How leadership insecurity influences hiring and
demotion decisions
Lotte Swank∗ September 2, 2019
Abstract
Leaders can experience a threat of replacement from their subordi-nates. For example, a political party leader can fear being replaced by his running mate and losing the potential to become famous. A manager can be afraid that one of his coworkers takes over his posi-tion, leading to a decrease in his wage and power. Previous research has shown that this threat, defined as leadership insecurity, disturbs leaders’ hiring decisions. One proposed solution for this problem is delegating complete authority to the leader, which gives the leader the certainty of not being fired. In this paper, I show that this commitment is costly because it devastates the opportunity to replace incompetent leaders. I investigate under which circumstances it is beneficial to fully authorize the leader. Lastly, I propose some other solutions that decrease the hiring inefficiencies, while disturbing demotion efficiency less.
Keywords — leadership insecurity, hiring efficiency, demotion efficiency, organizational structure
1
Introduction
In 2012, Diederik Samsom was promoted to the position of party leader of the Dutch Labour Party (PvdA), which was at the time the largest left-wing party of the Netherlands. He invited his political friend Lodewijk Asscher to join the PvdA and to become his deputy leader. Asscher joined and a few years of productive collaboration followed. In 2016, however, the
∗
PvdA started losing parliamentary seats and the critics towards Samsom became more negative. As a result, the party members requested a vote of confidence. Samsom lost this vote and was replaced by Asscher in December 2016.
Situations like this are not uncommon in politics. For example, in 2019, Theresa May, the former Prime Minister of the United Kingdom, received negative feedback from the public after not being able to pass Brexit-related legislation. As a result of this lack of support, she had to resign. Lately, Boris Johnson has been appointed as her within-party successor.
Also within corporate firms, it became more common that leaders — i.e. managers and CEOs — got replaced by their subordinates. Over the past decades, job protection of top executives in OECD countries decreased, meaning that it got easier for firm owners to fire or demote managers and CEOs (Jenter and Kanaan, 2015). Additionally, the percentage of top execu-tives that are hired from the existing employee pool increased (Dewatripont and Tirole, 1994; Tsoulouhas et al., 2007). In sum, since high position insecurity increased and the occurrence of internal promotions got more common, corporate leaders can experience a threat from their subordinates. In this paper, the threat that a leader experiences as a consequence of fearing replacement by subordinates is defined as leadership insecurity. I derive a theoretical model where a leader has to perform a task. Beforehand, the leader has to decide whether he wants to fulfill this task on his own or whether he wants to collaborate with another agent. Good collaboration increases task performance but also leads to the possibility that the leader gets replaced by his coworker. I show that this leadership threat influences the leader’s hiring decisions.
employee. The owner delegates the authority over the cooperation decision to the leader because the leader has better information about the ability of the potential employee. The leader, however, does not have complete authority. If the leader hires an employee, the owner still has the power to replace the leader by the new employee.
The model has the following main characteristics. First, leaders and (po-tential) employees differ in ability. Additionally, the ability of the leader and the employee are assumed to have a complementary effect on performance; a good leader makes his subordinate more productive and vice versa. Sec-ond, hiring an employee is costly because the leader has to spend part of his time in explaining the employee the main processes within the organization. Therefore, the owner prefers the leader to only cooperate with high ability employees. Third, the owner wants the most able agent to get the leadership position. If the leader hires an employee, the owner receives an imperfect signal about the employee’s ability. Based on this signal, the owner decides whether she replaces the leader by the new employee.
I define and investigate the following concepts. First, let hiring efficiency be defined as the extent to which the owner is satisfied with the hiring de-cision of the leader. Thus, full hiring efficiency occurs if the leader makes the same decisions as the owner would have made. If the leader’s hiring de-cisions diverge from the owner’s wishes, hiring efficiency decreases. Second, let demotion efficiency be defined as the measure of the owner’s satisfaction with the division of the agents over the leadership position and the regu-lar employee position. Thus, with full demotion efficiency, the leader gets always demoted if the applying employee is more able than the leader.
she does not have to be afraid to be replaced by his or her subordinates. I show that this First Benchmark Model is characterized by absolute hir-ing efficiency. Demotion efficiency, however, is low in the First Benchmark Model.
The Second Benchmark Model is of more theoretical nature. It investi-gates the outcomes in a situation where the owner has sufficient information and power to make all the decisions. Thus, the Second Benchmark Model describes a situation where the sole role of the leader is to perform the task but he does not have any authority; the hiring and the replacement deci-sion are both made by the owner. I show that both hiring efficiency and demotion efficiency are maximized in the Second Benchmark Model. There-fore, the owner’s expected utility level is as high as possible in this Second Benchmark Model.
The Main Model lies in between the two extremes of the benchmark models. Now, the leader has the authority to decide whether he wants to cooperate with the potential employee. The owner, however, has the power to replace the leader by the new employee, if an employee is hired. As a consequence, the leader can experience a threat to his position if he hires high ability employees. I show that the insecurity of the leader can lead to a decrease in hiring efficiency. Yet, demotion efficiency can get more efficient at the moment that leadership insecurity is introduced.
I compare the equilibrium outcomes of the three models. I find that the Second Benchmark Model is as efficient as possible. Yet, in practice, leaders have usually more information about the organization’s core processes than the owners have. Therefore, it is hard to run an organization without given the leader any authority. Hence, a choice between an organizational struc-ture where the leader has complete authority — i.e. the First Benchmark Model — and an organizational structure where the leader has limited au-thority — i.e. the Main Model — has to be made. I derive under which organizational structure hiring and demotion decisions are disturbed least.
owner power, respectively. In section 6, the Main Model with leadership in-security is analyzed. In section 7, the implications of the derived equilibria are discussed. Finally, section 8 concludes.
2
Literature review
This paper is not the first to focus on the disturbance in hiring decisions as a consequence of leadership insecurity. All models in the previous papers (which are described below) show that leadership insecurity completely dis-turbs the leader’s hiring decision. Even though some of the papers propose solutions that can (partly) restore hiring efficiency, the result that, in ab-sence of interventions, the leader’s strategy always harms the owner, seems rather extreme.
First, South and Matejka (1990) theorize that bad management rein-forces itself. They claim that weak performing managers are reluctant to hire individuals who may harm their status. As a consequence, the new reg-ular employee positions and, with that, future management positions, are also occupied by weak agents, who make bad selection decisions in turn.
Just as in my paper, Friebel and Raith (2004) describe a three-layered model. In their model, a manager works for a principal and has to hire a worker. After being hired, the new worker has, to some extent, the op-portunity to signal his ability to the principal. Afterwards, the principal can replace the manager by the worker. The authors define the extent to which the worker can signal his ability as a hierarchical communication pa-rameter. They show that hiring decisions get more disturbed if the degree of hierarchical communication is higher. Furthermore, they state that the only way to overcome the disturbance in hiring decisions is to cut off all communication streams between the owner and the employee.
academic position. Every period, the university fires the worst-performing incumbents. The remaining incumbents have to fill the vacant places with new entrants. The author shows that the incumbents have incentives to not hire the most competent entrants. This is due to the fear of being fired in future periods. Lastly, it is shown that if universities grant incumbents tenure, their fear of being replaced decreases. As a consequence, the selection of entrants becomes more efficient.
The effect of position insecurity on hiring decisions has been investigated also in political situations. A theoretical model of Egorov and Sonin (2011) describes the fear of coups that dictators experience. The authors assume that the probability that a coup occurs increases in the ability of the asso-ciates of the dictator. As a consequence, dictators do not dare to collaborate with competent associates. The authors claim that the reason that dictator-ships are inefficient is not that the dictators are not competent themselves but that they are unwilling to collaborate with decent associates.
