Description
Managerial economics as defined by Edwin Mansfield is "concerned with application of the economic concepts and economic analysis to the problems of formulating rational managerial decision
Project Characteristics, Organizational Structure, and
Managerial Incentives
+
Ajay Subramanian
Dept. of Risk Management and Insurance
J. Mack Robinson College of Business
Georgia State University
Email: [email protected]
Anand Venkateswaran
Dept. of Finance and Insurance
College of Business Administration
Northeastern University
Email: [email protected]
Richard Fu
Dept. of Accounting and Finance
College of Management
San Jose State University
Email: [email protected]
June 8, 2010
We thank Paul Bolster, Robert Gertner, Emir Kamenica, Canice Prendergast, Luis Rayo, Haresh Sapra, Lars
Stole, and seminar participants at the University of Chicago Booth School of Business, Georgia State University, and
Northeastern University for valuable comments. Subramanian gratefully acknowledges …nancial support from the
Research Program Council at the Robinson College of Business. The usual disclaimers apply.
Project Characteristics, Organizational Structure, and
Managerial Incentives
Abstract
We develop a model to show how agency con‡icts between shareholders and managers,
manager synergies, and the threat of expropriation by managers interact to a¤ect a …rm’s
internal organizational structure and the incentives of its managers. Our theory provides a
novel explanation based on agency considerations for two empirical regularities; …rms typically
have pyramidal organizational forms; and the pay-performance sensitivities of managers increase
with their hierarchical level. We also derive a number of implications that relate characteristics
of a …rm’s pool of projects—their risk and pro…tability— to its internal organization and the
incentives of its managers. (i) The optimal breadth of a …rm’s organization increases with the
risk of the …rm’s projects. (ii) The optimal height of the organization declines with risk and
increases with pro…tability. (iii) The pay-performance sensitivities of top managers increase
with pro…tability. Our results explain recent empirical evidence that …rms “‡atten” over time.
Broadly, our study contributes to the literature on the theory of the …rm by providing insights
into sources of unobserved heterogeneity among …rms, the organization of managers within …rms,
and the design of their incentives.
Key Words: Organizational Structure, Hierarchy, Incentives, Uncertainty, Synergies, Bargain-
ing Power
1 Introduction
A large and growing body of literature is devoted to the investigation of the determinants of the
size and internal organization of …rms. Bolton and Dewatripont (2005) identify the core issues at
the heart of the problem. “Why do hierarchies exist? How are e¢cient hierarchical organizations
designed? What determines the number of layers (or tiers) in a hierarchy, the pay structure, and
employees’ incentives along the hierarchical ladder?” A number of studies examine these issues
from di¤erent perspectives: (i) the e¤ects of uncertainty and learning about the quality of a …rm’s
projects (e.g. Sah and Stiglitz, 1986); (ii) information processing and communication costs (e.g.
Radner, 1992, Bolton and Dewatripont, 1994, Garicano, 2000); (iii) the link between organizational
structure and incentives (e.g. Qian, 1994, Maskin et al, 2000, Mookherjee and Reichelstein, 2001);
and (iv) the e¤ects of bargaining and expropriation (Stole and Zwiebel, 1996, Rajan and Zingales,
2001).
We develop a framework that integrates features of models analyzed in these relatively inde-
pendent strands of the literature to show how characteristics of a …rm’s external environment—the
risk and pro…tability of its projects—interact with its internal characteristics—agency con‡icts,
manager synergies and expropriation/bargaining power—to a¤ect its organizational structure and
the incentives of its managers. Our theory o¤ers a novel explanation based on manager–shareholder
agency con‡icts for two empirical regularities: …rms usually have pyramidal organizational forms;
and higher-level managers receive higher-powered incentive compensation than lower-level ones.
We also derive a number of testable predictions that relate project characteristics to organizational
structure—the number of hierarchical levels, the number of managers at each level, and the varia-
tion of managerial incentives across levels. Our results highlight the e¤ects of latent variables such
as manager synergies and expropriation/bargaining power on a …rm’s internal organization and,
thereby, executive compensation. We, therefore, shed light on some of the sources of unobserved
heterogeneity among …rms, and the channels through which they in‡uence executive compensation.
To motivate the model, consider the following illustrative examples. Fidelity wants to select
a portfolio in which to invest capital provided by investors. Asset managers in the lowest tier of
the organization gather information on potential investments. Higher tiers …lter this information
1
to narrow down the set of potential investments and determine the appropriate capital allocations.
Finally, the highest tiers of portfolio managers select the portfolio. The case of Walmart seeking
to open outlets in a new country o¤ers a similar example. Lower level managers conduct initial
feasibility studies to determine market conditions such as demand and the extent of competition,
and submit reports to their superiors. Higher tier managers conduct further analyses to identify
potential locations and outlet sizes. Finally, top-level managers select the locations and the scale
of operations.
Although the above examples describe …rms in di¤erent industries, they share certain key fea-
tures. The preliminary tasks in the development of the …rm’s “project” are delegated to agents at
the lowest level of the organization. At this stage, there is a high degree of uncertainty about the
“quality" of the project that determines its payo¤. For example, in the case of Walmart, when
agents in the lowest tier conduct initial feasibility studies, there is signi…cant uncertainty about
the scale and pro…tability of Walmart’s foreign operations. The agents in the lowest tier exert
e¤ort to develop the project, and then pass it on to the next tier in the organization that, in turn,
further develops the project before passing it on to the higher level. As the project …lters up the
organization, the uncertainty about its quality diminishes. For instance, in the case of Fidelity,
the uncertainty about the composition of its portfolio is resolved as information about potential
investments moves up the organization. The project is implemented after it passes through the
highest tier.
We attempt to capture the essence of the above examples in a model of the internal organi-
zational structure of a …rm. Speci…cally, we consider an organization with multiple hierarchical
levels and multiple decision–makers or managers at each level. Our focus is on the development
of a project rather than its …nal implementation. The total number of managers, the number of
levels in the organization, the number of managers at each level, and their incentives are endoge-
nously determined by the characteristics of the …rm’s pool of projects (hereafter, the project). As
illustrated by the above examples, we analyze the development of a project by successive tiers of
managers.
We model the implementation payo¤ of the project at any level, which is the value of future
2
earnings from the project if it is hypothetically immediately implemented without further develop-
ment by higher-level managers. The implementation payo¤ evolves as a Gaussian process and is
observable. The change in the implementation payo¤ of the project at each hierarchical level has
three components—the project’s intrinsic quality, the output generated by the managers’ e¤ort,
and the project’s intrinsic risk. The project’s intrinsic quality represents the component of the
expected change in the implementation payo¤ that is independent of the managers’ e¤ort. There is
imperfect, but symmetric, information about the project’s intrinsic quality. Agents have normally
distributed prior assessments of the intrinsic quality and update their assessments based on ob-
servations of the project’s implementation payo¤s. The variance of agents’ posterior assessments
of the project’s intrinsic quality at any date is its transient risk. The transient risk is resolved
over time as successive levels of managers develop the project and generate information about its
intrinsic quality. The variance of the project’s implementation payo¤, which is constant over time,
is the intrinsic risk of the project. We incorporate positive synergies among managers at each
level so that multiple managers complement each other. Manager synergies cause the marginal
productivity of each manager to be enhanced by the e¤ort exerted by other managers at the same
level. The output generated by the managers is increasing and concave in their e¤ort.
Managers can expropriate a proportion of the output generated at each level. Only the imple-
mentation payo¤ net of rents expropriated by managers is veri…able (by a third party such as a
court of law) and, therefore, contractible. Manager incur personal costs from expropriation. Man-
agers are risk-averse with preferences that display constant absolute risk aversion (CARA), and
receive explicit contracts from the …rm’s owners (shareholders) that are publicly observable.
The risk-neutral shareholders of the …rm choose its organizational structure (the number of
levels and the number of managers at each level) at the initial date and o¤er each manager a
contract conditional on the current information about project quality. We show that it is optimal
for shareholders to prevent manager expropriation by designing each manager’s contract so that it
guarantees her a “certainty equivalent” reservation payo¤ that is proportional to the output gen-
erated by all managers at her level. Hence, the threat of expropriation by managers endogenously
provides them with signi…cant bargaining power vis-a-vis shareholders. Further, the total propor-
3
tion of the output generated at any level that the managers receive as compensation increases with
the number of managers. The managers’ e¤ort choices are determined in Nash equilibrium of the
game in which each manager rationally anticipates the e¤ort exerted by other managers.
