Firm Specific Human Capital Investment in an Agency

Relationship

Anthony M. Marino ∗

Marshall School of Business

Department of Finance and Business Economics

University of Southern California

Los Angeles, CA 90089-1422

E-mail: amarino@usc.edu

May 5, 2017

Abstract

This paper considers the twofold problem of compensation contracting and the design of a

human capital investment scheme. Before contracting the principal and the agent can engage

in a joint stochastic production process of exerting effort to raise the agent’s productivity in

the firm. The principal can employ synchronous effort exertion or either actor can assume

a leadership role. We determine which organizational design is best for the principal at the

endogenously optimal compensation contract, depending on how the efforts interact. We also

determine when it is optimal for the principal to subsidize the agent to improve profitability.

JEL Code: L20, L21, L22, and L23

Key Words: Firm Specific Human Capital, Investment Design, Agency

∗Thanks go to my colleagues Odilon Camara, Chad Kendall, John Matsusaka, Joao Ramos, Sandra Rozo, andYanhui Wu. The author also thanks the Marshall School of Business for generous research support.

1

1. Introduction

Organizations in general and firms in particular routinely invest in the firm specific human capital of

affiliated agents. In addition to providing a monetary compensation contract, it is fairly common

for principals to provide training or other support resources to the agent with the purpose of

increasing the future productivity of the agent in the firm. In many cases, this is a team or joint

production process between the agent and the firm. That is, the firm must exert effort to train and

the agent must exert effort to learn. Apart from training, if there is an agency problem between the

principal and the agent, then the firm must devise an optimal compensation contract to alleviate

that problem, and, in addition, it must design an optimal human capital investment scheme. The

present paper studies this twofold problem of optimal organizational design and compensation. Our

focus is on the organizational design of the team training process involving the principal and the

agent.

The importance of corporate training programs can be seen in recent expenditure numbers as

well as through educational initiatives by prominent firms. In the United States, corporate training

grew to over 15% in 2013 to over 70 billion in terms of total dollars spent. Worldwide, in this

same period over 130 billion dollars were allocated to corporate training. Key examples abound.

Companies such as General Electric, Motorola and Philips have increased their budgets on virtual

learning for their employees.1 Apple maintains its Genius training manual which leaves nothing to

the imagination for its trainees, in that it specifies how a customer is to be greeted, probed, provided

a solution, listened to and given a farewell on exit.2 Apple University also provides classes that are

tailored to employees’ positions and backgrounds.3 GoogleEDU is offering more classes than ever to

its employees, with about a third of its over 33000 employees taking their in house training. Google

1See Bersin (2014).2See Gallo (2012).3See Chen (2014)

2

offers tailored classes based on an employee’s work area (e.g., engineering or sales) and career

stage (e.g., junior developer or senior manager).4 Moreover, Google employees are encouraged

and coordinated by management to take the initiative and offer classes to their colleagues, in

their Googler-to-Googler model. Hewlett Packard employs on the job learning, relationship based

coaching and formal learning through face to face interaction and web based instruction.5

Note that we see a variety of different human capital investment schemes being utilized and a

great deal of resources being devoted by firms to improve the productivities of their employees. In

many of the above real word examples, the firm (the teacher) and the employee must each exert

effort in order to train the employee, and one of the two parties may take the initiative in the

process. This paper will focus on the question of whether it is more profitable to have the firm

or the employee take the initiative in a training process. Further, under what conditions should

neither take the initiative so that simultaneous interaction is best?

We begin our analysis by constructing a hidden action agency problem, where the agent can take

on a high or a low level of productivity (human capital) in producing cash flow. We characterize the

optimal agency contract conditional on that level of productivity. Next, we model a pre-contract

human capital investment technology wherein the principal and the agent jointly invest effort or

resources in trying to improve the agent’s future productivity in the firm. In many real world

instances, the principal devotes resources, including money, time or other inputs, to this process,

but we broadly term this "effort" by the principal. Both the firm and the agent benefit, from the

agent becoming more productive, through an expected increase in cash flow during the endogenous

contracting and cash flow production stage. The human capital investment process is modeled as

a stochastic joint production process, between the principal and the agent, which increases the

4See Walker (2012).

5This information is available on the HP web page, www.hp.com.

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probability that the agent will be of the high productivity type.

The principal’s and the agent’s efforts can be strategic complements or strategic substitutes in

this investment process. In the strategic complement case, the firm’s investment effort increases the

marginal productivity of the agent’s effort. While this might be thought of as the most prevalent

example, it is also possible that the firm’s investment effort decreases the agent’s marginal produc-

tivity of effort. This is the case of strategic substitutability between the principal’s and the agent’s

investment efforts. In each of these cases, we assume that the two efforts have positive marginal

productivities in terms of the total productivity of learning. The nature of the task that the agent

is hired to implement for the organization and the nature of the learning associated with that task

will determine exogenously the substitute or complement relationship between the efforts of the

principal and the agent. Given this relationship, the principal chooses the design of the training

process.

The principal is the organizational designer in that he/she chooses the basics of how the two

actors will interact in the human capital investment game. We consider three modes of interaction.

One mode is that each actor adopts a Nash strategy with respect to his or her investment effort.

This is a learning process involving synchronous effort exertion by the principal and the agent. The

other two modes involve either the agent or the principal taking a leadership role in the investment

process, by precommitting to a particular effort and allowing the other actor to optimally respond.

Our results indicate that if the actors’ efforts are complements, then it is best for the firm to

have the agent or to have itself be the leader in the investment game.6 If the actors’ efforts are

substitutes, then it is best for the firm to be the leader in the investment game. In each of these

substitutes and complements cases, the Nash organizational form is always dominated, in terms of

what is best for the firm, by one of the leadership solutions. That the firm can always benefit if a

6We provide necessary and sufficient conditions under which it is optimal for the agent to lead or for the firm to

lead in the complements case.

4

leadership role is taken by either actor is an interesting result, especially when the agent leads. We

present a quadratic effort cost example to illustrate these results and to present the comparative

statics of the three equilibria.

In a final section of the paper, the question of whether the firm should provide subsidies to the

agent is studied. We examine the two dominant organizational designs where either the principal

leads or where the agent leads. In each case, we ask whether the principal can improve profit

by giving the agent an optimal subsidy which comes from the principal’s expected surplus during

the later cash flow stage. The firm’s subsidy choice variable takes the form of a fraction of the

principal’s expected incremental cash flow resulting from the principal’s investment in the agent’s

human capital.7 We characterize necessary conditions under which a subsidy from the firm to the

agent is optimal.

There are key examples of the three modes of interaction considered here. Nash interaction

corresponds to a teaching-learning process in which agents collaborate simultaneously in order to

improve the productivity of the agent in their organization. An example, would be in person or

hands on tutorial type training between the principal and the agent. Other examples would be

the interaction between a principal and an apprentice, or a seminar setting wherein an agent is

being trained by a principal. Each of these examples can be modeled as a Nash type game in

efforts, where the principal exerts teaching effort and the agent simultaneously exerts "learning"

effort to assimilate the material being presented. Generally in a (Nash) classroom setting, the

principal prepares a presentation, the agent studies teaching materials, and they meet to present

and assimilate.

An agent, taking the leadership role, would precommit to an investment effort with knowledge

of how the principal will follow up with a corresponding optimal investment effort in response.

7Educational investment efforts are observable but non-contractible due to the inability to verify such efforts.

Thus, we study a surplus sharing scheme which is observable and verifiable.