A common conclusion of the aforementioned papers is that leaders’ fear, as a consequence of position insecurity, always harms the principal because hiring decisions get disturbed. Yet, a few questions remain open. First, why does leadership insecurity still exist in practice, despite the negative conse-quences for the principal? In other words, why do principals not commit to retaining their original leaders? Second, the former models about leadership insecurity describe an inevitable aversion for hiring high ability employees. But what happens with those most competent employees if all leaders do not want to hire them? Last, if having a high ability is a disadvantage rather than an advantage in the job market, why do people still decide to acquire formal education?
expected utility. This way, I can give a second explanation for why leadership insecurity is still present in practice and why owners do not transfer complete authority to the leaders. Furthermore, my model can answer the other two stated questions.
Economic research has already investigated one reason for why organi-zations with structures that lead to leadership insecurity are still present in practice. This reason is that leadership insecurity can be beneficial to the principal because it can motivate the leader and the employee to exert effort. Usually, in economic models, firm profitability increases when agents exert effort. However, exerting effort is costly for the employee herself. Therefore, the principal has to incentivize the agents (Sappington, 1991).
One of the first discussed ways to encourage employee effort is a pay-for-performance scheme. With schemes like these, the wage of the agent is positively dependent on her performance. Performance, in turn, is a positive function of the agent’s effort level. Thus, the agent is rewarded for exerting effort. By choosing the piece-rate wage properly, the principal can induce the agent to choose optimal effort levels (Lazear, 1986, 2000).
Also empirically, it has been shown that these pay-for-performance schemes motivate agents to work harder (Lavy, 2009) and lead to higher firm profits (Gielen et al., 2010). However, if agents are risk-averse or if there is no perfect relation between performance and effort levels, pay-for-performance schemes are not efficient (Milkovich and Wigdor, 1991). Furthermore, it can be costly to measure the individual performance of all agents that are working for a firm (Weibel et al., 2009).
Thus, if pay-for-performance schemes do not work because of one of the aforementioned reasons, a contest can be used to solve those problems. For example, in a contest, only the agent’s performance relative to the other agents has to be determined. Therefore, contests often have less adminis-trative expenses than pay-for-performance schemes (Lazear and Oyer, 2007; Oyer and Schaefer, 2010). Furthermore, contests are robust for common risk effects, which benefits risk-averse agents (Green and Stokey, 1983). Last, contests are more efficient than pay-for-performance schemes if the employ-ees are risk-averse (Nalebuff and Stiglitz, 1983). Due to these advantages, the popularity of contests has increased over time (Ehrenberg and Bognanno, 1990).
Also, it has become more common to implement contest reward struc-tures where the tournament prize is a promotion to a higher position (Baker et al., 1988; Boudreau et al., 2016). Bognanno (2001) showed that contests with a prize in the form of promotion also lead to optimal effort levels. DeVaro (2006) showed empirically that productivity rises in firms that use contest reward structures with promotion prizes.
By introducing leadership insecurity, a situation that is similar to a con-test arises. The only difference with a regular concon-test structure is the asym-metry. In my model, the employee strives for a promotion while the leader’s objective is to not be demoted. Yet, both the leader and the employee are induced to increase their effort levels in the hope of occupying the leadership position.
In this paper, I derive a second reason for why leadership insecurity is still present in practice. That is, organizational structures with leadership insecurity might be beneficial to the owner because they maintain the op-portunity to replace the leader. Usually, leaders have more influence on firm profitability or organizational efficiency than regular employees (Or-tega, 2003). Therefore, if an employee is observed to be more competent than the leader, the owner can increase her expected utility level by al-ternating the positions. Thus, even though leadership insecurity can harm employee selection efficiency, it can enhance demotion efficiency.
variable. That is, both the leader and the employee can either have a high or a low ability. In these type of models, only one equilibrium — with complete hiring disturbance — exists. This paper distinguishes itself from earlier work on position insecurity by also focusing on the positive effects of leadership insecurity — i.e. a potential increase in demotion efficiency — and compare those with the negative effects — i.e. the decrease in hiring efficiency. By modelling ability as a continuous variable instead of a binary variable, I derive multiple equilibria. One of these equilibria has the same characteristics as the former models and predicts a complete disturbance in hiring decisions. However, I demonstrate two other equilibria where hiring efficiency is harmed less. In those two equilibria, leadership insecurity is, under certain circumstances, beneficial to the owner. As such, I add an additional explanation for the existence of leadership insecurity.
3
The model
The model in this paper describes the behaviour of two agents, of whom one performs a task for a principal. This relationship can be found in society in multiple places. First, the agent could be a manager who runs a firm. The firm is owned by shareholders, who then form the principal. Second, the agent could be a political party leader, who represents the interests of the party members or voters — i.e. the principal. Principal-agent relationships like these can, however, also be found at a smaller scale — e.g. a household that hires a construction worker to renovate its house. In this paper, I refer to the agent — i.e. the manager, the political party leader or the construction worker — as the leader (referred to by index ”l” and being a male). The principal — i.e. the shareholders, the political party members or the household — is called the owner (referred to by index ”o” and being a female).
can appoint a running mate. A construction worker can hire a coworker. In this paper, I refer to the potential collaborator as the potential employee (referred to by index ”e” and being a female).
There are two strategic players in the model, who make active decisions: the leader and the owner. The third player, the potential employee, does not engage in active decision making.
The first main characteristic of the model is that both leaders and em-ployees can differ in ability. The ability of the employee (”ae”) comes from
the uniform distribution on [0, 1]. Also the ability of the leader (”al”) is
uni-formly distributed on [0, 1]. The leader’s ability and the employee’s ability are independently distributed.
I assume that the performance on the given task is increasing in the ability of the leader and the employee. Hiring an employee is costly for the leader and for the owner since the leader has to spend a part of his time in giving the employee explanations about the work processes and organizational strategies. Therefore, the owner wants the leader to only cooperate with employees with sufficiently high ability.
The second main characteristic of the model is leadership insecurity, which is modelled as follows. I assume that the leader has a greater influence on task performance than the regular employee. Therefore, the owner prefers that the most able person occupies the leadership position. If the leader decides to collaborate — i.e. he hires an employee — the owner receives an imperfect signal about the ability of the employee. Based on this signal, the owner can decide to alternate the positions of the leader and the employee. The timing of the model is as follows. The model is described in four stages, Stage 1 to 4.
Stage 1. Nature determines the leader’s ability — al — and the employee’s
ability — ae. The leader’s ability is common knowledge and observed by
everyone. Only the leader and the employee observe ae.
Stage 2. The leader decides whether he hires the potential employee, h ∈ {0, 1}. This decision is dependent on his belief about the probability that the owner replaces him if he hires an employee with ability ae. If the leader
payoffs are realized and the game ends. If h = 1, the employee is hired. Then, the game continues in Stage 3.
Stage 3. The owner observes that h = 1 and receives an imperfect signal about the employee’s ability. The owner makes an estimation about the employee’s ability.
Stage 4. The owner makes her replacement decision, d ∈ {0, 1}. If d = 0, the leader and the employee retain their original positions and payoffs are realized. If d = 1, the owner demotes the leader to the regular employee position and promotes the employee to the leadership position. Afterwards, payoffs are realized.
Given the values of ae and al, the leader’s utility function is represented
by:
Ul(h, d) = h · (ae− c) + (1 − d) · W. (1)
If h = 1, then the leader hires an employee. Then, the leader’s utility is increasing in the ability of the employee. Furthermore, hiring an employee is costly. These costs, in terms of the time that the leader has to spend in informing the employee about the procedures and strategies, are represented by c < 1. The leader’s extra benefit of occupying a leadership position is represented by W . This extra benefit includes the benefits of having a leadership position that are not present in a regular employee position. This includes, for example, becoming famous as a political party leader, receiving a wage premium as a manager and having the power to make decisions about, for instance, working schedules. If the leader is not replaced — i.e. d = 0 — the leader receives these leadership benefits W . For the sake of mathematical simplicity, I do not include the effect of the leader’s ability on his utility level directly. However, W can be an increasing function of the leader’s ability.
