Divide et Impera: Managerial Career Concerns in...

Divide et Impera:

Managerial Career Concerns in

Organizations∗

Hendrik Hakenes†

University of Bonn

Svetlana Katolnik‡

Leibniz University of Hannover

August 3, 2012

Abstract

We develop a three-tier hierarchy model, where a principal has the control

rights over managerial replacement and where a manager has the authority

over organizational structure. By deciding on the optimal span of control the

manager is governed by career concerns. Although the probability of employ-

ing a star performer increases in the number of agents, a manager decreases

the probability of being replaced by a subordinate by delegating a task to a

sufficiently large number of agents. In a trade-off between decreasing costs of

the personal replacement on the one hand, but increasing labor coordination

costs on the other hand, the model derives predictions about when managers

tend to choose an excessively high span of control, creating inefficiencies at

the firm level.

Keywords: Career concerns, Task delegation, Span of control, Managerial

turnover, Promotions.

JEL-Codes: J21, L23, M12, M5.

∗We would like to thank Matthias Krakel for helpful comments.†University of Bonn, Institute for Financial Economics and Statistics, Adenauerallee 24-42,

D-53113 Bonn, [email protected]‡Leibniz University of Hannover, Institute of Financial Economics, Konigsworther Platz 1, D-

30167 Hannover, +49-511-762-5166, [email protected]

1 Introduction

In January 2010, Sara Mathew became CEO of Dun & Bradstreet Corporation, the

world’s leading provider of commercial information. This change was accompanied

by an increase of direct subordinates from six people under the direction of her

predecessor to sixteen direct reports. One of the main reasons for this suddenly

more than doubled span of control was that Sara Mathew “wanted to stay on top”.

Today, after gaining some experience and becoming more secure in her position,

Sara Mathew feels more “comfortable” with only seven direct reports.1

The example of Sarah Mathew shows that the span of control seems to be a very

strong instrument for protecting a manager’s position. By increasing the number

of direct reports a manager can reduce the influence as well as the visibility of each

subordinate and thus strengthen the personal power inside the firm. This can be

especially important for new managers, who are primarily confronted with a high

level of job uncertainty. Sarah Mathew indicates a general trend in the “being the

boss”-strategy of top executives. Also the findings of empirical analyses identify

that managers have largely expanded their span of direct control during the past

two decades, strengthening their power at the top hierarchy level (see, for example,

Rajan and Wulf (2006) and Guadalupe, Li, and Wulf (2012)).

In his seminal work Mintzberg (1973) highlights that, per definition, a manager has

the formal authority over organizational decisions. In this paper we go back to the

roots of this definition and explore theoretically possible consequences of allocating

this authority to managers, when they face career concerns. Career concerns are

associated with a manager’s biased actions focusing on the objective of not being

replaced. We examine the nature of these incentives by endogenizing the question

of the optimal span of control of a manager, when a risk-averse principal assigns

1Description based on the Harvard Business Review, April 2012.

1

sovereignty over task delegation to him. We argue that by choosing a sufficiently

large span of control, a manager can obscure the abilities of his subordinates and thus

raise the probability of keeping his working position. Finally, the optimal span of

control results from a trade-off between decreasing costs of the personal replacement

on the one hand and increasing labor coordination costs on the other hand.

By that, our model provides a novel theoretical explanation for the empirical ques-

tion, why managers, although being pressed for time, more and more broaden their

span of direct control.2 Given our choice to focus on managerial career concerns

and their optimal span of control, our model mostly applies to hierarchical company

structures with teams of varying sizes, where a principal has to delegate part of

his sovereignty to a manager. Thereby, our results can be used for different indus-

tries and comprise both the private and public sectors of the economy and therefore

represent a universally applicable solution. Furthermore, not only in the field of

economics, but also in the sphere of politics, since its famous implementation by

Julius Caesar, the “Divide et impera”-strategy can play an important role for the

protection of personal power of top-ranking politicians.3

In the theoretical model, a principal hires a manager, who chooses the number of

ex-ante identical agents to perform a task. While the manager has the authority over

organizational decisions, the principal holds power over replacement of the manager

2According to empirical findings, the span of control of a CEO is positively related to time

expenditure - and therefore also to costs - for interaction with his employees (see, for example,

Bandiera, Prat, Sadun, and Wulf (2012)).3For example, in the German political system, the number of Parliamentary State Secretaries

is not limited by law and, hence, is elected by the respective minister. Although the scope of work

does not change, their quantity is subject to a widely used increase, most recently since Angela

Merkel took office in 2005. The example of the Federal Minister of Health, Daniel Bahr, who former

was Parliamentary State Secretary by the Federal Minister of Health, shows that Parliamentary

State Secretaries can replace incumbent ministers and, therefore, represent a potential threat to

their positions.

2

and selects the agent with the highest ex-post expected ability. Even though the

possibility of hiring a star performer is increasing in the number of agents, each

agent carries out a smaller fraction of the overall task and thus learning about the

agents’ individual abilities occurs more slowly. The model shows that a manager

decreases the probability of being replaced, when he delegates a task to a sufficiently

large number of agents.

Although increasing the span of control can raise the tenure and with it the employee

benefits of managers, task coordination becomes more costly. Hence, the manager’s

optimal span of control results from a trade-off between decreasing costs of the per-

sonal replacement on the one side and rising costs of labor coordination on the other

side. Therefore, an increase in the number of subordinates leads to various sources

of inefficiencies: This involves not only direct inefficiency costs of task coordination,

but also indirect costs of possible suboptimal internal recruitment decisions. Conse-

quently, managerial career concerns and the choice of the optimal span of control are

not only a private concern of the manager, moreover they represent a fundamental

source of inefficiencies in the firm’s global context.

We provide comparative static results on the formation and the extent of these in-

efficiencies. The complex interactions between important exogenous variables and

their impact on the manager’s endogenous team size choice generate new insights

concerning the provision of managerial incentives. One such insight is that, for a

comparatively low cost factor as well as a sufficiently high risk aversion concerning

the agents’ unknown abilities, the optimal span of control is inversely related to the

manager’s ability level and positively related to task complexity. Hence, by allocat-

ing managers of high quality to low cost sectors and by assigning complex tasks to

them a principal can reduce inefficiencies resulting from managerial organizational

choice.4

4According to our example of Sarah Mathew, the result accounts for the effect that managers

3

The work relates to other fields of the existing literature. First, it refers to the

literature on managerial incentives in the presence of career concerns (e.g. Fama

(1980), Holmstrom (1999), Holmstrom and Ricart i Costa (1986), Gibbons and

Murphy (1992), Dewatripont, Jewitt, and Tirole (1999a), Dewatripont, Jewitt, and

Tirole (1999b), Krakel and Sliwka (2009)). These papers primary focus on implicit

incentives arising from career concerns, its implications on effort choices, investment

decisions or also compensation contracts. In our work, we focus on consequences of

career concerns on the manager’s choice of organizational design that has not been

explored in the career concerns literature so far.

Second, there exists literature on the optimal allocation of authority (e.g. Williamson

(1967), Calvo and Wellisz (1978), Rosen (1978), Qian (1994)). It mostly explores

technological issues, such as the optimal span of control or wage scales at different

hierarchical tiers. In our work, the choice of the optimal span of control is not pri-

mary motivated by technological reasons. It rather represents a strategic instrument

of managers to secure the personal working position.

Third, there are parallels to the literature on task assignment and promotions

(e.g. Ricart I Costa (1988), Meyer (1991), Meyer (1994), Bernhardt (1995), Or-

tega (2003)). The models concentrate on technological questions of optimal task

assignment, its implications on learning about agents’ abilities, on wage levels as

well as on promotion decisions, including considerations about the external market

for labor. Our work differs from this approach, as it primary investigates ineffi-

ciencies following from task assignment of managers. Hence, task assignment is

considered as a managerial instrument to influence learning about the subordinates’

abilities.

reduce their span of control after acquiring sufficient firm-specific or managerial human capital

in order to reduce costs of labor coordination. Although our model does not explicitly includes

changes or learning effects of managerial ability, it focuses on the difference between the expected

abilities of workers and the manager’s ability instead.