5

An example would be the case where a sales agent (or a junior executive) goes through a dry run

of a sales pitch (or corporate presentation) in front of a principal and the principal responds by

making critical comments. A second set of important examples would be cases where the agent

designs a professional development plan which is intended to enhance the agent’s human capital

and the principal responds with resources which would accommodate that plan. Resources could

include paid time off, flexible hours, and partial to full subsidization of fees associated with the

proposed regime. For example, my university provides this type of arrangement for clinical faculty

seeking professional development. A proposal is submitted by the faculty member and it is accepted

or rejected. If it is accepted, resources are provided including travel expenses, fees and possibly

course relief. Also, tenured faculty can apply for summer research funding by proposing a research

agenda which is subjected to scrutiny by a committee. If the proposal is accepted, resources are

provided. Many corporations have similar arrangements for professional development. Finally, in

many corporations, an employee can seek, on a voluntary basis, outside or inside course work and

this may be partially or totally funded. The employee moves first by choosing the school/course

work or the inside class and obtaining admission. The outside course work is sometimes provided by

an outside contractor providing custom employee education for that firm. The corporation responds

by providing resources which might include partial or full educational expenses and possibly time

off or flexible hours. All of the above examples could fall into the agent leadership mode.

A primary example of the firm taking the leadership role is in the area of online education.

The firm designs the learning module and precommits their effort or resources by putting the

instructional program online with the idea that the employee will optimally respond to that module.

The employee exerts effort to learn, given the program. Some mandatory training classes required

by organizations may also fit into this category of instruction, specifically where the mode of

instruction is straight lecture/indoctrination with limited teacher-student interaction and little

6

prior preparation by the employee. New employee training and orientation sessions are examples of

this case. My university now has program for new faculty where teaching issues specific to our school

and methods of instruction (e.g., instruction on the case method of teaching) are presented. New

employees do not have access to the materials before class. Corporations have similar programs

and sometimes use manuals to disseminate the orientation material. Note that as part of the

organizational design, any of the examples of the Nash mode of interaction given above could be

converted to one of the leadership modes, if the principal deems that conversion appropriate.

The organization of the paper is summarized as in the following description. Section 2 dis-

cusses related literature. Section 3 presents the basic hidden action problem and its solution.

Section 4 formulates the pre-contracting human capital investment process, characterizes the Nash

solution and characterizes the two leadership equilibria. Section 5 compares the principal’s profit

and the agent’s utility among the three equilibria under the assumptions that the two efforts are

complements or substitutes. Section 6 examines a quadratic effort cost example and conducts the

comparative statics of each of the equilibria of Section 4. Section 7 studies the issue of whether the

principal should cross subsidize the agent. Section 8 concludes.

2. Related Literature

The classic firm specific human capital papers of Becker (1962) and Hashimoto (1981) argue that

the cost of and the return to such investment will be shared by the employee and the firm. Sharing is

incentivized by the fact that it reduces the probability that either party will terminate employment

and impose a loss on the other actor. The classic analysis is conducted in the context of perfect

competition. Our paper casts the problem in a standard principal-agent setting where there is a

single employee and the firm or principal is a monopsonist. In this setting, each of the principal and

the agent will bear a cost of investment through effort exertion and each will share in the surplus

7

created by that investment as in the earlier literature.8

This paper is most related to the literature on the active principal in agency problems. By

active, we mean that the principal controls possible features of the agency relationship in addition

to the monetary compensation contract in the context of hidden action and/or hidden information

agency problems. Garcia (2014) and Bernardo et al. (2001) consider the allocation of capital

and the creation of a compensation contract by a principal who faces both hidden information

and hidden action problems. Hirchleifer and Suh (1992) and Sung (1995) study the case where

a principal contracts with an agent to control project choice as well as the exertion of effort by

the agent. Holmstrom and Milgrom (1991) study task assignment and optimal compensation by

the principal facing a hidden action agency problem with multiple task efforts to be exerted by

an agent. Feltham and Xie (1994) consider the design of the agent’s performance measure and

the optimal contract by a principal in the context of a hidden action problem. Harris and Raviv

(1996) study capital allocation by the principal in the context of a hidden information contracting

problem. Besanko and Sibley (1991) present a model with transfer pricing and compensation in

the presence of hidden action and information.

Marino (2011) examines sequential versus simultaneous transfer of knowledge between agents

working for a principal who incentivizes such transfer. The transfer of knowledge involves a team

production process of teaching effort and learning effort exerted by agents, as opposed to the present

case where we study knowledge transfer and assimilation between a principal and an agent. There

is no costly cash flow effort in this model and it features only teaching and learning efforts between

agents. Here, there is a team of two agents with all of the action being driven by states in which

one agent is informed of a complex procedure and the other is not. In these states, the informed

agent can become the teacher and explain to the uninformed agent, the learner, the procedure.

8The agent shares because, in equilibrium, the participation constraint is non-binding and limited liability holds.

8

Under the principal’s optimal contract which is based only on cash flow, neither agent gets a

payoff from teaching and learning efforts, unless knowledge is actually transferred to the learner

from the informed agent, in which case cash flow results. The informed agent or teacher can be

a Stackelberg leader in this knowledge game or agents can exert efforts as Nash players. In this

very different model, the principal’s equilibrium payoff is greater when the teacher leads if efforts

are complements and Nash strategies are better for the principal if efforts are substitutes. These

results differ from ours below, but in the present paper, the learner can lead and, in equilibrium,

cash flow can be generated regardless of whether or not the firm/teacher is successful in improving

the agent’s productivity.

The paper by Kvaloy and Schottner (2015) places costly motivation into a hidden action agency

problem and studies the interaction between the monetary contract and the principal’s optimal

motivation selection. The papers by Marino and Zabojnik (2008), Marino and Ozbas (2014) and

Marino (2015) consider cases where the principal controls, along with the monetary contract, an-

other non-monetary instrument. In Marino and Zabojnik (2008), this instrument is a work related

perk, in Marino and Ozbas (2014) it is the disclosure of information revealing the status of em-

ployees, and in Marino (2015) it is health and safety in the workplace. The above active principal

literature discusses many control variables in addition to the monetary contract, but the design of

training programs studied in this paper has not been analyzed.

This paper is also related to a game theoretic literature which studies first versus second mover

advantage in market games. Examples are Gal-Or (1985), Dowrick (1986), and Amir and Stepanova

(2006). Our paper differs because the role of each player is set by the principal and, of course, the

nature of benefits and costs are different in the agency setting from those in the market setting.

We discuss these points in Section 4.

Finally there is a related public finance literature which studies leadership by example wherein

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decisions are sequential and the choice made by the leader affects the actions of followers. See

Jack and Radcalde (2015) for a concise review of the theoretical literature and a Bolivian field

experiment examining the effects of leadership by example on voluntary public good provision.

While this literature discusses leadership and its effects on followers through information signaling,

social status and free riding, the context is public finance as opposed to a firm with agency problems

contracting with an agent. Finally, there is a public finance literature exemplified by Pavoni and

Violante (2007) which investigates optimal training and subsidization of unemployed workers as

a part of welfare-to-work programs. Our results in Section 7, which consider subsidization of the

agent by the principal, then relate to this literature, but again the setting is within a firm with

an agency problem and the motive is profit maximization as opposed to welfare maximization by

a social planner. We are interested in the necessary conditions for a profit maximizing firm to be

willing to share some of its surplus from productivity enhancement with the agent. The former

social planning literature is concerned with the sequencing over time and the duration of optimal

unemployment insurance, job search monitoring and social assistance.

3. The Agency Problem

Consider a simple principal-agent setting where an agent exerts effort which is unobservable to the

principal. Both the agent and the principal are risk neutral. The agent’s utility is

= − ()

where is the monetary compensation paid by the principal and is effort, with ∈ [0 1]. The

cost of effort satisfies

A.1 (0) = 0(0) = 0 and 0 00 0 and 000 ≥ 0 for 0

This assumption makes both the marginal and the total cost functions convex in effort. We assume

10

that the outside option of the agent is zero.

The principal’s technology for producing cash flow is modeled as a two outcome process, with

cash flow given by ̃ ∈ {0 } 0 The probability of the high cash flow is where is a

measure of the agent’s firm specific human capital. Productivity or human capital is either high or

low, ∈ { } ∈ (0 1) with the () = We will later model an investment

process carried out in an earlier stage game wherein the firm and the agent can invest their efforts

in raising the probability The firm’s expected cash flow is then given by In the contracting

stage, the principal observes the agent’s realized human capital.