1
In the
symmetric equilibrium, managers at the same level receive identical contracts and exert the same
e¤ort.
Organizational Structure; The Key Tradeo¤s: The synergies among managers at each
level create an incentive for the shareholders to increase the number of managers. However, the
managers’ compensation as a proportion of the output they generate also increases with the number
of managers. The optimal number of managers at each level re‡ects the tradeo¤ between the bene…ts
of synergies among managers, and the costs of compensating them.
Because successive tiers of managers enhance the project’s payo¤ through their e¤ort choices,
increasing the number of hierarchical levels has a positive e¤ect on the project’s payo¤. Further, as
illustrated by our motivating examples, managers at higher levels have superior information about
the project’s …nal payo¤ and, therefore, face lower transient risk. As a result, shareholders can
exploit the bene…cial e¤ects of better risk-sharing with higher-level managers. The project’s payo¤
is, however, lowered by …xed costs that increase with the number of tiers in the organization. The
shareholders choose the number of levels in the organization to trade o¤ the bene…ts of further
project development against the increased costs associated with a larger number of hierarchical
levels.
Pyramidal Organizational Form: The …rm’s optimal organizational structure has the
commonly observed pyramidal form, that is, the number of managers declines with the hierarchical
level. Because managers at higher levels face lower transient risk, they receive more powerful
incentives and generate greater output. The total proportion of the output generated by managers
that they receive as compensation, however, also increases with the number of managers. The
output the managers generate is, however, concave in their e¤ort so that the marginal e¤ect of an
additional manager declines with the number of managers. Consequently, the relative impact of
managerial compensation costs in determining the number of managers is greater at higher levels.
1
Because managers have CARA preferences and payo¤s are normally distributed, we follow Holmstrom and Mil-
grom (1991) in considering contracts that are a¢ne in the increments of the implementation payo¤.
4
Hence, the number of managers at higher levels is smaller than the number at lower levels, but
each manager is more productive.
If there is perfect information about projects (there is no transient risk), the number of managers
does not vary across levels so that the organizational structure has a matrix form. Uncertainty and
learning, therefore, play key roles in generating the commonly observed pyramidal form. Our theory
o¤ers a novel explanation for the pyramidal organizational form based on manager-shareholder
agency con‡icts. In this respect, we complement earlier studies that explain the pyramidal form
using information processing costs (Bolton and Dewatripont, 1994) or variations in the di¢culty
of tasks across managerial tiers (Garicano, 2000).
Project Characteristics and Organizational Structure: The number of managers at each
hierarchical level increases with the project’s intrinsic and transient risks. The output generated
by managers at each hierarchical level declines with the project’s (intrinsic and transient) risks
because it is costlier to provide incentives to the risk-averse managers so that they exert lower
e¤ort in equilibrium. Since the total managerial compensation at each level is proportional to the
output generated, its relative importance in determining the number of managers at that level
declines with risk. The optimal number of managers at each hierarchical level in the organization,
therefore, increases with the project’s intrinsic and transient risk.
Next, we show that the number of hierarchical levels declines with the project’s intrinsic and
transient risks. As discussed earlier, the optimal number of hierarchical levels trades o¤ the positive
e¤ects of further project development against the increased …xed costs associated with a larger
number of levels. As the output generated by each hierarchical level decreases with project risk,
the relative impact of the …xed costs in determining the number of levels increases. Hence, the
optimal number of hierarchical levels decreases with project risk.
The e¤ects of the project’s intrinsic and transient risks on the height and breadth of the …rm’s
organization are novel predictions of our analysis that arise from the interplay among costs of
risk-sharing, manager synergies and compensation costs. These implications are not obtained in
earlier models such as Qian (1994) in which all payo¤s are deterministic, or in studies that focus
on the e¤ects of information processing and communication costs abstracting away from incentive
5
considerations (e.g. Bolton and Dewatripont, 1994, Garicano, 2000).
We show that the number of levels increases with the project’s initial mean intrinsic quality
or pro…tability. Because the project’s expected payo¤ increases with its pro…tability, the marginal
e¤ect of …xed costs on the optimal number of levels declines. The number of hierarchical levels,
therefore, increases with pro…tability.
Our results imply that riskier …rms have ‡atter organizational structures, and more pro…table
…rms have taller organizational structures. Insofar as pro…tability is expected to decline, and risk
levels expected to increase, as industries mature and become more competitive, our results are
consistent with anecdotal evidence that …rms are “‡attening” over time.
Project Characteristics and Manager Incentives: The managers’ incentive contracts
are simultaneously and endogenously determined along with the …rm’s organizational structure.
The managers’ pay-performance sensitivities—the sensitivities of managers’ compensation to the
change in the implementation payo¤—increase with their hierarchical level, and decrease with the
project’s intrinsic and transient risks.
2
Pay-performance sensitivities increase with level because
costs of risk-sharing decline, which is consistent with evidence that higher-level managers receive
relatively greater equity-based compensation than lower level managers.
Because the number of levels increases with the project’s pro…tability, the pay-performance
sensitivities of top-level managers increase. In contrast, in traditional principal-single agent mod-
els, the pay-performance sensitivity of the agent is only a¤ected by risk (e.g. Holmstrom and
Milgrom, 1987). The positive e¤ect of the pro…tability of the …rm’s projects on top managers’ pay-
performance sensitivities is, therefore, a key distinguishing prediction of our theory, which arises
from the modeling of multiple managers in di¤erent hierarchical levels.
2 Related Literature
Previous literature examines the determinants of …rms’ internal organizational structures from
di¤erent perspectives. Sah and Stiglitz (1986) analyze the link between the NPV of a …rm’s projects
2
Empirically, the pay-performance sensitivity is typically measured as the fraction of stock and option holdings
by an employee, to the total market value of equity of the …rm. The higher the fraction, the greater is the pay-
performance sensitivity.
6
and its internal organization using a model in which there is imperfect information about the quality
of a …rm’s pool of projects, and there are two managers who have exogenously speci…ed probabilities
of accepting bad projects or rejecting good ones. Building on Calvo and Wellisz (1978), Qian
(1994) develops a more general model of a …rm’s organization with deterministic payo¤s in which
managers are imperfectly monitored and their e¤ort choices are endogenous. We complement these
studies by developing a model of the hierarchical organization of a …rm in which there is imperfect
information and learning about the quality of projects as well as risk in their payo¤s. We also
incorporate synergies among managers’ e¤ort choices (e.g. Athey and Roberts, 2001, Dessein et al,
2007). These distinguishing features of our model lead to implications for the e¤ects of project risk
and quality uncertainty on organizational structure, as well as the variation of managers’ incentives
(as distinct from their compensation levels) across the tiers of the …rm. Further, we endogenously
derive the commonly observed pyramidal organizational structure.
Another important stream of the literature investigates the e¤ects of information processing and
communication costs on …rms’ organizations (Keren and Levhari, 1983, Radner, 1992, 1993, Bolton
and Dewatripont, 1994, Van Zandt, 1999, Garicano, 2000, Dessein, 2002, Dessein and Santos, 2003,
Vayanos, 2003). We complement this literature by focusing on the link between organizational
structure and incentives. Bolton and Dewatripont (1994) and Garicano (2000) also endogenously
derive a pyramidal organizational form. In Bolton and Dewatripont (1994), a pyramidal structure
arises from the tradeo¤ between the returns to specialization and information processing costs. In
Garicano (2000), it arises from variations in the frequency and di¢culty of tasks handled by agents
across tiers. Because their focus is on the e¤ects of information processing and communication
costs on organizational structure, these studies abstract away from incentive considerations. In
our model, a pyramidal structure arises from the interactions among costs of risk-sharing, manager
synergies, and bargaining power. Incentive considerations, therefore, play a key role in generating
a pyramidal form in our analysis.
An independent strand of the literature investigates the e¤ects of bargaining and expropriation
on organizational structure (e.g. Stole and Zwiebel, 1996). Harris and Raviv (2002) develop a model
in which organizational structure re‡ects the tradeo¤ between the coordination of the expertise of
7
managers against the costs of compensating them. We complement these studies by considering
a general hierarchical organization with multiple levels and multiple managers at each level. In
our framework, a …rm’s choice of organizational structure depends on the presence of uncertainty
about project quality, and the resulting process of learning induced by managers’ e¤ort. Further,
managers are compensated through explicit contracts. These distinguishing features of our model
lead to new implications for the impact of risk and pro…tability on managers’ incentive structures
and their organization.