Given ae and al, the owner’s utility function is represented by:
Uo(h, d) = l · i + h · (e − c). (2)
Here, l ∈ {al, ae} represents the ability of the agent who occupies the
his position. If d = 1 — i.e. the new employee replaces the leader — l is equal to ae. Likewise, e ∈ {al, ae} represents the ability of the agent that
occupies the regular employee position. So, e = al if d = 1 — i.e. the leader
and the employee change positions — and e = ae if d = 0, meaning that
no position changes are made. If an employee is hired — i.e. h = 1 — the leader has to spend time in explaining the strategies to his coworker, leading to cost l · c, for l ∈ {al, ae}.
Variable i ≥ c represents the importance of the leader, relative to a regular employee. If i = c, the leader and the employee perform as equal team members. If i gets larger, the relative importance of the agent in the leadership position increases.
The utility of the owner is increasing in a complementary manner in the ability of the leader and the ability of the employee. If supervision is very bad — i.e. l is close to 0 — the subordinate cannot perform, regardless of his ability. The agent with the leadership position, however, can produce i on his own.
In the upcoming sections, I analyze this model in three different settings. First, in section 4, the First Benchmark Model without leadership insecurity is described. Here, the leader makes his hiring decision — h ∈ {0, 1} — while the owner cannot make any active decisions. Thus, the leader has complete authority, meaning that leadership insecurity does not exist. Second, in section 5, I analyze the Second Benchmark Model — where the owner has perfect information and has the power to make all the decisions. Thus, the hiring decision and the replacement decision are both made by the owner. This means that the leader does not have any authority. Finally, in section 6, my Main Model with leadership insecurity is analyzed. Here, the hiring decision is made by the leader and the replacement decision is made by the owner.
4
Benchmark I - Complete leadership authority
independent of his hiring decision. Therefore, in the First Benchmark Model, leadership insecurity does not exist.
One example of organizations without leadership insecurity is formed by monarchies. In a monarchy, the leader — i.e. the king or the queen — is appointed by birth. No one else, except for his/her family members after his/her death, can claim his/her position. Another example is the Catholic Church. After the pope is elected, he is expected to hold his position to the moment he dies. Additionally, the same lifetime appointment exists for the high cardinals that are appointed by the pope. Thus, just like monarchs, the high ranking Catholic officials do not experience any threats concerning their positions. In corporate firms, leaders — i.e. the managers or the CEOs — can also feel completely safe about their positions if employment protection is very strict. In such a situation, even when the owner — i.e. the shareholders — would prefer to fire the leader, they are prohibited to do so by law. Lately, employment protection has been most strict in Turkey, Portugal and Mexico (Boeri and Van Ours, 2013). Firms in countries like these are therefore most likely to fulfill the assumptions of the First Benchmark Model. The timing in the First Benchmark Model is as follows. Because the owner does not make active decisions, only Stage 1 and Stage 2 exist. Stage 1. Nature determines the leader’s ability — al — and the employee’s
ability — ae. The leader’s ability is common knowledge and observed by
everyone. Only the leader and the employee observe ae.
Stage 2. The leader decides whether he hires the potential employee, h ∈ {0.1}. If h = 0, then the potential employee is not hired, the leader’s and owner’s payoffs are realized and the game ends. If h = 1, then the leader hires the employee, the leader’s and owner’s payoffs are realized and the game ends.
Thus, in this First Benchmark Model without leadership insecurity, d = 0 always holds. Substituting d = 0 in (1) yields:
Ulbm1(h, 0) = h(ae− c) + W. (3)
without leadership insecurity, l = al and e = ae always hold. Substituting
this in (2) yields:
Uobm1(h, 0) = al i + h · (ae− c). (4)
This model is shown graphically in Figure 1.
L h = 1 h = 0 Ubm1 l (1, 0) = ae− c + W Uobm1(1, 0) = al(i + ae− c) Ulbm1(0, 0) = W Ubm1 o (0, 0) = ali
Figure 1: Subgame of the First Benchmark Model, starting from Stage 2. Let Abm1e contain the set of the types of employees with ability ae that
the leader hires. Thus, Abm1e contains all the values of ae for which h = 1
holds. Proposition 1 describes the equilibrium in the First Benchmark Model without leadership insecurity, the set Abm1e , and the owner’s expected utility level.
Proposition 1 The equilibrium in the First Benchmark Model satisfies the following. Let al be the ability of the leader and ae be the ability of the
employee.
b) The equilibrium is characterized by complete hiring efficiency.
c) The expected equilibrium utility level of the owner equals E[Uobm1] = ali +(1−c)
2
2 .
Proof. -Part a) In the First Benchmark Model, the leader knows that cannot get replaced — i.e. d = 0 always holds. The leader’s utility if he hires an employee with ability ae equals:
Ulbm1(1, 0) = ae− c + W. (5)
If the leader does not hire the employee, his utility equals:
Ulbm1(0, 0) = W. (6)
The leader hires the new employee with ability ae in the First Benchmark
Model iff Ulbm1(1, 0) ≥ Ulbm1(0, 0), i.e.
ae− c + W ≥ W ⇐⇒
ae≥ c. (7)
Hence, in the First Benchmark Model, the leader hires employees with ability in the set Abm1e = [c, 1].
-Part b) To determine whether the leader’s hiring decisions are efficient from the owner’s perspective — i.e. whether there is hiring efficiency — the model is solved for the values of ae that the owner would have hired if she
could make the hiring decision herself. Let Abm1oe be defined as the set of types of employees that the owner would have hired. After deriving the set Abm1oe , the set is compared with the leader’s hired set — i.e. Abm1e — to investigate any differences.
If the owner cannot replace the leader, the owner’s utility if h = 1, given ae and al, equals:
The owner’s utility if the employee is not hired, given al, equals:
Uobm1(0, 0) = ali. (9)
The owner would have hired the employee with ability ae iff Uobm1(1, 0) ≥
Uobm1(0, 0), i.e.
al(i + ae− c) ≥ ali ⇐⇒
al(ae− c) ≥ 0 ⇐⇒
ae≥ c. (10)
Hence, the owner prefers the leader to hire set Abm1oe = [c, 1]. Since Abm1oe = Abm1e , there is complete hiring efficiency and the leader’s hiring decisions are efficient from the owner’s perspective.
-Part c) The owner’s expected utility in the First Benchmark Model equals:
E[Uobm1(h, d)] = P r(h = 1) · E[Uobm1(1, 0)] + P r(h = 0) · Uobm1(0, 0) ⇐⇒
E[Uobm1(h, d)] = P r(h = 1)·al(i+E[ae|ae≥ c]−c)
+P r(h = 0)·ali. (11)
Here, the probability that h = 0 equals the probability that ae < c, which
equals c. The probability that h = 1 equals 1 − P r(h = 0) = 1 − c. Lastly, E[ae|ae≥ c] equals the middle of the interval of Abm1e :
E[ae|ae≥ c] =
1 + c 2 . Substituting these values in (11) yields:
Because this First Benchmark Model does not contain leadership insecurity, the leader does not experience any disadvantages of hiring high ability em-ployees. Therefore, the leader’s only reason to reject an employee arises if the employee’s ability is lower than the cost of hiring her. For the owner, this trade-off is the same. That is, the owner would have also hired the em-ployee if her ability is higher than the cost of hiring her. Therefore, there is complete hiring efficiency and the leader’s hiring decisions are efficient from the owner’s perspective.