The principal sets a contract of the form = + ̃ where is a fixed salary and is an

incentive payment. Given the two outcome process, this formulation is fully general. We assume

that the agent is subject to limited liability in the sense that

≥ 0 (LL)

The principal will choose and so as to maximize expected profit subject to (LL) and the agent’s

participation constraint,

+ − () ≥ 0 (P)

The agent’s problem is to choose as a solution to max{}

+ − () so that

= 0() and = 0−1()

Define () ≡ 0−1() and note that (0) = 0 0 0 and 00 ≤ 09 The second order condition

9The function 0 has 0 0 and 202 ≥ 0 Its inverse is so that satisfies 0 0 and 00 ≤ 0 Because0(0) = 0 we have 0−1(0) = (0) = 0

11

for this problem is met and we have

= 0 0 and = 0 0

Given this set up, we can prove10

Lemma 1. The agent’s participation constraint is non-binding and = 0 at an optimal contract

with 0.

The principal’s problem is now reduced to max{}

(1− )() The first order condition is

−() + (1− )0() = 0 (1)

The second order condition to the principal’s problem, −20 + (1 − )2200 0 is satisfied

under our assumptions. We have that

=−0 + (1− )0 + (1− )200

20 − (1− )2200

Because the second order condition is met, the sign of this expression is that of the numerator. While

(1−)200 ≤ 0 we can show that at a point where (1) is met, −0+(1−)0 ≥ 0 as 00 ≤ 011

It follows that has an indeterminate sign, without further information on the form of the

effort cost function. For example, if is of the form 1 then = 1 and = 0

Alternatively, if takes the form ()− (+ 1) then 0 with = 15 and ∈ (03 04)

The result −0 + (1− )0 ≥ 0 also implies that (1− 2)0 ≥ 0 so that ≤ 12 That, is

the incentive payment by the firm is always less than one half of the firm’s cash flow.

10All proofs are provided in the Appendix.

11We have that (1− )0 = so that the sign of this expression is that of −0+ Because 0 0 00 ≤ 0and (0) = 0 we have −0 + ≥ 0 as 00 ≤ 0

12

A greater human capital level does not necessarily lead to a greater incentive share to the agent.

However, as we shall see below, a greater human capital level does lead to a greater expected cash

payoff to both the agent and to the principal, at an endogenously optimal contract. It is this effect

that is necessary for the firm or the agent to have the incentive to invest in such capital.

4. Investment in Human Capital

Let () define the solution to the principal’s problem (1), for agent of type = Define

the firm’s expected profit in the human capital state as ≡ (1 − ())(()) and,

likewise for the agent, define his utility in the human capital state as ≡ ()(())−

((())) Given an optimal contract, the firm’s ex ante or pre-contracting expectation of profit

across different human capital types is

[(1− ())(())] = + (1− )

Similarly, the agent’s expectation of utility, given an optimal contract, is

[()(())− ((()))] = + (1− )

In what follows, we will make use of the following incremental constructs. Let

≡ − and ≡ −

represent the incremental profit and utility, respectively, when the actor attains the high level of

productivity as opposed to the low level, in the context of an optimal contract.

The firm and the agent can engage in a pre-contracting investment stage. In this stage, the two

13

actors can jointly exert costly effort or resources as a team to increase the probability that the

agent would have the high human capital level. Because this is a team or joint production learning

process, we assume that efforts taken by each actor are observable, but not verifiable. Thus,

contracting directly on effort is not possible. Likewise, we assume that although the principal

observes the agent’s human capital during the later contracting stage, contracting on realized firm

specific human capital is not possible again due to the inability to verify this by third parties. Each

actor exerts effort to help the agent become more productive in the organization, with the principal

deciding the architecture or the organizational structure of the human capital investment process.

The variable = denotes the effort of the agent and the principal, respectively, in enhancing

the agent’s human capital. The technology of human capital investment is given by

A.2 = + + where ∈ [0 1] ∈ [0 1] and ∈ [−1+1]− {0} =

We assume a sufficient condition to guarantee that the marginal productivities of are each

positive, in the case where 0

A.3 If 0 + 0 for =

The cost of is given by () = These cost functions satisfy

A.4 (0) = 0(0) = 0 0()

00 () 0 and 000 () ≥ 0 for 0 =

Under A.2 and A.3, each effort has a positive marginal productivity in enhancing the employee’s

total productivity in a stochastic manner, so that efforts are productive. The interaction term

can be positive or negative, representing the cases where the two efforts are strategic complements

or substitutes, under the terminology of Bulow et al. (1985). In the strategic complement case,

the firm’s investment effort increases the marginal productivity of the agent’s effort. While this

might be thought of as the most prevalent example, it is also possible that the firm’s investment

effort decreases the agent’s marginal productivity. In this case, the firm’s effort, in a sense, crowds

out the marginal productivity of the agent in learning. Too many instructions from the principal

14

could, for example confuse or stifle the agent in learning. Another unit of effort on the part of the

agent induces more learning if the firm’s effort is kept lower, as the agent is better off on the margin

working on his or her own. An example where the principal’s effort is strategically complementary

with that of the agent is the case where the principal is teaching the agent how to physically operate

a piece of machinery. More effort by the principal in demonstrating operations could enhance the

marginal productivity of the agent. A strategic substitute relationship could exist where the agent

is learning to use a computer software package, and, by its nature, the user must use it to learn

it or learn by "doing". In what follows, we drop the strategic terminology and simply refer to the

complements case as the case where 0 and the substitutes case as the case where 0

We build concavity into the optimization problems through the effort cost functions. In A.4, we

assume that both marginal and total costs of effort are convex with zero total and marginal costs

at the origin. In order to focus on the issue of who if either of the actors should take the initiative,

we have assumed that efforts are either global substitutes or complements in the technology of

teaching/learning and that this feature is determined by the nature of the knowledge that the

agent is to assimilate. Moreover, we assume that there are no changes in the basic technology with

changes in the principal’s design.12

The time line for decision making and information revelation is as in the following. In the

first stage, at time 0, the principal precommits to the mode of interaction in the human capital

investment process, and the firm and the agent exert investment efforts. Period 0 can be thought

of as an internship/training period which precedes cash flow production in period one. The agent

could be paid a nominal fixed payment during this period which is not modeled here. In the second

12Given the many moving parts in the present model, we abstract from this question. However, there could be

trade-offs between these technologies. One scenario is as in the following. The simultaneous mode could lead to a

scaled up due to ability of the teacher to locally adapt to the student, whereas the sequential mode could reduce

effort cost because the follower can asynchronously access the leader’s pre-committed template when the follower’s

effort cost is lowest. We leave this complication to future research.

15

stage, at time 1, the agent realizes his/her human capital level and this is public information to the

firm and agent. With this information in hand, the firm sets the contract for cash flow production

and the agent exerts cash flow effort. At time 2, cash flow results.

Given that the firm controls the mode of interaction with the agent, an interesting question

arises. If the firm has a choice among these plans, which would it choose as a part of its organiza-

tional design?13 That is, under what parameter values might one type of investment program lead

to more profit than another?

4.1. Nash Solution

First, consider the Nash type human capital investment process. The firm would solve max{}

+

− ( ) with first order condition

( + ) − 0 ( ) = 0 (2)

The second order condition is met by 00 0 The firm’s reaction function is

= 0−1 (( + ) ) ≡ () (3)

The agent solves max{}

+ − () with first order condition

( + ) − 0() = 0 (4)

13We assume that the firm controls the mode and chooses so as to maximize profit, because the firm is the

organizational designer. It might be true that the employee gains more from an alternative mode than that chosen by

the principal. However, in all of the equilibria, the employee gains more than he or she would under non-participation

in the program.

