.9) and (3.17), we have

W2' 2ff[A2 + p2f./fl]fdxdy

2pJf[ff/f]2fdxdy

2prf((mp/(l-p2)) [y-m-p(x-2)])2fdxdy

2p2(mp/(l-p2))2(E(y-m)2-2pE[ (y-m) (x-) ]+p2E(x-i)2

2p2(mP)2/(l-p2),

or p2 W2'((l-p2)/2(m0)2. (3.19)

Substituting A2 and P2 into (3.16), we have

M(x,y)- ([(W2+G)/-W2/2mp)(m-p2)] (pW2'/2mp)x + (W2'/2m2)y12

(A2 + B2x + C2Y}2, (3.20)

where A2=(W2+G)/2- (W'/2m) (m-pl), B2=-pW2'/2m# and C2=W2'/2mp.

Therefore, the expected compensation for the manager is

E[M(x,y)]-(1/2m#)2ff(A2+B2x+C2y)2fdxdy =((W2+G*)/2)2+ (W2/2m) 2(-p2).

(3.21)

The worker. From (3.12), (3.12') and the power utility function

assumption, we have

S(x,y)- [AI + /pllfi/f + p12zfa/f]3
- (At + pn(.'/(l-p2))[x-2-p(y-m)] +p 2(m.a/(l-p2))[y-m-p(x-I)] 3. (3.22)

Therefore, the manager's utility function can be written as

U(S(x,y))-3[AI + pifa./f + p12fz2/f]. (3.23)

Substituting (3.23) into (3.5) with the equality holding yields

WI+H* 3ff[A, + p1f /f + p10fa2/f]fdxdy

= 3Ai.

Therefore, A1=(WI+H")/3. (3.24)








From (3.7) and (3.23), we have

kiW' 3ff[AI + l fI/f + 112f2z/f.1 dxdy

= 3pnl(')2/(l-p2),

or pi = klWi'((l-p2)/3(1')2. (3.25)

From (3.8) and (3.23), similarly we have

A12 k2Wi'((l-p2)/3(m~2)2 (3.26)
Substituting A1, p~l, and Ai2 into (3.22), we obtain

S(x,y)= ((Wi+H)/3 + kiWi'(1-p2)/3(2')2(2'/(l-p2))[x-2-p(y-m)

+ k2Wi' (1-p2)/3(m)2m 2 /(l-p2)) [y-m-p(x-2)] 3

= ((Wi+H)/3 + [kiWi'/31' pk2Wi'/3m2] (x-2)

+ [-pkiWi'/3V' + k2Wi'/3m,2 ](y-m))3

= (Ai + Bix + Ciy)3 (3.27)

where A,= (Wi+H0)/3 (kW1i'/3') (2-pm) (k2W'/3m2 )(m-pI)

Bi= kiWi'/32' pk2W1'/3m,2

Ci= -pklW1'/31' + k2Wi'/3m a2

Therefore, the worker's expected compensation is

E[S(x,y)] = ((Wi+H*)/3)3

+ (Wi+HO)(Wi'/3 'm, 2)2(l-p2)[(kima )2+(k2l')2-2pkik2,n'm2]. (3.28)

The principal. The principal's expected revenue and profit can be

written as

E[p(y)x-c(x)] = E[(po+py)x-(co+cx)] = [(po+pm-c)2-co+pp] (3.29)

E[w7al,a2z,f] E[(po+py)x-(co+cx)] E[M(x,y)] E[S(x,y)]

- [(po+pm-c)1-co+pp] ((W2+G*)/2)2 (W2'/2m)2(l-p2) -(W1+H/3)3 -
(Wi+H) (Wi'/31' mn)2 (lZ) [(kima2 2+(k2' )2-2pkik2'm, 2]. (3.30)

The effort level. Since Wi and (Wi)2 are convex by assumption,

the principal's expected profit, E[(lar,a2,p], is a concave function of








al, a2, and P. Hence, the unique a,*, ar*, and that maximizes the
E[r] exist.

From (3.30), the optimal effort levels, al, a2, p are such that

Tl(al)- E. l[7al,a2,,]
(pm-c)l' ((WI+H*)/3)2(kW1')

-{[kiW' (-p2)/(3' m )2] [ (Wi) +2 1(Wi+H )W"]

[(k12) 2+(k21')2-2pklk2' mh }2

= 0. (3.31)

T11(al)m E 1[11Nalal2] < 0.
T2(a2)- Ea [.rIa,aZ2,]

(pAn ) ((W1+H*)/3)2(k2W1')

-([k2W' (1-p2)/(3e9'm% )2] [ (W1)2+2 (Wi+H )Wi"]

[(k m)2) 2+(k2' )z-2pklk2', m2 ]}

0. (3.32)

T22(a2)- E,,2a [n1a,a2,J] < 0.

T3(6) Ep[w aI,a2,P]
pimp (W2'(W2+G)/2)) (l-2)(W2'W2"/2(m,)2)

+ 2m2 (l-p2)(Wi+H )(Wi'/39')2(l/nm2)3[(kim1%)2+(k2' )2-2pklk2 2'ma]

2(l-p2)(Wi+HO)(Wl'/3'm M.)2[k m~ -pk2l' ]k mOz

= 0. (3.33)

T33(p)- EPP r1ai,a2,P < 0.

In order to facilitate the computation, it is assumed that

)(al)-=a, and m(a2,f)=a2+p. For the manager, from (3.20) and (3.21), we

have

aAz/8aa pW2'/2 < 0.

aA2/aa2 W2/2 < 0.








aA2/af -(W2(9+o2-pa*1))/2 < 0.

aB2/aal 0, aBz//a2 0, aB2/af -pW2"/2 > 0.

ac2/aa8 0, acz/a2 0, ac2/ap Wz"/2 > 0.

8E[M]/ac 0, aE[M]/aa2 0,

aE[M]/a8 (W2'(W2+G*)/2) + (W2'W2"(l-p2))/2 > 0.

For the worker, from (3.27) and (3.28), we have

aA1/aal = -k2W1"(aI-pa2-p6))/3 klk2W1"(a2-pa1+f))/3 + pk2W1'/3 < 0.

aAl/aa2 = -kk2W1i"(aI-paz-pP))/3 klk2W1 (a2-pa1+/))/3 + pkiW1'/3 < 0.

aA1/ap = pkiW,'/3 k2W1'/3 < 0.

aBi/laa = (k -pklk2)WI"/3 > 0.
1
a8B/8a2 (-pk2+klk2)Wl"/3 > 0.
2
aBla/# = 0.

acl/aal (-pk +klk2)W1"/3 > 0.
2
a8C/82 = (k2-pkk2)W11"/3 > 0.

acl/a3 = 0.

aE[S]/aa, = ((W,+H)/3)2klW1'+(l-p2)(k2+k2-2pklk2)(l/3)2 (k(Wl')2(W1+H*


aE[S]/8a2


)


+ 2kiW1'Wi"(W1+H)) > 0.

= ((W1+H)/3)2zk2W1 +(-p2) (k2+k2-2pklk2) (l/3)2k2(W1')2(WI+H)
1 2
+ 2k2W,1'W"(W1+H)) > 0.


aE[s]/ap = 0.

The effect of changing the quality mark-up price (p), the marginal

production cost (c), the manager's market reservation level (G), the

worker's market reservation level (H*), and the job-enlargement factor

(kI or k2) are discussed as follows.








The Effect of The Quality Mark-up Price (p)

Result 3.10. Both agents' efforts increase with an increase of

the quality mark-up price. Agents will be paid by contracts with lower

salary and higher commission rates. The principal's expected profit and

the agents' expected compensation are also increased.

Proof. Holding all other parameters fixed, only the effects of a change

in the quality mark-up price parameter (p) are considered.

dal/dp -(aTj/ap)/(T1/a8a1), sign[da1/dp]=sign[aT1/ap]-sign[a2+p] > 0.

da2/dp = -(aT2/ap)/(aT2/aj1), sign[da2/dp]=sign[aT2/ap]=sign[a1] > 0.

df/dp -(aT3/ap)/(8T3/a8), sign[ d3/dp]-sign[aT3/ap]-sign[[a] > 0.

dA1/dp = (aA1/ap)+(aA1/aal)(dal/dp)+(aAl/ac2)(daz/dp)+(aA1/a8) (dp/dp) < 0.

dBl/dp = (aB/app)+(B/ )(da/dp)+(aB/aa2)(da2/dp)+(aB1/a)(d8/dp) > 0.

dC1/dp (ac1/ap)+(ac1/aa1) (daj/dp)+(aC1/aa2) (daz/dp)+(ac/a8) (df/dp) > 0.

dE(S)/dp =(aE(S)/ap) + (aE(S)/aa,)(daj/dp) +(aE(S)/aa2)(da2/dp)

+ (aE(S)/ap)(dp/dp) > 0.

dA2/dp = (aA2/ap)+(8A2/aal) (dal/dp)+(aA2/aa2) (da/dp)+(aAz/a~) (dp/dp) < 0.

dB2/dp = (8B2/ap)+(aB2/aa) (dal/dp)+(8B2/c2)(daz/dp)+(aB2/a8) (dl/dp) > 0.

dC2/dp = (ac2/ap)+(ac2/aa8)(dal/dp)+(aC2/aa2)(da2/dp)+(ac2/a) (dO/dp) > 0.

dE(M)/dp = (aE(M)/ap) + (aE(M)/8a1)(dac/dp) + (aE(M)/aa2)(da2/dp)

+ (aE(M)/a8)(df/dp) > 0.

dE(w)/dp aE(r)/ap a1(az+i) > 0. Q.E.D.