4

Fourth, the model relates to the research on managerial turnover. The existing

literature focuses either on the probability of managerial turnover as a problem of

entrenchment (e.g. Shleifer and Vishny (1989)) or on the advantageousness and the

consequences of managerial turnover as a question of efficiency (e.g. Hoffler and

Sliwka (2003), Sliwka (2007)). We focus on the probability that a manager will be

replaced, not in terms of managerial entrenchment, but moreover influenced through

the manager’s choice of organizational design.

Fifth, there exist theoretical models which directly deal with managerial career con-

cerns and inefficiencies in organizational design as a question of task assignment and

delegation and of a general worker-manager interaction. Carmichael (1988) shows

that tenure plays a key role in creating incentives to hire people who might turn out

to be better than themselves. Prendergast (1995) presents a theory of responsibility

in organizations. He argues that when a manager collects skills by carrying out

tasks, he then delegates too few tasks to his subordinates and thereby hoards re-

sponsibility to increase his future wages. Related to this topic, Sliwka (2001) shows

that delegation reduces the power of middle managers, as subordinates become able

to show their ability. He focuses on the problem that managers, therefore, become

reluctant to delegate. In contrast, Pagano and Volpin (2005) present a theory, where

managers engage in a worker-manager alliance and thus act in the interests of both

parties. In their model, manager create inefficiencies by providing a generous em-

ployment policy for workers for the purpose of reducing a takeover-threat of their

company. This, in turn, helps to protect private benefits and high personal wages

of managers.

In this paper, we present a completely novel idea, how top executives can exploit

their personal working position in protecting themselves from being replaced. There-

with, we disclose a new source of inefficiencies associated with a manager’s author-

ity over task assignment. Finally, we are able to account for significant empirical

analyses drawing attention to the puzzling question of a remarkably rising span of

5

managerial control.5

The remainder of this paper is organized as follows. Section 2 develops the main

model, following by an equilibrium analysis in section 3. Section 4 provides compar-

ative static results for important exogenous parameters. In section 5 we conclude.

2 The Model

Consider a risk-averse firm (the principal), who employs a manager of ability A > 0.

The manager subcontracts with n ex-ante identical agents to complete a single one-

period task of the measure L > 0. The number of agents is endogenously chosen

by the manager at the beginning of the period, with n ≥ 1.6 The task is equally

divided by all agents and individual output measures are available. Output is the

sum of an agent’s standard normally distributed ability ai, with ai ∼ N(0, 1) and

cov(ai, ai+1) = 0 and of a normally distributed error term ϵi, with ϵi ∼ N(0, σ2) and

cov(ϵi, ϵi+1) = 0, with i = 1, ..., n, respectively. The resultant output levels serve as

signals to update the agents’ expected individual abilities.7 Manager and principal

have the same level of information at every point of time. Following preferences of a

mean-variance representation, the principal chooses to replace the manager by one

of the subordinates, when the manager turns out to be of lower ability.8 In order

5See, for example, the studies of Rajan and Wulf (2006) and Guadalupe, Li, and Wulf (2012).6We abstract from the problem of integrity, except that n ≥ 1.7Note that in the absence of effort incentives, the choice of individual performance metrics

produces the most informative signal about an agent’s ability. As aggregation reduces the infor-

mativeness of each agent’s signal by comparison, our results can be all the more applied for the

case of aggregate performance measures. For a broader comparison of individual and aggregated

performance evaluations see, for example, Arya and Mittendorf (2011).8The choice of the most able worker for the manager’s position can have various rationales.

For example, Meyer (1991) argues that the output in the manager’s job is more related to ability

than in the previous hierarchical position. Furthermore, also the promotion of an intra-firm agent

6

not to lose his private benefits associated with the job, the manager maximizes the

probability of keeping his position.9

As a motivating example, first suppose that the manager delegates the complete

task to only one agent (n = 1). This agent then produces a signal of

s1 = a1 +ϵ1√L.

Output is the sum of the agent’s ability a1 and of the measurement shock ϵ1, which

extent depends on the size of the task L. According to the central limit theorem, we

divide the variance of the error term σ2 by L. This corresponds to dividing the error

term by√L. Hence, increasing the task size is equivalent to increasing the number

of observations or, consequently, equivalent to increasing the precision of the signal

si.10

Now consider the case with multiple agents. Each agent completes a share of l = L/n

of the whole task. It follows that by working on a subtask of the measure l, each

agent’s signal equals

si = ai +ϵi√l

= ai +ϵi√L/n

. (1)

Now, the measurement error depends on the task size L as well as on the number of

agents n. Altogether, all subtasks have an identical marginal effect on the signal si.

The resultant output is a noisy indicator of ability and measurement becomes noisier,

when the number of agents grows. The intuition is that, when the manager increases

the number of agents, each agent carries out a smaller fraction of the overall task

can have advantages in comparison to an external worker: Ortega (2003), for example, models

an agent’s ability as the sum of general and specific human capital. As the market benchmark

wage only considers the general human capital of a worker, specific human capital represents an

additional costless benefit to a firm. Although we do not explicitly model these relationships, their

intuitions seem quite reasonable.9For simplification, private benefits are normalized to zero in the model.

10See, for example, Greene (2002) for a broader analysis of the central limit theorem.

7

and the variance of the measurement error increases. This weakens the precision of

each signal and in turn also the updating of an agent’s ability. A lower degree of

updating strengthens the weight on each agent’s prior. Hence, task complexity and

team size have an exactly opposite effect on the measurement precision. However,

there is one essential difference: The larger the number of agents, the lower is the

measurement precision, but, at the same time, the higher is the resultant probability

of hiring a star performer. Hence, by considering the probability of being replaced,

the manager has to take into account both the signal precision of an individual agent

and the overall number of signals produced.

By choosing the optimal span of control the manager incurs labor coordination costs

of

C(n) = c n√

L/n. (2)

Labor costs are subproportionally increasing in the size of an agent’s task L/n.

Moreover, total costs are increasing in the number of agents.11 The parameter c

represents a constant cost factor. Overall, the manager’s choice of the optimal

span of control depends on his 1) replacement probability, specified by managerial

replacement costs, and on the 2) labor costs of task coordination.

The timing of the model is as follows:

1. Principal employs a manager of ability A > 0.

11The consideration of labor coordination costs represents a standard approach in the team pro-

duction literature. In their seminal work Becker and Murphy (1992) emphasize that specialization

and the division of labor fundamentally depend on coordination costs that increase with the num-

ber of agents. Thereby, coordination costs can be associated with different sources of inefficiencies,

such as principal agent conflicts, problems of task coordination and monitoring or also communica-

tion difficulties. Given our choice not to directly consider the agents’ incentives, allows us to purely

focus on our key question of managerial inefficiencies associated with the choice of organizational

design. Nevertheless, the inclusion of labor coordination costs involves an indirect consideration

of the associated costs.

8

2. Manager chooses the number of agents with prior abilities ai ∼ N(0, 1) to perform

a task.

3. Production takes place, individual outputs si are realized to update the agents’

expected abilities.

4. Principal replaces the manager by the agent with the highest expected ability,

when ∃ i: U(A) < U(ai(si)). Otherwise, the manager can stay in the job.

