1. Out with the old? Team Composition and
Job Rotation
Dana Liu
October, 2013
Supervised by Dr. Andrew Wait
Thesis submitted in partial fulfilment of the requirements for Honours in the
Bachelor of Economics, University of Sydney.

2. Acknowledgements
I would like to thank Andrew for his continued advice, encouragement and assistance
throughout the year. I would also like to thank Vladimir for his advice with earlier
versions of my draft.

3. Statement of Originality
I hereby declare that this thesis is my own work and to the best of my knowledge
it contains no material previously published or written by another person, except
where acknowledgement is made in the text. Any and all contributions are explicitly
acknowledged and abides by the University of Sydney academic honesty policy.

4. Abstract
Team production is one of the most prevalent ways of organising a firm. Given this,
the size and composition of teams is an important consideration for a principal or
employer in many workplaces. While teams of familiar workers may create higher
surplus compared to unfamiliar workers for a given level of effort (that is, incumbent
workers could enjoy a Synergy), a team of newcomers may be induced to exert more
effort. I develop a model in which a Principal faces a trade-off between the natural
Synergy of incumbents and increased effort of newcomers. In doing so, this thesis
generalises some of the existing literature on workforce composition, notably Bel et
al. (2013), by endogenising both the size of the workforce and the Principal’s payoff
function.
Firstly, I model the Principal’s decision for team size and composition using an
outside-option bargaining model for distributing surplus and find that the Principal’s
decision is driven by the relative size of Synergy produced by incumbent workers
compared with the rent she can extract from newcomers. I also find that when the
equilibrium Synergy is increasing at an increasing rate with respect to the number of
incumbent workers hired, the Principal considers infra-marginal returns and makes a
binary-type hiring decision, hiring either all incumbents or all newcomers.

5. Secondly, I analyse the model using an alternative method of distributing sur-
plus (the Shapley value) and find that a Principal’s decision regarding team composi-
tion is highly dependent on the prevailing bargaining method. This finding suggests
that the result in Bel et al. (2013), which considers a two person example, can be
reversed if their specific assumptions regarding the bargaining method are relaxed.
Thirdly, through the addition of a time dimension, I extend my model to a job
rotation context and find that, consistent with the static case, both the evolution of
the payoff function and the bargaining method are crucial considerations. Specifically,
a Principal may rotate too much or too little depending on the bargaining method
applied. Lastly, I show that under a system of rotation, newcomers are indifferent
as to who makes the hiring decision whilst incumbent workers strictly prefer decen-
tralised decision making with respect to composition and rotation. These results have
implications for the organisational structure of different types of firms and production
processes which I discuss throughout the thesis.
2

6. Contents
1 Introduction 3
2 Literature Review 8
3 Team size and composition: worker outside-option 16
3.1 Model set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Bargaining Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Investment Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 Number and type of workers hired . . . . . . . . . . . . . . . . . . . . 23
3.5.1 Principal’s marginal return increasing in I . . . . . . . . . . . 26
3.5.2 Increasing returns; an example . . . . . . . . . . . . . . . . . . 30
3.5.3 Principal’s marginal return decreasing in I . . . . . . . . . . . 34
3.5.4 Decreasing returns; an example . . . . . . . . . . . . . . . . . 37
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1

7. 4 Team composition: inside-option bargaining 42
4.1 Bargaining Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Investment Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Number and type of workers hired . . . . . . . . . . . . . . . . . . . . 46
4.3.1 The change in equilibrium Synergy increasing in I . . . . . . . 46
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 Team implications 53
5.1 Job Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 The Principal’s choice and the second-best . . . . . . . . . . . . . . . 58
5.3 Agents’ preference for decision-making . . . . . . . . . . . . . . . . . 62
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6 Conclusion 66
Appendices 68
A Proof of Proposition 3. 69
B Shapley Value 71
2

8. Chapter 1
Introduction
Production often occurs in teams and given that team output is often necessarily
shared amongst team members, a fundamental issue of team production is an ineffi-
cient level of investment. Moral hazard, leading to agents exerting an insufficient or
sub-optimal level of effort, often occurs in workplaces because contracts are incom-
plete, and a principal or an employer cannot observe the amount of effort exerted by
individual agents (see Holmstrom 1982, for example).1
The environment assumed throughout this thesis will be one where contracts
are incomplete and investment by all agents is ex ante while surplus is realised ex
post, leading to an underlying hold-up problem. Developing the model in this direc-
tion directly relates to production in a variety of situations; for certain employment
1
Though several papers have considered sequential investment as a possible solution by allow-
ing agents to invest ex post when the contracting environment is more complete (e.g. Mai et al.
forthcoming and Zhou and Chen 2013), this may not always be desirable or even feasible. While
sequential investment can lead to unambiguously higher contributions, under certain conditions, it
may lead to overinvestment by either ex ante or ex post agents (see Mai et al. forthcoming).
3

9. relationships, all agents choose their effort simultaneously and the surplus and subse-
quent bargaining is realised afterwards. For example, music bands or Broadway casts
collaborate towards a song or a show, the artistic and financial successes of which are
realised ex post. Similarly sporting teams and teams of consultants work together
towards a common goal, the value of which is not realised until after they have played
a game or produced a recommendation.
While deciding on the composition of her workforce, the Principal must take
into account the underinvestment in effort due to moral hazard. While empirically
it has been found that agents sometimes internalise some of their co-workers benefits
(see e.g. Bandiera et al. 2012), it is not hard to imagine that peer effects may be
insufficient in inducing effort in many workplace environments where positive exter-
nalities from effort are not completely internalised. This is especially the case if one
assumes that part of the externality accrues to the employer. Nevertheless, the fact
that individuals often interact dissimilarly with different kinds of workers suggest that
team composition is an important workplace consideration.
To analyse this issue of underinvestment and team composition in the context
of a workplace, one can conceive of workers as simply being of different types, with
certain kinds of workers working better with each other. In this case, the Principal
will face a trade-off between fostering natural Synergies between incumbent workers,
and maintaining effort levels which will be lowered due to the presence of Synergy.
The resulting team assembled by the Principal will then depend on the interaction
4

10. between Synergy and effort, as well as the prevailing bargaining method.
While use of an inside-option bargaining model to distribute surplus is the most
common approach (e.g. Nash bargaining in Grossman and Hart 1986, or the Shap-
ley value in Hart and Moore 1990 and Mai et al. 2014), this thesis also considers
outside-option bargaining along the lines of Rubinstein’s (1982) infinite horizon bar-
gaining model as an alternative, and illustrates the effect of the bargaining method
on the Principal’s incentives, and ultimately, on the choice of team. Specifically, this
method of surplus distribution takes into account the switching costs that agents face
when searching for alternative employment and the ability of the Principal to extract
economic rent as a result.
As the Principal’s return depends on both the benefit she receives from her
agents as well as the cost of hiring them, team size will naturally be limited by the
cost of hiring additional workers. While existing papers on optimal team composition
or heterogeneity either limit their analysis purely to optimal composition (e.g. Prat
2002) or use an arbitrary number of workers (e.g. Bel et al. 2013), this thesis also
endogenises size as a consideration for the Principal.
These analyses of team composition suggest that in some situations, a firm can
benefit from instituting job rotation. The frequency with which a Principal wishes to
rotate her workers will depend on the evolution of worker interaction or equivalently,
the development of Synergy. As workers spend more time together, it is natural that
they should grow familiar with one another’s traits and develop an ability to work
5

11. together more easily, and hence be able to enjoy excess surplus for the same level
of effort. Job rotation is not an unexplored area and theoretical models on optimal
job rotation have been developed in the literature (see e.g. Li and Tian 2013 and
Azizi et al. 2010). However, to the best of my knowledge, no existing paper has
explored the idea of optimal job rotation with respect to the team composition of
heterogeneous workers, which this thesis attempts to do. Rather than assessing the
level of complementarity a worker has with his or her job, this thesis will explore
the importance of complementarity between workers, where rotation occurs through
moving workers into different teams and working on what is essentially the same job.
Practical examples of this kind of rotation can be seen in consulting firms and airlines.
Lastly, this thesis looks at the efficiency of rotation when the choice is made
by the Principal compared to the second-best optimal case, and how this is affected
by the bargaining method in place. If the Principal benefits predominantly from
the natural Synergy of incumbents, then she will be more likely to hire workers that
possess these Synergies. On the other hand, if effort is relatively more important to
the Principal then she may prefer to rotate more often. For example, the manager
of a music band may benefit greatly from the inherent chemistry between members
that is crucial to sales, whereas the owner of an airline may care much more about
the effort exerted in ensuring safety standards are met.
This thesis will proceed as follows: Chapter two provides an overview of the
literature, Chapter three sets up the model and analyses the choice of team size
6

12. and composition; Chapter four considers an alternative bargaining method; Chapter
five considers implications for job rotation and decision making; and Chapter six
concludes.
7

13. Chapter 2
Literature Review
The incomplete contracting environment that is assumed throughout this thesis leads
to a classic moral hazard in teams problem, and the resulting underinvestment is a
central issue throughout my analysis. Within this framework, I allow the Principal to
choose the composition of a team — the number of incumbents who have an intrinsic
Synergy working together, and the number of newcomers who have a propensity to
exert more effort — along with the size of the team. In this context, while a Principal
cannot eliminate the moral hazard problem, she can somewhat manipulate the team
structure to influence both investment and total output.
To provide some background and motivation for the modelling developed in
later chapters, I will first review the related literature on moral hazard and under-
investment in the context of the principal-agent problem; team heterogeneity in the
workplace; and the existing models of job rotation.
8

14. Moral Hazard and the Principal Agent Problem
The issue of moral hazard in team production leading to underinvestment has been
widely explored in the literature (see e.g. Holmstrom 1979, Lazear and Rosen 1981
and Holmstrom 1982). In the context of a workplace, moral hazard in effort occurs
because contracts are incomplete and a principal or an employer cannot observe the
amount of effort exerted by individual agents. Holmstrom (1982) demonstrated that
in a setting where a team’s actions jointly determine some outcome amount, there can
exist no Nash equilibrium where agents exert the Pareto optimal amount of effort.
This result relies on the assumption that agents do not completely internalise the
positive externalities their effort places on their fellow workers.
Empirically, it has been found that agents often internalise some of the bene-
fits of their colleagues and this can partially mitigate the free-riding problem which
causes underinvestment in effort. For example, Bandiera, Barankay and Rasul (2012)
conducted a field experiment on fruit farms in the United Kingdom and found that
agents gain non-pecuniary benefits from working alongside their friends and place
positive weights on their friends’ earnings. However, there are many workplace envi-
ronments where peer effects may not be sufficient in inducing effort because positive
externalities from effort are not completely internalised. This is likely to be the case
in situations where part of the externality accrues to the Principal or employer.
Several papers have also considered sequential investment as a possible solu-
9