Remark. The risk-neutral principal's marginal revenue increases with an

increase of the quality mark-up price. Therefore, he would like to

induce higher effort levels. For the risk-averse agents, the increase

of commission rates provides stronger incentive than the fixed salary to

induce efforts. Although the expected compensation for agents is






53

increased, the principal receives even a higher revenue. Therefore, the

expected profit is also increased.



The Effect of The Marginal Production Cost (c)

Result 3.11. The worker's production effort decreases with an

increase in the marginal production cost. Both agents' salaries are

increased, but the worker's commission rates are decreased. The

principal's expected profit and the worker's expected compensation are

both decreased, but the manager's expected payment remains unchanged.

Proof. Holding all other parameters fixed, only the effects of a change

in the marginal production cost (c) are considered.

dal/dc -(aTI/ac)/(aT/8aaI), sign[dal/dc]=sign[aTl/ac]=sign[-1] < 0.

daz/dc = (aT2/ac)/(8T2/a1), sign[da2/dc]-sign[aT2/ac] = 0.

do/dc = -(8T3/ac)/(aT3/8f), sign[dp/dc]=sign[aT3/8c] 0.

dAl/dc (8AA/ac)+(aA/aal) (dal/dc)+(aA/8aa2)(da2/dc)+(aA1/a) (dp/dc) > 0.

dBd/dc = (aB,/ac)+(aB1/aa) (dal/dc)+(aB/8aa2)(da2/dc)+(aB1/a9) (dp/dc) < 0.

dCj/dc (ac,/ac+(aC,/aa1)(dal/dc)+(8aC/aa2)(da2/dc)+(aCl/ap)(d//dc) < 0.

dE(S)/dc = (aE(S)/ac) + (aE(S)/aal)(da1/dc) + (aE(S)/aa2)(da2/dc)

+ (aE(S)/ap)(dP/dc) < 0.

dA2/dc (aA2/ac)+(aA2/aa) (dal/dc)+(aA2/aa2)(da2/dc)+(8A2/8a) (d//dc) > 0.

dB2/dc = (aB2/ac)+(aB2/aal)(dal/dc)+(aB2/aa2)(da2/dc)+(aB2/a8) (d/dc) 0.

dC2/dc = (ac2/ac)+(ac2/aal)(dal/dc)+(ac/aa)(da/d)+(a2/a) (d/dc) 0.

dE(M)/dc (aE(M)/ac) + (aE(M)/aal)(dal/dc) +(aE(M)/aa2)(da2/dc)

+ (aE(M)/a8)(dp/dc) 0.

dE(w)/dc = aE(r)/ac = -a, < 0. Q.E.D.









Remark. While the marginal production cost is increasing, the

principal's expected marginal revenue is decreasing. On the one hand,

the principal would like to induce lower production effort. A contract

with lower commissions reduces the worker's desire to produce. On the

other hand, the principal does not want to reduce the quality standard.

Thus, a higher salary will offset the negative effect so that the

manager's effort and expected compensation remain unchanged.



The Effect of The Manager's Market Reservation Utility (G)

Result 3.12. The manager's effort and commission rates decrease

as his market reservation level increases. Both agents' salaries are

increased. The principal's expected profit is decreased with this

change. The manager's expected compensation is decreased provided that

his marginal disutility is greater than J2. The worker's expected

compensation remains unchanged.

Proof. Holding all other parameters fixed, only the effects of a change

in the manager's market value are considered.

dal/dG = -(aT1/aG)/(aT1/aa1), sign[dai/dG ]-sign[8T1/aG"]-0.

da2/dG = -(aT2/aG)/(aT2/aa1), sign[da2/dG0 ]-sign[8T2/aG ]=0.

dp/dG0= -(aT3/aG0)/(aT3/a>), sign[df/dG]-=sign[8T3/aG]=sign[-W2'/2]<0.

dA1/dG (BA1/aG0 )+(8Ai/aaI) (dal/dG )+(aA1/aa2) (da2/dG )+(aA1/a8) (dp/dG* )

> 0.

dB1/dG* (aB1/aG")+(aB1/aa) (dal/dG")+(aB1/aa2) (da2/dG*)+(aB1/a8) (dp/dG)

0.

dC,/dG* (aC1/aG)+(ac1/aa1) (daj/dG)+(aC1/aC2) (da2/dG)+(aC1/a~) (dp/dG)

0.









dE(S)/dG- (aE(S)/aG*) + (aE(S)/aa1)(dal/dG) + (aE(S)/8a2)(da2/dG)

+ (BE(S)/8a)(df/dG) 0.

dA2/dG* = (aA2/aG )+(8A2/aa1) (da1/dG )+(aA2/8c2) (da2/dG )+(aA2/a~) (dp/dG )

> 0.

dB2/dG (8B2/aG )+(aB2/aa~) (daj/dG)+(aB2/ac2) (da2/dG)+(aB2/a8) (d/3/dG)

< 0.

dC2/dG0 (ac2/aG0)+(C2/aa1) (daj/dG)+(a8C/a2) (dag/dG0)+(ac2/8a) (dp/dG)

< 0.

dE(M)/dG = (aE(M)/aG*) + (aE(M)/8aj)(daj/dG0) +(aE(M)/a82)(da2/dG0)

+ (aE(M)/a8)(df/dG) < 0 (if W2'>J2).

dE(r)/dG* = aE(w)/aG* -(W2+G)/2 < 0. Q.E.D.

Remark. When the manager's market value increases, it is more costly to

induce a given level of effort. The principal's marginal revenue

remains unchanged. Thus, the principal would like to induce less effort

from the manager by reducing the commission rates. In the meantime, the

principal does not want to reduce production. A higher salary keeps the

worker's efforts at the same levels. Thus, the worker's expected

compensation remains unchanged.



The Effect of The Worker's Market Reservation Utility (H0)

Result 3.13. The worker's production and quality-enhancing

efforts along with the commission rate decrease with an increase in his

market reservation level. Both agents' salaries are increased. The

principal's expected profit is decreased and the manager's expected

compensation remains unchanged.









Proof. Holding all other parameters fixed, only the effects of a change

in the worker's market value are considered.

dal/dH (aTj/aHO)/(aTj/aa1), sign[dat/dH* ]-sign[Ti/aH ]

sign[(-2/3)((W,+H*)/3) (kW,')- (2kjWi'Wj"(l-p2) (k2+k2-2pklk2)/9) ]<0.
1 2
da2/dH -(8T2/aH)/(aT2/aa1), sign[da2/dH ]-sign[8T2/8H0]

sign[(-2/3)((W1+H*)/3)(k2W1')-(2k2WI'Wi"(l1-p2)(k2+k- 2pklk2)/9)]<0.
1 2
dp/dH0 -(8aT3/8H)/(aT3/8a), sign[dP/dH*]-sign[aT3/8H0] 0.

dA,/dH (aA,/aH)+(8A/a) (i/d(dda/dH )+(aA1/8ac)(da2/dH0)+(8A/8O) (d /dH )

> 0.

dBj/dH (aBj/aH)+(aBj/aac) (daj/dH )+(aB1/8a2) (da2/dH )+(aBa/8p) (dfi/dH)

< 0.

dC,/dH = (acj/aH)+(acj/8a,) (da/dH)+(ac8/8a2) (da2/dH)+(8C/8f) (d3/dH)

< 0.

dE(S)/dH (aE(S)/IH) + (aE(S)/8aa)(da1/dH) + (8E(S)/8a2)(da2/dH)

+ (aE(S)/af)(dp/dH) = ?

dAz/dH = (aA2/aH )+(aA2/aal) (dal/dH)+(aA2/aa2) (da2/dH )+(aA2/a3) (df/dH)

> 0.

dB2/dH = (aB2/aH0 )+(aB2/aa1) (da/dH0 )+(aB2/8a2) (d2a/dH )+(,B2/8,) (d/dH )

= 0.

dC2/dH (8a2/aH )+(ac2/aaC) (dai/dH )+(8C2/8a2) (daz/dH0 )+(8Cz/Bf) (df/dH)


dE(M)/dH0



dE(wr)/dH*


0.

- (8E(M)/aH) + (aE(M)/aai)(daj/dH) +(aE(M)/aa2)(da2/dH)

+ (8E(M)/a8)(d/3/dH) = 0.

- 8E(w)/8H* = -((W2+H)/3 )2 (k2+k -2pkk2) < 0.
1 2
Q.E.D.