3 Equilibrium Analysis

In order to determine the manager’s optimal span of control, we will initially focus on

the manager’s probability of replacement. First of all, we will analyze the problem

set from the standpoint of a representative agent. As all agents complete an equal

share of the overall task, the results are valid for any subordinate. Thereby, we will

investigate determinants of the probability of managerial replacement, given the

signal of one single agent, without primary considering that an increase in team size

simultaneously raises the number of signals. Based on these results, we will extend

our analysis by incorporating the signals of all team agents and their impact on

the manager’s replacement costs. Hence, this second analysis accounts both for the

effects of 1) the signal of each single agent and of 2) the number of signals produced.

Finally, we will complete our results by taking the costs of task coordination C(n)

into the manager’s consideration of the optimal span of control.

According to equation (1), the distribution of each signal si results from the distri-

butions of ai ∼ N(0, 1) and ϵi ∼ N(0, σ2

L/n). Accordingly, updating follows from

si = ai + ϵi ∼ N

(0, 1 +

σ2

L/n

). (3)

The variance of each signal is the sum of the variance of the prior and of the variance

of the error term. It follows that the variance of a signal always exceeds the variance

9

of an agent’s ability, because it additionally incorporates the measurement error,

which is increasing in the number of agents and decreasing in task complexity.

Consequently, measurement variance is decreasing in the size of the subtask L/n.

The signal si determines the updating of an agent’s ability according to the Bayes’s

theorem for normally distributed random variables.12 It follows that an agent’s

ex-post expected ability equals

ai(si) =n σ2

L + n σ2︸︷︷︸=:α

0 +L

L + n σ2︸︷︷︸=:1−α

si =L

L + n σ2si. (4)

The posterior variance of an agent’s ability results from the precisions of the prior

variance and of the conditional variance and equals α. Therewith, the a posteriori

distribution of an agent’s ability, given the signal si, equals

ai | si ∼ N

(L

L + n σ2si,

n σ2

L + n σ2

). (5)

Proof: See the appendix.

The ex-post expected ability of an agent ai(si), consists of a weighted average of

the prior ai (which equals zero) and of the signal si. The weight on the prior equals

α and corresponds to the variance of the ex-post estimation of an agent’s ability.

The weight on the output measure equals 1 − α. Both weights sum up to 1 and,

hence, an increase in the measurement precision reduces the weight on the prior

estimation of ability and vice versa. Equivalently, a smaller weight on the prior

corresponds to a smaller variance of the ex-post estimation of ability and therefore

to a larger weight on the signal si. Furthermore, learning decreases the variance of

an agent’s expected ability in comparison to the variance of the prior(

n σ2

L+n σ2 < 1)

and makes the estimation of ai more precise. Updating is increasing in the number

of observations, represented by the task size L, and is decreasing in the team size n

as well as in the variance of the error term σ2. The opposite holds for the variance of

12See, for example, the standard result of Greene (2002).

10

the a posteriori distribution of an agent’s ability. Therefore, updating can decrease

or even eliminate the manager’s advantage over his subordinates and thus threaten

his tenure. From the standpoint of a single signal si, the manager prefers to increase

the number of agents to minimize updating and to decrease the probability that this

single agent turns out to be of superior ability.

To incorporate the uncertainty associated with the expectation of an agent’s ability,

we use an exponential utility function to evaluate the abilities of manager and agents.

The cut-off rule for managerial replacement derives from individual comparisons of

the utility of the manager and the ex-post expected utility of an agent. Consequently,

when U(A) ≥ E[U(ai(si))], the manager can stay in the job. The condition is

fulfilled, when

−e−ρA ≥∫ ∞

−∞−e−ρ a f(a) da. (6)

The parameter ρ represents a constant absolute risk aversion coefficient. The vari-

able a measures the agent’s unknown ability. Incorporating equation (5), we can

rearrange condition (6) to

−e−ρA ≥ −e−ρ (2 L s−n ρ σ2)

2 (L+n σ2) . (7)

Consequently, condition (7) is fulfilled, when

A ≥ 2Lsi − n ρ σ2

2 (L + nσ2). (8)

Rearranging equation (8) yields s as the maximal possible value of a signal si, for

which the manger will retain his job, given an exogenous n. We obtain the following

dismissing rule:

Proposition 1 When ∃ i : si > s, with

s = A +n (2 A + ρ) σ2

2 L, (9)

then also ∃ i : E[U(ai(si))] > U(A) and the principal replaces the manager by the

agent with the highest ex-post expected ability ai(si).

11

The critical signal s is increasing in A, ρ, σ2 and n and decreasing in L. The

higher the manager’s ability A, the higher is also the buffer range for the manager

and s positively relates to A. Equation (9) is also increasing in the risk aversion

parameter ρ: In consideration of the risk aversion, the exchange of the manager

occurs only when the expectation of an agent’s uncertain type significantly outweighs

the manager’s type. The higher the risk aversion of the principal, the larger the

manager’s ability has to be outbuilt by an agent. Hence, risk aversion increases the

benefits of a manager concerning his objective of not being replaced. The variance

of the error term σ2 increases the uncertainty of performance measurement and

therefore decreases the updating of an agent’s ability. The lower the weight on the

signal si, the higher is also the maximal possible s. The team size n has the a similar

direction of influence on s: When the number of agents increases, the precision of

each signal decreases and consequently also the level of updating. For an infinite

number of agents, s also converges to infinity. In contrast, s is decreasing in the

task size L. Task complexity positively relates to the size of the subtask carried out

by each agent and therefore strengthens the task precision. This, in turn, decreases

the maximal s.

The distribution of an agent’s signal follows si ∼ N(0, 12 + σ2

L/n

). Let Φ(s) be the

distribution function of the standard normal distribution. Then, for an average

observation of s, the distribution function of one signal equals

F (s) = Φ

(s

√1

1 + σ2 n/L

). (10)

To implement the effect that the number of signals is increasing in the span of man-

agerial control, we use the cumulative distribution function technique to derive the

distribution of the maximum of n signals. Thereby, the joint distribution function of

stochastically independent variables results from the product of the single distribu-

tion functions.13 Accordingly, for identically distributed continuous signals (si = s),

13See, for example, Poirier (1995).

12

the maximum of n signals has the common distribution of

F (s)n = Φ

(s

√1

1 + σ2 n/L

)n

. (11)

We substitute s of equation (9) into equation (11) to evaluate the probability that

the maximum of the n signals will fall below the critical s. It is convenient to define

the converse probability 1 − Φ(s)n, which is the probability that the manager will

be replaced, as managerial costs of the personal replacement X, with

X = Pr{max s > s} = 1 − Φ

((A +

n (2 A + ρ) σ2

2 L

)√ 1

1 + σ2 n/L

)n

. (12)

Now, it is important analyze the team size effect on the manager’s costs of the

personal replacement.

Proposition 2 For n = 0, the manager’s replacement costs X are zero. For n →

∞, X tends to zero. There exists a single peak of X at some strictly positive value

of n. As, per definition, n ≥ 1, the manager’s replacement probability is minimal

for n → ∞.


As exemplified in figure 1, the replacement probability is not monotonically decreas-

ing in the number of agents. The slope of the replacement cost curve results from a

basic trade-off between a rising probability of hiring a star performer and a decreas-

ing probability of identifying one. On the one hand, when the manager increases the

number of agents, the probability of employing an agent of superior ability increases,

which in turn raises the probability of managerial replacement. On the other hand,

when the manager divides the task between a large number of agents, each agent

can carry out only a small fraction of the overall task. This decreases the updating

of each agent’s ability from the initial expected value of zero and complicates the

13

0.5 1.0 1.5 2.0 2.5 3.0n

0.01

0.02

0.03

0.04

0.05

0.06

X

Figure 1: Costs of managerial replacement

The figure shows the manager’s costs of the personal replacement X, represented in the probability

of being replaced, in dependency on the team size n. Parameters are L = 2, ρ = 8.5, σ = 1 and

A = 0.2. For n = 0, replacement costs X equal zero and are increasing (decreasing) in n for

n < n (n > n). Replacement costs are maximal at the not feasible point of n = 0.336, with

X |n=n= 0.068. The vertically dotted line shows the minimal team size frontier of n = 1. As

n ≥ 1, the manager optimally hires as many agents as possible.

identification of a star performer. The probability of managerial replacement de-

creases. While this first effect dominates for small team sizes (n < n), for large team

sizes (n > n) the effect of the increased number of agents can no longer outweigh

the effect of the smaller signal precisions and replacement costs negatively relate to

team size.