15. tion for alleviating the issue of moral hazard. Mai, Smirnov and Wait (forthcoming),
and Zhou and Chen (2013) both find that when allowing for sequential investment,
ex post agents can invest under a more complete contracting environment and the
contribution by agents is unambiguously higher. However, Mai, Smirnov and Wait
(forthcoming) also emphasise the conditions under which sequential investment may
lead to overinvestment by either ex ante or ex post agents, and is therefore not al-
ways desirable. Hence the environment assumed throughout this thesis will be one
where contracts are incomplete and all investment is made simultaneously by agents
ex ante, while surplus is realised ex post. Examples of this kind of contract are evi-
dent in employment relationships where all agents choose their effort simultaneously
and the surplus and subsequent bargaining is realised afterwards. For example, when
music bands work together to compose new songs or practise for a performance, it
is often very difficult for a manager to observe the effort exerted by individual team
members and the level or quality of production (which may be measured in sales) will
not be realised until after the collaborative effort is complete.1
Team Composition and Size
Since individuals often interact dissimilarly with different kinds of workers, team com-
position is an important consideration for a Principal, especially when the contracting
environment is incomplete. One way of modelling the heterogeneity of workers is sim-
1
While use of sequential investment would complicate some of the interactions between efforts of
agents, it would not affect the basic trade-off presented in my model.
10

16. ply conceiving of workers as being of different types, with certain kinds of workers
working better with each other.
Uzzi and Spiro (2005) measured the financial and artistic success of Broadway
musicals and found that the positive effect of working with familiar team members
persisted only up to a certain point, after which it reversed and the most successful
teams were those that contained some newcomers. This is because in deciding on
team composition, the Principal faces a trade-off between the Synergy produced by
incumbent workers and the greater effort exerted by newcomers. In many cases, the
energy and vitality of newcomers, expressed in the form of greater effort, mean that
despite the natural advantage of incumbents, a mixed team is preferable. Hence, this
thesis analyses the conditions under which a Principal would wish to employ a mixed
team, a team of only incumbent workers, and even a team of only newcomers.
Relatedly, Bel, Smirnov and Wait (2013) develops a model where certain workers
benefit from the familiarity of working with others of the same type and find that the
reduction in effort in these workers can be so strong that an employer may want to
forgo their Synergy and hire new workers. While Bel et al. (2013) considers optimal
team composition through a two-player case, this thesis attempts to generalise the
hiring decision by endogenising the optimal size of the workforce, and analysing the
different effects of employing alternative bargaining methods. I show that the result
in Bel et al. (2013) can be reversed if assumptions on the bargaining method are
relaxed and an alternative bargaining method is applied.
11

17. Furthermore in terms of team composition, Prat (2002) developed a model to
look at the optimal degree of homogeneity of a firm’s workforce. Where agents share
a common prior but may receive different signals about the state of the world, the
paper finds that if agents’ actions are complements in the payoff function, then homo-
geneity is preferred, whereas if agents’ actions are substitutes in the payoff function,
then heterogeneity is preferred. This reflects the idea that activities for which good fit
between units is vital, one would prefer homogeneity in order to maximise coordina-
tion, whereas activities that involve the exploitation of new opportunities will benefit
from heterogeneity. Similar to Prat (2002), this thesis finds that the structure of
the payoff function is central to the Principal’s decision. However, while Prat (2002)
does not consider the effect of interaction between complementary agents’ actions on
equilibrium effort levels, this thesis incorporates a trade-off between complementarity
of homogeneous workers and increasing effort levels through heterogeneity, effectively
endogenising the payoff function.
Furthermore, since the Principal’s profit depends on both the benefit she re-
ceives from her agents as well as the cost of hiring them, optimal team size must also
depend on the cost of employing additional workers. Prat (2002) analyses optimal
team heterogeneity without considering the size of an optimal team, and Bel et al.
(2013) bases its findings on team composition using a specific, two-person example.
This thesis endogenises optimal team size by introducing a cost consideration for the
Principal.
12

18. Bargaining Methods
In order to create the most productive team possible, the Principal makes decisions
as to team size and composition, given the bargaining method in place. The Shapley
value is a core solution in game theory introduced by Lloyd Shapley in 1953 and
application of the Shapley value to cooperative games provides a unique distribution
of gains and costs that is fair relative to the contribution of each player. As a result,
the literature relating to distribution of surplus between principals and agents has
generally followed the Shapley value (see e.g. Hart and Moore 1990, and Mai et al.,
forthcoming).
However, it is possible that many situations more closely resemble an alterna-
tive form of bargaining. I generalise my analysis of optimal team composition and size
from the perspective of a Principal by considering the effects of alternative bargaining
strategies. Specifically, I consider both use of the Shapley value in my framework,
as well as the use of an outside-options model based on Shaked and Sutton’s (1984)
version of Rubinstein’s (1982) infinite-horizon bargaining model. The latter method
of distributing surplus incorporates an additional cost faced by agents when switch-
ing jobs, and the ability of the Principal to extract economic rent as a result. For
example, a college graduate who has just joined the workforce may find it particularly
costly to find alternative employment since they have not remained at their current
workplace long enough to develop Synergy with their co-workers, become an incum-
13

19. bent, and signal their value to the market.
Job Rotation
This thesis also extends the area of literature on team composition by allowing the
interaction between workers to evolve, so as to provide scope for the analysis of the
optimal team in a job rotation context. This extension is a novel consideration of this
thesis, and may be achieved through the addition of a time dimension to the model.
Job rotation is not an unexplored area and theoretical models on optimal job
rotation have been developed in the literature. Li and Tian (2013) modifies a stan-
dard directed search model to develop a model of intrafirm job rotation which allows
workers to be matched to their most compatible positions. They find that despite the
costs associated with rotation, large firms benefit from their ability to rotate workers
as this allows them to partially overcome the cost of mismatch. Furthermore, Azizi,
Zolfaghari and Liang (2010) develops a mathematical algorithm applicable to man-
ufacturing workers that captures the idea that job rotation forgoes some benefits of
specialisation but becomes strictly preferable when lowered effort due to boredom is
substantial.
However, neither of the above models explore the idea of optimal job rotation
with respect to the team composition of heterogeneous workers, and to the best of
my knowledge no existing paper has attempted to do so. Overall, it seems that the
literature on job rotation focuses on the analysis of the costs and benefits of rotating
14

20. workers through different jobs. Rather than assessing the level of complementarity
a worker has with his or her job, this thesis extends the existing literature on job
rotation towards new, unexplored directions by analysing the importance of comple-
mentarity between workers, working on what is essentially the same job.
In doing so, I also consider the situations under which the Principal may choose
to rotate too much or too little relative to the optimal, how this is driven by the bar-
gaining method in place, as well as when agents may favour a decentralised decision
making process for rotation.
Summary
This thesis analyses team size and composition decisions in a way which generalises
the results from existing literature through endogenising team size and the payoff
function, and relaxing assumptions on the bargaining method. Furthermore, this
thesis extends the current literature on both team composition and job rotation by
applying its results on optimal team composition of heterogeneous workers to the
context of job rotation.
15

21. Chapter 3
Team size and composition: worker
outside-option
This chapter sets up the process for determining the size and composition of a work-
force when some workers enjoy an excess surplus (Synergy) when working together,
relative to other workers. The resulting trade-off between Synergy and effort, as well
as the binding outside option that some workers are assumed to possess, determines
the nature of the Principal’s return and hence informs her hiring decision.
3.1 Model set-up
I first consider the static case, where firms attempt to maximise their profits by
choosing an optimal proportion of workers. Consider a firm that can choose to hire
two kinds of workers: incumbents and newcomers. Incumbent workers are those that
16

22. enjoy a positive Synergy when working together, that is, their joint output from
working together is greater than the sum of their individual inputs. This could be
because they have worked on the same kind of job before, or because they are familiar
with the people they are working with. More formally,
S(eI
av) = vI
(eI
i ,
I
j=1
eI
−i) −
I
i=1
vI
i (eI
i )
Where S(eI
av) is the total Synergy produced by incumbent workers; eI
av is the average
effort exerted by all incumbent workers, specifically, eI
av =
I
i=1
eI
I
; vI
(ei,
I
j=1
eI
−i) is
the joint output of incumbent workers; and
I
i=1
vI
i (eI
i ) is the sum of the independent
outputs of these incumbents. Newcomers, on the other hand, do not enjoy any positive
Synergy so that
S(eN
) = 0.
Consider, T = I + N, where T is the total number of workers hired by the firm, I is
the number of incumbent workers and N is the number of newcomers.
Assumption 1. Each type of worker in the team faces the same production function.
Output produced by all workers is non-negative and increasing in effort i.e. vi(ei) ≥
0, vi(ei) > 0.1
1
For simplicity in this section, no assumptions will be made on the second-order condition of
production.
17

23. Hence, total output by all incumbent workers, I, is given by:
V I
= Ivi(eI∗
) + S(eI
av)
where vi(eI∗
) is each incumbent worker’s output at the equilibrium level of effort,
which is determined by the number of incumbent workers, and S(eI
av) is the total
synergy of incumbent workers. Total output by all newcomers, N, is given by
V N
= Nvi(eN∗
).
Assumption 2. Workers incur a cost of exerting effort and the cost function, Ci(ei),
is non-negative and strictly convex in effort i.e. Ci(ei) ≥ 0, Ci(ei) > 0 and Ci (ei) > 0.
Assumption 3. Synergy is decreasing in effort (that is, dS
deI
av
< 0), but is increasing
in I for a given level of effort i.e. dS
dI
> 0, eI
av > 0
The assumption that Synergy and effort are negatively related describes the following
scenario. Incumbents have a Synergy without effort, on the other hand, newcomers
can make up for their lack of inherent chemistry by putting in effort. That is, where
potential Synergy exists between two agents familiar with each other and can hence
communicate effectively with less effort, the more effort exerted, the less synergies to
be enjoyed. One can also interpret this assumption as being the difference in relative
18