Remark. The intuition behind this result is similar to that behind

result 3.12. The worker's compensation cannot be determined, due to the

conflicting effects of salary and commissions.



The Effect of Job-Enlargement Factor (k1 or k2)

Result 3.14. The worker's effort (al, a2) devoted to the

production and quality-enhancing processes decreases with a decrease in

the job-enlargement factor (i.e. an increase in k, or k2). The

manager's fixed salary is increased, but, his expected compensation and

effort level remain unchanged. And, the principal's expected profit is

decreased.

Proof. Holding all other parameters fixed, only the effects of a change

in the job-enlargement factor k, are considered.

dai/dk = -(aT1/akl)/(8Tl/aa), sign[dai/dkl]-sign[aT1/akl]

= sign(-2((Wi+Ho)al/9)kl(Wl')2 ((W1+Ho)/3)2(WI1+k1a1W1")

((1-p2) (WI'+klaWI")/9) [(W')2+2W" (W+H ) ] (k2+k2-2pklk2)
1 2
(kWi' (1-p2)/9) [ 2aIWi'W1"+2aI(Wi" )2] (k2+k 2 2pklk2)
1 2
-(klWI'(1-p2)/9) ((WI' )2+2W"(W+H=)][2kl-2pk2]} < 0.

daz/dki -(T2/aki)/(8T2/aa1), sign[da2/dk ]-sign[ aT/ak1]

= sign(-2((Wi+Ho)ai)/9)k2(W' )2 ((Wi+HO)/3)2(k2aiWI")

((1-p2)k2ziW"/9) [(WI' )2+2Wi" (W+H) ] (k2+k2 2pkk2)
1 2
-(k2Wi' (1-p2)/9) [2aiW1'W"+2a(W")2] (k2+k2- 2pkk2)
1 2
-(k2W,' (-p2)/9) (W)2 +2W"(Wi+Hi)][2k1-2pk2]) < 0.

df/dk: = -(aT3/akl)/(aT3/af), sign[d8/dkl]=sign[aT3/aki] 0.









dA1/dki (aA1/akl)+(aA1/aal) (dal/dkl)+ (A1/Cz2) (da2/dkl)+(AA/8p) (d//dk)

?.

dB,/dkI (aB1/akj)+(aB1/aaj) (dai/dkl)+(aB/8a2) (da2/dk)+(OB1/O8) (dp/dkl)

= ?.

dC1/dk1 (aC1/apc+(aC1/8aa) (/dk)+( /8 da2/dk)+(aC/) (d,/dk1)



dE(S)/dkl (aE(S)/akl) + (aE(S)/aal)(dal/dk1) + (8E(S)/aa2)(da2/dk1)

+ (aE(S)/a8)(d3/dkj) ?.

dA2/dki (aA2/ak )+(aA2/aa1) (daj/dkl)+(aA2/aa2) (da/dk)+(aA2/a8) (df/dkl)

> 0.

dB2/dki (aB2/akk)+(aB2/aa1) (dal/dk)+(aB2/aa8) (da2/dkj)+(aB2/ap) (dp/dkj)

0.

dC2/dk1 (ac2/akl)+(ac/aa) (da/dkadk)+(C2 2(da/dk)+(aC2/a) (d/dki)

0.

dE(M)/dkl = (aE(M)/ak1) + (aE(M)/8ao) (dal/dk1) +(aE(M)/aa2) (daz/dk)

+ (aE(M)/8 ) (dp/dk) = 0.

dE(r)/dki = aE(r)/aki

S-Wl'al((Wx+H")/3)2_ (I -p2) (W1, (W11/3)2 (k2+k 2-2pklk)
1 2
+ 2((Wi+H)Wl')/9)ac'Wl"(kz+k2-2pklk2)+(Wi+Ho)(Wi'/3)2(2ki-2pk2)
1 2
< 0.

Q.E.D.

Remark. When the worker treats the new responsibility as a burden

rather than enlarged interest, his efforts decrease. Although only one

of the factors changes, both production and quality-enhancing efforts

decrease due to the negative correlation between the two states of

nature.









Summary



In this chapter, a hierarchical agency model was proposed to

simulate the integration of quality assurance and the production process

as observed in a JIT environment. The bottom-line worker is assigned

dual responsibilities. The worker is better off with this arrangement,

not only through the increased compensation, but also through the

enlarged job interest. The results obtained in this model are quite

consistent with that in Holmstrom (1979), Basu et al. (1985) and Suh

(1988), although different settings and assumptions are considered.

Both general contracts structure and more specific comparative

statics were discussed in the chapter. The first-best solution cannot

be reached because of the agents' moral hazard problems. Basically, the

contract is an increasing function of its attributes, and the mean and

variance of the attributes provide a common standard to evaluate the

final performance. Several assumptions were made in an effort to derive

the comparative statics results. This is a general restriction for

agency theory. As Grossman and Hart (1983) point out, even the simplest

results require certain restrictive assumptions. On the whole, the

principal is better off to reduce the uncertainty caused by the negative

correlation between quantity and quality. The principal also prefers

the worker who is more interested in enlarged job assignments. When

alternative job opportunities for the agents become more attractive, it

is suggested that the principal should modify the compensation plan with

an increased salary and lowered commission rates. Improvements in the

marginal production cost and the quality mark-up price are also







60
encouraged by the principal. Extensions and further research of this

model are discussed in chapter 6.













CHAPTER 4
PROFIT SHARING AND TARGET SETTING IN PROCUREMENT MANAGEMENT



A clear understanding of the relationship between procurement and

production management is critical in today's business environment.

Procurement management is all of the management functions related to the

acquisition process, which includes deciding which vendors to use,

negotiating contracts, expending delivery, and acting as liaison between

vendors and other company departments; it is not limited to placing and

tracking orders. Procurement is of strategic importance and must

satisfy the company's long-term supply needs to support the company's

production capabilities. Recently that importance is increasing because

of two factors: the tremendous impact of materials costs and quality on

profits, and the increasing prominence of automated manufacturing.

The costs of purchased materials are substantial and growing. The

typical U.S. manufacturer spent 40 percent of its total income from

sales on purchased materials in 1945. The proportion rose to 50 percent

in 1960, 60 percent in late 1980, and is expected by top executives to

climb even higher (Miller and Roth, 1988). The proportion spent on

purchased materials varies from industry to industry. For example,

automobile manufacturers spend about 60 percent of their revenues on

material purchases, farm implement manufacturers about 65 percent, food

processors about 70 percent, and petroleum refining firms over 80









percent, yet the pharmaceutical industry spends only 25 percent

(Gaither, 1990, p. 587). There are also some variations by country.

Japanese firms, as an example, spend on average 7 percent more of their

income on materials than do firms in North America and Europe due to the

lack of natural resources (Krajewski and Ritzman, 1990, p. 394).

Despite such variations, most firms fall within the 45 to 65 percent

range, giving materials procurement great cost-reduction potential.

As the automation of manufacturing continues today, labor costs

represent only about 10 to 15 percent of production costs in many mass

production industries. Some estimate that labor costs will decline to

about 5 percent of production costs by the year 2000 (Gaither, 1990, p.

587). Therefore, labor costs will become less significant and material

costs will become the central focus in the control of production costs

in some industries. Automation, especially geared toward Just-In-Time

production, requires rigid control of design, delivery schedules, and

quality of purchased materials. Hence, procurement management is

crucial to establish and maintain vendor relations ensuring that

materials of right design and of good quality are delivered in the right

quantities at the right times.

Because of the significant amount of capital tied up with

materials, it is particularly important to control production costs.

Japanese industries, as an example, import most of their materials, yet

still make considerable profits and possess large market share. This

case deserves our consideration. In this chapter, a hierarchical agency

model is proposed to discuss the interaction of procurement and

production departments under target-setting and profit-sharing









compensation plans. It is shown that carefully designed compensation

schemes can be used as a tool to provide motivation to reduce production

costs.

Profit-sharing plans, in which an individual's compensation is

tied to the overall performance of the firm, have received wide

acceptance. Kruse (1987) reports that twenty percent of the United

States labor force, about 22 million employees, participate in over

400,000 workplace profit-sharing plans, and that the number of profit-

sharing pension plans has increased by 19,000 per year since 1970. A

recent New York Exchange Survey indicates that 70 percent of firms with

profit-sharing plans report that they lead to improved productivity

(Baker, et al., 1988). Schuster's (1984) study shows that the success

of a profit-sharing plan usually generates a productivity improvement of

5 to 15 percent in the first year, better work attitudes, product design

improvements, and improved product quality. From the workers'

viewpoint, there is greater employment stability in a difficult and

turbulent environment.

The Scanlon plan is one of the best-known profit-sharing examples.