On the one hand, as n ≥ 1, the manager minimizes his replacement costs X by

choosing an infinite number of agents. On the other hand, the costs of labor coor-

dination C(n) of equation (2) are strictly increasing in the number of agents and

therefore limit the manager’s optimal span of control. Overall, according to the

equations (2) and (12), the manager chooses a team size, denoted n∗, which mini-

mizes his total costs as the sum of the 1) costs of the personal replacement X and

of the 2) labor coordination costs C(n):

minn

(X + C(n)) = Pr{max s > s} + C(n)

14

= 1 − Φ

((A +

n (2 A + ρ) σ2

2 L

)√ 1

1 + σ2 n/L

)n

+ c n√

L/n. (13)

This objective function reveals the two costs, X and C(n), the manager has to

trade-off. Moreover, as X is not monotonically decreasing in n, the manager has to

account both for the effect of a rising probability of hiring a star performer, which

dominates for small team sizes, and for the effect of the decreasing visibility of each

agent’s ability, which dominates for large team sizes. In detail, it is important to

analyze, in which cases the manager chooses n∗ = 1 compared to the scenario of

n∗ > 1. Obviously, as the single agent equilibrium represents a technically feasible

option, the manager creates inefficiencies in terms of additional labor costs, when

he chooses n∗ > 1. Consequently, it is important to derive conditions on that the

manager is incentivized to create these inefficiencies.

Proposition 3 For n = 0, total costs X +C(n) equal zero. For n → ∞, total costs

tend to infinity. For a large cost factor c, the manager optimally chooses n∗ = 1.

For c sufficiently small, there exists a unique optimal team size n∗ > 1, which is

characterized by the following implicit function:

∂X

∂n= −∂C(n)

∂n= −1

2c√L/n∗.


Figure 2 exemplifies the interaction of replacement and labor costs for the two

results of n∗ > 1 and n∗ = 1. Logically, a local minimum of total costs can only

be located in the range of n > n, where X is decreasing in n. Here, the effect of

a decreasing signal precision outweighs the effect of a rising probability of hiring

an agent of superior ability. In this range, the manager has to trade-off increasing

costs of labor coordination and decreasing replacement costs. When the labor cost

factor is comparatively small, marginal labor costs are low and the manager is

incentivized to increase his span of control. As exemplified in the left picture of

15

0.5 1.0 1.5 2.0 2.5 3.0n

0.01

0.02

0.03

0.04

0.05

0.06

0.07

X, CHnL, X+CHnL

n*

X+CHnL

CHnL

X

0.5 1.0 1.5 2.0 2.5 3.0n

0.1

0.2

0.3

0.4

0.5

0.6

0.7

X, CHnL, X+CHnL

n*

X+CHnL

CHnL

X

Figure 2: Managerial costs of replacement and labor costs

The figures show the relationship between a manager’s costs of the personal replacement X (blue

line), his labor costs C(n) (brown line) and his total costs X + C(n) (thick line) in dependency

on the team size n. Again, the dotted vertical line gives the minimal team size frontier of n = 1.

In the left figure, parameters are L = 2, ρ = 8.5, σ = 1, A = 0.2, as before, and c = 1/150. Total

costs are minimal for n = 2.029, with X+C(n) |n=2.029= 0.014. Furthermore, X+C(n) |n=2.029=

0.014 < X + C(n) |n=1= 0.033 and the manager optimally chooses n∗ > 1. In the right figure,

c = 0.3, with all other parameters remaining constant. Total costs are increasing in n and therefore

are minimal for n∗ = 1, with X + C(n) |n∗=1= 0.448.

figure 2, the manager then optimally chooses n∗ > 1. Contrary, a large cost factor

can prevent the manager from increasing the number of subordinates and force him

to choose n∗ = 1. As in the right picture of figure 2, this is the case when the labor

cost increase from hiring an additional agent outweighs the associated decrease in

replacement costs in the range of n > n and total costs are strictly increasing

in n. Alternatively, high marginal labor costs can cause higher total costs at the

minimum of the labor cost curve than for n = 1. And finally, a high labor cost

factor can shift the minimum of the total cost curve to the not feasible range of

0 < n < 1, so that the manager’s optimal choice is n∗ = 1. Overall, labor costs

represent a key instrument in the question of inefficiencies resulting from managerial

organizational design. By assigning managers to different coordination cost sectors,

the principal can indirectly influence the occurrence as well as the extent of these

16

inefficiencies.14 Therefore, an efficient organizational design assigns managers facing

a high probability of replacement to high coordination cost sectors and vice versa.

4 Comparative Static Results

First, when the manager’s ability increases, he faces a greater buffer range and is

able to afford a smaller team of agents.

Proposition 4 The manager’s optimal team size n∗ is decreasing in A, when

(i)

ρ > 2

√L( Lσ4

+2

σ2+

1

2 L + σ2

)(14)

and

(ii)

c <1

8

√(−8 L3 − 20 L2 σ2 + 2 L (ρ2 − 6) σ4 + ρ2 σ6)2

L3 (L + σ2)4. (15)


As exemplified in figure 3, the result indicates that high quality managers prefer

smaller teams, as for them the probability of being replaced is lower compared to

low quality managers. Thus, low quality managers are more harmful in their organi-

zational choice, especially when labor costs are comparatively low. The intuition is

that, when managerial ability increases, replacement costs X decrease and a smaller

team size is sufficient to keep the former replacement probability. As coordination

costs C(n) are increasing in n, the manager can afford to choose a lower level of

14For example, coordination costs can be low in companies, where divisions are located geo-

graphically close and production can be coordinated easily.

17

0.2 0.3 0.4 0.5A

1.8

1.9

2.0

2.1

n*

Figure 3: Comparative static with respect to A

The figure shows the comparative static result of proposition 4. Parameters are L = 2, ρ = 8.5,

σ = 1 and c = 1/150, as before. The corresponding conditions for the risk aversion parameter of

ρ > 5.797 and for the cost factor of c ≤ 0.949 are satisfied. The red dot shows the initial optimal

team size of figure 2 of n∗ = 2.029 for the manager’s ability level of A = 0.2.

the replacement probability in comparison to his previous equilibrium. In contrast,

when A decreases, the manager has to increase his team size to keep the former

replacement probability. However, labor costs increase with the team size and both

cost functions act against each other. Overall, the manager increases his team size,

as long as the cost factor c does not exceed the threshold of proposition 4 (ii).

Besides the upper limit for the cost parameter c, proposition 4 is valid for a minimal

value of the risk aversion ρ. The intuition is that, when the risk aversion concerning

the agents’ unknown abilities is relatively small, the manager faces a high probability

of replacement. In this case, the replacement cost curve has a negative slope only for

comparatively large team sizes. In consideration of labor costs, which are increasing

in n, total costs can be minimal for n∗ = 1. An increase in the parameter A cannot

further reduce the optimal team size, as, per definition, n ≥ 1. In the opposite

case of a decrease in A, the replacement cost curve shifts upwards and the point

of its maximum shifts rightwards. This increases the team size at the new local

minimum of the total cost curve. When the required conditions for the comparative

static results are not fulfilled, managerial costs are too high to choose n∗ > 1, either

18

because a very high team size is required (small ρ) or because marginal labor costs

are too high (large c). Consequently, the optimal team size remains at n∗ = 1.