24. returns to effort between incumbents and newcomers, where both incumbent workers
and newcomers have Synergy but newcomers enjoy higher returns for exerting effort.
For example, two unfamiliar strangers may have much more to gain by exerting extra
effort to communicate when working together than two familiar workers. Consider,
for example, co-pilots and dance partners. If this is indeed the case, then one can
normalise a newcomer’s Synergy return to effort as being 0, and hence in relation to
that, an incumbent’s return is negative.
3.2 Timing
The game has the following timing (as shown in Figure 3.1). At time 0, the firm
chooses the number of incumbents and newcomers it wishes to employ and hires them
simultaneously. Those workers then choose their level of effort at time 1. Following
Hart and Moore (1990), effort is non-contractible and similarly an ex ante surplus
sharing agreement is not feasible. At time 2, workers are paid their marginal product
and incumbent workers bargain for their excess surplus (Synergy) with the Principal.
Figure 3.1: Timing of the game.
19

25. 3.3 Bargaining Solution
Given that contracts are incomplete, the parties negotiate ex post over the distri-
bution of surplus.2
To model the distribution of surplus, a version of Shaked and
Sutton’s (1984) outside-offer, alternating-move bargaining game is adopted. In this
game, a modification of Rubinstein’s (1982) infinite-horizon bargaining model is used,
where a party can take up an outside offer after they reject an offer. In equilibrium,
the presence of an outside option only matters if it is binding, and here newcomers will
receive their full outside option. On the other hand, incumbent workers will receive a
payoff higher than their outside option. Specifically, they capture some of the Synergy
of working inside the team on top of their outside possibilities. This modelling ap-
proach has the advantage of possessing the realistic implication that incumbents get
paid different amounts in equilibrium and also has the theoretical advantage of pre-
senting an alternative to the often-used Shapley value method of distributing surplus.3
Assumption 4. An incumbent worker receives their marginal product excluding
Synergy i.e. vi(eI
), as their base wage and bargains with the Principal over their
excess surplus (Synergy). Newcomers receive their marginal product minus a fixed
amount R which the Principal extracts from them as economic rent i.e. they receive
vi(eN
) − R.
2
An alternative, isomorphic assumption could be made, as in Aghion and Tirole (1997), that
these non-contractible investments and surpluses are over-and-above payments (and efforts) that
are made relating to a standard principal-agent incentive contract.
3
The Shapley value method of distributing surplus is used in the next chapter.
20

26. One can think of vi(eI
) and (vi(eN
) − R) as being the workers’ outside options. If
an incumbent leaves, they forgo their share of Synergy and risks only receiving their
marginal product, vi, as a newcomer elsewhere. The outside option for a newcomer is
lower than their marginal product since it is assumed that a newcomer faces additional
costs in finding new employment. One can think of this from the perspective of a
college graduate. Since the graduate has not stayed long enough at their current
workplace, they cannot signal their potential attractiveness as an employee to the
market and hence would incur a greater cost in acquiring alternative employment
compared to an incumbent who can be thought of as an experienced hire.
In the case of the incumbent who generates higher surplus inside the team,
their outside option is not likely to be binding. To capture this, the model allows
the incumbent to share in some of the Synergy. Given the essential nature of the
Principal to production, I assume here that the Principal captures 50 percent of the
Synergy, with the rest shared by all incumbents.4
Assumption 5. Half of the excess surplus is distributed to the Principal and the
remaining half is evenly distributed between each incumbent worker i.e. the Principal
receives 1
2
S(eI
av) and each incumbent worker receives, 1
2I
S(eI
av).
4
As argued in Hart (1995), a firm must centre around some physical asset (even if this asset is
something non-traditional such as a patent or client list). If the Principal provides this asset then
she will play an essential role in the production process.
21

27. It is worth noting that while not innocuous, the qualitative results hold for a range
of alternative (equivalent) assumptions regarding the distribution of surplus, that is,
the exact share of Synergy that the Principal receives does not affect the nature of my
results. I also further explore team composition under an inside-option bargaining
model in the next chapter.
3.4 Investment Decisions
Given Assumptions 2, 4 and 5, incumbents face the following payoff:
πI = vi(eI
i ) +
1
2I
S(eI
av) − ci(eI
i ) (3.1)
And solves the F.O.C.:
vi(eI
i ) = ci(eI
i ) −
1
2I
S (eI
av) (3.2)
Newcomers face the payoff:
πN = vi(eN
i ) − R − ci(eN
i ) (3.3)
And solves the F.O.C.:
vi(eN
i ) = ci(eN
i ) (3.4)
Since Synergy is decreasing in effort (Assumption 3), S (eI
av) < 0 and, vi(eI∗
i ) >
22

28. vi(eN∗
i ), hence:
eI∗
< eN∗
(3.5)
Proposition 1. In equilibrium, incumbent workers have a higher marginal return to
effort and exert a lower level of effort compared to newcomers.
The intuition for this result is that newcomers enjoy greater returns to effort than
incumbents, and parallels the result found in Bel et al. (2013) in their two-player
example. While incumbents have a higher overall surplus, the marginal return they
receive from exerting more effort is relatively low and in this case can even be negative,
depending on the size of Synergy. Newcomers on the other hand, experience strictly
non-negative returns to effort since vi(ei) is strictly increasing in ei and they do not
produce any Synergy. This result is also consistent with many employment situations
where new hires must exert more effort, both to familiarise herself with the dynamics
of her team and to demonstrate to the Principal her value as an employee. In some
ways, I have modelled effort as a substitute for intrinsic Synergy; this sets up the
foundation for many of the results in this thesis.
3.5 Number and type of workers hired
In deciding on the number and composition of workers hired, a Principal wishes
to maximise her profit or net gain. Since the Principal pays every newcomer their
23

29. marginal product minus R and every additional incumbent their marginal product
minus a portion of their Synergy, the Principal effectively incurs no marginal cost in
hiring an additional worker other than any cost incurred in administering them.
Assumption 6. The marginal cost incurred by the Principal consists only of a posi-
tive management cost associated with administering workers, which is increasing in T.
This administrative cost is modelled to be increasing in T since it is likely that as the
number of workers increase, it will be more difficult to administer them. This accords
with models of firms that suggest managers have limited attention and managing a
larger organisation becomes increasingly difficult.5
Given Assumption 4, the Principal’s marginal benefit for hiring an additional
newcomer,
dπN
P
dN
, is constant at R. Furthermore, the Principal’s marginal benefit of
hiring an additional incumbent worker,
dπI
P
dI
, will depend only on the marginal effect on
total Synergy, of which the Principal receives half. This marginal benefit is ambiguous
and depends on the functional form of Synergy.
To illustrate the relevance of the functional form of Synergy, it should be recog-
nised that there are two effects that increasing I will have on the level of Synergy.
Let the overall change in the equilibrium level of Synergy due to a change in I be
denoted as dS∗
dI
, after incorporating any changes in equilibrium effort caused by the
5
For example, Radner and Van Zandt (2001) argues that despite information being costless, the
administrative cost of a firm as increasing in firm size since managers must spend time processing
information.
24

30. change in I. For a given level of effort, an increase in I will increase total Synergy,
i.e. dS
dI
> 0. If Synergy increases at an increasing rate with I (i.e. d2S
dI2 > 0), then as
I increases, each worker receives a greater share of S which should decreases optimal
effort and further increases S (Assumption 3). Hence dS∗
dI
> 0, when d2S
dI2 > 0. How-
ever, when Synergy increases with I at a decreasing rate (i.e.d2S
dI2 < 0), the effect on S∗
is a priori ambiguous. This is because every additional incumbent worker adds less
to total Synergy than the previous one and hence lowers the the amount of Synergy
received by each individual worker, leading to an increase in optimal effort and an
overall ambiguous effect on S∗
since effort and Synergy are substitutes.
It should also be noted that since the Principal’s only return from hiring incum-
bent workers is half their total stock of Synergy (i.e. the Principal receives no share
of vi(eI
)), the Principal will never hire incumbent workers when the equilibrium level
of Synergy is strictly decreasing in I i.e. dS∗
dI
< 0. This is because profit would be
decreasing for every incumbent hired.
Proposition 2. A Principal will never hire additional incumbent workers if the re-
turn to doing so is negative i.e. dS∗
dI
< 0.
In order to analyse the Principal’s choice in deciding the size and composition of a
workforce in a meaningful way, in the following sections I consider instances where
hiring additional workers lead to positive returns. That is, I consider the situations
25

31. when both
dπI
P
dI
(equivalently, dS∗
dI
) and
dπN
P
dN
(= R) are positive.
3.5.1 Principal’s marginal return increasing in I
The Principal’s return from hiring additional incumbent workers is positive when
dS∗
dlI
> 0, and is increasing if d2S∗
dI2 > 0. In this case, the Principal’s return is positive
and increasing because the equilibrium change in Synergy is positive and increasing
in I.
When an increase in I leads to an increase in dS∗
dI
, the Principal’s decision is
binary. Either the Principal hires only incumbent workers or only newcomers and
does so by comparing the overall benefit of hiring newcomers with the overall benefit
of hiring incumbents. To see this, consider the illustration of the Principal’s choice
shown in Figure 3.2.
Figure 3.2: Principal’s return is increasing in I
26

32. The point where the marginal benefit of hiring an additional newcomer (
dπN
P
dN
)
intersects the marginal cost function (dC
dT
) is the optimal number of newcomers the
Principal can hire when choosing newcomers, hence it can be denoted as N∗
. Simi-
larly the point where the marginal benefit of hiring an additional incumbent worker
(
dπI
P
dI
) intersects the marginal cost function, is the optimal number of incumbents the
Principal can hire when choosing incumbents, hence it can be denoted as I∗
.
The benefit of hiring incumbents can be found by calculating the area under
dπI
P
dI
but above the cost curve up to point I∗
, namely CI∗
2
. Similarly, the benefit of hiring
newcomers can be found by calculating the area under
dπN
P
dN
but above the marginal
cost curve up to point N∗
, namely N∗(D+C)
2
.
The Principal will only hire incumbents (newcomers) if
CI∗
2
> (<)
N∗
(D + C)
2
,
which can be simplified to
I∗
> (<)
N∗
(D + C)
C
or
I∗
> (<)
N∗
[R − C (0)]
πI
P (0) − C (0)
.
27