5
Benchmark II - complete owner power
In the Second Benchmark Model, the leader does not have any authority. That is, the sole role of the leader is related to production. The leader is important to the owner because he performs the task and generates profits. However, in the Second Benchmark Model, the owner has perfect informa-tion and can make the hiring decision herself. Furthermore, the owner has the option to replace the leader by the new employee.
In politics, the characteristics of the Second Benchmark Model are not often found in practice. It is unlikely that the owners — i.e. the members of a political party — have sufficient knowledge to be engaged in all the hiring decisions of the party. Also in large corporate firms, it is uncommon that the shareholders make all the hiring decisions. Thus, the Second Benchmark Model is of more theoretical nature. However, organizations with structures that converge to the characteristics of the Second Benchmark Model do exist. One example is an amateur sports club. In most amateur sports clubs, all daily decisions are made by or need to be approved by the members in General Member Meetings. In these meetings, new employee selections are usually discussed. Furthermore, the members have power over the board composition.
The timing of the Second Benchmark Model is as follows. The game is described in three stages.
Stage 1. Nature determines the leader’s ability — al — and the employee’s
knowledge and thus observed by everyone.
Stage 2. The owner makes the hiring decision, h ∈ {0, 1}. If the owner chooses h = 0, the employee is not hired. Then, the leader’s and the owner’s payoffs are realized and the game ends. If h = 1, the employee is hired. Then, the game continues in Stage 3.
Stage 3. The owner makes the replacement decision, d ∈ {0, 1}. If d = 0, the leader and the employee retain their original positions and payoffs are realized. If d = 1, the owner demotes the leader to the regular employee position and promotes the employee to the leadership position. Afterwards, payoffs are realized.
In the Second Benchmark Model, the utility of the owner, given al and
if h = 0 — i.e. no new employee is hired — equals:
Uobm2(0, 0) = ali. (13)
If h = 1 and d = 0, then the owner hires a new employee but does not alternate the positions of the leader and the employee. In that case, the owner’s utility, given ae and al, equals:
Uobm2(1, 0) = al(i + ae− c). (14)
If h = 1 and d = 1, then the owner hires the employee with ability ae, demotes the leader to the regular employee position, and promotes the
employee to the leadership position. In that case, the owner’s utility equals: Uobm2(1, 1) = ae(i + al− c). (15)
This model is shown graphically in Figure 2 for given values of al and ae.
Let Abm2e contain the set of the types of employees with ability ae that
the leader hires. Thus, Abm2
e contains all values of aefor which h = 1 holds.
O O h = 1 d = 0 d = 1 h = 0 Uobm2(1, 0) = al(i + ae− c) Uobm2(1, 1) = ae(i + al− c) Uobm2(0, 0) = ali
Proposition 2 The equilibrium in the Second Benchmark Model satisfies the following. Let al be the ability of the leader and ae be the ability of the
employee.
a) If al < ae, then the owner hires an employee with ability ae iff ae ∈
Abm2.1e = [ ali
i+al−c, 1
i
. Afterwards, the owner replaces the leader by the new employee.
b) If al ≥ ae, then the owner hires an employee with ability ae iff ae ∈
Abm2.2e = [c, 1]. Afterwards, the owner lets the leader retain his posi-tion.
c) If al < ae, then the owner’s expected equilibrium utility level equals
E[Uobm2.1] = i+al−c−ali
i+al−c ·
i+al−c+ali
2 +
(ali)2
i+al−c.
d) If al ≥ ae, then the owner’s expected equilibrium utility level equals
E[Ubm2.2
o ] = ali + (1−c)
2
2 .
Proof. See Appendix A.
The owner replaces the leader by the new employee iff the employee is more able than the leader. Therefore, if the draw of ae is lower than al (see
part b)), the owner’s only reason to hire an employee is to increase the productivity of the organization without changing positions, which occurs iff ae > c. Hence, the owner’s hiring decisions are then the same as the
hiring decisions in the First Benchmark Model in the case that ae ≤ al.
Thus, in that case, the owner’s expected utility in the Second Benchmark Model is equal to her expected utility in the First Benchmark Model.
If the potential employee’s ability is higher than the leader’s ability (see part a)), the owner’s hiring strategy diverges from the First Benchmark Model. If the leader’s ability is lower than c — i.e. the threshold for hiring in the First Benchmark Model — the owner can have an incentive to hire employees with al < ae < c in the Second Benchmark Model. In this
owner’s expected equilibrium utility in the Second Benchmark Model is now higher than her expected equilibrium utility in the First Benchmark Model. This is caused by two reasons. First, demotion efficiency increases. That is, the owner can now raise her expected utility by replacing the leader by the more able employee. Second, the owner changes the hiring strategy since the option of replacement is available now.
Because the owner has more decision power in the Second Benchmark Model than in the First Benchmark Model, the owner’s expected utility level is higher in the Second Benchmark Model. The owner can always make the same decisions as in the First Benchmark Model, leading to equal expected utility levels, but can sometimes improve the situation. If al ≥ ae, the
hiring decisions from the Second Benchmark Model are identical to the hiring decisions in the First Benchmark Model. That is, there is absolute hiring efficiency. However, in contrast to the First Benchmark Model, the Second Benchmark Model is also characterized by demotion efficiency. Therefore, in the Second Benchmark Model, the owner’s expected utility if al < ae is
higher because the owner replaces the leader by the employee. If al < ae,
the hiring strategies of the owner in the Second Benchmark Model are also not the same as the leader’s hiring decisions in the First Benchmark Model. Now, the owner hires some types of employees with ability lower than c, such that the new employee can replace the lower able leader. Therefore, the owner’s expected utility in the Second Benchmark Model is higher than or equal to her expected utility in the First Benchmark Model if alis smaller
than c.
6
Analysis - the Main Model
The two benchmark models describe two extreme situations. In the First Benchmark Model, the owner is not involved in the firm decisions at all. That is, she does not have sufficient information or sufficient power to replace the leader by the new employee. In contrast, in the Second Benchmark Model, the owner is fully involved in all the firm decisions. She can perfectly derive the ability of the employee and, based on this, decide whether to hire the potential employee and whether to replace the original leader by the new employee or not.
The Main Model describes a situation that lies in between those two ex-tremes. In the Main Model, the owner is partly involved in decision making. She is not able to observe the potential employee’s ability at the moment the potential employee applies. Therefore, the owner delegates the hiring decision to the better-informed leader. Because the leader has complete knowledge about the daily procedures of the firm, he can observe the poten-tial employee’s ability at the moment the potenpoten-tial employee applies. If the leader hires the potential employee, the owner receives an imperfect signal about the employee’s ability. Based on this signal, the owner can decide whether to alternate the leader’s and employee’s positions. Thus, the leader has some authority in the Main Model. That is, he can make the hiring decision but can be demoted by the owner.
division of top-level positions between employees. Thus, if the shareholders believe that an employee is more able than the manager, they can replace the manager by the employee.
The model can also represent a political party in a democratic society. The party is run by a political party leader and ”owned” by the party members or the party voters. The main political party leaders are oftentimes well-known to their voters because of their frequent television, radio and newspaper appearances. Therefore, voters can form a reliable estimation of the party leader’s ability. However, since the regular voter is not familiar with the lower level applicants, they can usually not influence the hiring decisions. Yet, at the moment that a party member stands out and reaches the national news with exceptional performance, the voters receive this as an imperfect signal about this member’s ability and can express their wishes through a vote.
The timing of the Main Model is as follows. The model is described in four stages, Stage 1 to 4.