16

The second order condition is again met under 00 0 The agent’s reaction function is

= 0−1 (( + )) ≡ ( ) (5)

We wish now to present conditions under which a Nash equilibrium exists and is unique and

stable. In our set up, the only reward for either actor in making a human capital investment is

the expected additional revenue in the firm’s cash flow process generated by the agent possessing

the greater human capital level. This is given by for the firm and for the agent. Because

0−1 (0) = 0, a necessary condition for an interior solution in to the first order conditions (2) and

(4), is that 0 for each case.14 Given that the contract is endogenous, the result that 0 is

not obvious, but we show that, in fact, this is true under our assumptions. We have

Lemma 2. Let A.1-A.4 hold. Then 0 =

Thus, given the optimal contract, it is always desirable, for both the principal and the agent,

when the agent assumes the higher human capital level. Each actor then has an incentive to invest

in making that state of the world more probable. What is important for this result is that the

incremental value of the higher ability, be positive for each actor in order to incentivize positive

investment in the agent’s firm specific human capital.

The results of Lemma 2 can be used to study how changes in the distribution of possible ability

levels impacts the incremental values , through the optimal contract. Given an endogenously

optimal contract, the results of Lemma 2 imply that

0 and

015

14Note that (0) = 0−1 () whereas 0−1 (0) = 0

15The proof of Lemma 2 establishes these sign conditions.

17

Increases in the base productivity level (implying a decrease in the variance or spread but an increase

in the mean human capital), say due to a better educated population of workers, will decrease the

incremental benefits to both the principal and the agent for human capital investment. Moreover,

increases in the high productivity level (implying an increase in the variance or spread and the

mean), say due to better recruiting screening, will increase the incremental benefits to both the

principal and the agent for human capital investment. How changes in impact investment levels

will in turn depend on the investment equilibrium, the contract, and the complement-substitute

relationship between human capital efforts. We will illustrate these comparative statics in Section

6.

Next, let us examine the curvature properties of the above reaction functions. Computing

derivatives we have

0() = 00 and 00 () = −()2(00 )−3000 (6)

Observation of (6) allows us to state

Lemma 3. Let A.1-A.4 hold. Then () is increasing (decreasing) and concave in if 0

(if 0)

Assuming that a Nash equilibrium exists, for stability of that equilibrium, it suffices that

|−10 ( )| |0( )| for all ∈ (0 1) Using (6), this stability condition is given by

A.5 00(( ))00 ( )− 2 0 for all ∈ (0 1)16

We will term an investment effort feasible if ∈ [0 1] In the following, we define the largest value

taken on by the function −1 ( ) for ( −1 ( )) in the unit square or the feasible region

≡ argmax {−1 ( ) | ( −1 ( )) ∈ [0 1]× [0 1]}

16An equivalent statement of this stability condition is 00()00 ( ())− 2 0 for all ∈ (0 1)

18

For existence of a Nash equilibrium, we can state a set of sufficient conditions in

Proposition 1. Let A.1-A.5 hold. Sufficient conditions for the existence of a unique and stable

Nash equilibrium in the feasible region are as follows for the cases of substitute and complement

investment efforts.

A.6 0 : (0) = 0−1 () ∈ (0 1)

A.7 0 : ( ) −1 ( )

Assumption A.6 insures that the difference between ( (0)) and −1 ( (0)) is positive and

that the difference between ( (1)) and −1 ( (1)) is negative, in the substitute case. Using the

intermediate value theorem and the decreasingness and concavity of () the equilibrium exists

and is unique. Assumption A.7 performs a similar function in the complements case.

4.2. The Agent and Firm Leadership Solutions

If the agent is the leader in the investment process, then the agent will solve max{}

[ ()++

()] + − () with first order condition

( + ()) + ( + )0 ()− 0() = 0 (7)

The second order condition can be written as

2200 − 00 + ( + )

00 0 (8)

From (6) and Lemma 3, we have that 00 ≤ 0 Thus, the second order condition is met if

A.8 00()00 ( ())− 22 0 for all ∈ (0 1)17

Condition A.8 implies the weaker condition A.5 which is sufficient for the stability of the Nash

17This condition can be equivalently written as 00(( ))00 ( )− 22 0 for all ∈ (0 1)

19

solution.

If the firm is in the leadership role, then the firm solves max{}

[ + ( ) + ( ) ] +

− ( ) with first order condition

( + ( )) + ( + )0( )− 0 ( ) = 0 (9)

As in the agent leadership solution, the second order condition

2200 − 00 + ( + )

00 0

is met if A.8 is met. As pointed out previously, A.8 is equivalent to the condition 00(( ))00 ( )−

22 0 for all ∈ (0 1)

Conditions for existence of the agent and firm leadership solutions are given in

Proposition 2. Let A1-A.4 and A.6-A.8 be met. Moreover let

A.9 min {( + 0−1 ( )) +

00(0−1( ))

0 ( + 0−1 ())

+

00(0−1 ())

} 0

If, in addition,

A.10 ( + 0−1 ( ( + ))) + ( + )

00(0−1( (+)))

− 0(1) 0

then the agent leadership solution exists and is unique; and if, in addition,

A.11 ( + 0−1(( + ))) + ( + ) 00(0−1((+)))

− 0 (1) 0

then the firm leadership solution exists and is unique.

Condition A.8 guarantees strict concavity of the agent’s and the principal’s leadership objective

functions. Conditions A.9 guarantee that the limits of (7) and (9) are positive as tend to zero, for

= respectively. Conditions A.10 and A.11 suffice to make the first order derivatives (7) and

(9) negative in sign as tends to 1, for = respectively. Combining these results with strict

20

concavity of the objective functions, we obtain existence and uniqueness through the intermediate

value theorem. If we combine the results of Propositions 1 and 2, taking into account that A.8

implies A.5, we obtain

Corollary to Propositions 1 and 2. Let A.1-A.4 and A.6-A.11 hold. Then there is a unique

and stable Nash equilibrium and unique solutions to the agent and firm leadership problems in the

investment game.

5. Comparisons

We now turn to the organizational design question of how should the firm construct the investment

process so as to obtain the greatest profit? Does this design decision change with a change in

parameters? We think about the answers to these questions by considering the two cases where

is either positive or negative, corresponding to the two investment efforts being either substitutes

or complements. Let ex ante expected profit at the firm leadership, agent leadership, and Nash

solutions be denoted () = respectively. Likewise, let () represent the agent’s

expected utility at each of the solutions, = Investment equilibria will be denoted ()

for = In view of the Corollary to Propositions 1 and 2, we will maintain assumptions

A1-A.4 and A.6-A.11 throughout.

5.1. The Substitute Case

Let be negative, so that the two investment efforts are strategic substitutes. In this case, each

reaction function is downward sloping in the feasible region () ∈ [0 1]× [0 1] with the absolute

value of the slope of −1 ( ) greater than that of ( ) at each ∈ (0 1)

Consider the firm leadership solution. The firm maximizes its profit ( ) = [ + +

] + − ( ) subject to − ( ) = 0 over a selection of and . The iso-profit

21

contours of the firm’s ( ) have slope

|( )= constant =

−( + − 0 ( ) ) +

(10)

in ( ) space. This is zero along the firm’s reaction function, it is positive to the right of the

firm’s reaction function and negative to its left. Further, as one moves to the northwest along

−1 ( ) the firm reaches greater levels of profit because () = ( + ) 0 by A.3,

and () = 0 along −1 ( ) (See Figure 1.). The firm leadership solution occurs where the

constraint function ( ) and the iso-profit contour are tangent

−( + − ( ) )

+ =

00= 0( ) 0 (11)

so that ( ) is at a point where the iso-profit contour has a negative slope, whereas, at the Nash

equilibrium, along ( ) that slope is zero. It then follows that (

) is to the northwest of

( ) (We have ( + − ( ) ) 0) and that the firm leadership solution achieves a

greater profit than at the Nash solution, ( ) (). The latter is true, because the Nash

solution was available to the firm and it was not chosen. The agent’s utility is increasing as we move

to the southeast on the function ( ) because it is true that () = ( + ) 0

and () = 0 along ( ) Thus, we have that ( ) () The Nash solution confers

greater utility on the agent than does the firm leadership solution, while the firm leadership solution

dominates Nash in terms of the firm’s profit.