33. The Principal is otherwise indifferent, i.e. if
I∗
=
N∗
(R − C (0))
πI
P (0) − C (0)
.
Equivalently, what the Principal is comparing is areas X and Y as shown on Figure
3.2 above. If area X is greater than area Y, then the Principal will hire only newcom-
ers. Alternatively, if area Y is greater than area X, then the Principal will hire only
incumbent workers.
Proposition 3. If Synergy is increasing in I, then the Principal’s hiring decision is
binary and will hire either all incumbents or all newcomers.
Proof. See Appendix.
This binary hiring decision, where the Principal hires only incumbents or new-
comers, occurs because in the case where Synergy is increasing in I, the Principal’s
marginal benefit is non-negative in hiring either incumbents or newcomers. Hence in
order to maximise total benefit, the Principal no longer undertakes the usual marginal
analysis typically employed in economics, by considering the contribution of the last
incumbent (or newcomer) hired. Instead, the Principal’s hiring decision is affected by
the infra-marginal contribution of each each worker. Given the Principal captures half
of the Synergy, she considers how hiring an additional incumbent affects the Synergy
28

34. she captures from all incumbents, which is affected by infra-marginal workers (and
not just the additional worker used).6
Thus as Synergy is increasing in I, the Principal considers the total return from
all incumbents or all newcomers and a ‘corner solution’ results. A parallel result can
be found in the wage-bargaining model of Stole and Zwiebel (1996) in which the
contribution of infra-marginal workers affects the total number of employees a firm
wishes to hire. 7
Alternatively, one can think of this result occurring because when Synergy
is increasing in I, the Principal may tolerate lower returns at lower levels of I if
returns are high enough at greater levels of I. The Principal’s decision depends on
the size overall of Synergy, relative to the rent it can extract from newcomers. This
is consistent with the empirical observation that skilled workers are less likely to
be laid off during times of economic downturn. If a role requires a lot of learning
(equivalently, there is much potential Synergy to be developed), then a Principal is
more likely to favour a team of incumbents. On the other hand, for roles where there
is little learning involved (or equivalently, there is little to be gained from hiring
incumbents) then a Principal may prefer to hire newcomers.
Clearly, in situations where there are huge benefits to hiring incumbents, that
6
The same argument applies to when the two areas are equal. When X and Y are equal, the
Principal’s total benefit is equal when hiring N∗
or I∗
, and the Principal is indifferent between hiring
newcomers and incumbents. However the Principal’s decision will remain binary and she will hire
either one or the other.
7
Here, the infra-marginal worker needs to be considered of their effect on the total level of Synergy
created. In contrast, in Stole and Zweibel (1996), infra-marginal workers are important because of
the impact that have on the wage-bargaining power of employees.
29

35. is, there are extremely high levels of potential Synergy, incumbents will always be
favoured.
Proposition 4. Assuming a linear marginal cost of hiring additional workers, if dS∗
dI
is strictly convex in I then the Principal will only hire incumbent workers.
This is a special case of Proposition 3 and occurs because when equilibrium Synergy
is convex in I,
dπI
P
dI
is increasing at an increasing rate and it is thus always beneficial
for the Principal to hire more incumbent workers (her marginal benefit is increasing
at an increasing rate), so that I∗
would equal infinity. However, assuming that firms
do not enjoy increasing returns to labour, this situation is unlikely to occur as one
would expect that the team cannot be expanded to infinity. Thus, from here on I will
not consider this case of equilibrium Synergy being convex in I, as it is unrealistic
and its absence does not detract from the subsequent analysis.
3.5.2 Increasing returns; an example
To provide some additional intuition, consider the following example. Consider the
case where the production function of all workers is vi(ei) = ei but where incumbent
workers jointly produce an excess surplus of Synergy which takes the functional form,
S = (I + 1)I1/2
(A − eI
av) where A > 1. All workers face the cost function ci = 1
2
e2
.
Also assume in this simple example that the fixed cost of hiring workers is zero and
30

36. that the total cost function takes the form βT.
When optimising return with respect to effort, incumbent workers solve the
following problem,
max
e
πI = eI
i +
α(I + 1)I1/2
2I
(A − eI
av) −
1
2
(eI
)2
(3.6)
We can rearrange to find equilibrium level of effort:
eI∗
= 1 −
α
2
√
I
−
α
2I3/2
(3.7)
Since the Principal receives half of all synergies, the Principal’s return is:
πI
P =
1
2
α(I + 1)I1/2
(A − eI
av) =
1
2
α(I + 1)I1/2
(A − 1 +
α
2
√
I
+
α
2I3/2
)
πI
P =
α
2
(I3/2
+ I1/2
)A −
α
2
(I3/2
+ I1/2
) +
α2
4
(I + 1) +
α2
4
+
α2
4I
(3.8)
To find the effect of increasing I on the Principal’s marginal return, take the derivative
of πI
P with respect to I to get,
dπp
dI
=
3
4
αI1/2
(A − 1) +
1
4
αI−1/2
(A − 1) +
α2
4
−
α2
4I2
(3.9)
Here,
dπI
P
dI
is positive, and increasing,
d2πI
P
dI2 > 0, but at a decreasing rate since
d3πI
P
dI3 < 0.
31

37. Hence the firm calculates the area bounded by
dπN
P
dN
and dC
dT
up to the point N∗
and
compares it to the area bounded by
dπI
P
dI
and dC
dT
, up to the point I∗
(Figure 3.3 below).
The Principal is essentially comparing the net benefit from hiring incumbent workers
with the net benefit of hiring newcomers. The Principal will hire only incumbent
workers (newcomers) if
πI
P ≥ (≤)πN
P
I∗
0
(
3α
4
√
I(A − 1) +
α
4
I−1/2
(A − 1) +
α2
4
−
α2
4I2
− βI)dI ≥ (≤)
N∗
R
2
For simplicity and without affecting any qualitative results, assume that A = 2 and
β = 1 and integrate to yield,
α(I∗3/2
+ I∗1/2
) +
α2
2
(I∗
+ I∗−1
) − I2
≥ N∗
R
Hence the Principal will only hire incumbent workers (newcomers) if
R ≤ (≥)
α(I∗3/2
+ I∗1/2
) + α2
2
(I∗
+ I∗−1
) − I2
N∗
(3.10)
Here, Synergy is an increasing function of I but a decreasing function of effort.
However, equilibrium effort increases as I increases (Equation 3.7), hence an increase
in I leads to a positive marginal return to the Principal that is also increasing (at
a decreasing rate). Furthermore, α increases the likelihood of a Principal hiring
incumbent workers as it has a scale effect on Synergy, that is, the higher α is, the
32

38. Figure 3.3: Synergy increases with I at a decreasing rate
higher the level of Synergy adding an incumbent worker will provide. From Equation
3.7 one can also see that equilibrium effort is decreasing in α, which means that
the higher α is, the less equilibrium effort will be exerted by incumbent workers
(Assumption 3), and hence the more Synergy extracted by the Principal.
Hence, as α increases, the Principal is more likely to hire incumbent workers
(Equation 3.10). This accords with the idea that the higher the potential Synergy
relative to R, the more a Principal is likely to prefer incumbent workers. For example,
a music band may have a lot to benefit from working with familiar team members
with whom they can easily collaborate (effectively having a higher α). Alternatively,
a cashier at a supermarket may face little scope to develop Synergy with her fellow
cashiers. Hence a Principal is more likely to favour hiring a group of incumbents
when faced with a team of musicians so as to benefit from their potential Synergy,
compared to a team of cashiers.
33

39. 3.5.3 Principal’s marginal return decreasing in I
Now consider the case when the Principal’s marginal return from hiring an additional
incumbent worker is positive (
dπI
P
dI
> 0) but decreasing (
d2πI
P
dI2 < 0). This occurs when
the change in equilibrium Synergy is positive and decreasing i.e. dS∗
dI
> 0; d2S∗
dI2 > 0.
This follows the idea that an additional incumbent worker leads to an increase
in total Synergy but adds less than the previous incumbent worker. Recall that for
a given level of effort, an increase in I will increase total Synergy (Assumption 3),
however since each incumbent worker adds less than the last, each worker receives a
smaller absolute amount of Synergy as I increases. One must then take into account a
second effect where a smaller absolute amount of Synergy leads to incumbent workers
exerting more effort at equilibrium (as Synergy and effort are substitutes), which
works in the direction of decreasing Synergy.
When the Principal’s return is positive but decreasing in the number of incum-
bent workers hired, the Principal’s decision is no longer necessarily binary and the
usual marginal analysis applies. In the standard way, a Principal will hire incumbent
workers up to the point where her marginal return from doing so (
dπI
P
dI
), is equal to
her marginal return from hiring newcomers (
dπN
P
dN
). This critical point is given by I∗
.
This is because at any point prior to I∗
,
dπI
P
dI
>
dπN
P
dN
. At any point after I∗
the Princi-
pal’s marginal return from hiring incumbents is lower than her marginal return from
hiring newcomers, hence the firm will then hire newcomers up to the point where
dπN
P
dN
intersects the marginal cost curve dC
dT
at N∗
.
34

40. Figure 3.4: Firm hires incumbents and newcomers
Proposition 5. When the Principal’s marginal return from hiring an additional in-
cumbent worker is positive but decreasing, the Principal hires incumbent workers up to
the point where
dπI
P
dI
=
dπN
P
dN
and then hires newcomers up to the point where
dπN
P
dIN
= dC
dT
.
This general case where equilibrium Synergy is increasing in I but at a decreasing rate
is an intuitive one. One can think of this situation as being one where the Principal
uses incumbent workers as a labour input, and that each of these workers adds to the
Principal’s total benefit, but at a decreasing rate. Hence this situation is akin to one
where a firm faces diminishing returns to labour, except in this case the Principal is
deciding between an input with high but diminishing returns and one with lower but
constant returns.
Note that in the above figure, it is no longer profitable for the Principal to
35