Stage 1. Nature determines the leader’s ability — al — and the employee’s
ability — ae. The leader’s ability is common knowledge and observed by
everyone. Only the leader and the employee observe the employee’s ability. Stage 2. The leader decides whether he hires the potential employee, h ∈ {0, 1}. If h = 0, the employee is not hired. Then, the leader’s and owner’s payoffs are realized and the game ends. If h = 1, the employee is hired. Then, the game continues in Stage 3.
Stage 3. The owner observes that h = 1 and receives an imperfect signal about the employee’s ability. The owner makes an estimation about the employee’s ability.
Stage 4. The owner makes her replacement decision, d ∈ {0, 1}. If d = 0, the leader and the employee retain their original positions and payoffs are realized and the game ends. If d = 1, the owner demotes the leader to the regular employee position and promotes the employee to the leadership position. Afterwards, payoffs are realized and the game ends.
Let Ae contain the set of the types of employees with ability ae that
Furthermore, let ae be defined as the lowest value of ae in set Ae. Likewise,
let ae be defined as the highest value of ae that is part of Ae. Hence,
Ae= [ae, ae]. The equilibrium boundaries of set Aeare determined in section
6.6.
If Stage 3 is reached, the owner receives an imperfect signal about ae.
This signal, P (ae), is defined as follows:
P (ae) =
1 2ae+
1
2ε, (16)
where ε describes random noise that influences the signal that the owner receives. This noise variable is distributed as ε ∈ U [ae, ae]. Furthermore, I
assume that corr(ae, ε) = 0. 1
Based on the signal P (ae) and the owner’s belief about set Ae, the owner
decides whether she replaces the leader or not, d ∈ {0, 1}. Finally, payoffs are realized. As before, the leader’s and owner’s utility functions, given ae
and al, are represented with (1) and (2) respectively:
Ul(h, d) = h · (ae− c) + (1 − d) · W,
Uo(h, d) = l · i + h · (e − c).
If the leader does not hire a new employee, the leader’s and owner’s utility levels, given al, equal Ul(0, 0) = W and Uo(0, 0) = ali, respectively. If the
leader hires a new employee and the owner does not alternate their positions, the utility levels are equal to Ul(1, 0) = ae−c+W and Uo(1, 0) = al(i+ae−c),
given al and ae. Lastly, if a new employee is hired by the leader and the
owner alternates the positions, the utility levels equal Ul(1, 1) = ae− c and
Uo(1, 1) = ae(i + al− c), given ae and al. This model is shown graphically
in Figure 3 for given values of ae and al.
1
The assumption that the set of possible ε equals the set Ae, is rather extreme. The
assumption is made such that signal P (ae) always makes sense and cannot take a value
L O h = 1 d = 0 d = 1 h = 0 Ul(1, 0) = ae− c + W Uo(1, 0) = al(i + ae− c) Ul(1, 1) = ae− c Uo(1, 1) = (ae(i + al− c) Ul(0, 0) = W Uo(0, 0) = ali
6.1 Approach
This model is solved for a perfect Bayesian Nash Equilibrium (PBE). Using backward induction, I start with deriving the owner’s strategy given that h = 1 in Stage 4. The owner has beliefs about the leader’s hiring range, defined as Aboe . Let aebo be defined as the lowest value of set Aboe . Likewise, aebo is
defined as the highest value of Aboe . Hence, Aboe = [aebo, aebo]. Furthermore,
the owner receives a signal about the employee’s ability, called P (ae). Based
on Aboe , signal P (ae), and the observation that h = 1, the owner makes
an estimation about the employee’s ability in Stage 3. Because P (ae) is
dependent on the draw of ε, there is a certain probability that the leader gets demoted if he hires an employee with ability ae. This probability is defined as
q(ae). Let qbl(ae) be defined as the leader’s belief about the probability that
an employee with ability aereplaces him if he hires this employee. Based on
this belief, the leader forms his hiring strategy, formalized as set Ae, in Stage
2. If ae∈ Ae, the leader hires the potential employee. Otherwise, the leader
decides to continue on his own. In the PBEs, the players’ strategies are optimal responses for eachother, given the players’ belief. Furthermore, the players’ beliefs are consistent, meaning that Abo
e = Ae and qbl(ae) = q(ae).
I only focus on equilibria where Ae comes from a single closed interval.
Based on the owner’s and leader’s strategies, four candidate equilibria, where Ae is bounded between ae and ae, are proposed. Afterwards, the existence
of those four candidate equilibria is investigated.
6.2 The owner - Stage 3 and Stage 4
If h = 1, the owner has to make her replacement decision, d ∈ {0, 1}. As before, the owner prefers to replace the leader if Uo(1, 1) > Uo(1, 0). Thus,
given ae and al and if the owner would know ae, she would prefer d = 1 iff:
ae(i + al− c) > al(i + ae− c) ⇐⇒
ae(i − c) > al(i − c) ⇐⇒
However, the owner is not able to observe ae. Therefore, she has to make an
estimation of the value of ae. Because the owner is risk neutral and prefers
d = 1 over d = 0 iff ae> al, she prefers to demote the leader if the expected
value of ae exceeds al.
6.2.1 The owner’s belief - Stage 3
The owner’s estimation of aeif h = 1 is based on two pieces of information.
First, the owner observes that h = 1, leading to the belief that ae ∈ Aboe
(which will be determined later). Second, the owner receives the signal P (ae) = 12ae+ 12ε. Based on these two pieces of information, the owner
makes an estimation of ae. This estimation is defined as E[ae|P (ae)] and is
described in Lemma 1.
Lemma 1 Consider h = 1, so the owner receives an imperfect signal about the employee’s ability, called P (ae). Furthermore, observes that h = 1,
leading to the belief that ae ∈ Aboe . Then, the owner’s expectation of the
employee’s ability, E[ae|P (ae)], equals the value of the signal, P (ae).
Proof. Because the owner observes that h = 1, leading to the blief that ae
has a value on the interval [aebo, aebo]. Furthermore, the owner receives a
signal P (ae) about the employee’s ability. As a consequence of this signal,
ae cannot come from the full set Aboe anymore. Therefore, the interval from
which the realized value of aecan come from gets smaller. Let aHe be defined
as the belief of the owner about the highest value that ae can have,
condi-tional on h = 1 and the value of P (ae). Likewise, aLe is defined as the belief
of the owner about the lowest value that ae can have, conditional on h = 1
and the value of P (ae). Because ae is uniformly distributed, the owner’s
estimation of ae, given h = 1 and P (ae), equals the middle of the interval
[aLe, aHe ]:
E[ae|P (ae)] =
aLe + aHe
2 . (18)
Thus, to determine E[ae|P (ae)], aHe and aLe need to be derived.
of their possible values, conditional on P (ae), are the same. Let the lowest
possible value of ε, conditional on P (ae) and Aboe , be defined as εL. Likewise,
the highest posible value of ε, conditional on P (ae) and Aboe , is defined as
εH. Because of the aforementioned symmetry, εH = aHe and εL = aLe. Furthermore, the owner knows that ae= aHe if ε = εL. Likewise, ae= aLe if
ε = εH. Rewriting (16) yields the following two conditions:
aHe = 2P (ae) − εL, (19)
aLe = 2P (ae) − εH. (20)
We must have aHe = εH ≤ aebo and aLe = εL ≥ aebo. Two different cases
have to be considered, depending on which of the two inequalities is binding. Case 1. Let aHe = εH = aebo and aLe = εL > aebo (see Figure 4 for a
graphical representation). aebo P (ae) aebo= aHe = εH aebo− P (ae) aebo− P (ae) aLe = εL Figure 4: aHe = εH = aebo and aLe = εL> aebo.
In the first case, P (ae) is closer to aebo than to aebo. In that case, aLe is
reached if ε = εH = aebo. Then, the distance between P (ae) and aLe is equal
to aebo− P (ae). It follows that aLe = P (ae) − aebo− P (ae) = 2P (ae) − aebo.