An independent strand of the literature investigates the e¤ects of bargaining and expropriation
on organizational structure (e.g. Stole and Zwiebel, 1996 a, b, Rajan and Zingales, 2001). We
complement these studies by considering a general hierarchical organization with multiple levels
and multiple managers at each level. In our framework, a …rm’s choice of organizational structure
depends on the presence of uncertainty about project quality, and the resulting process of learning
induced by managers’ e¤ort. Further, managers are compensated through explicit contracts. These
distinguishing features of our model lead to new implications for the impact of risk and pro…tability
on managers’ incentive structures and their organization.
3
3 The Model
We develop a model of the internal organization of a …rm in which agents are organized in multiple
hierarchical levels. Our model could also be viewed as that of a division within a large …rm. As in
Sah and Stiglitz (1986), Geanakoplos and Milgrom (1991) and Radner (1992), we focus on higher-
level decision-making activities such as project development. We hereafter refer to the agents
in the organization as “managers.” The total number of managers, the number of levels in the
organization, the number of managers at each level, and their incentive structures are determined
endogenously by the characteristics of the …rm’s pool of projects or investment opportunity set.
Our focus is on the development of a project rather than the generation of the initial “idea”
for the project or its eventual implementation. For instance, in the example of Walmart in the
3
A body of literature analyzes hierarchical organizations as a general “mechanism design” problem (see Martimort,
2005, Mookherjee, 2006 for recent surveys). Another strand of the literature compares several aspects of unitary (U-
form) and divisional (M-form) organizations (e.g. Aghion and Tirole, 1995, Maskin et al, 2000, Qian et al, 2006).
8
introduction, top managers (or, indeed any other agent) might conceive (or suggest) the initial
idea of opening outlets in a new country. The development of this “project” (the establishment of
Walmart’s operations in the country), however, requires e¤ort by successive tiers of managers in
Walmart’s organizational structure. After top management makes the …nal decisions on the scale of
Walmart’s operations, the actual implementation of the project (the construction of outlet stores,
warehouses, etc.) is carried out. Similarly, the composition of Fidelity’s …nal investment portfolio
requires the e¤ort of successive levels of portfolio managers who gather and process information on
several potential investments and recommend appropriate capital allocations. After top portfolio
managers act on the recommendations and decide on the …nal composition of portfolio, the tasks
of actually allocating capital to the various investments are carried out. The illustrative examples
highlight the following aspects of the model: (i) the focus on project development rather than
idea conception or project implementation, and (ii) project development involves actions and the
transmission of information by successive tiers of managers upwards through an organization (see
Milgrom and Roberts (1991, Chapter 4)).
3.1 The Structure of the Organization
There are · managers, 1 levels in the organization, and ·(i) managers at a level i, where “1” is
the lowest level and “1” the highest. ·, 1, ·(i) are later determined endogenously. Managers are
identical ex ante, but are di¤erentiated ex post by their positions in the organization. This allows
us to focus on the incentive e¤ects of organizational design without obfuscating the implications
with those arising due to manager heterogeneity. (Our results are unaltered if we assume that
managers at the same hierarchical level have the same ability, but ability increases with hierarchical
level.) Consistent with the preceding discussion, and as in studies such as Sah and Stiglitz (1986),
Geanakoplos and Milgrom (1991) and Radner (1992), managers at the lowest level or tier of the
organization exert e¤ort to develop the project before passing it to the next tier in the hierarchy that,
in turn, develops the project before passing it on further, and so on. The project is implemented
and its payo¤ is realized after it passes through all levels in the organization. Hence, with 1 levels
in the organization, projects’ payo¤s are realized after they have passed through all 1 levels. The
9
…nal implementation of the project generally involves more routine tasks that are carried out by
lower-level employees such as production workers. Because our focus is on the development of the
project rather than its implementation, these additional agents are not explicitly modeled as in Sah
and Stiglitz (1986), Geanakoplos and Milgrom (1991) and Radner (1992).
3.2 The Implementation Payo¤ Process
We focus on a representative project that requires an initial capital investment that is hereafter
normalized to 1. There are multiple periods with equally spaced dates ¦0, 1, 2, ...¦. Managers
at hierarchical level i work on the project during period [i ÷ 1, i]. The key state variable is the
implementation payo¤ 1(.) of the project at any date. The implementation payo¤ 1(i) at date i is
the total payo¤ (present value of future earnings) from the project if it is hypothetically implemented
after passing through the …rst i levels of the organization without further development by managers
at higher levels. The implementation payo¤ process 1(.) is observable. As we discuss in more detail
later, the implementation payo¤ is non-veri…able because managers can expropriate a portion of
the output they generate. Consequently, only the implementation payo¤ net of rents expropriated
by managers is veri…able and, therefore, contractible.
At each level of the organization, managers exert unobservable e¤ort to develop the project.
Their e¤ort increases the project’s implementation payo¤. There are positive externalities created
by the possibility of synergies among managers at each level. Synergies among managers could,
for example, arise from the fact that managers have limited information processing capacities so
that di¤erent managers could carry out complementary tasks involved in the development of a
project. In reality, synergies could exist within a hierarchical level as well as across di¤erent levels.
We restrict consideration to synergies within each hierarchical level to simplify the model. The
project’s …nal payo¤ is, however, generated by the e¤ort of successive levels of managers. Further,
as we discuss later, the output generated by lower-level managers lowers the uncertainty faced by
higher-level managers. Consequently, they are more productive as they can be provided with more
powerful incentives. In these respects, we do capture at least some aspects of “cross level” synergies.
Suppose manager , at level i exerts e¤ort -
)
i
. We use subscripts to denote the hierarchical level
10
and superscripts to denote the particular manager. The change in the project’s implementation
payo¤ over period [i ÷1, i] is
1
¦.
¡
.
¦
(i) ÷1(i ÷1) =
_
¸
¸
¸
¸
¸
_
Intrinsic Quality
¸..¸
j +
Managers’ Output
¸ .. ¸
.(i)
)=1
_
_
)
_
_
.(i)
I=1
-
I
i
_
_
q(-
)
i
)
_
_
+
Intrinsic Risk
¸..¸
c
i
_
¸
¸
¸
¸
¸
_
. (1)
In (1), 1(i ÷ 1) is the project’s implementation payo¤ after it has passed through i ÷ 1 levels.
1
¦.
¡
.
¦
(i) is the project’s implementation payo¤ after it has passed through the i
th
level; the subscript
explicitly indicates its dependence on the e¤ort of the managers at level i.
Manager Output
From (1), the contribution of manager , at level i to the change in the project’s implementation
payo¤ is )
_
.(i)
I=1
-
I
i
_
q(-
)
i
). We assume that )(.) and q(.) are both strictly increasing so that each
manager’s marginal productivity is enhanced by the aggregate e¤ort exerted by all managers at the
same level.
4
De…ne
_
-
1
i
, -
2
i
, ..., -
.(i)
i
_
=
.(i)
)=1
_
_
)
_
_
.(i)
I=1
-
I
i
_
_
q(-
)
i
)
_
_
, (2)
which is the output generated by the managers at level i.
Assumption 1
_
-
1
i
, -
2
i
, ..., -
.(i)
i
_
is strictly increasing in each of its arguments, strictly concave
in
_
-
1
i
, -
2
i
, ..., -
.(i)
i
_
, and twice continuously di¤erentiable.
The above assumption captures the fact that, although there are positive synergies among
managers arising from the bene…ts of the coordination of activities across multiple managers, there
are decreasing returns to scale in such coordination.
The Project’s Intrinsic Risk
In (1), c
i
is a normal random variable with mean 0 and variance :
2
. The random variables
¦c
i
; i = 1, ..., 1¦ are independently and identically distributed. The variance :
2
of the noise variables
¦c
i
¦ is the project’s intrinsic risk.
4
We could alter the model to allow for the marginal productivity of each manager to be enhanced by the e¤ort
exerted by the other managers at her level (that is, excluding her e¤ort). This modi…cation complicates the notation
and analysis without altering any of our implications.
11
The Project’s Intrinsic Quality and Transient Risk
The parameter j in (1) is the intrinsic quality of the project, which determines the true expected
change (the expected change from the standpoint of the hypothetical omniscient agent) in the
project’s implementation payo¤ independent of the managers’ e¤ort. All agents have imperfect,
but symmetric information about j . The common initial prior distribution of j is normal with
mean : and variance (o
0
)
2
, that is,
j ~ ·
, (o
0
)
2
) (3)
Hereafter, we refer to the parameter : as the project’s initial mean quality.