41. hire any workers after N∗
, hence it is the level of R that determines the optimal
size of the team. However, if R is very low in comparison to Synergy, such that
dπN
P
dI
intersects with the marginal cost curve after
dπN
P
dN
, then the Principal will only hire in-
cumbent workers and the number of workers is determined by I∗
. See figure 3.5 below.
Proposition 6. When Synergy is very high or equivalently when R is very low, the
optimal team size is determined by I∗
. Alternatively, when R is relatively high, the
optimal team size is determined by N∗
.
Figure 3.5: Firm hires only incumbents
In this case, the Principal hires only incumbent workers because the marginal product
of adding an incumbent worker is greater than the marginal product of adding a
newcomer when it intersects with the marginal cost curve at I∗
. One can think of
this kind of situation occurring when jobs require a high level of interaction between
36

42. workers in order to create or deliver a final product and hence the level of Synergy
is relatively high. For example, television performers such as The Wiggles or Hi-5
yield extremely high levels of Synergy whilst performing together. Music bands (such
as the Backstreet Boys or the Beatles) experience a similar Synergy as a group that
would be greatly diminished if a Principal were to replace one member for a newcomer.
Judas Priest and Iron Maiden are both successful rock bands that experienced a fall in
popularity after replacing their lead singers. For example, After Judas Priest replaced
their lead singer Rob Halford with Timmy Owens, they produced two poorly received
albums despite fans’ initially positive reactions to the addition of Owens.
3.5.4 Decreasing returns; an example
Again, to provide some further intuition, consider the following example. Consider the
case where the production function of all workers is vi(ei) = ei but where incumbent
workers jointly produce an excess surplus of Synergy which takes the functional form,
α
√
I(A − eI
av). All workers face the cost function ci = 1
2
e2
. Recall that,
I
i=1
eI
I
When optimising return with respect to effort, incumbent workers solve the following
problem,
max
e
πI = eI
i +
α
√
I
2I
(A − eI
av) −
1
2
eI2
(3.11)
37

43. Rearranging to find equilibrium level of effort,
eI∗
= 1 −
α
2I3/2
(3.12)
Since the Principal receives half of all synergies, the Principal’s return is:
πI
P =
1
2
α
√
I(A − eI
av) =
1
2
α
√
I(A − 1 +
α
2I3/2
)
πI
P =
α
2
√
IA −
α
2
√
I +
α2
4I
(3.13)
To find the effect of increasing I on the Principal’s marginal return, take the derivative
of πI
P with respect to I to get,
dπI
P
dI
=
α
4
√
I
[A − 1] −
α2
4I2
. (3.14)
Since α < 1 and A > 1,
dπI
P
dI
is positive, but is decreasing i.e.
d2πI
P
dI2 < 0, and at an
increasing rate (
d3πI
P
dI3 > 0). Similar to the general example, if R is comparatively high
or equivalently, if Synergy is sufficiently low, then the firm will hire a combination of
incumbent workers and newcomers. The firm will hire incumbents up to I∗
and then
newcomers up to N∗
. See figure 3.6 below.
Here, though S∗
is increasing in I, the substitution effect that increasing I has
on effort leads to a smaller and smaller increase in overall S∗
and hence πI
P . In this
particular case, if I becomes too large, the substitution effect of an increase in effort
38

44. Figure 3.6: Firm hires incumbents and newcomers
will overwhelm the scale effect and lead to very low (here, even negative) marginal
returns to I.
This kind of mixed team outcome is consistent with situations where there is
some level of Synergy the Principal would like to foster, but it is not large enough
justify hiring only incumbent workers, since the rent the Principal is able to extract
form newcomers is still relatively high. For example, fast food giants such as KFC
Ltd. are known to hire young, unskilled workers often still in school, as they attract
a much lower wage.8
However, these companies also keep a portion of their workers
until they are much older and hence receive a higher wage. This is because there is
a benefit to be gained in hiring older workers (incumbents) who work well together
and are capable of teaching newcomers. But the Principal only benefits substantially
8
KFC Australia hires workers as young as 15. Between 2001 and 2009, a 15 year old worker at
KFC earned 40 percent of the adult award wage, and only 90 percent of that 40 percent (effectively,
36 percent of the adult wage) in the first six months of employment (Industry Source, KFC National
Enterprise Award 2001).
39

45. from a part of the workforce being incumbents, after which it becomes much more
lucrative to hire young, unskilled newcomers at a steep discount.
3.6 Summary
The preceding chapter illustrates the decision making process of the Principal in
deciding the size and composition of her workforce. Under the bargaining method
employed in this chapter, the Principal’s return does not depend on effort, except
where it affects the total equilibrium level of Synergy produced by incumbents. Hence,
the Principal’s decision is driven by the level of Synergy relative to the rent she can
extract from newcomers. When equilibrium Synergy is increasing at an increasing
rate with I, the Principal will consider the infra-marginal returns of each worker
and hire either all incumbent workers or all newcomers. Alternatively, if equilibrium
Synergy is increasing at a decreasing rate, then the Principal will hire incumbents
until the marginal return to doing so falls below the marginal return from hiring
newcomers. As a result, unless the level of economic rent the Principal can extract
from newcomers is very low, the optimal team size is determined by R.
As the distribution of surplus in this chapter has been modelled in such a
way that the Principal extracts economic rent from newcomers and extracts Synergy
from incumbents, her incentives are primarily driven by the interaction of Synergy,
equilibrium levels of incumbents and the number of incumbent workers. It is arguably
important then, to consider alternative bargaining solutions so as to see that these
40

46. results are driven by the inherent trade-off between Synergy and effort, and not by
this specific bargaining model. An alternative bargaining solution is thus considered
in the next chapter.
41

47. Chapter 4
Team composition: inside-option
bargaining
The Principal’s decision in hiring workers will always be driven by her incentives.
Hence, as noted before, the way surplus is distributed in a firm is an important con-
sideration. While in the last chapter the Principal received none of the non-Synergy
return of her agents, in this chapter, I develop an alternative model of optimal worker
composition, where the level of effort of both incumbents and newcomers directly
affect the Principal’s payoff. Specifically, I follow Grossman and Hart (1986), Hart
and Moore (1990), and Mai, Smirnov and Wait (2014, forthcoming), in considering
an inside-option (Shapley value) distribution of surplus.
42

48. 4.1 Bargaining Solution
Consider the case of a single Principal (without whom, no production is possible), I
incumbent workers and N newcomers.
Assumption 7. Assume that workers and the firm bargain over total output i.e.
incumbent workers bargain with the firm over vi(eI∗
) and SI
, and newcomers bargain
with the firm over V N
.1
Let δ be the vector of players (P, N1, N2, ...NN , I1, I2, ...II) and i ∈ δ. The shares of
surplus is given by the Shapley value which is defined below,
Bi =
M|i∈M
p(M)[v(M|δ) − v(M|δ)]
where p(M) = (|M|−1)!(|N|−|M|)!
(|N|)!
For details, see Hart and Moore (1990).
When the Principal hires an additional newcomer, each newcomer’s absolute share
of output does not change since they each contribute a constant amount (vN ) with
no Synergies. Hence regardless of how many incumbent workers and newcomers the
1
As opposed to Assumptions 4 and 5 in Chapter Three.
43

49. Principal hires, each newcomer will receive
1
2
vN . (4.1)
Now consider the process with incumbents. Since the Principal decides how
many people to hire and hires all of them simultaneously, when there is SI
to be
gained, SI−1
is zero and each incumbent agent contributes the same amount to SI
(For details, see Appendix A). Hence, each time the Principal hires an additional
incumbent worker, each incumbent worker receives a smaller share of total Synergy,
as well as half their non-Synergy output. Namely, each incumbent worker receives
1
2
vI +
SI
(eI
av)
I + 1
. (4.2)
Lastly, the Principal receives
N
2
vN +
I
2
vI +
SI
(eI
av)
I + 1
. (4.3)
While in the last chapter, the Principal’s return depended only on a fixed
amount R extracted as rent from newcomers and a fraction of Synergy from in-
cumbents, here the Principal’s return depends on both the effort levels of incumbent
workers and newcomers, as well as a fraction of Synergy from incumbents.
44

50. 4.2 Investment Decisions
Here, the investment decisions are similar to the outside-option case, except that
agents now only receive half their non-Synergy output. Incumbent workers receive
the payoff:
πI =
1
2
vI
i (eI
i ) +
S(eI
av)
I + 1
− cI
i (eI
i ) (4.4)
And solves the F.O.C.:
1
2
v I
i (eI
i ) = ci(eI
i ) −
S(eI
av)
I + 1
(4.5)
Newcomers face the payoff:
πN =
1
2
vN
i (eN
i ) − cN
i (eN
i ) (4.6)
And solves the F.O.C.:
1
2
v N
i (eN
i ) = ci(eN
i ). (4.7)
Since Synergy is decreasing in effort (Assumption 3), S (eI
av) < 0 and vI
i (eI∗
i ) >
v N
i (eN∗
i ), leading to eI∗
< eN∗
. Hence Proposition 1 continues to hold under bargain-
ing and at equilibrium; incumbent workers have a higher marginal return to effort
and exert a lower level of effort compared to newcomers.
45

51. 4.3 Number and type of workers hired
As in the previous case, the Principal’s marginal benefit from hiring a newcomer,
dπN
P
dN
is constant at R = 1
2
vN
. Similar to before, when Synergy increases at a decreasing
rate with I, incumbent workers exert greater levels of effort at equilibrium since they
are receiving a smaller share of Synergy. And alternatively when Synergy increases
at an increasing rate with I, incumbent workers will exert less effort at equilibrium.
However, the Principal’s marginal benefit of hiring an additional incumbent
worker,
dπI
P
dI
, will now depend on both the effect on vi(eI∗
i ) and on total Synergy. That
is, the Principal now has to take into consideration that while hiring an additional
newcomer does not affect the level of effort exerted by all other newcomers, hiring
an additional incumbent worker with the effect of increasing total Synergy will come
at the cost of decreased non-Synergy return. In other words, the Principal now faces
a direct trade-off between Synergy and effort since increased Synergy leads to lower
effort and hence lower non-Synergy output, of which the Principal receives a share.
4.3.1 The change in equilibrium Synergy increasing in I
In the previous chapter (3.5.1) it was established that when the change in equilibrium
Synergy is positive and increasing, i.e. d2S∗
dI2 > 0, the Principal’s marginal return would
be positive (since she simply received half of total Synergy) and increase in I and
hence her hiring decision would be binary. As a result, the Principal’s decision was
driven by the size of Synergy relative to the rent it could extract from newcomers.
46