Every value of aein between aLe and aHe is equally likely to occur. It follows
Case 2. Let aLe = εL = aebo and aHe = εH < aebo (see Figure 5 for a graphical representation). aebo= aLe = εL P (ae) aebo P (ae) − aebo P (ae) − aebo aHe = εH Figure 5: aLe = εL= aeboand aHe = εH < aebo.
In the second case, P (ae) is closer to aebothan to aebo. In that case, aHe
is reached if ε = εL = aebo. Then, the distance between P (ae) and aHe is
equal to P (ae) − aebo. It follows that
aHe = P (ae) +
P (ae) − aebo
.
Every value in between aLe and aHe is equally likely to occur. It follows that E[ae|P (ae)] = aLe + aHe 2 = aebo+ P (ae) + P (ae) − aebo 2 = P (ae).
So, in both cases, E[ae|P (ae)] = P (ae) holds.
6.2.2 The owner’s strategy - Stage 4
As described in (17), under perfect information, so the owner knows the value of ae, she replaces the leader if ae > al. However, the owner is not
able to observe ae. Therefore, the owner’s replacement decision is based
on her estimation of ae. The owner’s replacement decision is described in
Lemma 2.
Lemma 2 Consider h = 1. The owner then chooses d = 0 if E[ae|P (ae)] =
Proof. The owner replaces the leader iff E[Uo(1, 1)] > E[Uo(1, 0)], i.e. iff:
E[ae|P (ae)](i + al− c) > al(i + E[ae|P (ae)] − c) ⇐⇒
E[ae|P (ae)](i − c) > al(i − c) ⇐⇒
E[ae|P (ae)] > al. (21)
In Lemma 1, E[ae|P (ae)] = P (ae) is derived. Substituting this in (21) yields:
P (ae) > al.
If h = 1, the owner replaces the leader if she thinks that the new employee is more able than the leader, which occurs if P (ae) > al. If the owner’s
estimation about the employee’s ability is lower than the leader’s ability, occurring if al≥ P (ae), the owner chooses to let the leader and the employee
retain their original positions.
6.3 The probability that the leader gets demoted - Stage 2
Because the draw of ε is not known at the moment that the leader makes the hiring decision, it is a priori not always sure whether E[ae|P (ae)] = P (ae)
exceeds al and thus not sure whether the leader will get demoted if he
hires the potential employee. Let q(ae) be defined as the probability that
the leader gets replaced if he hires an employee with ability ae, given the
realized equilibrium values ae and ae. Thus, q(ae) = P r
E[ae|P (ae)] > al
. Lemma 3 describes this probability.
Lemma 3 Consider h = 1, so the leader hires an employee with ability ae.
Then, the probability that the leader will be replaced is given by:
Proof. To determine q(ae) = P r E[ae|P (ae)] > al = P r P (ae) > al , the distribution of P (ae), given ae, needs to be determined first. Recall from
(16) that P (ae) is given by:
P (ae) =
1 2ae+
1
2ε, (23)
where ε ∈ U [ae, ae] and cor(ae, ε) = 0.
The highest possible value of P (ae), given ae and Ae, occurs if ε takes on
its highest value, which equals ae. Substituting this in (23) yields the highest
possible value of P (ae) for a specific ae and given Ae, which is defined as
P (ae):
P (ae) =
1
2(ae+ ae).
Likewise, the lowest possible value of P (ae), given ae and Ae, defined as
P (ae), occurs if ε takes on its lowest value, ae. Substituting this in (23)
yields:
P (ae) =
1
2(ae+ ae).
Due to the uniform distribution of ε, every value in between P (ae) and
P (ae) is equally likely to occur, for a specific ae and given Ae. Thus,
P (ae) ∈ U [
1
2(ae+ ae), 1
2(ae+ ae)]. (24)
If al is lower than P (ae), then the leader gets replaced inevitably,
irre-spective of the draw of ε. In that case, q(ae) = 1. If alis higher than P (ae),
then the leader never gets replaced, irrespective of the draw of ε. Thus, then q(ae) = 0 holds. If P (ae) > al > P (ae), then the leader gets replaced with
the probability that P (ae) exceeds al, i.e.
1 2(ae+ ae) − al 1 2(ae− ae) = ae+ ae− 2al ae− ae . In sum: q(ae) = 0 if al≥ 12(ae+ ae), ae+ae−2al ae−ae if 1 2(ae+ ae) > al≥ 1 2(ae+ ae), 1 if 12(ae+ ae) > al. (25)
6.4 The leader - Stage 2
The leader’s strategy consists of forming the set Ae = [ae, ae]. This sets
contains all the values of ae for which h = 1 holds. This occurs if an
employee has an ability for which E[Ul(1, d)] ≥ Ul(0, 0) holds.
Let qbl(ae) be defined as the leader’s estimation about the probability
that he gets replaced if he hires an employee with ability ae. Because the
leader knows that he gets replaced iff the owner’s estimation about the employees ability exceeds his own ability — i.e. E[ae|P (ae)] > al — the
leaders estimation about the probability that he gets replaced, qbl(ae), equal
the probability that P (ae) exceeds his own ability, al.
The leader’s hiring decision is described in Lemma 4.
Lemma 4 Tthe leader chooses h = 1 if ae ≥ qbl(ae) · W + c. If ae <
qbl(ae) · W + c, then the leader chooses h = 0.
Proof. The leader hires a new employee if E[Ul(1, d)] ≥ Ul(0, 0).
Further-more, the leader believes that he gets replaced with probability qbl(ae) if he
hires an employee with ability ae. Thus, the leader chooses h = 1 iff
1 − qbl(ae) · Ul(1, 0) + qbl(ae) · Ul(1, 1) ≥ Ul(0, 0). (26)
Ul(0, 0) = W . Substituting these values in (26) yields:
1 − qbl(ae) · (ae− c + W ) + qbl(ae) · (ae− c) ≥ W ⇐⇒
ae− c ≥ qbl(ae) · W ⇐⇒
ae≥ qbl(ae) · W + c. (27)
6.5 Candidate equilibria
Now that the owner’s and leader’s strategies are derived, I propose four can-didate equilibria. For each cancan-didate equilibrium, I derive the equilibrium strategy of the leader (Stage 2) — i.e. the set Ae = [ae, ae] for which h = 1
holds — given d P (ae). Furthermore, I derive the owner’s replacement
strategy (Stage 4) — i.e. d P (ae) — given h = 1, ae and ae. Afterwards,
the conditions under which these equilibria exist are checked. I propose the following candidate equilibria:
• The Never Demoted Equilibrium. In this candidate equilibrium, the set Ae is chosen such that d P (ae)
= 0. Thus, given h = 1, q(ae) = 0 holds for all values of P (ae), meaning that the leader never
gets replaced.
• The Always Demoted Equilibrium. In this candidate equilibrium, the set Ae is chosen such that d P (ae)
= 1. Thus, given h = 1, q(ae) = 1 holds for all values of P (ae), meaning that the leader always
gets replaced.
• The Sometimes Demoted Equilibrium. In this candidate equilib-rium, the set Ae is chosen such that d P (ae) sometimes equals 1 and
sometimes equals 0, depending on the value of P (ae). The probability
that the leader gets demoted if he hires an employee with ability ae,
• The Never Hired Equilibrium. In this candidate equilibrium, the set Ae is empty. That is, the leader never hires a new employee.
Because the owner then never makes a decision, the leader always retains his position.
6.6 Equilibria
6.6.1 The Never Demoted Equilibrium
The Never Demoted Equilibrium is described in Proposition 3.