First developed in the 1950s by Joe Scanlon, an ex-steelworker and

steelworkers' union official who was later lured to the M.I.T. faculty,

the Scanlon plan involves a combination of group and organization-level

pay incentives, an employee suggestion system, and a participative

approach to accessing and evaluating suggestions (Kanten, 1987; Arnold

and Feldman, 1986). Production committees are set up for each

department in the organization. These committees consist of the

supervisor or senior manager in the department and representatives of









employees, who may either be elected or appointed by the union. These

production committees screen suggestions for productivity improvement

from both employees and managers. The cost savings generated by

suggestions that are accepted are paid to everyone in the department

originating the suggestion. In addition, under the Scanlon plan, gains

resulting from productivity improvements are paid in the form of a

monthly bonus to all employees. Everyone receives a share in proportion

to his wage or salary level. Overall, the record of the Scanlon plan is

quite positive. The most frequent benefits associated with the plan are

greater organizational efficiency, increased participation in decision

making, a greater willingness on the part of employees to accept change,

and an improved climate of union-management relations (White, 1979).

Weitzman (1980), MacDonald (1984) and others argue that the so-

called "optimal compensation schemes" of the agency problem are at a

rather high level of abstraction, and provide little in the way of

operational hypotheses. Therefore, the analysis of this chapter is

based on a linear profit-sharing plan similar to the Scanlon plan, with

a target or standard of the level of production costs that must be

attained before the agents receive any share of profits. Linear

contracts introduce sufficient simplification to provide interesting

results without abstracting beyond reality. The linear contract has

been in wide-spread use in practice (Weitzman, 1980; Cummins, 1977;

Atkinson, 1978; Berhold, 1971; Schern, 1964). Atkinson (1978) prefers a

procedure of standard-setting in conjunction with a profit-sharing plan

as opposed to a pure profit-sharing plan. His reasons are as follows.

First, the standard-setting incentive structure allows for direct









incorporation of the principal's beliefs and thus provides a specific

basis for reward and control purposes. Second, under the standard-

setting plan, the contractual risk for the agent is based on the effect

of the agent's marginal return and not on the entire risk of the agency,

hence the agent's risk will probably be lessened.

In what follows in this chapter, the next section introduces the

hierarchical agency model to be utilized in this presentation. It will

follow the Scanlon/Atkinson concept. Then, the characterization of the

first-best and the second-best profit-sharing compensation plans are

discussed and compared. Detailed comparative statics of exogenous

parameters are analyzed. The chapter then concludes with a summary of

the findings.



The Model



The interaction between the principal and the agents is viewed as

a game where the principal declares linear profit-sharing contracts with

target costs levels to the procurement and production managers (the

agents). The procurement manager and the production manager, based on

the compensation plans, decide on the effort levels they plan to devote

to quality materials procurement and production costs reduction,

respectively, and then report their achievements. In this model both

production costs (C) and material quality (q) are observable by three

parties. This information is called "hard" which means that the agents'

reports are verifiable by the principal, and false reports are not

possible (Tirole, 1986). The agents must convey their achievement in a









credible way. For example, the principal can look at the sampling

result of an incoming lot and the receipts of various costs during

production to convince himself that the managers have truly accomplished

the jobs that they declare to have accomplished.

Production costs (C) are influenced by the production manager's

effort (a), the material quality level (q), and a random environmental

factor such as machine breakdowns. Production costs can be reduced when

the production manager expends more effort or the materials quality is

improved. Material quality (q) is assumed to have only two levels, high

or low. For example, given an acceptable quality level (AQL) at 5

percent, a lot that has only a .5 percent defective rate can be

considered to be at a high level, and a lot that has exactly a 5 percent

defective rate can be considered to be at a low level. Material quality

can also be defined as the combination of defective rate and delivery

lead times. The chance of getting high-quality materials (Ph(p)) is

increasing in the procurement manager's effort 6. It is further assumed

that the production cost C is normally distributed with mean u(C) and

variance ac2, and u(C)=c-c1a-c2q where c is the average production cost

without the managers' efforts, cl is the marginal cost reduction by the

production manager's effort, and c2 is the marginal cost reduction by

improving materials quality.

The benefits of other economic activities of the principal (in

terms of profits) are denoted by a random variable R. For example, R

can be the returns on investment in conducting research and development

or in training employees. It is assumed that R is normally distributed

with mean u(R) and variance or2, and u(R)=r-rlq where r is the return









rate when materials quality is at the minimum acceptable level, and r,

is the marginal decrease of return due to the additional capital spent

on purchasing quality materials. In general, higher quality material

demands a higher purchasing price. Each dollar tied up in procurement

is a dollar unavailable for investment in new products, technological

improvements, or other beneficial economic activities. Nevertheless,

the principal is still assumed to prefer high-quality materials, i.e.

c2>ri. It is also assumed that R and C are negatively correlated, i.e.

Cov(R,C)=parac<0. Increasing production costs will reduce the profit

margin and thus hinder the principal investing in other activities.

Callen's (1988) paper can be considered as a special case of this model

when the covariance term equals zero.

The production manager's compensation Sq(C) is based on the

following formula:

Sq(C)=A-B(C-cq), q-h,l,

where A is his base salary, B is the sharing ratio (0BI51), C is the

actual production cost, and cq is the target cost at each quality level

and Ch
Scanlon plan, the procurement manager will receive a share in proportion

to his salary level. Therefore, the procurement manager's compensation

Mq(C) = A2-B2(C-c) = A2-kB(C-cq) = k(A-B(C-cq)) = kSq(C),

where k-A2/Ai~O is called the departmental differentiation factor.

The production manager's utility, U(Sq(C)-di(a)), is defined over

his payment and his personal effort; and d1(a) is a convex function.

Similarly, the procurement manager's utility function, V(Mq(C)-d2(f)),

is defined over his payment and his personal effort, and d2(f) is a







68

convex function. Both managers are risk avoiders. It is the principal

who decides B and k. The production level is assumed at a fixed volume

and price, therefore, its revenue can be dropped out from the

principal's profit function. The principal's utility function, I(w), is

defined over his profit w, where w R-M,(C)-Sq(C)-C R-(k+l)Sq(C)-C.

Thus, the agency model can be established as follows. The principal

seeks to

MAX E[I(r)] = Z Pq(#)I[R-Mq(C)-S,(C)-C] (4.1)
B,k q

Subject to EPq(3)U(Sq(C)-d1(a)) > U(S) (4.2)
q

Max ZPq(3)U(Sq(C)-d1(a)) (4.3)
a q

EPq(p)V(Mq(C)-d2(9)) > V(M) (4.4)
q

Max EPq(W)V(Mq(C)-d2(f)) (4.5)


EPq(#) = 1,
q
where S* and M* are the managers' reservation welfare levels.

In addition, the principal and the managers are assumed to have

exponential utility functions that exhibit constant absolute risk

aversion (CARA). Let n, n1, and n2 be the Arrow-Pratt measure of

absolute risk aversion for the principal, the production manager, and

the procurement manager, respectively. As is well known, the normally

distributed random variable permits one to employ a mean-variance model

(Tobin, 1958; Feldstein, 1969; Bamberg, 1986; Epps, 1981). The

popularity of the mean-variance approach is due to the property that

risk is fully described by the variance of the random variable. Thus,









along with the CARA utility function, the principal's objective can be

written as

MAX E Pqg()((r-riq)-(k+l)A+(k+l)B[c-c1a-c2q-Cq]-(c-c1a-c2q))
B,k
-(n/2) (a2-2[1-(k+l)B]paora+[l-(k+l)B] 2c2).

The maximization of the two managers' expected utilities is also

identical to maximization of the associated certainty equivalents as

follows:

MAX Pg[A-B(c-cIa-c2q-cq) ]-d1(a)-(n1/2)B2c2,
a q

and MAX E Pqk[A-B(c-cla-c2q-Cq))]-d2() -(n2/2)k2B2c2.
P q

Substituting (4.3) and (4.5) with their corresponding first-order

conditions, the above agency model can be written as

[MOD]

MAX E Pq())((r-riq)-(k+l)A+(k+l)B[c-cla-c2q- q]-(c-cia-C2q))
B,k q
-(n/2) { 2-2[1- (k+l)B] pae+[l- (k+l)B]2a2) (4.6)

Subject to

E Pq[A-B(c-cla-c2q- q)]-d((a)-(nl/2)B22 S (4.7)

BCi-d1'(a) = 0 (4.8)

E Pqk[A-B(c-cla-c2q-cq)]-d2(6)-(n2/2)k2B2ac2 > M (4.9)

BkPh'(C)[c2(h-1)+h-c^l]-d2'(8) 0 (4.10)

2 Pq(3) = 1, q=h,l

0 < (k+l)B < 1, and k > 0.









Characterization of Optimal Compensation



In the absence of constraints (4.8) and (4.10), the above [MOD]

program determines the first-best solution which serves as a benchmark

to be compared with the second-best solution.



The First-Best Solution

The first-best solution implies that the principal could

costlessly monitor the managers' effort levels or decisions. For

example, in a highly-centralized firm, the managers have to choose the

effort levels that maximize the principal's objective rather than

maximize their own objectives. Consequently, the principal has no

incentive to offer more than each manager's reservation welfare.