52. Here, however, the Principal’s decision is affected by a direct trade-off between effort
and Synergy, hence it does not necessarily follow that a positive and increasing level
of equilibrium Synergy means increasing marginal returns.
Now, when dS∗
dI
is positive and increasing in I, for every additional incumbent
worker hired the Principal gains from increasing returns to Synergy but at the same
time, forgoes some non-Synergy output due to lower equilibrium effort levels. Using
the same example as in section 3.5.1, consider the case where the production function
of all workers is vi(ei) = ei, and incumbent workers jointly produce an excess surplus
of Synergy of S = (I + 1)I1/2
(A − eI
av), A > 1. All workers face the cost function
ci = 1
2
e2
, the fixed cost in hiring workers is zero, and the marginal cost function is
βT. When optimising return with respect to effort, incumbent workers now solve the
following problem,
max
e
πI =
1
2
eI
i +
α(I + 1)I1/2
I + 1
(A − eI
av) −
1
2
(eI
)2
(4.8)
Rearranging to find equilibrium level of effort,
eI∗
=
1
2
−
α
√
I
. (4.9)
The Principal’s return is:
πI
P =
I
2
(
1
2
−
α
√
I
) +
α(I + 1)I1/2
I + 1
(A −
1
2
+
α
√
I
)
47

53. πI
P =
1
4
I −
1
2
αI1/2
+
1
2
α(A −
1
2
) + α2
. (4.10)
To find the effect of increasing I on the Principal’s marginal return, take the derivative
of πP with respect to I to get,
dπp
dI
=
1
4
−
1
4
αI−1/2
+
1
2
αI−1/2
(A − 1). (4.11)
Here,
dπI
P
dI
is positive, and decreasing,
d2πI
P
dI2 < 0, at an increasing rate (
d3πI
P
dI3 > 0).
Hence, in contrast to the predictions of the previous chapter, rather than an upward
sloping
d2piI
P
dI2 and a binary hiring decision to hire only incumbent workers, the Principal
hires incumbents up to I∗
and then newcomers up to N∗
i.e. the team is mixed. See
figure 4.1 below.
Figure 4.1: Firm hires incumbents and newcomers
This downward sloping curve reflects that despite the incumbent workers en-
joying the same level of Synergy as in 3.5.1, the Principal’s marginal return from
48

54. hiring incumbent workers is now relatively lower at higher levels of I, compared to
the previous chapter. This is because, while in the last chapter, the Principal’s re-
turn from hiring incumbent workers was simply half of all total Synergy. Here, the
Principal’s return also incorporates the non-Synergy output of incumbent workers,
which is lowered by the presence of Synergy. Hence despite the change in equilibrium
Synergy being positive and increasing in I, the Principal’s marginal return is positive
but decreasing in I.
This means that under the bargaining solution of this chapter, the Synergy
produced by incumbent workers must be larger in order to induce the Principal to
hire incumbent workers since the Principal must now also consider this additional
trade-off between Synergy and non-Synergy surplus.
Proposition 7. When the Principal’s payoff depends on both the overall level of Syn-
ergy and the overall level of effort, she faces an additional tradeoff between Synergy
and effort and is less likely to hire incumbent workers than in the case where her
payoff depends only on equilibrium Synergy.
A mixed team resulting from this kind of mechanism also has practical implications.
Consulting teams or teams of screenwriters are examples where a mixed team may be
optimal (rather than a team of incumbents), due to the trade-off between effort and
Synergy. Whereas incumbents may know all the tricks of the trade and be familiar
49

55. with a particular problem (e.g. market entry) or genre (e.g. Romantic Comedy),
hiring an additional incumbent worker may be less valuable than hiring a newcomer
with new ideas and a fresh perspective. Empirically, television shows often have
teams of writers which are rotated through for different seasons and consulting teams
are often comprised of members with different tenures. Broadway productions have
also been shown to be more successful when containing a team of incumbents and
newcomers (see e.g. Uzzi and Spiro 2005). One can think of this as being a balance
of the natural synergy of incumbents and the energy and effort of newcomers.
Notably, in this case of when the Principal’s return from hiring incumbent work-
ers is positive but decreasing, while it is possible to have a team consisting only of
incumbent workers 2
, it is never possible to have an optimal team consisting of only
newcomers. This is because, by definition, incumbent workers enjoy an excess sur-
plus above their vi. Since all workers face the same production function (Assumption
1), Synergy requires at least two incumbents, and the Principal receives half of the
non-Synergy output of all workers, the Principal’s marginal return from hiring in-
cumbent workers cannot intersect the vertical axis (see Figure 4.1) at a point below
her marginal return from hiring newcomers.
Proposition 8. It can never be optimal for the Principal to hire only newcomers
when surplus is distributed using the Shapley Value.
2
A team consisting of only incumbent workers occurs when Synergy is extremely high, resulting
in
dπI
P
dI intersecting the marginal cost curve at a point later than
dπN
P
dN . Namely, I∗
is greater than
N∗
50

56. This result differs from the previous chapter where the Principal’s return from hiring
incumbent workers depended only on equilibrium Synergy. In the previous chapter,
despite incumbent workers enjoying positive Synergy above vi, a Principal’s decision
was essentially to compare R (the level of economic rent a Principal could extract from
newcomers due to switching costs) relative to the benefit of experience (Synergy).
Hence, for roles which have little scope for developing Synergy, a Principal would
opt to hire only newcomers. Here, however, the Principal’s return depends on both
the Synergy and non-Synergy output of both types of workers, and by definition,
incumbents are capable of producing more at the same level of effort. It is the
inclusion of vi (non-Synergy output of incumbents) that leads to Proposition 8.
4.4 Summary
In the last chapter, the outcome for team composition was driven by the relative sizes
of Synergy and R. This chapter demonstrated that the prevailing bargaining solution
affects the Principal’s decision in hiring a workforce, by using an alternative mecha-
nism to drive the Principal’s decision. Under the inside bargaining model employed
in this chapter, effort plays a more prominent role in the Principal’s payoff function
and hence the Principal is more likely to hire newcomers. A parallel result can be
found in the property rights literature (see e.g. Chiu 1998, De Meza and Lockwood
1998). The difference in the Principal’s hiring decision in this chapter compared to
51

57. the previous one also generalises the result found in Bel et al. (2013) which finds
that a Principal may prefer to hire a team of newcomers too often, by showing that
analysis on the choice of team is highly dependent on the bargaining solution used.
However, the underlying trade-off between effort and Synergy has not changed.
The Principal is more likely to hire newcomers because vi features more prominently
in her payoff function but the basic trade off between promoting the natural Synergy
between incumbent workers and exploiting the higher effort levels of newcomers is
intuitively evident in both cases.
52

58. Chapter 5
Team implications
The preceding chapters demonstrated the importance of the bargaining method for
the Principal’s hiring decisions and the trade-off that exists between Synergy and
effort when deciding on team size and especially team composition. In this chapter, I
consider the implications on job rotation, in particular when a Principal may rotate
team members too often or too little. I also consider situations where agents would
prefer to make the team composition decision themselves.
5.1 Job Rotation
Extending the previous static models to a model of rotation is straightforward. Using
the same intuition as in the static cases, the pivotal question is how the marginal
return to the Principal changes over time. This will in turn depend on how S∗
changes over time, as well as the relevant bargaining method. For a given, pre-
53

59. established optimal level of I, the Principal may choose to never break up teams of
incumbents because they become increasingly more productive in a way that benefits
her, or she may find it optimal to frequently rotate in newcomers in order to stimulate
effort levels.
One can think of this kind of job rotation occurring under two situations. One
where workers are of inherently different types and a firm can hire teams of incumbents
whose Synergy will evolve over time, or hire teams of newcomers who will never
develop any Synergy. Alternatively, one can think of all workers as being identical
and newcomers are simply workers who have not previously worked with each other.
Then, upon being paired with another worker, they will develop Synergy over time
and hence become incumbents.
While the benefit of rotation in the former case of inherently different workers
can be informed by the existing literature on team composition (e.g. Prat’s (2002)
analysis of the optimal level of workforce heterogeneity). I will explore job rotation
in the context of the latter case, and show that even in a framework where workers
are inherently similar, there can be a case made for rotation between workers. This
kind of analysis is realistic in many employment settings, especially for jobs requiring
a high level of team work, communication and coordination. Two very highly skilled
workers with years of experience may enjoy very little excess surplus in the form of
Synergy when first working together, simply because they are unfamiliar with the
other person. However, over time they may develop familiarity and increase their
54

60. natural Synergy.
Consider a simple case of a firm that places its homogeneous workers in teams
of two. These workers start off as newcomers, and slowly develop Synergy over time,
which is denoted as t. It is reasonable to assume that these workers enjoy an increas-
ing Synergy the longer they work together. To illustrate increasing total Synergy over
time, I introduce the following assumption.
Assumption 8. The equilibrium Synergy between two workers is given as a function
of time and effort, S∗
(t, e), and is an increasing function of time. i.e. S(0, e) = 0,
dS∗(t,e)
dt
> 0.
Recall that the Principal’s benefit, πI
P , will depend on the equilibrium change in
Synergy (after taking into account a change in equilibrium effort), as well as the bar-
gaining method in place (Proposition 7).
Proposition 9. If
dπI
P
dt
is strictly increasing, then the optimal team always consists
of incumbents.
55

61. Figure 5.1: No rotation
Figure 5.1 illustrates the case when the marginal return to the Principal is always
increasing when two workers are left to stay together, compared to the case when the
marginal return to the Principal is constant when workers remain newcomers (either
because they work independently or are rotated so often that they have not had a
chance to develop Synergy) i.e.
dπI
P
dt
≥
dπN
P
dt
. In this case, it is never optimal for the
Principal to break up the incumbent pair and rotate in a new worker. Such a situation
is more likely to occur if the Principal cares only about the total level of Synergy and
derives no direct benefit from vi(eI∗
), as in Chapter three.
Alternatively, Figure 5.2 illustrates the case where the marginal return to the
Principal increases first and then decreases over time as two workers stay together.
Since upon rotation, the new team yields the same return to the Principal as workers
working independently, it is optimal for the Principal to allow the two workers to
stay together as long as
dπI
P
dt
≥
dπN
P
dt
, and rotate them when
dπI
P
dt
≤
dπN
P
dt
. Since Synergy
56