All agents rationally update their assessments of the project’s intrinsic quality in a Bayesian
manner based on their observations of the project’s implementation payo¤s, which serve as signals
of the project’s intrinsic quality. It follows that managers at higher levels in the …rm have more
accurate information about the project’s intrinsic quality than those at lower levels. The variance
of the posterior distribution of j at any date is the project’s transient risk. Because the uncertainty
about project quality is resolved over time due to Bayesian learning by agents, the project’s transient
risk declines over time.
In the context of the Fidelity example in the introduction, the uncertainty about the composition
of the …nal investment portfolio declines as successive tiers of portfolio managers gather information
on potential investments. In the case of Walmart, the uncertainty about the scale and pro…tability
of Walmart’s operations in the foreign country declines as one moves up the organization. As
another example, consider the case of Bank of America that wishes to screen a large initial pool of
applicants for loans. Loan managers at each level screen loan applications. The uncertainty about
the average quality of the pool of loans declines as successive tiers of loan managers screen the
applications until the …nal loans are sanctioned by top management.
12
3.3 Preferences, Manager Expropriation and Contracting
Managers have identical CARA preferences with multiplicative disutilities of e¤ort described by
the Von Neumann-Morgenstern utility function
l(r, c) = ÷exp
_
÷¸(r ÷cc
c
)
_
, c 2 (4)
where r is the manager’s state-dependent payo¤, and c is her e¤ort. The risk aversion coe¢cient
¸ and the e¤ort disutility function cc
c
are identical across managers. The …rm’s shareholders are
risk-neutral. The discount rates of the shareholders and managers are normalized to zero.
Managers receive explicit contracts from shareholders. By (1), the managers at level i directly
in‡uence the change in the implementation payo¤ over period [i ÷1, i]. Without loss of generality,
we assume that managers at a particular level i receive explicit contracts at date i ÷1 before they
work on the project, that is, managers at successive levels are recursively o¤ered contracts as the
project moves up the organization.
5
Managers at each level can expropriate a proportion of the output they generate, but incur
personal costs from expropriation. Only the implementation payo¤ net of rents expropriated by
managers is veri…able by a third party (e.g. a court of law) so that contracts can only be contin-
gent on the implementation payo¤ net of expropriated rents. In Appendix B, we explicitly model
manager expropriation and show that it is optimal for the shareholders to prevent manager expro-
priation by o¤ering each manager a contract that guarantees her a “certainty equivalent” payo¤
that is proportional to the output generated at her level (we describe this more precisely later). To
avoid complicating the notation and exposition, we assume this result in the following discussion
and refer the reader to Appendix B for the endogenous derivation of managers’ certainty equivalent
payo¤s.
5
Suppose instead that all managers receive contracts at date 0: Because information about the project’s intrinsic
quality is revealed over time, optimal contracts for managers at any level i would be contingent on the information
revealed at date i 1, and would be identical to the contracts we derive in our analysis.
13
3.4 The Information Structure and Bayesian Learning
The information …ltration ¦1
i
¦ of the underlying probability space (, 1, 1) , which is shared by
all agents in the economy, is generated by the implementation payo¤ process 1(.). To simplify
notation, we drop the subscripts indicating the dependence of the implementation payo¤ on the
managers’ e¤ort whenever there is no danger of confusion.
Suppose the equilibrium e¤ort exerted by manager , at level i is c
)
i
. In equilibrium, the e¤ort
exerted by each manager is correctly anticipated by all other agents. From standard Bayesian
updating formulae (Oksendal, 2001), the posterior assessment of the project’s intrinsic quality
after it has passed through levels 1, 2, ..., 1 (which is shared by all agents in equilibrium) is normal
with mean :
1
and variance (o
1
)
2
where :
1
, (o
1
)
2
satisfy
:
1
=
)
2
:
11
+ (o
11
)
2
_
1(1) ÷1(1 ÷1) ÷
_
c
1
1
, ..., c
.(1)
1
__
(o
11
)
2
+)
2
, :
0
= :, (5)
(o
1
)
2
=
(o
11
)
2
)
2
(o
11
)
2
+)
2
. (6)
In (5), the function (.) is de…ned in (2). We note that (o
)
)
2
< (o
)1
)
2
, that is, the project’s
transient risk declines as it moves up the organizational hierarchy. Note that a manager’s informa-
tion di¤ers from that of other agents “o¤ equilibrium” because it includes knowledge of her own
e¤ort. In equilibrium, however, because managers’ e¤ort choices are correctly anticipated by all
agents, all agents’ information …ltrations coincide.
3.5 Contract Structure
The implementation payo¤ process is normally distributed, the managers have CARA preferences,
and the shareholders are risk-neutral. Accordingly, we follow studies such as Gibbons and Murphy
(1992) by restricting consideration to renegotiation-proof contracts that are a¢ne functions of the
change in the implementation payo¤ (see also Section 4.2 of Bolton and Dewatripont, 2005). More
14
precisely, a manager , at level i is o¤ered a contract that has a payo¤
Q
)
i
= a
)
i
+/
)
i
[1(i) ÷1(i ÷1)] . (7)
The above incorporates the results of the analysis in Appendix B that it is optimal for share-
holders to prevent manager expropriation by o¤ering each manager a contract that guarantees her
a certainty equivalent payo¤ (to be described shortly) that is a proportion of the output generated
by all managers at her level. Each manager’s contract is, therefore, directly contingent on the
change in the implementation payo¤.
In (7), a manager’s contract is contingent on the output generated at her level. If the contract
were also contingent on the output generated at higher levels, then it would impose additional risk
on the manager even though the manager does not contribute to the output generated at higher
levels. Such a contract could, therefore, be altered by an ex post Pareto improving renegotiation
that eliminates the manager’s exposure to the output generated at higher levels, that is, the contract
would not be renegotiation-proof. By restricting consideration to renegotiation-proof contracts, we
can therefore assume that the contract has the form (7).
Given the observable contracts o¤ered to all managers, their e¤ort choices are determined in
Nash equilibrium of the game in which each manager rationally anticipates the e¤ort exerted by
all other managers in response to their incentives. By (1), each manager only a¤ects the output at
her level. Therefore, it is not di¢cult to see that the manager’s e¤ort choice only depends on her
contract and her inferences about the e¤ort choices of other managers at her level.
More precisely, suppose that a particular manager , in hierarchical level i (hereafter denoted
as manager (i, ,)) anticipates that the e¤ort choices of other managers at her level are c
)
i
=
¦c
c
i
; : ,= ,¦ . By (4), given her contract parameters a
)
i
, /
)
i
, manager (i, ,) exerts e¤ort c
_
a
)
i
, /
)
i
, c
)
i
_
that solves
c
_
a
)
i
, /
)
i
, c
)
i
_
(8)
= arg max
c0
÷¸cc
c
+¸1
)
i1
_
a
)
i
+/
)
i
[1(i) ÷1(i ÷1)]
_
÷
1
2
¸
2
\
)
i1
_
a
)
i
+/
)
i
[1(i) ÷1(i ÷1)]
_
.
15
In (8), 1
)
i1
is the mean and \
)
i1
is the variance conditional on the information 1
)
i1
of manager
(i, ,), which di¤ers from 1
i1
"o¤ equilibrium" as it includes knowledge of her own e¤ort.
From (1) and (2),
c
_
a
)
i
, /
)
i
, c
)
i
_
= arg max
c0
÷¸cc
c
+¸1
)
i1
_
a
)
i
+/
)
i
_
j +
_
c, c
)
i
_
+c
i
__
(9)
÷
1
2
¸
2
\
)
i1
_
a
)
i
+/
)
i
_
j +
_
c, c
)
i
_
+c
i
__
= arg max
c0
÷¸cc
c
+¸1
)
i1
_
a
)
i
+/
)
i
_
j +
_
c, c
)
i
_
+c
i
__
÷
1
2
¸
2
(/
)
i
)
2
_
(o
i1
)
2
+)
2
_
.
The second equality above follows from (5) and (6), as well as the fact that c
i
is normally dis-
tributed, and manager (i, ,)’s posterior assessment of j is also normal. It follows from (9) that
the optimal e¤ort exerted by the manager depends only on her pay-performance sensitivity /
)
i
and
the anticipated e¤ort of other managers at the same hierarchical level. Henceforth, we drop the
argument a
)
i
of the optimal e¤ort. It follows that, in Nash equilibrium of the game for a given
set of observable contracts o¤ered to all managers, each manager’s e¤ort only depends on the
pay-performance sensitivities of all managers at the same hierarchical level.