Next consider the agent leadership solution. The agent maximizes () ≡ [ + +

] + − () subject to − −1 ( ) = 0 through a choice of and . The iso-utility

22

contours of the agent’s utility function have the slope

|() = constant = [

−( + − 0()) +

]−1 (12)

in ( ) space. Such slope is +∞ at each point along the agent’s reaction function and it is

positive above and negative below the agent’s reaction function. As we move to the southeast

along ( ) utility is increasing, as noted above. At the agent leadership solution, the iso-utility

contour is tangent to the constraint function −1 ( ) such that

[−( + − 0())

+ ]−1 = [

00]−1 = −10 ( ) 0 (13)

with ( + − 0()) 0 It then follows that ( ) is to the southeast of (

) Utility

to the agent is greater than that at the Nash solution, because the Nash solution is available to

the agent and not chosen. The firm’s profit is less than at the Nash solution, again because, as

noted above, as we move to the northwest along −1 ( ) profit is increasing and the equilibrium

is to the southeast of the Nash solution. Thus, while () () ( ) we have that

() () Figure 1 illustrates the three solutions with the firm leadership solution denoted

as the Nash solution denoted as , and the agent leadership solution is denoted as .

These results then point out that the firm will choose to be a leader, if the two investment

efforts are strategic substitutes. For the firm, we have

Proposition 3. Let A1-A.4 and A.6-A.11 hold and let 0. We have that ( ) ( )

()

23

5.2. The Complement Case

If the two efforts are strategic complements, then the two reaction functions are upward sloping

with the slope of −1 ( ) everywhere greater than that of ( ) in ( ) space. Along the

firm’s reaction function, the firm’s iso-profit contours have a zero slope as described by (10) with

( + − 0 ( ) ) = 0 To the right of the firm’s reaction function ( + − 0 ( ) ) 0

so that the iso—profit contours are positively sloped. To the left of the firm’s reaction function,

iso-profit contours are negatively sloped, by ( +− 0 ( ) ) 0. From (12), the agent’s iso-

utility contours have a slope of +∞ along the agent’s reaction function, such slope is positive below

( ), by (+ − 0()) 0 and it is negative above ( ) by (+ − 0()) 0

(See Figure 2.).

The firm leadership solution is described by (11) but with a positive sign for the slope, implied

by ( + − 0 ( ) ) 0 This solution lies on ( ) at a point to the northeast of the Nash

solution. For the firm, greater profit is generated than at the Nash solution, because it lies to the

northeast of Nash and because Nash was available to the firm and not chosen. For the agent, utility

is also greater than at Nash, because utility is increasing as one moves to the northeast from Nash

along ( )

The agent leadership solution also lies to the northeast of the Nash solution along −1 ( ) For

the agent, it is preferred to the Nash solution, () () because the Nash solution is not

chosen but yet it is available to the agent. However, it is not clear whether the firm is better off

at the agent leadership solution or at the firm leadership solution. For the firm, there could be an

advantage to being a second mover, but the answer as to whether the firm would want to move

first or second is not obvious.

To analyze this interesting question, we begin by considering the firm leadership equilibrium

which lies on the upward sloping ( ) at a point to the northeast of the Nash solution, where the

24

firm’s iso-profit contour is tangent to ( ). Such iso-profit contour intersects the firm’s reaction

function to the northeast of the Nash solution at a point defined as ≡ ( () ). This point

generates profit equal to that of the firm leadership solution, by definition. As pointed out above,

moving northeast on () from this point will increase the firm’s profit and moving southwest

will decrease the firm’s profit. Thus, the agent’s leadership solution dominates the firm’s leadership

solution in terms of profit if()

| () is positive and conversely if this derivative is negative. It

can be shown that

sign [()

| ()] = sign [

0 ( ())

0 (

) + ( + (

))−

0()

]

Necessary and sufficient conditions for the firm to prefer to follow or lead are then given in

() T ( ) iff0 ( (

))

0 (

) + ( + (

))−

0()

T 0 (14)

Figure 2 illustrates the three solutions and point , with indicating firm leadership, and 0

indicating alternative agent leadership solutions and indicating Nash. The point depicts the

case where the agent leadership dominates for the firm and the alternative point 0 depicts the

case where the firm leadership dominates for the firm.

Condition (14) compares the costs and benefits accruing to the agent leader as a result of another

unit of along the firm’s reaction function. The condition points out that a sufficient condition

for the dominance of agent leadership mode is that the two marginal costs of investment effort be

different and that the marginal cost generated by inducing an increase in the firm’s investment effort

be greater than that of increasing the agent’s investment effort. Alternatively, for dominance of the

agent leadership solution, it is necessary and sufficient that the marginal cost of inducing an increase

in the firm’s investment effort plus the marginal revenue of a unit of the agent’s investment effort, the

25

firm’s net marginal cost of effort, be greater than the marginal cost of the agent’s investment effort.

Basically, greater firm marginal cost of investment effort and greater marginal benefit of the agent’s

investment effort along with low marginal cost of the agent’s investment effort, lead to dominance

of the agent leadership solution. Only if the agent’s marginal cost of investment effort exceeds both

the induced marginal cost of the firm’s investment effort plus the marginal benefit of the agent’s

effort, i.e., the firm’s net marginal cost, is the firm leadership solution optimal in the complement

case. This case is possible where the firm’s marginal cost of investment is very low. However, even

this condition does not suffice for domination of firm leadership, because we show below that if

investment effort costs are quadratic and of the form 2 2 then under our assumptions A1-A.4

and A.6-A.11, the agent leadership solution dominates for all for which these assumptions are

met. If we use the general quadratic specification = 2 then =

(+ )

(2−2 ) =

(+(5) )

(2−2 ) Under our assumptions, this suffices to guarantee that (14) is met with positivity

and () ( ) The firm leadership solution can dominate when at least one of the marginal

investment effort costs are strictly convex (000 0) and when the marginal cost of inducing an

increase in the firm’s investment effort plus the marginal revenue of a unit of the agent’s investment

effort is less than the the marginal cost of the agent’s investment effort. A numerical example

illustrating this case is where = exp( )−(1+ ) = (19)(exp()−(1+)) = = 05 and

= 11 We can show that (14) is met with , ( ) = (079 068) (

) = (04335 04296)

and ( ) = 069 () = 047

For the complement case, we conclude that ( ) () () and that the ranking of

() and ( ) depends on (14). We summarize as

Proposition 4. Let A1-A.4 and A.6-A.11 hold and let 0. We have that ( ) ()

() and () T ( ) iff0( (

))

0 (

) + ( + (

))− 0()

T 0

Our comparisons point out that as long as the principal’s and the agent’s efforts interact in

26

the human capital investment process, the firm always benefits by designing a process which either

gives it or the agent the leadership role. The technology of investment is such that each effort

improves the total productivity of the other actor’s effort, but it can increase (complements) or

decrease (substitutes) the marginal productivity of the other agent’s effort.

If the two efforts are complements, then in cases where the investment effort costs do not differ

markedly ((14) is positive) the firm does not have a first mover advantage, because the leader-agent

internalizes the boost in the marginal product of his effort that is brought about by the increase in

the principal’s effort in reaction to the agent increasing effort. The boost in the agent-leader’s effort,

caused by this increase in marginal productivity, benefits the follower-principal such that profit is

greater than under a leadership strategy. The principal would prefer to have the agent choose along

the principal’s reaction function and have the agent make the first move as opposed to the converse.

Similar results with a second mover advantage in market games under complementarity have been

discussed in Gal-Or (1985), Dowrick (1986), and Amir and Stepanova (2006). Besides the fact that

we consider an agency setting with a contract, what also distinguishes our situation is that, when

there is a second mover advantage for each player in our model, the firm as the organizational

designer, can dictate that the agent lead. Otherwise, the second mover advantage would lead to

a stalemate and there would be no equilibrium. Alternatively, if the marginal costs of investment

effort are strictly convex and marginal cost of inducing an increase in the firm’s investment effort

plus the marginal revenue of a unit of the agent’s investment effort is less than the marginal cost

of the agent’s investment effort ((14) is negative), then the firm benefits more in the complements

case by moving first with the benefits described as above in the agent leadership case.