Proposition 3 Given al, the Never Demoted Equilibrium has the following
characteristics:
a) For the Never Demoted Equilibrium to exist, c < al and W ≥ (1−c)
2
2(al−c)
must hold.
b) The leader hires an employee iff her ability is in the set Ae = Aboe =
[c, al].
c) If h = 1, the owner chooses d P (ae) = 0 for all values of P (ae).
d) The owner’s expected equilibrium utility equals E[Uo(h, d)] = ali + al(al−c)2
2 .
Proof. See Appendix B
In the Never Demoted Equilibrium, the leader hires potential employees with ability in the range [c, al]. If the draw of the potential employee’s ability is
outside this range, the leader continues to work on his own. If an employee is hired, the owner always infers that the employee’s ability is lower than the leader’s ability. Therefore, the owner never alternates the positions between the leader and the new employee.
conclusion is based on the following reasoning. If ae < c, the leader is
always better off if he does not hire the potential employee. Even though the leader knows that he does not get demoted, the costs of hiring are in this case higher than the benefits. Hence, only leaders with al> c can hire
some types of potential employees while being sure they do not get demoted. Second, for the Never Demoted Equilibrium to exist, the leader must value his position sufficiently. That is, he must be better off if he does not take any risk of being replaced. This condition is fulfilled when W ≥ 2(a(1−c)2
l−c) (see
Appendix B for the derivation).
The expected utility level of the owner in the Never Demoted Equilibrium equals ali +al(al−c)
2
2 (for derivation, see Appendix B). This is lower than in
the Second Benchmark Model for two reasons. First, hiring efficiency is lower in the Never Demoted Equilibrium than in the Second Benchmark Model. Recall from Proposition 2, part b), that the owner in the Second Benchmark Model hires all employees with ability in the range [c, 1] iff al < ae (and in
the case that she demotes the leader, the lower threshold for hiring is even lower than c (see part a) of Proposition 2)). If al < 1, then the most
able employees are not hired in the Never Demoted Equilibrium, which decreases the owner’s expected utility relative to the Second Benchmark Model. The second reason for why the expected utility level is lower in the Never Demoted Equilibrium than in the Second Benchmark Model is that demotion efficiency in the Never Demoted Equilibrium is very low. Since the leader does not hire an employee that is more able than himself, the owner cannot profitably demote the leader.
In the First Benchmark Model and the Never Demoted Equilibrium, efficient demotion opportunities never occur. However, the leader is not re-luctant to hire high ability employees in the First Benchmark Model because he knows that he does not get replaced anyway. Due to the leadership in-security in the Never Demoted Equilibrium, the leader is not willing to hire employees with higher ability than his own. Therefore, when al< 1, hiring
efficiency, and consequentially the owner’s expected utility level, is lower in the Never Demoted Equilibrium than in the First Benchmark Model.
De-moted Equilibrium is lower than or equal to her expected utility in both benchmark models. However, if al is close to the maximum, the difference
in expected utility levels between the Never Demoted Equilibrium and the two benchmark models decreases, due to two reasons. First, if alis high, the
owner is less likely to want to replace the leader by the potential employee. Thus, demotion inefficiency decreases. Second, high ability leaders can hire employees with higher ability than leaders with a low ability can while still being sure to not be demoted. So, also hiring inefficiency decreases if al is
high.
6.6.2 The Always Demoted Equilibrium
The Always Demoted Equilibrium is described in Proposition 4.
Proposition 4 Given al, the Always Demoted Equilibrium has the following
characteristics:
a) For the Always Demoted Equilibrium to exist, al < W + c < 1 and
W ≥ 12(1 − al) must hold.
b) The leader hires an employee iff her ability is in the set Ae = Aboe =
[W + c, 1].
c) If h = 1, the owner chooses d P (ae) = 1, for all values of P (ae).
d) The owner’s equilibrium expected utility level equals E[Uo(h, d)] = (1 −
W − c) 1+W +c 2 (i + al− c) + (W + c)ali.
Proof. See Appendix C.
For the Always Demoted Equilibrium to exist, the leader must be willing to give up his position under certain circumstances. That is, getting replaced must not be too costly for the leader. This occurs if hiring an employee is not too expensive — i.e. c is low — and if the leader does not experience a big loss in his utility when he gets demoted — i.e. W is low. If W + c < 1, this condition is fulfilled for some values of ae (for the derivation, see Appendix
C).
In the Always Demoted Equilibrium, the owner’s expected equilibrium utility equals (1 − w − c)1+w+c2 (i + al− c) + (W + c)ial(see Appendix C for
the derivation). This expected equilibrium utility level is lower than in the Second Benchmark Model for two reasons. First, in the Second Benchmark Model, iff al > c the owner hires all employees with ability higher than c
(see part b) of Proposition 2) (and in the case that she prefers to replace the leader, the lower threshold for hiring gets even lower than c (see part a) of Proposition 2)). Potential employees with ability in the range [0, c + W ] are not hired in the Always Demoted Equilibrium because hiring them does not yield enough benefits to make the leader willing to give up his position. Thus, there is some hiring inefficiency in the Always Demoted Equilibrium. Second, since al < c + W in the Always Demoted Equilibrium, some types
of employees with an ability that is higher than the leader’s ability are not hired. Since, in this case, the leadership position is not occupied by the more able employee, there is some demotion inefficiency in the Always Demoted Equilibrium.
The owner’s expected equilibrium utility level in the Always Demoted Equilibrium could be either higher or lower than her expected utility in the First Benchmark Model. In the First Benchmark Model, the threshold for hiring is lower than in the Always Demoted Equilibrium and, thus, hiring efficiency is higher. Demotion efficiency, however, is higher in the Always Demoted Equilibrium as the employee’s ability is always higher than the leader’s ability. Therefore, if a new employee is hired, the leadership position is occupied by the most able agent.
small. In this case, demotion and hiring decisions in the Always Demoted Equilibrium are more efficient. The set of employee types that the leader hires gets close to the set that the owner also wants to hire — i.e. set Abm2e — which increases hiring efficiency. Additionally, demotion decisions are more efficient since more employee types that replace the leader are hired. Thus, if the leader does not highly value his position, the Always Demoted Equilibrium becomes more efficient.
6.6.3 The Sometimes Demoted Equilibrium
The Sometimes Demoted Equilibrium is described in Proposition 5.
Proposition 5 Given al, the Sometimes Demoted Equilibrium has the
following characteristics:
a) For the Sometimes Demoted Equilibrium to exist, 12c < al must hold.
b) The leader hires an employee iff her ability is in the set Ae = Aboe =
h
max{2al− c − W, 0}, min{2al− c, 1}
i .
c) If h = 1, the owner chooses d P (ae) = 1 if P (ae) > al. Otherwise,
d P (ae) = 0.
d) The owner’s expected equilibrium utility equals E[Uo(h, d)] = (1 + ae−
ae) · ali + (ae−al)(a2 e+al)(i + al− c) + (al− ae) · al(i + al+a2 e − c) for
ae∈ max{0, 2al− c − W } and ae∈ min{1, 2al− c}.
Proof. See Appendix D.
In the Sometimes Demoted Equilibrium, the leader hires potential employees with ability in the range [2al− c − W, 2al− c]. However, 2al− c − W could be
smaller than 0. In that case, the lower bound for hiring equals 0. Likewise, 2al− c could exceed 1. In that case, the upper bound for hiring equals 1.
leader’s ability, the owner lets the leader retain his position. Because the owner’s information about the employee’s ability is imperfect, she sometimes makes a wrong decision. That is, sometimes she replaces the leader when the leader’s ability exceeds the employee’s ability. In other cases, the leader is not replaced while his ability is lower than the employee’s ability. In expectation, however, the owner’s decisions are efficient.