Constraints (4.7) and (4.9) should be solved as qualities:

E Pq[A-B(c-ca-c2q-q)] dl(a)+(nl/2)B2 +S (4.11)

z Pqk[A-B(c-c1a-C2q-Cq)] d2()+(n2/2)k2B2oc2+Mo. (4.12)

substituting qualities (4.11) and (4.12) into (4.6) yields

MAX {(r-rl) (c-cla-c21)+Ph(c2-rl) (h-l) -d1(a) -d2(0)
B,k, a,

-(1/2)B2a2 (n1+n2k2) -S-M)

(n/2)(aZ2-2[1-(k+l)B]paor+[l-(k+l)B] 2ac2), (4.13)

which is a concave function given nonnegative degrees of risk aversion.

A contract (A,B,k,cq) is first-best optimal if and only if (B,k)f

maximizes (4.13), subject to the restrictions k-0 and O0(k+l)B1l.

The first-order conditions of (4.13) with respect to k and B are:

Ik B(Bao2(n2k+n(k+l))-n(ao2-poac)) = 0 (4.14)







71

and I Bc2(ni+n2k2+n(k+l)2)-n(k+l) (c2-parac) 0. (4.15)

Solving (4.14) and (4.15) simultaneously yields

kf = n/n2 (4.16)

and B, nn2z/(nln2+nn1+nn2) a2 (4.17)

where z a2_-parac.

Therefore, the procurement manager's optimal sharing ratio (kB)fb, and

the first-best sharing pool ((k+l)B),f (i.e. the sum of the two

managers' sharing ratios) are:

(kB), = nnlz/(nln2+nn1+nn2) a2.

and ((k+l)B), = n(n1+n2)z/(nln2+nni+nn2)oc2 (4.18)

In the first-best case, the principal has centralized control

power. By maximizing the principal's objective function (4.13) with

respect to a and P, the managers' effort levels are assigned to satisfy

the following conditions:

I.= c1-d1'(a) = 0, and (4.19)

Ip = Ph'(p)[(c2-rl)(h-l)] d2'(0) = 0. (4.20)

From (4.19), the production manager's marginal expenditure of effort

equals the marginal contribution to cost reduction. From (4.20), the

procurement manager's marginal expenditure of effort equals the marginal

changes to the principal's profits.

The characterization of the first-best solution is discussed in

the following results.

Result 4.1. The optimal first-best departmental differentiation

factor, kf, depends only on the managers' degree of risk aversion.

Proof. This result follows directly from examining equation (4.16).

Q.E.D.









Remark. When the principal can monitor each manager's decision choice

(effort) perfectly, the departmental differentiation factor does not

depend on the nature of responsibilities at each position. Therefore,

no matter which job is assigned, each manager is paid the same contract.

This is more likely in a highly-centralized firm such as exists in a

socialist society. In practice, however, payment is usually tied to

one's position in the firm. Medoff and Abraham (1980), for example,

find that between-job-level earnings differentials are more important

than within-job-level differentials. Also, Murphy (1985) finds that

corporate vice presidents receive average pay increases of 18.8 percent

upon promotion to another vice-presidential or higher position, compared

to average pay increases of only 3.3 percent in years when they remain

in the same position. This phenomenon is explained later in the second-

best case.

Result 4.2. Both managers' optimal first-best sharing ratios, Bg

and (kB),f, and the sharing pool ((k+l)B), are:

(1) between zero and one, and in total less than one;

(2) independent of each manager's efforts;

(3) decreasing in the manager's own degree of risk aversion;

(4) increasing when there is more negative correlation between R and C;

(5) decreasing in the volatility of production processes;

(6) increasing in the degree of risk aversion of the principal. And

each manager's optimal sharing ratio increases when the other manager

becomes more risk averse.









Proof. (1): From (4.17) and (4.18), unless the covariance is "much

larger" than the variance of costs, Bf, (kB), and ((k+l)B)fb are likely

to be between zero and one.

(2): The sharing ratios are independent of managers' efforts since the

effort levels are determined by the principal directly from (4.19) and

(4.20).

(3)-(6): From (4.19) and (4.20), I and I. are not functions of B, k,

n, nj, n2, and ao2. Therefore, only Ik and Ig need to be considered.

Taking total derivatives of Ik and Ig with respect to a parameter t

yields

(aIk/at)+(aIk/aB)(dB/dt)+(Ik/ak) (dk/dt)+(aIk/aa)(da/dt)+(alk/8) (d>/dt)

= (aI/at)+[oc2(n2k+n(k+l))](dB/dt)+[BaO2(n+n) ](dk/dt) 0, and (4.21)

(alB/at)+(alB/aB)(dB/dt)+(alB/ak)(dk/dt)+(alB/aa)(da/dt)+(alB/a) (d8/dt)

= (alB/at)+ao2[nl-+n2k2+n(k+l)2] (dB/dt)+[2Bao2(n2k+n(k+l)) -nz] (dk/dt)

= 0. (4.22)

Solving (4.21) and (4.22) simultaneously,

dB/dt = -J((Ik,IB)/(t,k))/J((IkIB)/(B,k)), and

dK/dt = -J((Ik,IB)/(B,t))/J((Ik,IB)/(B,k)),

where J(.) is the notation of the Jacobian.

It is straight forward to see that J((Ik,IB)/(B,k)) = -nn2a2z <0.

Therefore, sign[dBfb/dt] sign[J((Ik,B)/(t,k))],

sign[dkf,/dt] = sign[J((Ik,I,)/(B,t))],

sign[ d(kB) ,/dt]-sign[kdBf/dt+Bdkfb/dt], and

sign[d((k+l) B) /dt ]=sign[d(kB) /dt+dB/dt ] .

Hence, results (3)-(6) are obtained by examining the following signs:

sign [dBb/dn] sign [(k/n)(n2Ba 2)2] >0.








sign [dBaf/dnl] sign [-(n+ng)(Bu,2)2] <0.

sign [dBfb/dn2] sign [nk(ao2)2] >0.

sign [dBfb/do2] sign [-nnzBacz] <0.

sign [dB,/d|Cov(R,C)I] sign [nnzBac2] >0.

sign [d(kB),f/dn] sign [(k2/n)(n2Ba 2)2] >0.

sign [d(kB)n/dn1] sign [n(o 2)2] >0.

sign [d(kB)b/dn2] sign [-(n+nl)k(o2)2] <0.

sign [d(kB),/dac2] sign [-knn2Bacz] <0.

sign [d(kB)a,/dlCov(R,C) ] = sign [nn2Bka c2 >0.

sign [d((k+l)B)f/dn] >0.

sign [d((k+l)B)f/dnl] = sign [-n2(ac2)2] <0.

sign [d((k+l)B)fb/dn2] = sign [-nlk(a,2)2] <0.

sign [d((k+l)B)b/dao2] <0.

sign [d((k+l)B),f/dlCov(R,C)|] >0. Q.E.D.

Result 4.3. If the principal is risk neutral, then he bears all

risk, and the managers are paid a fixed salary.

Proof. Result 4.3 follows directly from examining equations (4.14) and

(4.15). Q.E.D.



The Second-Best Solution

The managers would maximize their own objectives rather than

follow the principal's instructions when their decisions (efforts) are

not observable by the principal. The second-best optimal compensations

can be obtained by first solving (4.8) and (4.10) for the optimal effort

levels as implicit functions of the sharing ratios and the departmental

differentiation factor, i.e. a=a(B,k) and p=6(B,k). Then, solve the








following function by substituting a(B,k) and P(B,k) along with (4.7)

and (4.9) into equation (4.6).

MAX (r-rll) (c-cla(B,k) -c2)+Ph(C2-rl) (h-l)-d1(a(B,k))-d2(9(B,k))

-(1/2)B2ac2(n1+nzk2 )-S-Mo -(n/2) ar2-2[l-(k+l)B]parac+[l-(k+l)B] 2a2),
(4.23)

which is a concave function of B and k, given nonnegative degrees of

risk aversion. A contract (A,B,k,cq) is second-best optimal if and only

if (B,k)sb maximizes (4.23), subject to the restriction k-O and

0O(k+l) Bl.

The first-order conditions of (4.23) with respect to k and B are:

[c1-di'(a)](8a/ak) + [Ph'(c2-rl)(h-l)-dz' (p)](ap/ak) + Bn(ac2-prac)

B2ac2(n2k+n(k+l)) = 0, and (4.24)

[c1-d1'(a)](aa/8B) + [Ph'(c2-rl)(h-l)Ph'(p)-d2( )](af8/aB)

+ n(k+l)(ac2-parac) Ba,2(n+n2k2+n(k+l)2) = 0. (4.25)

Implicit differentiation of equations (4.8) and (4.10) yields:

aa/ak = 0 (4.26)

aa/8B cl/d1"(a) > 0 (4.27)

a8/ak = -(Ph'B[c2(h-l)+Ch-C )/{Ph"Bk[c2(h-l)+Ch C]-d2") > 0 (4.28)

8i/8B = -{Ph'k[c2(h-l)+Ch-c l]/{Ph"Bk[c2(h-l)+ch j-d2") > 0 (4.29)

With no loss of generality, let Ph"(P) equal zero. Substituting (4.26)-

(4.29) into equations (4.24) and (4.25) yields:

L, = a2+nz-(n2k+n(k+l))Ba 2 = 0, (4.30)

and LB = a1+ka2+n(k+l)z-(nl+n2k2+n(k+l)2)BaC2 0, (4.31)

where al=c(cl-dl' )/dl", (4.32)

a2=Ph'xy/d2", (4.33)

x= c2(h-l)+Ch-Cl, and y= (c2-rl)(h-l)Ph'-d2'.