For future reference, let
c
)
i
= c
)
i
(/
1
i
, ..., /
.(i)
i
), (10)
denote the equilibrium e¤ort exerted by manager (i, ,) given the pay-performance sensitivities,
(/
1
i
, ..., /
.(i)
i
), of all managers at level i. By (2), the corresponding output generated by the managers
at level i is (c
1
i
, ..., c
.(i)
i
).
As a consequence of the analysis in Appendix B, there is no expropriation by managers if and
only if each manager’s contract guarantees her a “certainty equivalent” payo¤ that is a proportion
of the maximum output generated at her level. More precisely, a manager (i, ,)’s contract is feasible
if and only if it guarantees her a “certainty equivalent” payo¤ that is a proportion ¸(·(i)) ¸ (0, 1)
of the maximum output, (c
1
i
, ..., c
.(i)
i
), generated by all managers at level i (given their observable
16
contracts). For future reference, de…ne
j(·(i)) = ·(i)¸(·(i)), (11)
which is the total proportion of the output guaranteed to all managers at level i. In Appendix
B, we show that, if a manager’s personal costs from expropriation increase with the number of
managers at her level, which is consistent with the notion that monitoring managers at each level
becomes more di¢cult as their number increases, then j(.) is strictly increasing. That is, the total
proportion of the output generated by the managers at any level that they receive as compensation
increases with the number of managers. We therefore assume the following.
Assumption 2 j(.) is strictly increasing.
3.6 The Payo¤ and Objectives of Shareholders
As in studies such as Keren and Levhari (1983) and Qian (1994), there are …xed costs associated
with each tier in the organization that reduce the shareholders’ payo¤. These costs could arise from
operating costs, depreciation, information processing costs, etc. The level i in the organization is
associated with a …xed cost `
i
.
Assumption 3 `
i
is deterministic, strictly increasing, and strictly convex in i.
The payo¤ to the …rm’s shareholders net of the compensation paid out to managers and the
…xed costs ¦`
i
; i = 1, ..., 1¦ is
1
)
= 1(1) ÷
1
i=1
.(i)
)=1
_
a
)
i
+/
)
i
[1(i) ÷1(i ÷1)]
_
(12)
=
_
j1 +
1
i=1
_
c
1
i
, ..., c
.(i)
i
_
÷
1
i=1
`
i
+
1
i=1
c
i
_
÷
1
i=1
.(i)
)=1
_
a
)
i
+/
)
i
[1(i) ÷1(i ÷1)]
_
.
The managers’ e¤ort choices are given by (9) and the second equality above follows from (1).
In equilibrium, the risk-neutral shareholders choose the number of managers, the organizational
structure, and a feasible set of contracts for the managers, to maximize their expected payo¤
17
rationally anticipating the managers’ ex post e¤ort choices in response to their incentives. The
organizational structure parameters 1, ¦·(i)¦ (the number of hierarchical levels and the number
of managers at each level) are chosen at the initial date, whereas the managers’ contracts are
dynamically determined based on the current information about the project’s quality.
For a given organizational structure (1. ¦·(i)¦) , the optimal contractual parameters solve
_
a
)
i
, /
)
i
_
= arg max
¦o
¡
.
,b
¡
.
¦
1
i1
_
_
_
_
_
j1 ÷
1
a=1
`
a
+
1
a=1
_
c
1
a
, ..., c
.(a)
a
_
+
1
a=1
c
a
_
÷
1
a=1
.(a)
j=1
[a
j
a
+/
j
a
[1(r) ÷1(r ÷1)]]
_
_
_
_
, (13)
subject to the constraints that each manager (i, ,)
0
’s certainty equivalent payo¤ is a proportion
¸(·(i)) of the output generated at level i, that is,
1
i1
_
a
)
i
+/
)
i
(1(i) ÷1(i ÷1))
_
÷
1
2
¸\
i1
_
a
)
i
+/
)
i
(1(i) ÷1(i ÷1))
_
÷c
_
c
)
i
_
c
(14)
_ ¸(·(i))
_
c
1
i
, ..., c
.(i)
i
_
, , = 1, ..., ·(i)
In (13) and (14), 1(i) is the implementation payo¤ after the project has passed through i levels.
Using (5) and (6), we can rewrite the constraints (14) as
1
i1
_
a
)
i
+/
)
i
(1(i) ÷1(i ÷1))
_
÷
1
2
¸
_
/
)
i
_
2
_
(o
i1
)
2
+)
2
_
÷c
_
c
)
i
_
c
_ ¸(·(i))
_
c
1
i
, ..., c
.(i)
i
_
.
(15)
If \ (1, ¦·(i)¦ denotes the shareholders’ expected payo¤ as a function of the number of levels
1 and the managers at each level ¦·(i)¦, then they choose these at date 0 to solve
(1
, ¦·(i)¦
) = arg max
(1,f.(i)g)
\ (1, ¦·(i)¦) (16)
4 The Equilibrium
Since managers at each hierarchical level are identical, we focus on the symmetric equilibrium in
which all managers at the same level receive identical contracts and exert the same e¤ort. Fix the
organizational structure (1, ¦·(i)¦. Suppose that managers at level i receive the contract ¦a
i
, /
i
¦.
18
The e¤ort c
i
exerted by each manager at level i must satisfy the following …rst order conditions:
÷cc (c
i
)
c1
+/
i
0
0c
)
i
[
c
1
.
=c
2
.
=...=c
^(.)
.
= c
.
= 0; , = 1, ..., ·(i). (17)
From (13) and (15), the shareholders optimally choose the pay-performance sensitivities of
the managers, and adjust the …xed components of their compensation so that their participation
constraints are binding. Substituting (15) (with equality) in (13), and using (17), the shareholders’
optimal choice of the pay-performance sensitivity of the managers at level i solves
/
i
= arg max
b
.
(1 ÷j(·(i)))(c
i
(/
i
), ..., c
i
(/
i
)) ÷·(i)
_
c (c
i
(/
i
))
c
+
1
2
¸
_
(o
i1
)
2
+)
2
_
(/
i
)
2
_
,
(18)
where c
i
(/
i
) solves (17) (the argument explicitly indicates its dependence on the pay-performance
sensitivity /
i
), and j(.) is de…ned in (11).
To facilitate our subsequent analysis, we consider a class of tractable functional forms for the
functions )(.) and q(.) that determine the e¤ects of the managers’ e¤ort on output in (1).
Assumption 4
)(r) = .r
c
; q(j) = j
o
; . 0, c +, < 1. (19)
The following proposition explicitly characterizes the optimal e¤ort choices of managers for a given
organizational structure and incentive contracts.
Proposition 1 (E¤ort Choices for Given Organizational Structure and Incentives)
Suppose the organizational structure is (1, ¦·(i)¦), and managers at level i receive a contract
(a
i
, /
i
).
(i) The e¤ort exerted by managers at level i is
c
i
(1, ¦·(i)¦ , /
i
) =
_
.(c +,)(·(i))
c
cc
_ 1
¿oc
(/
i
)
1
¿oc
, (20)
(ii) The optimal number of managers at hierarchical level i, ·(i)
, and their pay-performance
19
sensitivities, /
i
, solve
(·(i)
, /
i
) = arg max
.(i),b
.
i
(·(i), /
i
), where (21)
i
(·(i), /
i
) = ¹(1 ÷j (·(i))) (·(i))
o
(/
i
)
t1
÷1(·(i))
o
(/
i
)
t
÷
1
2
¸·(i)
_
(o
i1
)
2
+)
2
_
(/
i
)
2
(22)
¹ =
_
.
¿
o+c
(c +,)
cc
_
o+c
¿oc
, 1 =
_
.(c +,)
c
o+c
¿
c
_ 1
¿oc
(23)
o =
cc +c ÷c ÷,
c ÷c ÷,
, t =
c
c ÷c ÷,
(24)
By (22), the number of managers at level i and their optimal pay-performance sensitivities do
not depend on the number of hierarchical levels 1 in the organization. In (21) and (22), the function
i
is the output generated by managers at level i net of their compensation. Since this accrues to
shareholders, we refer to it as the shareholder surplus at level i.
We now characterize the optimal number of managers at each hierarchical level and their pay-
performance sensitivities, which solve (21). To ease the subsequent notation and exposition, we
allow for the number of managers, ·(i), at level i to take non-integer values. We assume that
the …rst order conditions of the optimization program (21) are necessary and su¢cient to identify
(·(i)
, /
i
) .