Next, we turn to the comparative static with reference to ρ. When the risk aversion

concerning the workers’ uncertain expected abilities increases, the manager faces a

greater buffer range: The ex-post expected ability of an agent has to surpass the

manager’s ability by far, so that the probability to stay in the job increases for the

manager. Hence, ceteris paribus, he can decrease the number of subordinates to

reduce labor costs.

Proposition 5 The manager’s optimal team size n∗ is decreasing in ρ, when

(i)

ρ > 2

√L (L + σ2) (4 L + 3 σ2)

2 L σ4 + σ6(16)

and

(ii)

c <1

8

√(−16 L3 − 28 L2 σ2 + 2 L (ρ2 − 6) σ4 + ρ2 σ6)2

L3 (L + σ2)4. (17)


As exemplified in figure 4, a high risk aversion increases the job security of managers.

This can prevent managers from taking advantage of their personal working position

for the purpose of protecting tenure. Therefore, risk aversion serves not only as a

safety factor for the principal, but moreover also for the manager and helps to reduce

his organizational inefficiencies. Also the comparative static result of proposition 5

is valid for a minimum value of the risk aversion ρ as well as a maximum value of

the cost factor c. Here, the intuition of proposition 4 holds.

The following comparative static analyzes the impact of σ2 on n∗. A manager,

who delegates tasks with a high variance of performance measurement, can benefit

19

8.0 8.5 9.0 9.5Ρ

1.9

2.0

2.1

2.2

2.3

2.4

n*

Figure 4: Comparative static with respect to ρ

The figure shows the comparative static result of proposition 5. Parameters are L = 2, A = 0.2,

σ = 1 and c = 1/150, as before. The corresponding conditions for the risk aversion parameter of


team size of figure 2 of n∗ = 2.029 for a risk aversion of ρ = 8.5.

from the decreased informativeness of the agents’ signals. Hence, organizational

inefficiency is low, because the optimal number of subordinates is decreasing in the

variance of the measurement error.

Proposition 6 The manager’s optimal team size n∗ is decreasing in σ2, when

(i)

ρ > 2

√L (L + σ) (8 L2 + 8 L σ + 3 σ2)

σ2 (2 L + σ)2(18)

and

(ii)

c <1

18

√(−32 L4 − 64 L3 σ + 4 L2 (ρ2 − 11) σ2 + 4 L (ρ2 − 3) σ3 + ρ2 σ4)2

L3 (L + σ)4 (2 L + σ)2. (19)


Figure 5 exemplifies the comparative static result of proposition 6. On the one

hand, a high variance of performance evaluation reduces the measurement precision

20

0.95 1.00 1.05 1.10 1.15Σ

2

1.9

2.0

2.1

2.2

n*

Figure 5: Comparative static with respect to σ2

The figure shows the comparative static result of proposition 6. Parameters are L = 2, A = 0.2,

ρ = 8.5 and c = 1/150, as before. The corresponding conditions for the risk aversion parameter of


team size of our numerical example of figure 2 of n∗ = 2.029 for a variance of the error term of

σ2 = 1.

of the agents’ abilities. This, for example, can reduce the effectiveness of internal

recruitment decisions, as it hinders the principal from identifying agents of superior

ability. On the other hand, measurement variance increases the job security of

managers and therefore reduces their organizational inefficiencies. Again, besides

the minimum threshold of the aversion ρ, the comparative static result of proposition

6 is valid for a maximum value of the cost factor c and the intuition of proposition

4 holds.

Next, we turn to the comparative static result with respect to L. When L rises,

the size of each agent’s subtask increases and therewith the precision of the ex-post

estimation of each agent’s ability. This, in turn, raises the replacement probability

of the manager offering him incentives to increase the optimal team size.

Proposition 7 For c sufficiently small, the manager’s optimal team size n∗ is in-

creasing in L, when

ρ > 2√

3

√L (L + σ2)

σ4. (20)

21


1.8 2.0 2.2 2.4L

1.8

2.0

2.2

2.4

n*

Figure 6: Comparative static with respect to L

The figure shows the comparative static result with respect to L. Parameters are A = 0.2, ρ = 8.5,

σ = 1 and c = 1/150, as before. The corresponding condition for the risk aversion parameter of

ρ > 8.485 is satisfied. The red dot shows the initial optimal team size of our numerical example of

figure 2 of n∗ = 2.029 for a task size of L = 2.

Also the influence of the task size L on n∗ crucially depends the extent of the risk

aversion ρ as well as of the cost parameter c. Like in the previous propositions, a

high risk aversion ensures that the replacement cost curve is decreasing in n already

for sufficiently small team sizes, which enables the equilibrium of n∗ > 1 and also

the validity of the comparative static result of proposition 7. In contrast, in this

comparative static result a variation in L directly influences not only the replacement

costs, but also managerial labor costs. Graphically, both the replacement cost curve

and the labor cost curve shift upwards, when L rises. While from the standpoint

of the replacement costs, the optimal team size has to increase to retain the former

replacement probability, from the standpoint of the labor costs, an increase in n

results in a two-sided increase in labor costs: First, labor costs increase due to

an increase in n and, second, due to an increase in L. Moreover, both factors

have a multiplicative effect on C(n), which can keep the manager from increasing

his span of control, when the cost factor c is high. Contrary, when L decreases,

the manager can decrease his team size and at the same time retain his former

22

replacement probability. A decrease in both L and n reduces the pressure of the

labor costs and the manager can afford to increase his team size and a the same

time retain his former labor costs. This can prevent the manager from a decrease

in n∗ in case when the cost factor c is high. Therefore, when c is sufficiently low,

an increase in L should increase n∗ and vice versa. Nevertheless, because of the

double-sided pressure of the labor costs C(n) the conditions for c (and ρ) have

to be more limiting in comparison to the previous exogenous parameters. While

this economic intuition holds, no general expression for the cost factor c can be

derived because of the implicit function governing the optimal team size (see the

appendix). However, figure 6 shows the comparative static result of proposition

7 for an example, where the existence of this equilibrium is ensured. In this case,

placing the responsibility for complex tasks on the manager can increase inefficiencies

arising from task delegation of managers, especially when marginal labor costs are

low (small c). To rule out these inefficiencies, the principal should assign complex

task to managers who initially face a small probability of replacement.

Interestingly, a change in the exogenous parameters moves the manager away from

his previous level of the replacement probability. Additionally, when the compara-

tive static results are fulfilled, the following result holds: When the replacement cost

curve shifts upwards, labor costs contain a too large increase in n∗: The manager

absorbs part of the labor costs increase through an increase in his replacement prob-

ability. Contrary, when the replacement cost curve shifts downwards, the manager

can afford to decrease his replacement probability in comparison to his previous

equilibrium by attenuating his labor costs decrease. Both subcases yield a parallel

aligned change in X and C(n). Again, coordination costs play a key role when

considering the inefficiencies and their changes resulting from managerial organi-

zational choice. Labor costs smooth changes in n∗ subject to a variation in the

exogenous parameters. While the parameters A, ρ and σ2 have a direct influence on

replacement costs and only an indirect influence on labor costs through a variation

23

of n∗, the task size L always has a direct influence on both X and C(n). This,

in turn, implies, that, when considering the four exogenous parameters A, ρ, σ2

and L, the task size represents the most powerful variable to restrict organizational

inefficiencies of managers. As task size L and marginal labor costs c have a com-

plementary effect on overall labor costs, labor costs and organizational inefficiencies

can be minimized by allocating managers facing a low probability of replacement,

like for example managers of high ability, to low cost sectors and assigning complex

tasks to them and vice versa.15 This not only reduces organizational inefficiencies

in terms of additional labor costs, but moreover can yield an improvement of the

quality and with it of the efficiency of internal recruitment decisions.