When the two efforts are substitutes, there is the typical leadership advantage to the principal,

because the leader-principal internalizes an extra increase in his marginal productivity of effort

caused by the agent decreasing effort in reaction to an increase in the principal’s effort. This is a

27

relative benefit to the principal and as such the principal chooses the leadership role. The substitute

relationship is the usual case in games of firm interaction where there is a first mover advantage.

The implication of these results is that it is optimal, in terms of the firm’s profit, for one of the two

actors to take the initiative and precommit to a given level of effort to which the other actor will

optimally respond.

5.3. Discussion of the Substitute and Complement Cases

Consider the substitutes case. Assume that the principal’s effort is employed to create instructional

materials for the agent which serve to teach the agent a software program. In this case, it is likely

that more effort on the principal’s part in improving the instructional program increases the agent’s

total productivity in learning, but on the margin such extra effort decreases the agent’s marginal

productivity in learning through over documentation and detail in the instructions. This is the case

of substitute efforts and our results point to the recommendation that the principal should lead

by creating such appropriate instructional materials for the agent and the agent should follow by

using that instructional framework or template to learn by doing. This situation would be a perfect

candidate for the optimality of online instruction or a straight lecture or indoctrination type class

in which the principal leads and the agent follows without prior preparation. Indeed our intuition

would seem to tell us that strategic substitute relationships between efforts would call for online

or similar type firm leadership education. Moreover, we see that many real world software training

programs are offered online.

Consider the complements case wherein the agent wants to learn a new skill or refresh an old

one. Whether the firm or the agent should lead depends on (14). If the agent’s marginal cost of

effort is lower than the principal’s net marginal cost of effort, with strictly convex marginal costs

of effort, then the agent should lead and the principal might employ a self selection professional

28

development mode of learning where the agent selects an instruction regime and the principal

provides the resources. In fact, many actual professional development programs involve the agent

leading and they emanate from situations where the agent’s marginal effort cost is smaller than

that of the principal in implementing a training regime. Such is the case if the agent is the one

who detects a deficiency in a skill or a desire to improve a skill and then takes steps to embark on

training. The agent better knows than the principal the best the type of training suited to increase

productivity and, thus, might have a lower effort cost. Alternatively, if the principal’s net marginal

cost of effort is low, relative to the agent’s cost of effort due to say a very high agent opportunity

cost, then the principal should lead and employ say online instruction or one of the other firm

leadership methods. Both of these real world prescriptions would seem to make sense.

6. An Example: Quadratic Cost of Investment Effort

In this section, we explore the case where the investment effort costs take on the quadratic form

so as to exclude third order effects emanating from effort cost. For this case, we can conduct a

complete comparative static analysis. Let

() = 2 2 = (15)

Assumptions A.2-A.3 are independent of these functions, and A.4 is satisfied by (15) (Recall that

A.8 implies A.5.). We then only need to check A.6 through A.10 for this specialized case.

Assuming that a Nash solution exists, we can solve the quadratic versions of reaction functions

(3) and (5)

= −1 (( + ) ) and = −1 (( + )) (16)

29

Next solving these reaction functions for the Nash solution we obtain

= ( + )

( − 2 ) and =

( + )

( − 2 ) (17)

Assuming existence, conditions (7) and (9) along with the reaction functions can be used to

solve for the firm leadership solution

= ( + 2)

− 22and =

( + )

− 22(18)

and the agent leadership solution

=( + 2 )

− 22and =

( + )

− 22 (19)

Next, consider the specialization of our maintained assumptions. Assumption A.6 becomes

A.60 if 0

Assumption A.7 is specialized as

A.70 ( − 2 ) ( + ) if 0

where ≤ 1 Assumption A.8 is

A.80 − 22 0

in the quadratic case. Assumptions A.9-A.11 are given by

A.90 If 0 min { + 2

+2

} 0

A.100 − 22 ( + 2 ) and.

A.110 − 22 ( + 2)

Next, we consider condition (14) for the quadratic example. Note that, in the complements

case where 0 it is clear that from (18) and (19). Because the iso-profit contour,

30

tangent to ( ) at (

) is upward sloping between the two reaction functions, we have that

It follows that is to the northeast of

so that, as noted above, the firm’s profit

is greater at the agent leadership solution as compared to the firm leadership solution, if 0.

6.1. Comparisons in the Quadratic Case

The difference between the firm’s profit at the firm leadership solution and the agent leadership

solution is

( )−() =−2 (2 + 2 ( + )

2( − 2 )2 (20)

This expression takes on the sign of the numerator. If 0 then by (20), ( ) − () 0

and the agent leadership solution is best for the firm with the ranking () ( ) ( )

If is negative then (20) takes on the sign of

(2 + 2 ( + ) = ( + ) + ( + 2 ) (21)

Under A.90, (21) is positive. Thus we have that ( ) − () 0 and the firm leadership

solution is best for the firm. The conclusion for the quadratic case is that if investment efforts

are complements, then it is optimal for the firm to dictate that the agent takes the initiative and

leads, but if the efforts are substitutes then, in the quadratic and in all other more general cases

considered under our assumptions, it is optimal for the firm to lead.

6.2. Comparative Statics in the Quadratic Case

Let us begin with the Nash solution. Through direct differentiation of the closed form solutions, we

can determine the comparative statics of the equilibrium. These results are summarized in Table

31

1.18 Each cell shows the sign of the partial derivative of with respect to an indicated parameter.

¥

+ + − (−1) +

+ + (−1) − +

Table 1: Comparative Statics of the Nash Equilibrium

All of these effects are fairly intuitive. The direct productivity parameters for consisting of

− and each increase , whereas the effect of each of the cross productivity parameters,

6=, − 6=or 6=, on takes on the sign of This cross effect is positive if the two efforts are

complements and negative if they are substitutes. In the substitute case, for example, an increase

in the expected future cash flow for the agent, through an increase in or − will actually

decrease the optimal amount of investment effort by the principal. The principal would increase

investment effort under these assumptions if the two efforts are complements. Thus, in organizations

where the efforts are substitutes, we are likely to see less investment in the education of the agent

by the principal in reaction to greater productivities or lower costs to the agent for investment

in his/her human capital. In the complementary case, investment by the principal will increase if

the agent experiences an increase in his/her expected gain through a productivity increase or cost

reduction. Recall that increases in are driven by increases in while decreases in are implied

by increases in through the optimal contract

In the Nash case, 0 for both parties, regardless of whether the two efforts are

complements or substitutes. However, this result has a different economic interpretation depending

on whether the action variables are complements or substitutes. In the complements case, greater

18Comparative static derivatives are derived in the Appendix.

32

means greater strength of interaction, and the proposition that greater strength of interaction

results in greater investment by both agents is intuitive in this case. Here, an increase in raises the

total profitability of as well as the total utility of , so that the direct effects are positive. The

indirect cross effects are also positive in that an increase in will increase the marginal benefit of

6= In the substitute case, a greater means less strength of interaction because it is the absolute

value of which measures interactive strength. Therefore, the result 0 tells us that

greater strength of interaction results in both agents cutting back on human capital investment for

the agent. In the substitute case, the above direct effect is still positive, but the indirect cross effect

is negative. However, the direct effect dominates. These results are interesting because they say

that in firms where there is a strategic substitute relationship between principal and agent human

capital investments, greater interactive dependence of these efforts will produce less investment

effort by both agents than in an identical firm with a complementary relationship between these

efforts.

The comparative statics of the agent and firm leadership solutions are shown analogously in

Table 2.

¥

+ + − (−1) +

+ + (−1) − If 0+; If 0 ?

+ + − (−1) If 0+; If 0 ?

+ + (−1) − +

Table 2: Comparative Statics of the Leadership Solutions

33

Consider the leadership solutions. The effects of the direct and cross productivity parameters

are the same as in the Nash solution. While increases in the marginal rewards for human capital

investment for either agent will induce both agents to increase investment if efforts are complements,

this is not true in the substitute case. Here, if the leader becomes more productive at enhancing

human capital, then the follower will decrease investment, and conversely.