Solving (4.30) and (4.31) simultaneously yields:

kb [(n+nl)a2-nal+nnz ] / [(n+n2)al-na2+nn2z],

if this expression 0, (4.34)

0, if the above expression is < 0.

Bsb "- [(n+n2)al-na2+nn2z /(nln2+nn1+nn2)a c2

if this expression is in [0,1], (4.35)

1, if the above expression is > 1,

0, if the above expression is < 0.

Recalling from (4.8) and (4.10), the managers' effort levels are

selected as follows:

1. Bcl-dl'(a) 0, and (4.8)

Lp = BkPh'x-d2'(p) =0. (4.10)

Suppose that interior solutions exist, some results regarding the

characteristics of the second-best solution are discussed as follows.

Result 4.4. The optimal second-best departmental differentiation

factor, ksb, depends not only on the managers' risk attitudes, but also

on each position's contribution to the overall profits.

Proof. From (4.32) and (4.33), a, and a2 can be treated as the marginal

contribution to the overall profits by increasing a and P, respectively.

The result follows directly. Q.E.D.

Remark. When the principal cannot monitor the managers' efforts, the

departmental differentiation factor is tied to the position in the

hierarchy. The same person might be paid differently because of

different job titles. Lack of perfect monitoring schemes can explain

this common phenomenon in practice.







77

Result 4.5. The production manager's second-best effort level is

increasing in the sharing ratio. The procurement manager's second-best

effort level is increasing in the sharing ratio or the departmental

differentiation factor. Each manager's second-best effort level is less

than his first-best effort level (i.e. aab
Proof. Comparing equations (4.8) with (4.19) and (4.10) with (4.20),

since d1(a), d2(O) and Ph(p) are increasing in a or f, and Beb is between

zero and one, the result follows directly. Q.E.D.

Result 4.6. The second-best sharing pool is greater than the

first-best sharing pool (i.e. [(k+l)B]b>[ (k+l)B]fb). The second-best

effects on separate ksb and Bsb, however, are in the opposite direction.

Proof. From (4.18), [(k+l)B]fb= n(n1-n2)z/(n1n2+nn1+nn2)c2.

From (4.34) and (4.35),

[(k+l)B],b =[nla2+n2a +n(n1+n2)z]/(nln2+nni+nn2)ac2 (4.36)

Therefore, [(k+l)B]sb- [(k+l)B]fb = [nla2+n2a1]/(n1n2+nnl+nn2) 1 2 > 0.

From (4.16) and (4.34), sign[ksb-kfb,] sign[(n1n2+nn1+nn2)(a2-al)]. (4.37)

From (4.17) and (4.35), sign[Bsb-B,]= sign[n2a1+n(a1-a2)] (4.38)

If al-a220, then Bsb fb and ksb
and kb>kfb. Q.E.D.

Remark. Results (4.5) and (4.6) reveal the important so called "agency

costs," which measure the deviation from Pareto-optimal risk sharing,

from the principal's point of view. Let us define agency costs (AC) as

AC = E[I(k,,Bf)-I(ksb,Bsb)]. (4.39)

Then, agency costs as a measure of distance can be presumed to give an

estimate of how much (in terms of utility) the second-best contracts

could be improved if there were monitoring of the managers' efforts. In









fact, the nature of agency costs could be treated as an expected value

(in terms of utility) of perfect information. The intuition of the

opposite effect on k and B separately is as follows. The comparison of

aI and a2 indicates the relative importance of these two departments.

If the profit contribution from cost-reducing efforts is greater than

that from quality-improving efforts (i.e. a>>a2), then the production

manager should be compensated more to induce his efforts, and vice

versa. By and large, the principal has to pay more from the sharing

pool and the managers actually expend less effort when the principal

lacks a costless monitoring mechanism.

Result 4.7. Suppose the principal is risk neutral. Unlike the

first-best solution, the principal does not bear the entire contractual

risk.

Proof. Given n-0, n1>0 and n2>0, Bfb-0. At the second-best case,

ksb-nla2/n2a>0 and Bsb=al/nlacz>0. The risk-averse managers still have to

share some risk. Q.E.D.

Result 4.8. In both first-best and second-best cases, profit-

sharing compensation is preferred to fixed salary.

Proof. From (4.18) and (4.36), [(k+l)B]in and [(k+l)B]sb are both

positive. Result 4.8 follows directly. Q.E.D.

Remark. This result shows that it is better for the principal to offer

profit-sharing compensation rather than fixed salaries whether he can

monitor the managers' efforts or not.









The Comparative Statics



Until now this analysis has concentrated on optimal compensation

for a given environment (parameters). It will be interesting to

investigate the effects on contract designs of changing environments.

In the following results, several factors are discussed: the effect of

changing the participants' risk attitudes (n, nl, and n2), the

production cost volatility (ao2), the covariance (Cov(R,C)), the

production manager's cost-reducing coefficient (cl), the procurement

manager's cost-reducing coefficient (c2), the principal's return-

reducing coefficient (rl), the gap between cost targets (Ch-c ), the

difference between attainable material quality (h-l), the marginal

probability of getting high-quality materials (Ph'), and the degree of

effort aversion (-dl"/dl' and -d2"/d2') on (1) the production manager's

sharing ratio (Bsb), (2) the procurement manager's sharing ratio (kB)sb,

and (3) the principal's sharing pool [(k+l)B],,. Hereafter, analysis

focuses on the second-best case. The subscript (sb) has been dropped

for convenience.

Taking total dirivatives of L4, Lg, L, and L. with respect to a

parameter t yields

(8aL/at)+(aLk/aB) (dB/dt)+(8L/ak) (dk/dt)+(8Lk/8a) (da/dt)+(a4L/a8) (df/dt)

= (8Lk/at) [2(nZk+n(k+l) ](dB/dt) [Ba(n+n)] (dk/dt)+O(da/dt)

-xPh'(dp/dt) = 0, (4.40)

(aL/at)+(aL/aB) (dB/dt)+( LB/ak) (dk/dt)+(LB/aa) (da/dt)+(aLB/a/) (d/dt)

= (aLB/at)-o,2[nl+n2k2+n(k+l)2] (dB/dt)- [Bac2(n2k+n(k+l)) ] (dk/dt)

-cl(da/dt)-kxPh'(d3/dt) = 0, (4.41)







80

(aVat)+(a1/laB) (dB/dt)+(al/ak) (dk/dt)+(aL/aa) (da/dt)+(ai,/fl9) (d /dt)

(aI /8t)+c1(dB/dt)+0(dk/dt)-d1"(da/dt)+O(df/dt) 0, and (4.42)

(aLy/at)+(aLp/aB) (dB/dt)+(aLo/ak) (dk/dt)+(aLp/aa) (da/dt)+(aLp/a3) (df/dt)

(aLp/at)+kxPh'(dB/dt)+BPh'x(dk/dt)+0(da/dt)-d2"(dp/dt) 0.(4.43)

Solving equations (4.40) to (4.43) simultaneously,

dB/dt -J(( Ia,,L,,L)/(t,k,a,))/J((L,,LBL)/(B,k, )), and

dk/dt -J((I4,L,,La,)/(B,t,a,P))/J((LI,LB,I,Lp)/(B,k,a,P)), where J(.)

is the notation of the Jacobian. It is straight forward to see that

J((L,LB,, L, L)/ (B,k,a, ))

--Bcl2[ (xPh' )+d2" c2 (n+n2) ] -dl"Ba2 [d2"a,2(nln2+nnl+nn2)+(n+nl) (xPh' )2] <0.

Therefore, sign[dB/dt] = sign[J((Lk,LB,L ,L,)/(t,k,a,p))],

sign[dk/dt] = sign[J((LB,L, ,L#)/(B,t,a,p))],

sign[d(kB)/dt] sign[k(dB/dt)+B(dk/dt)], and

sign[d((k+l)B)/dt] = sign[d(kB)/dt+dB/dt].



The Effect of Participants' Risk Attitudes (n, nl. nz)

Result 4.9. Each manager's sharing ratio increases with an

increase of his own degree of risk aversion or an decrease of the other

manager's degree of risk aversion. The sharing pool decreases when the

managers become more risk averse.