5 Conclusion

In this paper, we disclose a new source inefficiencies resulting from managerial dis-

cretion over organizational decisions. Although the probability of hiring a star per-

former increases in the number of agents, the opposite holds for the probability of

identifying one. Our model shows that a manager reduces the probability of being

replaced by a subordinate, when he delegates a task to a sufficiently large number

of agents. However, because labor costs are increasing in team size, the manager’s

optimal span of control results from the trade-off of decreasing costs of the personal

replacement and increasing costs of labor coordination.

Of particular note, our work derives the following implications: In settings with low

marginal labor costs team sizes will be larger, especially when the manager faces a

comparatively high probability of replacement. Moreover, for a sufficciently high risk

15The first of these results is consistent with the findings of Becker and Murphy (1992) that

worker with higher human capital should be assigned to low cost sectors, when higher labor coor-

dination costs lower the marginal product of human capital.

24

aversion and a relatively small labor cost factor, the optimal team size of managers

negatively relates to managerial ability, the risk aversion concerning the agents’

unknown abilities as well as the uncertainty of performance measurement. Contrary,

the optimal team size is increasing in task complexity. Therewith, our work yields

clear predictions on the optimal internal design of organizations. Accordingly, an

efficient organizational structure allocates managers facing a high probability of

replacement, so, for example, managers of high ability, to low costs sectors and

assigns complex tasks to them and vice versa.

The results of our work provide several hypotheses that may guide future research.

First, extending the model by jointly considering the agents’ incentives in production

and wages could yield additional insights concerning the consequences of managerial

actions. Second, organizational authority of managers is related to complex ineffi-

ciencies, not only in terms of direct costs of task coordination, but furthermore also

through the quality of internal recruitment decisions, whose costs have not been

explicitly considered so far. Third, the span of control can be used as an instru-

ment of pressure for the manager itself. The investigation of its effectiveness and

its comparison to other instruments could yield interesting insights concerning the

question of the optimal provision of managerial incentives.

By endogenizing the manager’s choice of the optimal size of a team our model

provides conditions on the formation as well as on the extent of the associated

inefficiencies. Moreover, it derives important predictions about the impact of envi-

ronmental variables on the manager’s optimal span of control. Interestingly, their

complex interactional effects yield surprisingly clear implications concerning the op-

timal allocation of managers to organizations. Finally, we are able to account for

significant empirical research observing a remarkable move to greater spans of man-

agerial control during the past two decades.16

16See, for example, the studies of Rajan and Wulf (2006) and Guadalupe, Li, and Wulf (2012).

25

Appendix

Derivation of the posterior distribution of an agent’s ability. The proof

makes use of the standard results of Greene (2002). In the general case of the

normal distribution with a prior distribution of a ∼ N(µa, σ2a), the posterior, given

the observation s, equals a | s ∼ N(µa(s), σ2a(s)), with µa(s) = µa + (s − µs)

σ2a

σ2a+σ2

s.

The posterior variance of an agent’s ability is the sum of the prior variance and of

the conditional variance on the basis of their inverses: σ2a(s) = 1

1/σ2a(s)

= 11/σ2

a+1/σ2s.

In our example, with ai ∼ N(0, 1), and si = ai + ϵ, with ϵ ∼ N(0, σ2

L/n), it follows

that the ex-post expected value of the ability equals ai(si) = 0 + (si − 0) 1

12+ σ2

L/n

=

LL+n σ2 si. The variance of the posterior equals σ2

a(s) = 1

1/12+1/(

σ2

L/n

) = n σ2

L+n σ2 .

Consequently, the posterior of an agent’s ability is distributed according to ai | si ∼

N( LL+n σ2 si,

n σ2

L+n σ2 ).

The posterior estimation of an agent’s ability results from a weighted average of the

prior a and the signal s. The weight on the prior equals its precision 1/σ2a

1/σ2a+1/σ2

s, while

the weight on the output signal corresponds to the measurement precision 1/σ2s

1/σ2a+1/σ2

s.

In our example, the weight on the prior equals 1/12

1/12+1/( σ2

L/n)

= n σ2

L+n σ2 and corresponds

to the variance of the ex-post estimation of an agent’s ability. The weight on the

output measure equals1/( σ2

L/n)

1/12+1/( σ2

L/n)

= LL+n σ2 . �

Proof of Proposition 2. For n = 0, we receive X = 0 and for n > 0, we obtain

1 > X > 0. Hence, ∂X∂n

> 0 for n < n. The first derivative of X with respect to n

equals

∂X

∂n= (1 −X)

(log[2] − log[2 (1 −X)

1n ]

−f(n) n σ2 (2 A (L + n σ2) + ρ (2 L + n σ2))

L√

2 π (L + n σ2)√

1 + n σ2/L

), (21)

26

with

f(n) =e− (n ρ σ2+2 A (L+n σ2))2

8 L (L+n σ2)

Φ((

A + n (2 A+ρ) σ2

2 L

)√1

1+σ2 n/L

) =e− (n ρ σ2+2 A (L+n σ2))2

8 L (L+n σ2)(1 + erf [

A+n 2 (A+ρ) σ2

2 L√2+2 n σ2/L

])

=e− (n ρ σ2+2 A (L+n σ2))2

8 L (L+n σ2)

2 (1 −X)1n

. (22)

For equation (22) applies the range of 0 < f(n) ≤ 1. Furthermore, f(n) is increasing

in L and decreasing in ρ, σ, n, A. The denominator represents the probability that

the critical s will be undercut and the manager will retain his job, given a single

signal s. This probability is decreasing in L and increasing in ρ, σ, n, A. Accordingly,

this relationship is oppositional for the whole equation (22), because its numerator

is positive.

To see that X has a unique maximum at some strictly positive value of n, note that,

according to equation (21), ∂X∂n

> 0 when

log[2] >f(n) n σ2 (2 A (L + n σ2) + ρ (2 L + n σ2))

L√

2 π (L + n σ2)√

1 + n σ2/L+ log[2 (1 −X)

1n ]. (23)

The second term of the right-hand side gets arbitrarily close to log[2] for large values

of n. Therefore, this inequation is not met, when n is large enough and ∂X∂n

< 0

holds for n > n. Hence, for n → ∞, X tends to zero. �

Proof of Proposition 3. According to equation (13), the manager’s total costs

depend both on X and C(n). The costs C(n) are always positive: They equal zero

for n = 0 and are strictly increasing for n > 0. X equals zero for n = 0 and is

strictly positive for n > 0. The run of the replacement cost curve is concave, with

∂X∂n

> 0 for n < n and ∂X∂n

< 0 for n > n.

27

For n = 0, we receive X = C(n) = 0. For large values of n, according to the proof

of proposition 2, X tends to zero, when n → ∞. As C(n) is strictly increasing in n,

C(n) → ∞ and therefore also X + C(n) → ∞, when n → ∞.

For n < n both ∂X∂n

> 0 and ∂C(n)∂n

> 0 and therefore we receive ∂X+C(n)∂n

> 0. For n >

n, we receive ∂X∂n

< 0 and ∂C(n)∂n

> 0. In this range, there can exist a local minimum of

total costs at some strictly positive value of n, where ∂X∂n

= −∂C(n)∂n

= −12c√L/n∗,

thus where the positive slope of the labor cost curve equals the negative slope of

the replacement cost curve. This corresponds to the result of our implicit function

g(n, y) = ∂X+C(n)∂n

= 0, derived in the proof proposition 4, equation (24).

Next, we will determine in which cases the manager optimally chooses n∗ = 1, so

that we can derive conditions required for n∗ > 1. First, for comparatively large

values of c, marginal costs are largely increasing in n and total costs can be minimal

in the not feasible range of 0 < n < 1. In this case the manager optimally chooses

n∗ = 1. Second, when c is comparatively high, total costs at the local minimum

can be higher than for n = 1: X + C(n) |n=nmin> X + C(n) |n=1, so that the

manager optimally chooses n∗ = 1. Third, for extremely large values of c, we receive

∂C(n)∂n

> ∂X∂n

for n > n. In the range, where a minimum can be located, the labor

cost increase is always higher than a decrease in replacement costs. As exemplified

in the right picture of figure 2, in this case the total cost curve is strictly increasing

in n and no local minimum exists.