Proposition 2 (The Optimal Number of Managers at each Level and their Incentives)
The equilibrium number of managers at level i, ·(i)
, and their optimal pay-performance sensitiv-
ities, /
i
, solve the following equations.
0 = ¹(t ÷1)(1 ÷j (·(i)
))(·(i)
)
o
(/
i
)
t2
÷1t(·(i)
)
o
(/
i
)
t1
(25)
÷¸·(i)
_
(o
i1
)
2
+)
2
_
(/
i
) ,
0 = ¹o(1 ÷j (·(i)
))(·(i)
)
o1
(/
i
)
t1
÷1o(·(i)
)
o1
(/
i
)
t
(26)
÷
1
2
¸
_
(o
i1
)
2
+)
2
_
(/
i
)
2
÷oj
0
(·(i)
) (·(i)
)
o
(/
i
)
t1
.
We now determine the shareholders’ optimal choice of the number of hierarchical levels in the
20
organization. Let
i
=
i
(·(i)
, /
i
) (27)
The following proposition characterizes the optimal number of hierarchical levels in the organization.
Proposition 3 (The Optimal Number of Hierarchical Levels) The optimal number of hier-
archical levels, 1
, is …nite and solves
1
= arg max
1
_
:
0
1 ÷
1
i=1
`
i
_
+
1
i=1
i
(28)
It follows from Proposition 7 in the next section that the surplus generated at each hierarchical
level
i
is uniformly bounded. Since the …xed cost `
i
is increasing and strictly convex in i by
Assumption 3, the optimal number of hierarchical levels is …nite.
5 Organizational Structure and Incentives
In this section, we derive properties of the equilibrium. We analyze the e¤ects of project charac-
teristics on the …rm’s organizational structure and the managers’ incentives.
5.1 Organizational Form
We …rst examine the variation of the number of managers across hierarchical levels. The following
proposition shows that the optimal number of managers at a given level decreases as we move up
the organization, that is, the organization has the traditional pyramidal form.
Proposition 4 (Pyramidal Organizational Form) The optimal number of managers, ·(i)
,
at hierarchical level i decreases with i.
The above result arises from the subtle interplay between various forces. First, managers at
higher levels in the organization have better information about project quality and face lower
transient risk. This leads to lower costs of risk-sharing (that is, these managers can be provided with
more powerful incentives), which implies that, ceteris paribus, the output generated by managers
21
is higher as one moves up the organization. However, the total compensation of managers, as a
proportion of the output they generate, also increases with the number of managers. Since output
is concave in managers’ e¤ort, the marginal positive impact of the e¤ort of an additional manager
declines with the number of managers. Consequently, the marginal negative impact of the costs
of compensating managers on the optimal choice of the number of managers is greater at higher
hierarchical levels. As the hierarchical level increases, therefore, it is optimal to have fewer managers
who are more productive.
An important implication of the intuition above is that, in the presence of perfect information
about project quality (that is, there is no transient risk), there are an equal number of managers
at each level of the organization because there is no variation in the costs of risk-sharing across
levels. Hence, the organizational structure has a matrix rather than pyramidal form. The presence
of uncertainty about the quality of a …rm’s investment opportunities, and the resulting process
of learning induced by the e¤ort exerted by managers to develop the …rm’s projects, distort the
…rm’s organizational structure to the commonly observed pyramidal form. Earlier studies have
highlighted the e¤ects of imperfect monitoring (for example, Qian 1994), expropriation risk (Rajan
and Zingales 2001), and communication costs (Radner 1992, 1993, Van Zandt 1999, Garicano 2001)
on …rms’ organizational structure. Our framework suggests that the presence of agency costs of risk-
sharing, imperfect information, and the process of learning about project quality are also important
determinants of organizational design.
Bolton and Dewatripont (1994) and Garicano (2000) also endogously derive a pyramidal organi-
zational structure in their frameworks. In Bolton and Dewatripont (1994), a pyramidal form arises
from the tradeo¤ between the gains to specialization in processing certain types of information
and the costs of communicating information. In Garicano (2000), it arises from the fact that it is
optimal to assign agents at lower levels to easier, frequent tasks and those at higher levels to harder,
less frequent tasks. Because their focus is on the e¤ects of information processing and communi-
cation costs on organizational structure, these studies abstract away from incentive considerations.
In contrast, in our model, a pyramidal organizational form arises from the interactive e¤ects of
costs of risk-sharing, which decline as one moves up the organization, manager synergies, and the
22
possibility of expropriation. Incentive considerations, therefore, play a key role in generating the
pyramidal form in our analysis.
5.2 Breadth of the Organization
We now analyze the e¤ects of project characteristics on the number of managers at each hierarchical
level or the breadth of the organization.
Proposition 5 (Risk, Uncertainty and Organizational Breadth) The optimal number of man-
agers, ·(i)
, at any level i increases with the project’s intrinsic risk :
2
as well as its initial transient
risk o
2
0
.
The number of managers at each level is determined by trading o¤ the bene…ts of positive
synergies among managers and the costs of compensating them. The output generated by managers
declines with risk due to the higher agency costs of risk-sharing with the …rm’s shareholders. Hence,
the total compensation received by managers at each level also declines with the project’s intrinsic
and transient risk. The relative (or marginal) e¤ect of managers’ compensation in determining the
optimal number of managers at any level, therefore, declines with risk so that the …rm can support
a larger number of managers at the optimum. Proposition 5 leads to the following implication.
Implication 1 High-risk …rms have a larger number of managers at each hierarchical level than
low-risk …rms.
The intuition for Proposition 5 suggests that the predicted e¤ects of risk and uncertainty on
organizational breadth arise from the interactions among costs of risk-sharing, manager synergies
and bargaining power. These are novel implications of our model that are not obtained in the
frameworks of studies such as Qian (1994) in which all payo¤s are deterministic, or in studies that
focus on the e¤ects of information processing costs, but abstract away from incentive considerations
(e.g. Bolton and Dewatripont, 1994, Garicano, 2000).
5.3 The Surplus Generated by Each Hierarchical Level
We now investigate the surplus generated by managers at each hierarchical level.
23
Proposition 6 (The Surplus Generated by Each Hierarchical Level) The surplus,
i
(see
27) generated at each level i increases with i.
This result supports the intuition described earlier for Proposition 4. Managers at higher levels
face lower transient risk leading to better risk-sharing with the …rm’s shareholders. As a result,
managers at higher levels can be provided with more powerful incentives implying that the output
they generate and the surplus increases with hierarchical level.
Because the number of managers declines with hierarchical level by Proposition 4, the average
surplus per manager increases disproportionately with hierarchical level. Note that managers are
ex ante identical in our framework so that the positive relation between average surplus and level
arises purely due to the e¤ects of learning and the resolution of uncertainty.
The following result shows that the surplus generated at any given level decreases with the
project’s intrinsic risk and initial transient risk. The result follows directly from the fact that, as
discussed earlier, higher risk levels leads to greater costs of risk-sharing between the risk-neutral
shareholders and risk-averse managers.
Proposition 7 (Risk, Uncertainty, and the Surplus Generated by Each Level) The sur-
plus,
i
(see 27) generated at each level i decreases with the project’s intrinsic risk :
2
as well as
its initial transient risk o
2
0
.
Propositions 6 and 7 lead to the following implications.
Implication 2 The surplus generated at each hierarchical level increases (disproportionately) with
the level and declines with the project’s intrinsic and transient risks.
5.4 Height of the Organization
We now examine the e¤ects of project characteristics on the number of hierarchical levels or height
of the organization.
Proposition 8 (Pro…tability and Organizational Height) The equilibrium number of levels,
1
, increases with the project’s initial mean quality or pro…tability :.
24
As we examine the e¤ect of the project’s mean quality keeping its intrinsic risk and initial
transient risk …xed, it follows from Proposition 2 and (27) that the surplus generated at any given
level is …xed. By (28), as the project’s mean quality increases, the expected change in the project’s
implementation payo¤ increases. As a result, the relative (or marginal) impact of the …xed costs
¦`
i
¦ as well as the costs of risk-sharing with managers on the determination of the optimal number
of hierarchical levels declines. Therefore, the equilibrium number of levels increases with the mean
quality. Since the total expected payo¤ increases with the number of levels, the number of levels
in the organization and the expected payo¤ to the …rm’s shareholders are also positively related.
The above proposition leads to the following implication.
Implication 3 More pro…table …rms are taller than less pro…table …rms.