Proof. This result is obtained by examining the following signs.

sign[dB/dnl] sign[-Bac2d1"(d2"Ba2(n+n2)-B(xP' )2] < 0.

sign[dB/dn2] sign[d1"d2"n(Boc2)2] > 0.

sign[dB/dn] sign[-(Bao2(k+l)-z)] = ?

sign[dkB/dnl] = sign[nd,"d2"(Bac2)2] > 0.

sign[dkB/dn2] sign[-d2"k(Bac2) (d"acr2(n+nl)+ci2) ] < 0.









sign[dkB/dn] = sign[- (Bao2(k+l)-z) (d"a2nl-c12) ] ?

sign[d(k+l)B/dnl] sign[-n2d1"d2" (Ba2)2-dl" (Ba2Ph' )2] < 0

sign[d(k+l)B/dn2] sign[-kd2"(Bac2)2(dl"nlac2+cl2) ] < 0

sign[d(k+l)B/dn] sign[-(Ba.2(k+l)-z)B(d"(xPh' )2+dl"d2"ac22(n1+n2)

c2d2")] = ? Q.E.D.



The Effect of Volatility of Production Processes (a 2) and Covariance

(Cov(R.C))

Result 4.10. Each manager's sharing ratio and the sharing pool

increase when R and C are more negatively correlated. The effect of

increasing volatility of production depends on the value of a, and a2.

If the marginal profit of the production manager's effort is higher than

the procurement manager's effort, the sharing ratios and sharing pool

decrease with an increase of production risk.

Proof. This result is obtained by examining the following signs.

sign[dB/dCov(R,C)] = sign[d1"nB(d2"n2ac2+(xPh' )2)] > 0

sign[dkB/dCov(R,C)] sign[dl"dz"nnlBac2+d2r"c12] > 0.

sign[d(k+l)B/dCov(R,C)] > 0.

sign[dB/doc2] = sign[-d1"B(l/ac) (-2al+nporac) (-d2"ac2n2-

(xPh')2)+2d2"nBoc(al-a2)] < 0 if al>a2.

sign[dkB/dac2] = sign[ (1/,) (- 2a+npaoc) (d' Bk(d2"c 2 + h' )2)

kdl"(xPh')2) -2d' d2"nBkaU(a1-a2) +(l/ac) (-2az+porac) (c12d") < 0 most

likely.

sign[d(k+l)B/dca2]

- sign[(l/o) (-2al+npaac) (dl' B(k+l) (d2"ac2n2+ (Ph' )2) -kd" (xPh)2)

2d,'d2",nB(k+l)oa(ai-a2) +(I/ac)(-2a2+Porac)(c12d2")] < 0 if al>a2. Q.E.D.









Recalling from (4.32) and (4.33), ai-c1(cl-dl')/d1", ag-Ph'xy/dZ",

x- c2(h-l)+Ch-cl, and y- (cZ-rl)(h-l)Ph'-d2'.



The Effect of The Production Manager's Cost-Reducing Coefficient (c)l,

The Procurement Manager's Cost-Reducing Coefficient (c,), and The

Principal's Return-Reducing Coefficient (r)l

Result 4.11. The production (procurement) manager's sharing ratio

increases (decreases) with an increase of cl; decreases (increases) with

an increase of c2; and increases (decreases) with an increase of rl.

The sharing pool increases with an increase of c, or c2; decreases with

an increase of rl.

Proof. Result 4.11 follows by examining the following signs.

sign[dB/dc] = sign[2B(cl-dl')(d2"ac,2(n+n2)+(xPh')2)] > 0.

sign[dkB/dcl] = sign[-2Bnd2"aC 2(c-d' )] < 0.

sign[d(k+l)B/dcl] sign[2B(n2d2"ac2+(xPh' )2)(c1-d') ] > 0.

sign[dB/dc2] sign[-d1"d2"Bac2n(8a2/ac2)] < 0.

sign[dkB/dc2] = sign[ (dl"d2"oc2B(n+nl)+Bc12d2") (a2/a82) ] > 0.

sign[d(k+l)B/dc2] = sign[ (aa2/ac2) (di"d2"ac2Bni+Bc2d2") ] > 0.

sign[dB/drl] = sign[ (8a2/ar) (-d1"d2"cBa2n)] > 0.

sign[dkB/drl] sign[ (aa2/ar )(d1"d2'Bac2(n+n)+Bc12d2")] < 0.

sign[d(k+l)B/drl] sign[ (8a2/ar) (d1"d2"Bac2n1+Bc12d2")] < 0. Q.E.D.



The Effect of The Gap of Target Costs Ic-c_

Result 4.12. The production (procurement) manager's sharing ratio

increases (decreases) with an increase of the gap of target costs. The

overall effect decreases the sharing pool.









Proof. Result 4.12 follows by examining the following signs.
A A A A
sign[dB/dl c-c ] = sign[ (a2/a8 h-J ) (-d1"d2"Ba2n) > 0.

sign[dkB/d ch-cl] = sign[(8a2/a ch-c1 )(d"d2"Ban+n)+Bc1d2") ]< 0.

sign[d(k+l)B/d ch-c ] = sign[ (aa2/a I h-cJ ) (d1"d2"Ba cn1+Bc12d2")]< 0.

Q.E.D.

Remark. When the gap between cost targets is getting larger (i.e.
A A
smaller Ch or larger cl) to reach, the principal needs to provide more

incentive for the production manager.



The Effect of The Difference of Attainable Quality Levels and The

Marginal Probability of Getting High Quality Materials (h-l) and P,')

Result 4.13. The procurement (production) manager's sharing ratio

increases (decreases) with an increase of either the difference of

attainable materials quality or the marginal probability of getting the

getting high quality materials. The sharing pool, however, increases

with both changes.

Proof. Result 4.13 follows by examining the following signs.

sign[dB/d(h-l)] = sign[ (aa2/8(h-l))(-d1"d2"Bae2n)] < 0.

sign[dkB/d(h-l)] = sign[(8a2/8(h-l)) (dl"d2"Ba2 (n+n)+Bc12d2")] > 0.

sign[d(k+l)B/d(h-l) sign[ (8a2/a(h-l)) (d1"d2"Ba2n+Bc12d2") ] > 0.

sign[dB/dPh'] = sign[-2d1"Ba,2nxy] < 0.

sign[dkB/dPh'] = sign[2xyB(c12+dl"ac2(n+nl)] > 0.

sign[d(k+l)B/dPh'] sign[2xyB(c12+d1"2nl) ] > 0. Q.E.D.

Remark. When the attainable materials quality differs significantly,

the procurement manager is encouraged to expend personal effort to get

higher quality materials.









The Effect of The Degree of Absolute Effort Aversion (-d1"/d1' and

-d,/dzl

Result 4.14. Each manager's sharing ratio decreases with an

increase in his own degree of effort aversion, and increases when the

other person becomes more effort averse. The sharing pool decreases

when the managers are more effort averse.

Proof. Let v,--dl"/dl' and vz--d2"/d2'. Result 4.14 follows by examining

the following signs.

sign[dB/dvl] sign[(8al/a8v)Bd" (d2"ac2(n+n2)+(xP' )2] < 0.

sign[dkB/dvl] sign[-( al/avl)Bd1"d2"ac2n] > 0.

sign[d(k+l)B/dvl] sign[(8aa/avl)Bd" (d2"acZn2+(Ph' )2] < 0.

sign[dB/dv2] = sign[-(aa2/av2)Bd1"d2"Oc2n] > 0.

sign[dkB/dv2] sign[ (aa2/a2) (Bdl"d2"c2(n+n1)+Bc12d2")] < 0.

sign[d(k+l)B/dv2] sign[ (aa/av2) (Bd1"d2 "oc2n1+Bc12d2")] < 0.

Q.E.D.



Summary



In this chapter, a hierarchical agency model was proposed to

discuss the interaction of procurement and production departments under

linear profit-sharing and target-setting compensation plans. The

production manager is expected to expend effort to reduce production

costs. The procurement manager is expected to expend effort to increase

the chance of getting high-quality materials which tends to reduce

production costs. The principal would like to see the reduction of









production costs, provided that the higher price of better-quality

materials does not offset the production-cost reduction.

Several interesting results have been discussed. Managers'

compensations are tied to their positions when the principal lacks a

costless monitoring mechanism, which explains a common practice in the

real world. Comparing the second-best case with the first-best case, the

principal has to provide a greater sharing pool in order to induce the

managers' efforts, and each manager, in fact, works less hard. Agency

costs, defined as the deviation from Pareto-optimal risk sharing from

the principal's point of view, can be considered as the expected value

of getting a perfect monitoring mechanism. In both first-best and

second-best cases, profit-sharing compensation is always preferred to

fixed salary from the principal's viewpoint. This result can explain

why profit-sharing plans are getting more and more popular.

Detailed comparative statics were also discussed in this chapter.

Results are quite robust and satisfy intuitive conjecture. When the

attainable materials quality can be further improved or the marginal

chance of getting it is higher, the principal is willing to offer more

incentives. The principal has to bear more risk once the managers are

more effort averse or risk averse. The principal can also adjust the

target costs to trigger proper effort levels. Diversification

opportunities are always welcomed by the principal since the enhanced

diversification forces the managers to bear a higher share of the

volatile actual cost (i.e. a more negative Cov(R,C) corresponds to

higher sharing ratios). The principal should always improve the







86
technology and methodology of production processes in order to reduce

the processes' variances. Extensions and further research of this model

are discussed in chapter 6.