Consequently, for c sufficiently small, the manager optimally chooses n∗ > 1, while,

for c comparatively large, manager’s costs are minimal for n∗ = 1. �

Proof of Proposition 4. We use the Implicit Function Theorem to proof the

comparative static results. The first order condition for a minimum requires that

∂(X+C(n))∂n

= 0. The result represents our implicit function g(n, y) = 0, with y

standing for one of the exogenous parameters A, L, ρ and σ2. The comparative

28

static results follow from ∂n∗

∂y= −

∂g(n,y)∂y

∂g(n,y)∂n

. The second order condition for a minimum

requires that the sign of the denominator is positive: ∂g(n,y)∂n

= ∂2(X+C(n))∂n2 > 0.

Hence, as ∂g(n,y)∂n

= 0, we fulfill the conditions of the Implicit Function Theorem.

Now it is sufficient to consider the sign of the numerator −∂g(n,y)∂y

for each exogenous

parameter. When the first derivative is negative, then n∗ is increasing in y or vice

versa. Differentiating equation (13) with respect to n and equating to zero yields

our implicit function

g(n, y) =∂(X + C(n))

∂n= 0 =

1

2c√L/n + (1 −X)

(− f(n) n σ2 (2 A (L + n σ2) + ρ (2 L + n σ2))

L√

2 π (L + n σ2)√

1 + n σ2/L+ log[2] − log[2 (1 −X)

1n ]). (24)

As labor costs are increasing in team size, it follows that the labor cost curve has a

positive slope and ∂C(n)∂n

> 0, while the replacement probability curve has a negative

slope, with ∂X∂n

< 0 for n > n. The first order condition requires that −∂C(n)∂n

= ∂X∂n

.

This equals −12c√L/n∗ = ∂X

∂n. We make use of these results for the proofs of the

comparative static results for all four parameters A, L, ρ and σ2.

For the parameter A, proposition 4 is fulfilled, when ∂n∗

∂A= −

∂g(n,A)∂A

∂g(n,A)∂n

< 0 and conse-

quently when ∂g(n,A)∂A

> 0.

Differentiating equation (24) with respect to A, rearranging and simplifying terms

and using the result of −12c√

L/n∗ = ∂X∂n

yields the condition

0 < −4 n σ2 − 4 (L + n σ2)

(2 +

c√L/n n

1 −X

)+

(ρ +

n (2 A + ρ) σ2

L− L ρ

L + n σ2

)

n σ2 (2 A L + n (2 A + ρ) σ2 + 4 f(n) L√

2/π√

1 + n σ2/L). (25)

When condition (25) is fulfilled, then n∗ is decreasing in A. Next, we replace 1 ≤

1/(1 −X) < 2 by (1 −X)−1 = 2, as this term is in the negative part of the right-

hand side of equation (25). As this right-hand side overall has to be positive, this

29

replacement corresponds to our worst case for the fulfillment of the comparative

static result and hence represents a generally applicable solution. For the same

reasons, as f(n) is in the positive part of the right-hand side of equation (25), we

set f(n) = 0. This yields the following condition, required for an approval of the

comparative static result of proposition 4:

0 < SA(A) = −8 L(1 + c

√L/n n

)+

n2 (2 A + ρ)2 σ4

L

+n σ2

(− 4

(3 + 2 c

√L/n n

)+ (2 A + ρ)2 − L ρ2

L + n σ2

). (26)

It is convenient to define the right-hand side of equation (26) as SA(A). We complete

our proof of proposition 4 by deriving conditions, for which SA(A = 0) > 0 and

∂SA(A)∂A

> 0. Then the right-hand side of equation (26) is positive for A = 0 and is

increasing in A, hence is positive for any value of A.

For SA(A = 0) > 0, it is necessary that

ρ > 2

√L

(1

2 L + n σ2+

L + 2 n σ2

n2 σ4

)(27)

and

c <1

8

√(−8 L3 − 20 L2 n σ2 + 2 L n2 (ρ2 − 6) σ4 + n3 ρ2 σ6)2

L3 n (L + n σ2)4. (28)

From this it is easy to show that the first derivative with respect to n of equation

(27) is negative. It follows that this condition is decreasing in n. Hence, we can set

n = 1 and receive our final worst case condition for the risk aversion parameter ρ of

equation (14), proposition 4. Accordingly, the first derivative of equation (28) with

respect to n shows that the condition is constant or increasing in n, when condition

(14) is satisfied. Hence, by setting n = 1, we receive our final worst case condition

for c of equation (15), proposition 4.

Finally, the first derivative of equation (26) with respect to A equals

∂SA(A)

∂A=

4 n (2 A + ρ) σ2 (L + n σ2)

L> 0. (29)

30

As equation (29) is positive, n∗ is decreasing in A, when the conditions (14) and

(15) of proposition 4 are fulfilled. �

Proof of Proposition 5. For the parameter ρ, proposition 5 is fulfilled, when

∂n∗

∂ρ= −

∂g(n,ρ)∂ρ

∂g(n,ρ)∂n

< 0 and accordingly when ∂g(n,ρ)∂ρ

> 0.

Differentiating equation (24) with respect to ρ, rearranging and simplifying terms


L/n∗ = ∂X∂n


0 < −4 L

(3 +

4 c L√

L/n n

1 −X+

L

L + n σ2

)

+n σ2

(L + n σ2)2(2 L (A + ρ) + n (2 A + ρ) σ2)

(2 A L + n (2 A + ρ) σ2 + 2 f(n) L√

2/π√

1 + n σ2/L). (30)

When condition (30) is fulfilled, then n∗ is decreasing in ρ. Exactly as in the proof

of proposition 4, we replace 1 ≤ 1/(1 − X) < 2 by (1 − X)−1 = 2, as this term

is in the negative part of the right-hand side of equation (30). As this right-hand

side overall has to be positive, this replacement represents our worst case for the

fulfillment of the comparative static and hence is a generally applicable solution.

Accordingly, as f(n) is in the positive part of the right-hand side of equation (30),

we set f(n) = 0. This yields the following condition, required for an approval of the


0 < Sρ(A) = −4 L(3 + 2 c

√L/n n

)+n (2 A + ρ)2 σ2 +

L3 ρ2

(L + n σ2)2− L2 (4 + ρ2)

L + n σ2. (31)

We define the right-hand side of equation (31) as Sρ(A). We complete our proof of

proposition 5 by deriving conditions, for which Sρ(A = 0) > 0 and ∂Sρ(A)

∂A> 0. Then

31

the right-hand side of equation (31) is positive for A = 0 and is increasing in A,

hence is positive for any value of A.

For Sρ(A = 0) > 0, it is necessary that

ρ > 2

√L (L + n σ2) (4 L + 3 n σ2)

n2 σ4 (2 L + n σ2)(32)

and

c <1

8

√(−16 L3 − 28 L2 n σ2 + 2 L n2 (ρ2 − 6) σ4 + n3 ρ2 σ6)2

L3 n (L + n σ2)4. (33)

As in the proof of proposition 4, the first derivative with respect to n of equation

(32) is negative and hence is decreasing in n. Consequently, we can set n = 1

and receive our worst case condition for the risk aversion parameter ρ of equation

(16), proposition 5. Accordingly, the first derivative of equation (33) with respect

to n shows that the condition is constant or increasing in n, when constraint (16)

is satisfied. Hence, by setting n = 1 in equation (33), we receive our worst case

condition for c of equation (17), proposition 5.