62. Figure 5.2: Rotation
and effort are substitutes, this case is more likely to occur when the Principal also
cares about non-Synergy output (as in Chapter 4). This is because effort may decline
to a point where the trade-off between nurturing Synergy growth and limiting effort
loss means that it becomes beneficial for the Principal to forgo any more potential
Synergy and instead rotate in a new worker to stimulate effort.
Proposition 10. If
dπI
P
dt
is increasing up to a certain point, after which it is decreas-
ing, then it is optimal for the Principal to rotate in a newcomer after time X∗
.
Similar to the static cases, this result highlights the importance of taking into account
the kind of bargaining involved when considering whether or not is is optimal for a
firm to be rotating its workers, and if so, how often.
57

63. 5.2 The Principal’s choice and the second-best
If it were possible to write a complete contract and enforce effort levels, then the
first-best outcome would always entail using incumbent workers. This is because,
for any given level of effort, incumbent workers enjoy an additional surplus in the
form of Synergy and hence to maximise total surplus, incumbents would be used
and no rotation would ever occur. However, given that the environment assumed
by this thesis is one where contracts are incomplete, there may be some second-best
optimal level of rotation that will maximise total surplus. I will henceforth refer to
this second-best optimal as the social optimal.
Consider the case where the overall change in total output by the team, V I
, is
increasing with time at first but then decreases, as illustrated in Figure 5.3.
Figure 5.3: Optimal Rotation
In Figure 5.3, dV I
dt
illustrates how the total output of a team evolves over time,
58

64. and V N
is simply the total output if agents worked independently, or equivalently,
are rotated so often they always remain newcomers. The socially optimal time of
rotation is consequently at t = X∗
.
Now consider when the Principal’s return depends only on Synergy (in the same
vein as Chapter three), or even that it depends disproportionately more on Synergy
than on non-Synergy output. Then the Principal’s decision does not take into account
the decrease in non-Synergy output due to a decrease in effort, or at least does not
place enough weight on it, and her decision to rotate workers will be governed by
dπI
P
dt
in Figure 5.4 rather than dV I
dt
. This leads to the Principal rotating workers at time
X, where X < X∗
and the Principal engages in too little rotation. In extreme cases
where the Principal does not care about non-Synergy output and Synergy is very high,
leading to
dπI
P
dt
resembling that of Figure 5.1, then she will never rotate her workers
despite it being socially optimal to do so at X∗
. Hence, under the outside-option
bargaining solution considered in Chapter 3 where the Principal’s marginal return
from hiring incumbents depends only on change in equilibrium Synergy, a Principal
will care too much about increasing Synergy and will likely rotate her workers too
infrequently, or not at all.
Alternatively, consider when the Principal’s return depends disproportionately
on the level of non-Synergy output (such as the case in Chapter four where the
Principal receives half of all non-Synergy output but a share of Synergy divided
equally between each incumbent worker and the Principal). Then the Principal’s
59

65. Figure 5.4: Optimal Rotation
return and hence her decision is affected disproportionately by the decrease in effort
induced by increasing Synergy. As illustrated in Figure 5.5 the Principal’s decision is
governed by
dπI
P
dt
, and she will rotate workers at time X , where X < X∗
, and hence
the Principal rotates her workers too often compared to the social optimum. Thus,
under an inside-option bargaining solution such as that considered in Chapter four
(Shapley value) where the Principal’s marginal return from hiring incumbents depends
disproportionately on non-Synergy output, the Principal will care too much about
increasing effort and rotate her workers too frequently. This situation is summarised
in the following Proposition.
60

66. Figure 5.5: Optimal Rotation
Proposition 11. The Principal may not rotate workers often enough when her return
depends disproportionately on Synergy and she may rotate workers too often when her
return depends disproportionately on non-Synergy output.
This result generalises a parallel result found in Bel et al. (2013), that a Principal
tends to choose newcomers too often in their two player static analysis. While their
result always works in one direction, here the Principal may rotate too much or too
little (equivalently, choose newcomers too often or not enough), depending on the
bargaining method applied.
Since the bargaining solution affects the frequency with which a Principal ro-
tates her workers, these results may provide a foundation for empirically testing the
bargaining process at work in real-world studies of team rotation. Specifically, if a
Principal is under-rotating her workers it is likely that she faces a bargaining solution
61

67. akin to the outside-option model considered in Chapter three, and if she is over-
rotating her workers then it is likely that she faces something similar to the Shapley
value method of distributing surplus considered in Chapter four.
These results on rotation also have empirical implications for organisational
structure. If a Principal’s return is disproportionately dependent on excess surplus
generated by the chemistry of team members, then she is very unlikely to rotate her
workers. For example, the manager of a performing group who benefits from sales of
existing merchandise may prefer to maintain the existing team for as long as possible
in order to profit from their marketability as a group. In certain circumstances, it may
be desirable for a Principal to be able to commit to a certain frequency of rotation if
the group can benefit from new members.
Alternatively, if a Principal’s return is tied more heavily to effort then she will
likely rotate her workers more frequently, which can sometimes be desirable. For
example, pilots and flight attendants may be rotated frequently because effort in
ensuring certain safety standards are met may be much more important than any
potential Synergy.
5.3 Agents’ preference for decision-making
In the preceding chapters it has been found that for a given functional form of Synergy,
the Principal’s decision for team composition depends on the prevailing bargaining
solution. Similarly, it was found in this chapter that how often a Principal rotates
62

68. her workers is also tied to the bargaining method she faces and hence the observed
rotation may not be optimal and cannot be interpreted as such unless the Principal’s
incentives are perfectly aligned with that of all agents. There is scope then, to analyse
the preferences of agents, whether they prefer centralised or decentralised decision
making, and whether this depends on the bargaining method.
Note firstly, that newcomers have no preference as to team composition. This
is because their output (and hence return) is independent of both the number of
newcomers or incumbents hired.
Proposition 12. Conditioned upon being hired, newcomers are always indifferent as
to whether a Principal chooses the team composition or they chose it themselves.
Incumbents however, will always prefer to work with other incumbent workers. This
is because for any given level of effort, their return is higher when working alongside
other incumbents (Assumption 1). This is consistent with Bel et al. (2013), which
finds that despite incumbent workers and newcomers both enjoying Synergy, all agents
prefer to be paired with an incumbent worker.
Hence incumbent workers are more likely to prefer a decentralised decision mak-
ing structure (i.e. workers decide on team composition themselves). This is especially
the case when the Principal is more likely to hire newcomers, or more likely to rotate
workers i.e. when an inside-option bargaining method is employed.
63

69. Proposition 13. Incumbent workers always prefer to work with other incumbent
workers and hence prefer decentralised decision-making.
This result holds regardless of the bargaining method in place. Though the Principal
is more likely to hire incumbent workers and less likely to rotate her workers under
a bargaining method using an outside option (e.g. the method applied in Chapter
Three), incumbent workers will always prefer to work with more incumbent workers
since Synergy is always increasing in I for a given level of effort (Assumption 3).
Hence whenever a Principal hires any newcomers or chooses any frequency of rotation,
under any bargaining method, an incumbent strictly prefers a decentralised decision-
making process whereby only incumbents are hired. In other words, if left to decide,
incumbent workers will never accommodate the hiring of newcomers or engage in job
rotation.1
At first glance, Propositions 12 and 13 may seem restrictive in the static case
of team composition because it assumes (both descriptively and mathematically)
that there are no spillovers from incumbents to newcomers, that is, newcomers get
no additional benefit from working with other incumbents and are hence entirely
agnostic as to both team size and composition. However, these results are intuitive in
the framework of job rotation considered in this section. A newcomer gains very little
1
This result also parallels some of the findings in the literature in industrial organisations on
market entry and efficiency. For example, Mankiw and Whinston (1986) find that in homogeneous
product markets, free entry can lead to an excessive, inefficient level of entry.
64

70. initially from working with her unfamiliar team mates. However, over time, as she
grows familiar with her team and develops a certain level of chemistry and familiarity,
it begins to become more and more in her interest to maintain the Synergy that has
been built through prior experiences of working together. Hence she would strictly
prefer that no rotation occurs.
This finding is consistent with practical examples of labour union policies or
agreements. For example, collective bargaining agreements have often contained a
“last on, first off” clause which stipulates that those who are hired last are laid off
first (see e.g. Rogers, 1975).
5.4 Summary
This chapter extended the results found in Chapters three and four to analyse a
Principal’s decision of employing newcomers and incumbents in the context of job
rotation. In doing so, I have developed a very simple but novel model of optimal job
rotation for heterogenous workers working on what is essentially the same job. It was
found that, similar to the static case, depending on the prevailing bargaining method,
a Principal may rotate too often or too little relative to the optimal, generalising a
parallel result found in Bel et al. (2013) which was applicable to only one kind
of bargaining. Furthermore, it was found that incumbent workers strictly prefer a
decentralised decision-making system if the Principal will hire any newcomers or ever
rotate her workers.
65

71. Chapter 6
Conclusion
Given that team production often occurs in an incomplete contracting environment,
the size and composition of a team is of great concern to a Principal when different
types of workers interact dissimilarly with one another. While incumbent workers
often enjoy a natural Synergy, this can come at the cost of these workers exerting
less effort than a newcomer. The bargaining method is naturally important because
the Principal’s hiring decision will depend on the portion (and kind) of surplus she
anticipates she will receive. Hence I analyse the trade-off between Synergy and effort
under both an outside-option model and an inside-option (Shapley value) one.
In doing so, I generalised some of the existing literature on the team composition
of heterogeneous workers (e.g. Prat 2002) by endogenising the size of the workforce
and the payoff function. I also show that a Principal’s decision for team composition
is highly dependent on the prevailing bargaining method, suggesting that the result
66