CHAPTER 5
DEGREE OF SUPERVISION AND MORAL HAZARD IN HIERARCHICAL CONTROL


The study of hierarchical organizations was first developed by

Simon (1957), Simon and Ijiri (1977) and Lydall (1968). They attempt to

explain the observed skewness of the upper tail of the wages

distribution. Simon and Lydall both assume that (1) the employees of a

firm supervise employees at the immediate lower level, (2) the lowest-

level employees are the only production workers, (3) the span of

control, which is defined as the supervisor/supervisee ratio, is

constant across layers, and (4) the wage at any layer is a constant

multiple of the wage at the layer below. Studies by others use those

assumptions, in part. Simon and Lydall show that these assumptions

generate a Pareto wage distribution, but they do not give an economic

explanation of internal wage differentials and labor utilization

structure.

An endogenous explanation of the internal wage differentials is

proposed by Calvo and Wellisz (1979). Employees differ in quality

because of native ability, education, diligence, or other factors.

Calvo and Wellisz show that (1) the employees' quality and wage increase

with their hierarchical position, and (2) the imposition of a minimum

wage for production workers increases the quality and quantity of

production workers, and reduces the wage, quality and number of the

supervisors. Calvo and Wellisz's results help to explain the









hierarchical differentials in worker quality, wages and degree of

supervision. Production workers, however, are assumed to work at full

efficiency at all levels. That is, there is no interest conflict

between the supervisor and subordinates.

Stiglitz (1975) analyzed the role of incentives, risk, and

information in determining the structure of a hierarchy. In Stiglitz'

study, the wage is not a constant. The choice of a payment system

depends on attitudes toward risk, effort-supply elasticities, the

sources and magnitude of the uncertainties, and the nature of the

supervision used in the employment relationship. Stiglitz concluded

that it pays either not to monitor at all, or to monitor at a finite

level. The costs of supervision are not considered in his models.

From an empirical study, Rosen (1982) indicates that firm size and

earnings distributions follow similar functional forms and exhibit

similar general appearances. Williamson (1967) investigates the size of

firm theoretically using similar assumptions as in Simon and Lydall, and

concludes that limited firm size is due to a loss of control factor

which is given exogenously. William's conclusions have been challenged

by Mirrlees (1976) and Calvo (1977), who argue that the height of the

hierarchical pyramid does not set a limit to firm size unless ad hoc

considerations are introduced. Calvo and Wellisz (1978) justify that

hierarchical loss of control depends crucially on the nature of the

supervision process. If the employees are aware of the times at which

they are not being watched, loss of control may impose such a limit on

firm size.









These previous studies consider labor to be the only factor of

production, and assume that the hierarchy layers are established

exogenously. But, a recent survey conducted by the Institute of

Industrial Engineers indicates that 97 percent of the respondents

believe that a closer relationship between employees and managers is an

important factor for achieving increases in productivity.

Significantly, 88 percent strongly agree that lack of communication and

cooperation between departments reduces productivity (OR/MS Today,

1990). Supervisory input might therefore also be considered to be a

factor of production. Supervision is more than just monitoring; it also

serves as a communication tool and has certain direct effects on

production.

Stiglitz (1975) mentions various reasons for using supervisors:

supervisors as monitors of input, supervisors as monitors of ability,

supervisors as monitors of output, and supervisors acting as a

substitute for a payment system. It is the role of supervisors as

monitors of labor input that is elaborated in this chapter. Supervision

of the production process is the first level of management, involving

direct interactions and communication with production workers. Thurley

and Hamblin (1962) define the production supervisors as managers who

spent over ten percent of their time on the shopfloor, and who were

concerned with shopfloor control within different production systems.

The presence of a supervisory system also implies that the monitoring of

employees' efforts is less costly than the direct measurement of their

marginal product. That is, although the final outputs are observable by









the principal, it is hardly possible to differentiate production input

and supervisory input.

This chapter, unlike previous studies, focuses on the necessity of

production supervision and on the payoff of establishing an expanded

hierarchical structure. The employees have discretion to choose their

effort levels, and the supervisory efforts have direct effects on

production. There are five models discussed and compared in this

chapter. It is shown that, in most cases, the principal that is

involved in both production and other economic activities would like to

pay for supervisors to directly manage the production process. The

first model considers a two-layered hierarchy where the principal

doubles as a supervisor and the workers expend full effort. The second

model considers the same two-layered hierarchy except that the workers'

moral hazard problem exists. The third model considers a three-layered

structure where the principal mandates managers to supervise production

and the moral hazard problem does not exist in the organization. The

fourth model considers the same three-layered hierarchy except that a

moral hazard problem exists among the workers. Finally, the fifth model

considers the three-layered structure with the moral hazard problem at

both the production and supervisory levels. In what follows, these

models and their optimal solutions are discussed. Then comparisons of

these models are analyzed. This chapter concludes with a summary of the

findings.









The Models



Consider the following situation. The single principal (owner) of

an organization hires mi homogeneous production workers. The principal

can supervise the workers himself, or he can hire m2 homogeneous

managers to supervise the production line and then he, in turn, monitors

the managers. Output q is a function of effective labor and supervisory

inputs. I further assume the production function to be Cobb-Douglas

q k(mia)z'", (5.1)

where k is a scale parameter that measures the average productivity, m1a

stands for the effective labor force and aol, e and q are output

elasticities with 0
this production function, the workers cannot produce profitable products

without any input from their superiors. That is, the workers lack the

knowledge of managing and directing production themselves. If the

principal himself supervises the workers, then z is the principal's

effort, y, expended on production, i.e. z=- and 7y1. If the principal

hires m2 managers to supervise mi workers, then the supervisor inputs

z=m2l, where f is each manager's supervisory effort and p51. These

homogeneous m2 managers cannot jointly monitor the workers, but must

spend time with each worker to obtain useful outputs. Since the

supervisory input has its maximum limit, it is assumed that the

production function exhibits decreasing returns to scale, i.e. e+q
(Varian, 1984, p. 20). Therefore, by checking the Hessian matrix, the

production function is strictly concave for all positive values of m1a

and z (Henderson and Quandt, 1980, p. 72).







92

Monitoring, to be effective, requires the imposition of penalties

for substandard work. Let P1 be the probability of a worker's

performance being checked by his supervisor, which is called "the degree

of supervision" by Calvo and Wellisz (1979). If a woker is not checked,

he is presumed to have worked at maximum intensity (i.e. a-l) and he is

paid a wage S. If a worker's performance is checked, his effort level,

a, is revealed and he is paid aS, involving a penalty equal to (l-a)S.

Employees are unaware of the times at which they are being checked.

This payment scheme represents a mixture of a flat-rate scheme

(applicable to employee whose performance is not checked), and a piece-

rate scheme (applicable to employees whose work is checked). To gain in

clarity and simplicity mathematically, following Calvo and Wellisz

(1978), it is assumed that the penalties do not accrue to the principal

in the form of cost savings. The penalties, if any, are retained for

employee-related events such as social activities.

The worker's Von Neumann-Morgenstern utility function is defined

over his payment and his personal expenditure on effort; more

concretely, it is assumed that

U(a;S) = [S(1-Pi)+aSPI] (d)a)61

S[l-(1-a)Pi] (d)a)61 >0, (5.2)

where d, represents the disutility of effort. The worker's objective is

to choose his effort level to maximize his utility function (5.2). It

is assumed that the employee is risk neutral and effort averse. Thus

61>1. Since U(a;S) is concave in a, the first-order condition is both









necessary and sufficient for optimality. Hence the worker's optimal

effort level a* is equal to (SPi/61)11'(1-1)(/d1).

Examining the above function, if the wage S is too low, the worker does

not expend full effort even under perfect monitoring (P-l1). To avoid

this case, let S-S1di(6") (5.3)

The worker expends full effort (0a=l) under perfect monitoring. Thus

the worker's optimal effort level depends crucially on the degree of

supervision alone, i.e. a*-(Pi)1l(1-1). (5.4)

The worker's effort level increases with an increase of the degree of

supervision. From (5.2) and (5.3), U=6ldl(61-l)[l-(l-a)Pi-(dl/61)a61].

Since the minimum utility guaranteed to the worker is zero from (5.2),

it is assumed that 61dd>l. (61 needs to be further restricted to be

greater than two in the later part of this chapter.) This assumption

implies that only a certain type of employees will be attracted to work.

Substituting (5.4) in (5.2), at the optimum the worker's utility

function becomes

U*(S,C) S[1-Pi+(l-d1/61)P/1 61(1'-]

Taking the first derivative of U* with respect to P1 yields

au*/a8P = S[-l+((61-di)/(61-1))PI161-1'].

The worker's utility decreases with an increase of the degree of

supervision, given that 61>dl>l.

The degree of supervision is defined as a function of the number

of effective supervisors per supervisee (Calvo and Wellisz, 1979). At

the production workers' level, P1=hl(7/ml)5l if the principal conducts

supervision; P1-h1(pm2/ml)sl if the managers are hired. For simplicity,