Sρ(A)

∂A= 4 n (2 A + ρ) σ2 > 0. (34)

As equation (34) is positive, n∗ is decreasing in ρ, when the conditions (16) and (17)

of proposition 5 are satisfied. �

Proof of Proposition 6. For the parameter σ2, proposition 6 is fulfilled, when

∂n∗

∂σ2 = −∂g(n,σ2)

∂σ2

∂g(n,σ2)∂n

< 0 and accordingly when ∂g(n,σ2)∂σ2 > 0.

Differentiating equation (23) with respect to σ2, rearranging and simplifying terms


L/n∗ = ∂X∂(n)


0 < −16 L−9 c L

√L/n n

1 −X+

12 L n σ

L + n σ− 8 L n (2 A + ρ) σ

2 L (A + ρ) + n (2 A + ρ) σ

32

+n σ

(L + n σ)2(2 L (A + ρ) + n (2 A + ρ) σ)

(n ρ σ + 2 A (L + n σ) + 2 f(n) L

√2/π

√1 + n σ/L

). (35)

When condition (35) is fulfilled, then n∗ is decreasing in σ2. As in the proof of

proposition 4, we replace 1 ≤ 1/(1 − X) < 2 by (1 − X)−1 = 2, as this term is

in the negative part of the right-hand side of equation (34). As this right-hand

side overall has to be positive, this replacement represents our worst case for the

fulfillment of the comparative static and hence is a generally applicable solution.

Accordingly, as f(n) is in the positive part of the right-hand side of equation (35),

we set f(n) = 0. This yields the following condition, required for an approval of the


0 < Sσ2(A) = −6 L(2 + 3 c

√L/n n

)+ n (2 A + ρ)2 σ (36)

+L3 ρ2

(L + n σ)2+ L2

(16 (A + ρ)

2 L (A + ρ) + n (2 A + ρ) σ− 12 + ρ2

L + n σ

).

We define the right-hand side of equation (36) as Sσ2(A). We complete our proof

of proposition 6 by deriving conditions, for which Sσ2(A = 0) > 0 and∂Sσ2 (A)

∂A> 0.

Then the right-hand side of equation (36) is non-negative for A = 0 and is increasing

in A, hence is positive for any value of A.

For Sσ2(A = 0) > 0, it is necessary that

ρ > 2

√L (L + n σ) (8 L2 + 8 L n σ + 3 n2 σ2)

n2 σ2 (2 L + n σ)2(37)

and

c <1

18

√(−32 L4 − 64 L3 n σ + 4 L2 n2 (ρ2 − 11) σ2 + 4 L n3 (ρ2 − 3) σ3 + n4 ρ2 σ4)2

L3 n (L + n σ)4 (2 L + n σ)2.(38)

Again, it is easy to show that the first derivative with respect to n of equation (37) is

negative and, consequently, that the condition is decreasing in n. Hence, we can set

33

n = 1 and receive our worst case condition for the variance of the error term σ2 of

equation (18), proposition 6. In addition, the first derivative of equation (38) with

respect to n shows that the condition is constant or increasing in n, when constraint

(18) is satisfied. Hence, by setting n = 1, we receive our worst case condition for c

of equation (19), proposition 6.


∂Sσ2(A)

∂A= 4 n σ

(2 A + ρ− 4 L2 ρ

(2 L (A + ρ) + n (2 A + ρ) σ)2

). (39)

Given our constraint of equation (18), the result of equation (39) is positive. There-

fore, n∗ is decreasing in σ2, when the conditions of proposition 6 are satisfied. �

Proof of proposition 7. The optimal team size n∗ is increasing in L, when

∂n∗

∂L= −

∂g(n,L)

∂σ2

∂g(n,L)∂n

> 0 and accordingly when ∂g(n,L)∂L

< 0.

Differentiating equation (23) with respect to L, rearranging and simplifying terms


L/n∗ = ∂X∂n


0 > 4 L (L + n σ2) (2 L (A + ρ) + n (2 A + ρ) σ2)

(2 L− n σ2 + 4 (L + n σ2) +

c√L/n n (L + n σ2)

1 −X

)

−n (2 L (A + ρ) σ + n (2 A + ρ) σ3)2

(2 A L + n (2 A + ρ) σ2 + 2 f(n) L

√2/π

√1 + n σ/L

)

+L2 (L + n σ2)2

(4 c√

L/n√

2 π (L + n σ2)√

1 + n σ2/L

n f(n) (1 −X) σ2− 16 (A + ρ)

). (40)

When condition (40) is fulfilled, then n∗ is increasing in L. As in the proof of

proposition 4, we replace 1 ≤ 1/(1−X) < 2 by (1−X)−1 = 2, as this term is in the

34

positive part of the right-hand side of equation (40). As this right-hand side overall

has to be negative, this replacement is our worst case condition for the fulfillment

of the comparative static and therefore represents a generally applicable solution.

This yields the following condition:

0 > SL(A) = L2

(8 c√

L/n√

2 π (L + n σ2)√

1 + n σ2/L

n f(n) σ2− 16 (A + ρ)

)

+4 L(2 L

(3 + c

√L/n n

)+ n (3 + 2 c

√L/n n

)σ2)

(2 A + ρ +

L ρ

L + n σ2

)

−n σ2

(2 A + ρ +

L ρ

L + n σ2

)2 (2 A L + n (2 A + ρ) σ2

+2 f(n) L√

2/π√

1 + n σ2/L. (41)

We define the right-hand side of equation (41) as SL(A). The optimal team size n∗

is increasing in L, when SL(A = 0) < 0 and ∂SL(A)∂A

< 0. Then the right-hand side

of equation (41) is negative for A = 0 and is decreasing in A, hence is negative for

any value of A.

The expression in the first line of equation (41) is negative, when

c <√

2π

√f(n)2 n3 ρ2 σ4

(L+n σ2)3. As this term increases in n, we can set n = 1 and receive

our worst case condition of

c <

√2

π

√f(n)2 ρ2 σ4

(L + σ2)3. (42)

Additionally, for the remaining expression (second, third and fourth line) of equa-

tion (41) it is necessary that SL(A = 0) < 0. For the purpose of simplification,

we set f(n) = 0, as this term is in the negative part of our observed expression,

which represents our worst case condition. Consequently, the expression is negative,

35

when ρ > 2√

3√

L (L+n σ2)n2 σ4 and c < 1

8

√(2 L+n σ2)2 (12 L2+12 L n σ2−n2 ρ2 σ4)2

L3 n (L+n σ2)4. The

condition for ρ is decreasing in n, hence we can set n = 1 and receive our worst

case condition of equation (43), which corresponds to the result of proposition 7.

Accordingly, the condition for c is increasing in n, when condition (43) is satisfied

and we can set n = 1 for our worst case condition of equation (44), with

ρ > 2√

3

√L (L + σ2)

σ4(43)

and

c <1

8

√(2 L + σ2)2 (12 L2 + 12 L σ2 − ρ2 σ4)2

L3 (L + σ2)4. (44)

When the constraints of the equations (42)-(44) are fulfilled, then SL(A = 0) < 0

holds. This is the case, when c is sufficiently small and ρ sufficiently high.

The first derivative of equation (41) with respect to A equals

∂SL(A)

∂A= 8 L

(2 L

(3 + c

√L/n n

)+ n

(3 + 2 c

√L/n n

)σ2)

−16 L2 − 2 n σ2

(2 A + ρ +

L ρ

L + n σ2

)

(2 L (3 A + ρ) + 3 n (2 A + ρ) σ2 + 4 f(n) L

√2/π

√1 + n σ2/L

). (45)

When equation (45) is negative, then equation (41) is decreasing in A. Again, as

f(n) is in the negative part of equation (44), we can set f(n) = 0 to receive our worst

case condition. From that it is easy to show that equation (45) is negative, when

the constraints of the equations (43) and (44) are fulfilled. Overall, n∗ is increasing

in L, when the conditions of the equations (42)-(44) are satisfied. �

36

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