With complementary investment efforts, the impact of a change in is the same as that of

Nash in the respective leadership solutions for the leader and the follower, in that greater strength

of interaction leads to greater investment in human capital by both parties. The results change if

the investment efforts are substitutes. In this case, for leader and follower while 0

can be of either sign. If is positive then greater strength of interaction leads to less

investment effort by both parties as in the Nash case. For leader and follower , 0 is

more likely to be true if , and are large and and are small. In the new case where

0 greater strength of interaction lowers the leader’s investment in the follower’s human

capital, while it raises the investment of the follower in his/her own human capital. For leader

and follower , 0 is more likely to be true if , and are small and and are

large. Thus, if the follower’s marginal benefits of investment are relatively high and its marginal

costs low, while the marginal benefits of investment to leader are small, then greater degree of

interaction (negative becomes greater in absolute value) leads to more investment by the follower

and less by the leader.

7. Subsidies to the Agent

Let us examine the two dominant organizational designs where the principal leads (substitutes case

or complements case) and where the agent leads (complements case only). In each scenario, we ask

whether the principal can improve profit by giving the agent an optimal subsidy which comes from

34

the principal’s expected surplus. The firm’s subsidy choice variable, denoted ∈ [0 1) takes the

form of a fraction of the difference in the principal’s residual profit in the high human capital state

and the low human capital state, per unit of cash flow. At time zero, the principal commits to this

share in the investment stage as well as a menu of monetary contracts for period 1. For the agent,

the time zero expectation of this per unit subsidy is

() = [(1− ())(())− (1− ())(())]

The expected total subsidy at time zero is ( ) At time one, an agent of type solves

max{}

− () + [(1− ())(())− (1− ())(())]

which leads to the same solution as in the original agent problem = () This is true because

the total subsidy term

≡ [(1− ())(())− (1− ())(())]

is a constant. Likewise, the firm solves

max{}

(1− )()−

where the solution is () which is identical to that of the original firm problem. The subsidy and

the menu of contracts (), = which guide the agent in period 1, are chosen by the firm in

period 0, committed to at that time, and announced to the agent.

The firm optimally chooses the share at time zero. A corner solution, = 0 characterizes the

case where no subsidy is desirable to the firm. We seek to characterize necessary conditions for an

35

interior 0

Consider the firm leadership solution, with of either sign (substitutes or complements, with

(14) negative). Under the subsidy at time zero, the agent solves

max{}

( + + )( + ) + − ()

The solution is analogous to (5) with = 0−1 (( + )( + )) ≡ ( ) and ( + )

replacing Note that = [( + ) ](00) 0 and = ( + )(

00) takes on

the sign of . The firm solves

max{ }

[ + ( ) + ( ) ] (1− ) + − ( )

The first order condition for is analogous to (9)

(1− )( + ( )) + (1− )( + )( ) − 0 ( ) = 0

and the first order condition for is

− + (1− )( + )( ) = 0 (22)

The question of interest is under what conditions will be positive. Proposition 5 gives us necessary

conditions.

Proposition 5. Let the assumptions A.1- A.4 hold. In a firm leadership solution, 0 only if

At such an interior solution in we have that 12 and, in equilibrium, 12

(−1)

Both the firm and the agent expect to gain greater cash flow as a result of the agent becoming

more productive in the firm. The agent expects to gain and the firm expects to gain The firm

36

will not provide a subsidy to the agent in the training stage unless it gains more than the agent

from their joint investment in the agent’s human capital. Further, in equilibrium, the firm does

not share more than half of its expected surplus from human capital investment. In equilibrium,

there is an exogenous upper bound on the optimal which is given by 12

(−1)

The optimal

share of the firm’s surplus going to the agent must be less than half of the percentage divergence

of above one.

We only need to study the complements case where it can be optimal for the firm to ask

the agent to be a leader (Condition (14) is positive.). In the initial stage of this game, the firm

sets the share of its surplus going to the agent. Next, the agent chooses his or her investment

level . Finally, the firm responds to the agent’s choice of with its choice of through its

reaction function. The firm’s choice of then is described by = 0−1 (( + )(1 − ) )

≡ ( ) with (1 − ) replacing in (3). We have = [(1 − ) ](00 ) 0 and

= [− ( + )](00 ) 0 The agent solves

max{}

[ ( ) + + ( )]( + ) + − ()

and selects according to

() = ( + )( + ( )) + ( + )( + ) ( ) − 0() = 0 (23)

First order condition (23) defines = () The firm chooses as the solution to

max{}

[ (() ) + () + () (() )](1− ) + − ( (() )) (24)

Using the firm’s optimality condition for selection of the first order condition to (24) for an

37

interior simplifies to

() = − [ (() )+()+() (() )]+(1−) (+ (() ))

= 0

(25)

From (25), it is immediate that a necessary condition for 0 is that 0 Computing,

=

−2()2{(+ )+

( + )( + )2(1− )

000

(00 )3

+( + )

00[ (1−3)−2]}

(26)

If 2()2 0 all terms of (26) are positive except for [ (1 − 3) − 2] which is not

necessarily positive. We have

Proposition 6. Let the assumptions A.1- A.4 hold. Consider an agent leadership solution with

0 and where the second order condition 2()2 0 is met. If, in addition, the second

order condition to problem (24) is met for ∈ [0 1], we have that 0 only if |=0 0.

The latter condition is met if 2

In the complements case, we can show that, under our assumptions, the limit of condition

(25) as tends to one is negative. If all of the second order conditions are met, then a necessary

condition for there to be a fractional 0 at which (25) is zero is that |=0 0 using the

intermediate value theorem This is true because the first term of (25) is negative for all ∈ [0 1]

The necessary condition for the firm to want to subsidize the agent is that, starting with a zero

subsidy, a small positive subsidy increases the agent’s leadership investment in human capital. In

general, this need not be the case. However, if the firm’s private benefit from the agent’s increase

in productivity is at least twice as great as the agent’s private benefit from the same, then this

necessary condition for 0 is guaranteed to be met.

38

8. Conclusion

We consider the dual problem of compensation and the design of a human capital investment

scheme. Before contracting with the agent, the principal can design a joint production process

wherein the principal and the agent can exert efforts to improve the agent’s productivity in a

second stage cash flow process. In the latter process, the principal formulates an optimal contact

based on the agent’s realized human capital. While the agent’s optimal incentive share may or

may not increase with an increase in the agent’s human capital, both the principal and the agent

obtain a greater expected payoff with a greater human capital level at an endogenously optimal

contract. These expected gains then motivate each to invest effort in the pre-contract human capital

investment game.

We focus on which actor if any should take the initiative in the human capital investment

process. In the investment game, the principal should not rely on synchronous say teaching and

learning efforts by the principal and the agent. Under reasonable assumptions, if the efforts are

substitutes, the principal should assume a leadership role. If the efforts are complements, the agent

or the principal should lead, depending on marginal investment costs and the marginal revenue of

the agent’s investment effort. If the principal’s net marginal cost of effort is small and the agent’s

marginal cost of effort is large, then the principal should lead. If the agent’s marginal cost of effort

is less than the principal’s net marginal cost of effort, then the agent should lead. A salient point

is that for maximal profit, the principal should design the human capital investment process such

that one of the two actors takes the initiative and leads.

The quadratic effort cost example provides us with a set of comparative static results. At all

solutions, increases in the direct productivity parameters raise human capital efforts, but increases

in cross productivity parameters cause an investment effort to rise if the efforts are complements

and fall if they are substitutes. In the synchronous Nash solution, increases in the strength of

39

interaction of the two efforts leads to an increase in both efforts if they are complements and it

leads to decreases in efforts if they are substitutes. In the two leadership solutions, the effects of

changes in the direct and cross productivity parameters are as in the Nash case. Moreover, if the

two efforts are complements, the effect of an increase in the strength of interaction leads to greater

leader and follower efforts. However, if the two efforts are substitutes, the results can change. In

this case, greater strength of interaction causes the leader’s effort to fall but the follower’s effort

can rise or fall. The latter rises when the follower’s marginal benefits of investment are relatively

high and its marginal costs low, while the marginal benefits of investment to leader are small. It

falls when the follower’s marginal benefits of investment are relatively low and its marginal costs

high, while the marginal benefits of investment to leader are high.