72. in Bel et al. (2013), which considers a specific two person example, can be reversed
if assumptions on the bargaining method are relaxed. To show this, I demonstrate
that the mechanisms that drive the Principal’s choice of team composition under
the two bargaining methods differ. While in the outside-option case the Principal’s
decision is driven by the relative size of Synergy compared to the rent she can extract
from newcomers, under the inside-option (Shapley value) case, the Principal faces
an additional, direct trade-off between the natural Synergy of incumbents and the
greater effort of newcomers.
Furthermore, I extend the literature on both team composition and job rotation
in a novel direction by applying my model to the question of rotating newcomers
into (and incumbents out of) a team. Consistent with my findings in the static
case, both the evolution of the payoff function and the bargaining method are key
considerations that may induce a Principal to rotate too much or too little relative
to the social optimal. Lastly, I show that under a system of rotation newcomers
are indifferent as to who makes the hiring decision whilst incumbent workers strictly
prefer decentralised decision making.
There are however several limitations of the model that should be noted. Firstly,
the assumption that Synergy and effort are negatively related is a strong one despite
it being being consistent with the idea that newcomers receive a higher return to
effort. More general analysis could be undertaken where no restrictions are made on
the functional form of Synergy or its relationship with effort.
67

73. Moreover, though I have demonstrated the relevance of the prevailing bargain-
ing method and the characteristics of these methods that make a Principal more or
less likely to rotate her workers (equivalently, more or less likely to hire newcom-
ers), further comprehensive analysis could be undertaken on the effects of a broader
category of bargaining solutions.
Finally, I have considered only a very specific form of heterogeneity. Namely,
I analysed the case when workers are completely identical, except for some excess
surplus or Synergy that some workers have (or can be obtained over time) due to
previously working together. Though this has the benefit of demonstrating that
heterogeneity of workers can drive rotation even when workers are inherently the
same, it would be interesting to see in detail the interaction of workers who differ on
many levels.
68

74. Appendix A
Proof of Proposition 3.
Proof.1
Let p be the proportion of workers the firm hires which are incumbents and q be
the proportion of workers which are newcomers, 0 ≤ p ≤ 1; 0 ≤ q ≤ 1; p + q = 1.
Consider the case where X > Y (Figure 3.2) and the Principal only hires newcomers.
It must be the case if X > Y that
dπI
P
dI
intersects the vertical axis at a lower point
than
dπN
P
dN
(since otherwise as Synergy is increasing,
dπI
P
dI
>
dπN
P
dN
at every I) hence the
Principal cannot be made better off by a small deviation from q = 1 and p = 0 since
the first incumbent he or she hires faces a low
dπI
P
dI
. Furthermore, the Principal cannot
be made better off by replacing any number of newcomers with incumbents since if
there exists any q < 1 and p > 0 which will lead to a greater profit, then there must
exist a q < q and p > p which will lead to an even greater profit (since
dπI
p
dI
is
increasing) and so on until the greatest profit is made where p = 1, q = 0 which must
1
This proof refers to Figure 3.2.
69

75. mean that X > Y , which is a contradiction.
Now consider when Y > X and the Principal hires only incumbent workers, it
must be that
dπI
P
dI
intercepts the cost function at a point where
dπI
P
dI
>
dπN
P
dN
. Since
dπI
P
dI
is increasing and
dπN
P
dN
is constant, any deviation p < p and q > q, produces a profit
that can be made strictly better by increasing p and decreasing q until p = 1 and
q = 0 again.
Furthermore, consider when X = Y and the Principal is indifferent between
hiring incumbent workers or newcomers. Assuming that p = 1, if the Principal sub-
stitutes any portion p < 1 of incumbent workers for newcomers, the Principal can
be made better off by substituting all remaining incumbents for newcomers since
dπI
P
dI
is increasing while
dπN
P
dN
is constant and
dπI
P
dI
intercepts the vertical axis at a lower
point. Similarly assuming that q = 1, if the Principal substitutes any portion q < 1
of newcomers for incumbents, the Principal can be made better off by substituting
the remaining newcomers for incumbents since
dπI
P
dI
is increasing.
70

76. Appendix B
Shapley Value
Consider the case where there is a Principal, two incumbent workers and a newcomer,
v(P, I1, I2, N) = v1 + v2 + vN + SI
v(P, I1, I2) = v1 + v2 + SI
v(P, I1, N) = v1 + vN
v(P, I2, N) = v2 + vN
v(P, I1) = v1
v(P, I2) = v2
v(P, N) = vN
B1 = p(M)[v(P, I1, I2, N)−v(P, I2, N)]+p(M)[v(P, I1, I2)−v(P, I2)]+p(M)[v(P, I1, N)−
v(P, N)] + p(M)[v(P, I1) − v(P)]
71

77. B1 = 1
4
(v1 + SI
) + 1
12
(v1 + SI
) + 1
12
(v1) + 1
12
(v1)
B1 =
1
2
v1 +
1
3
SI
Similarly,
B2 =
1
2
v2 +
1
3
SI
BN = p(M)[v(P, I1, I2, N)−v(P, I1, I2)]+p(M)[v(P, I1, N)−v(P, I1)]+p(M)v[(P, I2, N)−
v(P, I2)] + p(M)[v(P, N) − v(P)]
BN = 1
4
(vN ) + 1
12
(vN ) + 1
12
(vN ) + 1
12
(vN )
BN =
1
2
vN
BP = p(M)[v(P, I1, I2, N)−v(I1, I2, N)]+p(M)[v(P, I1, N)−v(I1, N)]+p(M)[v(P, I2, N)−
v(I2, N)] + p(M)[v(P, I1) − v(I1)] + p(M)[v(P, I2) − v(I2)] + p(M)[v(P, N) − v(N)]
BP =
1
2
v1 +
1
2
vN +
1
3
SI
Now consider a case where there are three incumbent workers and one newcomer.
v(P, I1, I2, I3, N) = v1 + v2 + v3 + vN + SI3
v(P, I1, I2, I3) = v1 + v2 + v3 + SI3
v(P, I1, I2, N) = v1 + v2 + vN + SI2
72

78. v(P, I1, I3, N) = v1 + v3 + vN + SI2
v(P, I2, I3, N) = v2 + v3 + vN + SI2
v(P, I1, I2) = v1 + v2 + SI2
v(P, I1, I3) = v1 + v3 + SI2
v(P, I2, I3) = v2 + v3 + SI2
v(P, I1, N) = v1 + vN
v(P, I2, N) = v2 + vN
v(P, I3, N) = v3 + vN
v(P, I1) = v1
v(P, I2) = v2
v(P, I3) = v3
v(P, N) = vN .
B1 = p(M)[v(P, I1, I2, I3, N) − v(P, I2, I3, N)] + p(M)[v(P, I1, I2, I3) − v(P, I2, I3)] +
p(M)[v(P, I1, I2, N)−v(P, I2, N)]+p(M)[v(P, I1, I3, N)−v(P, I3, N)]+p(M)[v(P, I1, I2)−
v(P, I2)]+p(M)[v(P, I1, I3)−v(P, I3)]+p(M)[v(P, I1, N)−v(P, N)]+p(M)[v(P, I1)−
v(P)]
B1 = 1
5
(V1 + SI3
− SI2
) + 1
20
(v1 + SI3
− SI2
) + 1
20
(v1 + SI2
) + 1
20
(v1 + SI2
) + 1
30
(v1 +
SI2
) + 1
30
(v1 + SI2
) + 1
30
(v1) + 1
20
(v1)
73

79. B1 =
1
2
v1 +
1
4
SI3
−
1
12
SI2
Similarly.
B2 =
1
2
v2 +
1
4
SI3
−
1
12
SI2
B3 =
1
2
v3 +
1
4
SI3
−
1
12
SI2
BN = p(M)[v(P, I1, I2, I3, N)−v(P, I1, I2, I3)]+p(M)[v(P, I1, I2, N)−v(P, I1, I2, N)]+
p(M)[v(P, I1, I3, N)−v(P, I1, I3)]+p(M)[v(P, I2, I3, N)−v(P, I2, I3)]+p(M)[v(P, I1, N)−
v(P, I1)]+p(M)[v(P, I2, N)−v(P, I2)]+p(M)[v(P, I3, N)−v(P, I3)]+p(M)[v(P, N)−
v(P)]
BN = 1
5
vN + 1
20
vN + 1
20
vN + 1
20
vN + 1
30
vN + 1
30
vN + 1
30
vN + 1
20
vN
BN =
1
2
vN
BP = p(M)[v(P, I1, I2, I3, N) − v(I1, I2, I3, N)] + p(M)[v(P, I1, I2, I3) − v(I1, I2, I3)] +
p(M)[v(P, I1, I2, N)−v(I1, I2, N)]+p(M)[v(P, I1, I3, N)−v(I1, I3, N)]+p(M)[v(P, I2, I3, N)−
v(I2, I3, N)]+p(M)[v(P, I1, I2)−v(I1, I2)]+p(M)[v(P, I1, I3)−v(I1, I3)]+p(M)[v(P, I2, I3)−
v(I2, I3)]+p(M)[v(P, I1, N)−v(I1, N)]+p(M)[v(I2, N)]+p(M)[v(P, I3, N)−v(I3, N)]+
p(M)[v(P, I1)−v(I1)]+p(M)[v(P, I2)]+p(M)[v(P, I3)−v(I3)]+p(M)[v(P, N)−v(N)]
74

80. BP = 1
5
(v1 + v2 + v3 + vN + SI3
) + 1
20
(v1 + v2 + v3 + SI3
) + 1
20
(v1 + v2 + vN + SI2
) +
1
20
(v1 + v3 + vN + SI2
) + 1
20
(v2 + v3 + vN + SI2
) + 1
30
(v1 + v2 + SI2
) + 1
30
(v1 + v3 + SI2
) +
1
30
(v2+v3+SI2
)+ 1
30
(v1+vN )+ 1
30
(v2+vN )+ 1
30
(v3+vN )+ 1
20
(v1)+ 1
20
(v2)+ 1
20
(v3)+ 1
20
(vN )
BP =
1
2
v1 +
1
2
v2 +
1
2
v3 +
1
2
vN +
1
4
SI3
+
1
2
SI2
When the Principal adds an extra worker, it does not change their share of output
since they each contribute a constant amount (vN ) with no Synergies. Hence re-
gardless of how many incumbent workers and newcomers the Principal hires, each
newcomer will receive:
1
2
vN
Since the Principal decides how many people to hire and hires all of them
simultaneously, when there is SI
to be gained, SI−1
is zero and each incumbent agent
contributes the same amount to SI
. Hence, when the Principal hires an additional
incumbent worker, they get a smaller share of total Synergy. Namely, an incumbent
worker receives:
1
2
vI +
SI
I + 1
Lastly, the Principal receives:
N
2
vN +
I
2
vI +
SI
I + 1
.
75

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