If the leader’s ability is high, the probability that he gets replaced is limited. Therefore, a high ability leader is less reluctant to hire high ability employees. Hence, both the upper bound and lower bound for hiring are increasing in al.
If the leader highly values his position — i.e. W is high — he is more hurt when he gets replaced. Therefore, if W is high, the leader decreases the lower bound of the set Ae. This decision decreases the probability that
the leader gets replaced for all ae ∈ Ae, due to the change in the possible
values of the signal P (ae).
The expected equilibrium utility of the owner in the Sometimes Demoted Equilibrium equals E[Uo(h, d)] =
1+ae−ae
·ali+(ae−al)(ae+al)
2 (i+al−c)+
(al−ae)·al(i+ al+ae
2 −c), for ae∈ max{0, 2al−c−W } and ae∈ min{1, 2al−c}
(see Appendix D for the derivation). This expected equilibrium utility level is lower than in the Second Benchmark Model for two reasons. First, hiring decisions are disturbed in the Sometimes Demoted Equilibrium. The upper bound for hiring could be lower than the owner’s preferred value. The lower bound for hiring in the Sometimes Demoted Equilibrium can be either lower or higher than what the owner prefers. Under certain circumstances, the leader also hires employees who have an ability lower than c, and thus cost more than they benefit. This incentive arises because hiring employees with ae < c decreases the probability that the leader gets replaced for all types
of employees he hires.
because the owner sometimes makes the wrong decision. Because the owner does not have perfect information about the new employee’s ability, her replacement decision is not always efficient.
The owner’s expected equilibrium utility level in the Sometimes Demoted Equilibrium could either be higher or lower than her expected equilibrium utility in the First Benchmark Model. In the First Benchmark Model, the leader is not reluctant to hire high ability employees and never hires employ-ees who cost the organization more than they add to the organization. In the Sometimes Demoted Equilibrium, some high ability employees are not hired. Furthermore, sometimes, the leader hires employees who cost the or-ganization more than they serve. Therefore, hiring efficiency is higher in the First Benchmark Model. Because replacement is possible in the Sometimes Demoted Equilibrium, demotion efficiency is, in expectation, higher in the Sometimes Demoted Equilibrium than in the First Benchmark Model. 6.6.4 The Never Hired Equilibrium
The Never Hired Equilibrium is described in Proposition 6.
Proposition 6 Given al, the Never Hired Equilibrium has the following
characteristics:
a) For the Never Hired Equilibrium to exist, al< c and W ≥ 1 − c must
hold.
b) The leader always chooses h = 0. Thus, set Ae= Aboe is empty.
c) Because the owner can never make a demotion decision, the leader always retains his position.
d) The owner’s expected equilibrium utility equals E[Uo(h, d)] = ali.
Proof. See Appendix E.
For the Never Hired Equilibrium to exist, two conditions must be ful-filled. First, al < c must hold. If al would exceed c, the leader can hire
employees with al > ae > c who benefit the organization while he can still
be sure that he never gets replaced.
Second, the leader must always be worse off if he gets replaced in the Never Hired Equilibrium. Therefore, the leader must highly value his po-sition. If W > 1 − c, this condition is fulfilled (see Appendix E for the derivation).
The owner’s expected equilibrium utility level in the Never Hired Equi-librium equals ali (see Appendix B for the derivation). This is lower than
the owner’s expected equilibrium utility in the Second Benchmark Model for two reasons. First, since the leader never hires an employee, hiring efficiency is very low in the Never Hired Equilibrium. Second, demotion efficiency is also low since the owner can never make a demotion decision.
The owner’s expected equilibrium utility in the Never Hired Equilibrium is also smaller than in the First Benchmark Model. Demotion efficiency is equally low in the two models. Hiring efficiency, however, is high in the First Benchmark Model, and low in the Never Hired Equilibrium.
7
Implications
In the model of Friebel and Raith (2004), leadership insecurity always holds negative consequences for the owner. To overcome these disadvantages, the principal can decide to close the information streams. As a result, the owner does not know whether the leader or the employee has a higher ability and does not demote the leader. The leader anticipates this and hiring inefficiency decreases.
In the model of Carmichael (1988), the owner’s expected utility also always decreases due to professors fearing to be fired. The principal — the university — can reduce position insecurity by granting the agents — the professors — tenure. As a result, professors cannot get fired and make more efficient hiring decisions.
Friebel and Raith (2004) are theoretically equivalent. In both cases, the owner makes it impossible to replace the leader. Thus, by creating this commitment to not replace the leader, the organizational structure changes from a situation with leadership insecurity — i.e. my Main Model — to a situation with complete leadership authority — i.e. my First Benchmark Model. In section 6, however, I have shown that the owner is not always better off in the First Benchmark Model than in the Main Model. In this section, I investigate when committing to not demote the leader (in the form of cutting off information streams, granting tenure or other contractual arrangements) is beneficial to the owner. Afterwards, I propose some other solutions.
7.1 Committing to not demote the leader
As shown in section 6, hiring efficiency is lower in all equilibria of the Main Model than in the First Benchmark Model. Furthermore, in a Never De-moted Equilibrium and a Never Hired Equilibrium, the leader never gets replaced, just as in the First Benchmark Model. Therefore, the owner’s expected utility is always lower in those two equilibria than in the First Benchmark Model. In the Always Demoted Equilibrium and the Sometimes Demoted Equilibrium, however, the owner can sometimes make an efficient replacement decision. Therefore, the owner’s expected utility level can be higher in the Always Demoted Equilibrium and Sometimes Demoted Equi-librium than is the case with complete leadership authority.
In sum, committing to delegate complete authority to the leader is not always beneficial to the owner. Therefore, the owner has to determine what type of equilibria are likely to occur in the Main Model before committing to complete leadership authority. This decision is dependent on the leader’s job valuation, W . Additionally, the leader’s ability plays a role in this commitment decision. The possible values of al and W in each equilibrium
al
Never Demoted Equilibrium
al
Always Demoted Equilibrium
al
Sometimes Demoted Equilibrium 0 0 0 1 1 1 c W + c 1 2c 1 c 0 al
Never Hired Equilibrium
W Never Demoted Equilibrium
W Always Demoted Equilibrium
W Sometimes Demoted Equilibrium
0 0 0 (1−c)2 2(al−c) 1 2(1−al) 1 − c 1 − c 0 W Never Hired Equilibrium
Figure 8: Possible values of al and W in each equilibrium
If the leader has a low ability, the Never Hired Equilibrium and the Always Demoted Equilibrium can occur. In the Never Hired Equilibrium, the owner is always worse off than in the First Benchmark Model. In the Always Demoted Equilibrium, however, the owner’s expected utility can be higher than in the First Benchmark Model.
In case that the leader has a mediocre ability, the equilibria of the Main Model can be of the Never Demoted Equilibrium, the Always Demoted Equilibrium, or the Sometimes Demoted Equilibrium type. In the First Benchmark Model, the owner’s expected utility is always higher than in the Never Demoted Equilibrium. In the Always Demoted Equilibrium and the Sometimes Demoted Equilibrium, however, the owner’s expected utility can be either higher or lower than in the First Benchmark Model.
Lastly, if the leader’s ability is high, the Never Demoted Equilibrium and the Sometimes Demoted Equilibrium can occur. In that case, the owner’s expected utility can be either higher or lower than in the First Benchmark Model.
If W is low — i.e. the leader does not value his position a lot — only the Sometimes Demoted Equilibrium can occur. Then, the owner’s expected utility could be either higher or lower than with complete leadership author-ity.
If the leader has a mediocre job valuation, the equilibria of the Main Model can be of the Never Demoted Equilibrium, the Always Demoted Equi-librium, or the Sometimes Demoted Equilibrium type. Then, the owner’s expected utility can be either higher or lower than in the First Benchmark Model.