Finally, both the agent and the firm gain through human capital investment in the two domi-

nant leadership solutions. At these solutions, the firm can improve its expected equilibrium profit

position if it shares some of its expected gains from this investment with the agent. A necessary

condition for such a subsidy to be optimal is that the firm gains much more than does the agent

from improvement in the agent’s productivity.

Appendix

Proof of Lemma 1. The principal maximizes = −+(1−)+(+−())++

over a selection of and Divide the (P) constraint by 0 ()(−1) 0 and we have

(+ ()− ) = (+ 0 − ) ≥ 0 − 0 by 0 00 0 and (0) = 0. Thus = 0 The

first order condition for is −1 + + = 0 so that = 1 and = 0 ¥

Proof of Lemma 2. ( 0) It suffices to show that [(1 − ()(())] 0 This

40

derivative is given by

−+ (1− )+ (1− )20 + (1− )()20

= (1− )(+ 0) = (1− )(0) 0

using −+ (1− )0 from the principal’s problem.

( 0) It suffices to show that [()(()) − ((()))] 0 This derivative is

given by

( + )

if we use the first order conditions from the agent’s problem. First note that from the firm’s FOC

for we can solve for = (0 − )0 The term can be written as

=−0 + (1− )0+ (1− )200

20 − (1− )2200

The first two terms in the numerator of this expression can be simplified to −0 + 2. We have

=−0 + 2+ (1− )200

20 − (1− )2200

Combining this expression and the expression for

+

=(0 − )

0+−0 + 2+ (1− )200

20 − (1− )2200

Finally, placing these two expressions over a common denominator and simplifying, we obtain

+

=

(0)2

(0)(20 − (1− )2200)

41

Thus, from above,

[(1− ()(())]

=

20

20 − (1− )2200 0 ¥

Proof of Lemma 3. This result follows directly from (6). ¥

Proof of Proposition 1. ( 0) By assumption 1 (0) = 0−1 () ( (1)) 0 by

0 0 where (1) = (0−1 ( + )) Further, 1 = −1 ( (1)) Thus,

−1 ( (1)) ( (1)) We

have that 0 = −1 ( (0)) where (0) = (0−1 ( )) ∈ (0 1) Moreover, ( (0)) 0 Whence,

( (0)) −1 ( (0)) It follows that there is a Nash equilibrium in the interval ( (1) (0))

Under A.2, it is unique and stable.

( 0) By assumption ( ) −1 ( ) Next note that (0) = 0−1 ( ) and

−1 ( (0)) = 0 while ( (0)) = (0−1 ( )) 0 Whence, ( (0)) 0−1 ( (0)) It fol-

lows that there exists a Nash equilibrium in the interval ( (0) ) By A.2, it is unique and stable.

¥

Proof of Proposition 2. Both the agent-leader and firm-leader objective functions are strictly

concave, by A.8. For the agent leadership solution the →1

(7) is equal to the expression in A.10.

For the firm leadership solution, the →1

(9) is equal to the expression in A.11. We need only show

that →0

(7) 0 and →0

(9) 0 We have

→0

(7) = ( + 0−1 ( )) +

00 (0−1 ( ))

0 and

→0

(9) = ( + 0−1 ()) +

00(0−1 ())

0

under our assumptions. ¥

Comparative Static Derivatives in the Nash Solution and the Two Leadership Solu-

42

tions. Consider the derivatives of By A.80 we have = ( − 22 ) 0 and

= (−22 ) where the latter takes on the sign of The expression =

( + )( − 22 )2 0 by A.80 and A.90, and = ( +

)( − 22 )2 takes on the sign of under the same assumptions. Also, =

−( + )( − 22 )2 0 by A.80 and A.90, while = −( −

22 )2carries (-1) times the sign of by A.80. Finally, = ( ( + (2 +

)))( − 22 )2 Again by A.80, this carries the sign of the term ( + (2 +

)) If 0then this is positive. Let 0We rewrite as ( +2)+2 The

second term is positive and the first term is positive by A.90. The derivatives of are analogous.

Next consider the derivatives of in the firm leadership solution. We have =

( − 22 ) 0 by A.80, and = 2( − 22 ) takes on the sign

of , by A.80. Next, = −( + 2 )( − 22 )2 0 while =

−2 ( + )( − 22 )2 takes on the sign of (−1) by A.80 and A.90. Next, we

have = ( + 2)( − 22 )2 0 by A.80 and A.90, and =

2 ( + )( − 22 )2 takes on the sign of under the same assumptions. The

derivative = 2 (22 + ( + 2 ))( − 22 )2 carries the sign of

(22 + (+2 )) 0 by A.80 and A.90. Next we consider the derivatives of We

have that = ( − 22 ) 0 by A.80, and = ( − 22 )

which, by A.80, takes on the sign of The expression = −( + )( −

22 )2 0 by A.80 and A.90, and = − ( + 2)( − 22 )2 takes

on the sign of (−1) under the same assumptions. Next, = ( + 2 )( −

22 )2 0 by A.80 and A.90, and = ( + 2)( − 22 )2 which

takes on the sign of under the same assumptions. Finally, = ( (+2(2+

)))( − 22 )2 takes on the sign of ( + 2(2 + )), which is posi-

43

tive if is positive, by A.80 and A.90. If 0 then we have mixed terms with the first term

being positive and the second negative under A.90. The second term is more likely to domi-

nate the first when , and are small and and are large. To see this, consider

= ( + 2(2 + )) with 0 We have = 0 =

4 0 = + 22 0 = 2(2 + ) 0 by A.90 and

= 22 0 The term = + 4 is ambiguous. The signs of the

derivatives of follow from the derivations of the derivatives of ¥

Proofs of Propositions 3 and 4 are contained in the text.

Proof of Proposition 5. Substituting the expression for into the first order condition for ,

we obtain the following necessary condition for an interior solution in : (+ )[(+ )(1−)

00−

] = It follows that 00 − 0(1−)

(+ ) 0 using ( + ) =

0(+ )

By assumption A.4,

00

0 1 so that we have (1− ) (+ ) or that (1− 2)− 0 It then follows that

12 and (1− 2) or that 12(−1)

, as was to be shown. ¥

Proof of Proposition 6. First, note that condition (25) immediately implies that lim{→1}

(25) =

− [(1)] 0 by ((1) 1) = 0 Given that 2()2 0 it follows that a necessary

condition for 0 is that lim{→0}

(25) = − [ ((0) 0) + (0) + (0) ((0) 0)] + ( +

((0) 0))|=0 0 and the latter implies that

|=0 0 Note that setting = 0 yields the

default agent leadership solution considered previously. Finally, we can write

=

−2()2{(+ )+

( + )( + )2(1− )

000

(00 )3+( + )

00[ (1−3)−2]}

Evaluating at = 0

|=0 =

−2()

2|=0

{( + ) +()( + )

2000

(00 )3+

( + )

00( − 2)}

44

Under our assumptions,2()

2|=0 = [

2200− ( + )(

00)2000 ] − 00 0 so that all terms

are positive, if ( − 2) 0 ¥

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47

                za      

 

 

                                                                                                

                                                                                              E(π) = constant 

                                                                                               

                                                             F                                                   

                                                                                    N                         E(u) = constant 

 

                                                                                              A 

                                                                                                         

                                                                                                                                                                za(zf) 

 

 

                                                                                                                     zf(za)

 

 

                                                                                                                                                                                zf 

                                                                           Figure 1.  Substitutes Case 

48

                za      

 

                                                                                                                zf(za) 

                                                                                                                                           E(π) = constant                      

                                                                                                                                                                              za(zf) 

                                                                                                           E(u) = constant 

                                                                                          A                                       F                                                       

                                                                                       x                      

 

                                                                              A’ 

                                                                            N                                                                                    

                                                                                         E(u) = constant                                                                        

 

 

                                                                                                                     

 

 

                                                                                                                                                                                zf 

                                                                           Figure 2.  Complements Case 

49