The following proposition show that the optimal number of levels in the organization decreases
with the project’s intrinsic risk and the initial transient risk .
Proposition 9 (Risk, Uncertainty and Organizational Height) The optimal number of lev-
els, 1
, decreases with the project’s intrinsic risk :
2
as well as its initial transient risk o
2
0
.
The intuition for this result follows from the intuition for Proposition 7. An increase in the
intrinsic or transient risk of the project implies that the surplus generated at each hierarchical level
declines. Therefore, the marginal e¤ect of the depreciation costs ¦`
i
¦ in determining the optimal
number of levels increases. The optimal number of hierarchical levels, therefore, declines with the
project’s intrinsic and initial transient risk. Proposition 9 leads to the following implication.
Implication 4 High-risk …rms have a smaller number of hierarchical levels than low-risk …rms.
Implications 1, 3, and 4 together suggest that less pro…table and riskier …rms have a smaller
number of hierarchical levels and a larger number of managers at each level, that is, they are
“‡atter” than more pro…table and less risky …rms. Insofar as pro…tability is expected to decline,
and risk levels expected to increase as industries mature and become more competitive, these
predictions suggest that …rms “‡atten” over time, which appears to be consistent with anecdotal
evidence.
25
5.5 Manager Incentives
We now analyze the e¤ects of project characteristics on the incentive structures of managers. The
following result describes the variation of managers’ incentives across hierarchical levels.
Proposition 10 (Variation of Manager Incentives Across Levels) The pay-performance sen-
sitivity /
i
, of managers at level i increases with i.
The above result further supports the intuition that we described for our earlier results. Because
managers at higher levels face lower transient risk, their incentives can be better aligned with the
interests of the …rm’s shareholders, that is, the costs of risk-sharing are lower. It is important
to note, however, that the pay-performance sensitivity of a manager at any level is determined
endogenously along with the organizational structure; more speci…cally, on the equilibrium number
of managers at that level. Hence, it is not a priori obvious that pay-performance sensitivities increase
with the hierarchical level. Formally, we have the following implication, which is consistent with
anecdotal evidence that CEOs receive relatively greater equity-based compensation than lower level
managers.
Implication 5 The pay-performance sensitivity of a manager increases with her level in the or-
ganization.
We now show that the pay-performance sensitivity of a manager at a given level in the organi-
zation decreases with the project’s intrinsic and initial transient risk.
Proposition 11 (Risk, Uncertainty and Manager Incentives) The pay-performance sensi-
tivity /
i
, of managers at level i decreases with the project’s intrinsic risk :
2
as well as its initial
transient risk o
2
0
.
In contrast with traditional principal-single agent models such as Holmstrom and Milgrom
(1987), the organizational structure and the incentive structures of managers are simultaneously
and endogenously determined by the characteristics of the project. Proposition 11, therefore, does
not directly follow from the results of these models.
26
Implication 6 The pay-performance sensitivity of a manager decreases with the project’s (intrin-
sic and transient) risk.
The above implication is consistent with empirical evidence of a negative relationship between a
CEO’s pay-performance sensitivity and …rm risk (see Aggarwal and Samwick, 1999, Prendergast,
1999).
The following result, which follows immediately from Propositions 8, 9, 10, and 11, describes the
e¤ect of the project’s mean intrinsic quality or pro…tability on the incentives of top-level managers.
Proposition 12 (Incentives of Top-Level Managers) The pay-performance sensitivities of top-
level managers increase with the project’s initial mean intrinsic quality or pro…tability :.
The pay-performance sensitivities of top-level managers are simultaneously and endogenously
determined along with the number of levels in the organization. Hence, the modeling of the actions
of multiple managers in di¤erent hierarchical levels plays a key role in generating the positive
relation between top managers’ pay-performance sensitivities and the project’s initial mean quality
or pro…tability. In contrast, in traditional principal-single agent models with normally distributed
payo¤s, the pay-performance sensitivity of the agent is only a¤ected by risk (see Section 5 and
equation (32) in Holmstrom and Milgrom, 1987, Aggarwal and Samwick, 1999). We have the
following implication.
Implication 7 The pay-performance sensitivities of top-level managers increase with the prof-
itability of …rms’ investment opportunities.
The above implication suggests that there is a causal link between the pro…tability of the
…rm’s initial pool of projects and the incentives of its top-level managers, that is, pro…tability is
a determinant of top-manager incentives. The …nal payo¤s of the …rm’s projects are, however,
a¤ected by the e¤ort choices of managers at each hierarchical level that are, in turn, a¤ected by
their incentives. Consequently, managerial incentives and …rm performance are both endogenously
determined by characteristics of the …rm’s initial pool of projects that are di¢cult to directly
observe. The lack of a direct causal link between managerial incentives and …rm performance
complicates empirical analyses of their relationship (see Himmelberg et al, 1999).
27
6 Conclusions
We develop a theory of the impact of the characteristics of a …rm’s projects on the organization
of its managers and their incentives. By integrating aspects of models developed in relatively
independent strands of the literature, our analysis shows the interactive e¤ects of project risk,
quality uncertainty, information processing, manager synergies, and bargaining power on …rms’
organizational structures and the incentives of their managers.
The commonly observed pyramidal organizational form arises from the interplay among (i)
project quality uncertainty; (ii) information processing or learning; (iii) positive complementarities
among managers’ e¤ort choices; and (iv) managers’ bargaining power. The number of hierarchical
levels in a …rm increases with the pro…tability of its projects, and decreases with their risk. The
number of managers at each hierarchical level of the organization increases with risk. Insofar as
pro…tability is expected to decline, and risk levels expected to rise, as industries mature and become
more competitive, our theory provides an alternative explanation for recent empirical evidence that
…rms ‡atten over time.
Our model, in which organizational structure and manager incentives are simultaneously and
endogenously determined, also generates several implications for the design of executive compensa-
tion, and the link between executive compensation and organizational structure. First, managers’
pay-performance sensitivities (or “equity-based” compensation) increase with their level in the …rm.
Second, the pay-performance sensitivities of managers decline with the risk of the …rm’s projects
and the pay-performance sensitivities of top-level managers increase with the projects’ pro…tability.
Third, the height and breadth of the …rm’s organization decline with managers’ bargaining power.
Fourth, managers’ pay-performance sensitivities could increase with their bargaining power.
Our study contributes to the literature on the theory of the …rm by uncovering some of the
sources of unobserved heterogeneity among …rms, and showing the link between organizational
structure and executive compensation. We show how characteristics of a …rm’s external envi-
ronment – the risk and pro…tability of its projects – interact with its internal characteristics –
information processing costs, manager synergies, and bargaining power – to a¤ect the organization
and incentives of its managers.
28
Appendix A
Proof of Proposition 1
(i) The e¤ort c
i
exerted by each manager at level i (we drop the arguments indicating the depen-
dence on the organizational structure (1, ¦·(i)¦) and the pay-performance sensitivity /
i
to simplify
the notation) solves the …rst order conditions (17).
By (2) and (19),
(c
i
, ..., c
i
) = .·(i)
c+1
(c
i
)
c+o
(29)
Substituting (29) in (17) we have
/
i
.,(·(i))
c
(c
i
)
c+o1
+/
i
.c(·(i))
c
(c
i
)
c+o1
= cc (c
i
)
c1
. (30)
It follows directly from (30) that the e¤ort c
i
is given by (20).
(ii) By (16) and (18), the optimal number of managers at level i and their pay-performance
sensitivity /
i
solve
(·(i)
, /
i
) = arg max
.(i),b
.
(c
i
, ..., c
i
) ÷·(i)
_
c(c
i
)
c
+
1
2
¸
_
(o
i1
)
2
+)
2
_
(/
i
)
2
_
(31)
By (20) and (29), we have
(c
i
, ..., c
i
) = . (·(i))
c+1
(c
i
)
c+o
(32)
= . (·(i))
c+1
_
.(c +,)(·(i))
c
cc
_ o+c
¿oc
(/
i
)
o+c
¿oc
It follows from (31) and (32) that
(·(i)
, /
i
) = arg max
.(i),b
.
. (·(i))
c+1
_
.(c +,)(·(i))
c
cc
_ o+c
¿oc
(/
i
)
o+c
¿oc
÷·(i)c
_
.(c +,)(·(i))
c
cc
_ ¿
¿oc
(/
i
)
¿
¿oc
÷
1
2
¸·(i)
_
(o
i1
)
2
+)
2
_
(/
i
)
2
.
The above implies (21). Q.E.D.
Proof of Proposition 2
By (21) and Assumption ??, we have
0
i
(·(i)
, /
i
)
0·(i)
= 0;
0
i
(·(i)
, /
i
)
0/
i
= 0 (33)
(25) and (26) follow directly from (22) and (33). Q.E.D.
Proof of Proposition 3
By (16) and Proposition 2, the optimal number of hierarchical levels 1
solves
1
= arg max
1
\ (1, ¦·(i)
¦ .
29
By (1), (5), and (27),
\ (1, ¦·(i)
¦ = 1
0
[1(1) ÷1(0)] =
_
:
0
1 ÷
1
i=1
`
i
_
+
1
i=1
i
(28) follows directly from the above. By Proposition 7 that we prove later, the optimal surplus
generated at each hierarchical level
i
is bounded above by the surplus in the scenario where the
intrinsic and transient risks are both zero. It immediately follows from Assumption 3 that the
number of hierarchical levels is …nite. Q.E.D.
Proof of Proposition 4
The optimal number of managers at level i, ·(i)
and their pay-performance sensitivity solve (33).
De…ne
o
2
= (o
i1
)
2
+:
2
(34)
The proof proceeds by showing that
0.(i)
0(S
2
)
0. Since (o
i1
)
2
decreases with i by (6), it follows
that o
2
also decreases, which will then imply from the above that ·(i)
decreases with i. By the
implicit function theorem,
0·(i)
0(o
2
)
= ÷
0
2
.
0b0(S)
2
0
2
.
0b0.
÷
0
2
.
0.0(S)
2
0
2
.
0b
2
_
0
2
.
0b0.
_
2
÷
0
2
.
0b
2
0
2
.
0.
2
(·(i)
, /
i
) (35)
The denominator in the fraction on the right hand side above is negative by the second order
conditions for the maximum of the function
i
. To establish that
0.(i)
0(S
2
)
0, we need to show that
0
2
i
0/0 (o)
2
0
2
i
0/0·
÷
0
2
i
0·0 (o)
2
0
2
i
0/
2
(·(i)
, /
i
) 0, (36)
or alternatively
(/
i
)
2
·(i)
(o)
2
_
0
2
i
0/0 (o)
2
0
2
i
0/0·
÷
0
2
i
0·0 (o)
2
0
2
i
0/
2
_
0. (37)
By (22), to show (37) it su¢ces to establish that
÷/
i
·(i)
0
2
i
0/0·
÷
1
2
(/
i
)
2
0
2
i
0/
2
(38)
By (22) and (25), we have
/
i
0
i
0/
= ¹(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
÷1t(·(i)
)
o
(/
i
)
t
(39)
÷¸·(i)
_
o
2
_
(/
i
)
2
,
·(i)
0
i
0·
= ¹o(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
÷1o(·(i)
)
o
(/
i
)
t
(40)
÷
1
2
¸·(i)
_
o
2
_
(/
i
)
2
÷¹oj
0
(·(i)
)) (·(i)
)
o+1
(/
i
)
t1
.
30
Next, we note that
÷·(i)
/
i
0
2
i
0·0/
= ÷¹o(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
+1ot(·(i)
)
o
(/
i
)
t
(41)
+¸·(i)
_
o
2
_
(/
i
)
2
+¹o(t ÷1)j
0
(·(i)
)) (·(i)
)
o+1
(/
i
)
t1
By (33), /
i
0
.
0b
= 0 so that the right hand side of (39) is zero. Using this fact to simplify the right
hand side of (41), we obtain
÷·(i)
/
i
0
2
i
0·0/
= ¹o(t ÷1)j
0
(·(i)
)) (·(i)
)
o+1
(/
i
)
t1
(42)
÷(o ÷1)¸·(i)
_
o
2
_
(/
i
)
2
From (39), we have
÷(/
i
)
2
0
2
i
0/
2
= ÷¹(t ÷1)(t ÷2)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
(43)
+1t(t ÷1)(·(i)
)
o
(/
i
)
t
+¸·(i)
_
o
2
_
(/
i
)
2
= ÷¹(t ÷1)
2
(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
+¹(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
+1t(t ÷1)(·(i)
)
o
(/
i
)
t
+¸·(i)
_
o
2
_
(/
i
)
2
Since /
i
0
.
0b
= 0 by (33), it follows from (39) that
÷¹(t ÷1)
2
(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
+1t(t ÷1)(·(i)
)
o
(/
i
)
t
= ÷¸(t ÷1)·(i)
_
o
2
_
(/
i
)
2
Substituting the above in (43), we have
÷(/
i
)
2
0
2
i
0/
2
= ¹(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
÷¸(t ÷2)·(i)
_
o
2
_
(/
i
)
2
(44)
To establish (38), it follows from (42) and (44) that it su¢ces to establish that
¹o(t ÷1)j
0
(·(i)
)) (·(i)
)
o+1
(/
i
)
t1
÷(o ÷1)¸·(i)
_
o
2
_
(/
i
)
2
(45)
1
2
¹(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
÷
1
2
¸(t ÷2)·(i)
_
o
2
_
(/
i
)
2
By (33), ·(i)
0
.
0.
= 0. By (40),
¹o(t ÷1)j
0
(·(i)
)) (·(i)
)
o+1
(/
i
)
t1
= ¹o(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
(46)
÷1o(t ÷1)(·(i)
)
o
(/
i
)
t
÷
(t ÷1)
2
¸·(i)
_
o
2
_
(/
i
)
2
31
Substituting (46) and (45) and simplifying, we need to show that
¹o(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
÷1o(t ÷1)(·(i)
)
o
(/
i
)
t
(47)
÷(o ÷1 +
t ÷1
2
)¸·(i)
_
o
2
_
(/
i
)
2
1
2
¹(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
÷
1
2
¸(t ÷2)·(i)
_
o
2
_
(/
i
)
2
or
¹(o ÷
1
2
)(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
(48)
÷1o(t ÷1)(·(i)
)
o
(/
i
)
t
÷(o ÷
1
2
)¸·(i)
_
o
2
_
(/
i
)
2
0
Using the fact that the right hand side of (39) is zero by (33), we have
¹(o ÷
1
2
)(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
÷(o ÷
1
2
)¸·(i)
_
o
2
_
(/
i
)
2
(49)
= 1(o ÷
1
2
)t(·(i)
)
o
(/
i
)
t
Substituting (49) in (48), it remains to show that
_
o ÷
t
2
_
1(·(i)
)
o
(/
i
)
t
0,
which will follow if o
t
2
. By (24), to show o
t
2
, it su¢ces to show that cc +
c
2
c +,,
which follows from the fact that c 2, 0 < c, , < 1, and c +, < 1. Q.E.D.
Proof of Proposition 5
By the result of Proposition 4,
0.(i)
0(S
2
)
0 for any i where o
2
= :
2
+(o
i1
)
2
. Since o
2
increases
with :
2
, it follows that
0.(i)
0(c
2
)
0. By (6), (o
i1
)
2
increases with the initial transient risk (o
0
)
2
. Hence,
0.(i)
0((o
0
)
2
)
0. Therefore, the number of managers at each hierarchical level increases with
the project’s intrinsic risk and initial transient risk. Q.E.D.
Proof of Proposition 6
Consider any two hierarchical levels i and , with i < ,. By (22), we have
i
=
i
(·(i)
, /
i
) = ¹(1 ÷j (·(i)
) (·(i)
)
o
(/
i
)
t1
÷
1(·(i)
)
o
(/
i
)
t
÷
1
2
¸·(i)
_
(o
i1
)
2
+)
2
_
(/
i
)
2
¹(1 ÷j (·(,)
) (·(,)
)
o
_
/
)
_
t1
÷
1(·(,)
)
o
_
/
)
_
t
÷
1
2
¸·(,)
_
(o
i1
)
2
+)
2
_
_
/
)
_
2
¹(1 ÷j (·(,)
) (·(,)
)
o
_
/
)
_
t1
÷
1(·(,)
)
o
_
/
)
_
t
÷
1
2
¸·(,)
_
(o
)1
)
2
+)
2
_
_
/
)
_
2
=
)
(·(,)
, /
)
) =
)
.
The …rst two equalities above follow from the de…nition of the optimal surplus at level i. The …rst
inequality follows from the fact that (·(i)
, /
i
) maximize the surplus for managers at level i. The
32
second inequality follows from the fact that, by (6), (o
i1
)
2
< (o
)1
)
2
because i < ,. The last two
equalities follow from the de…nition of the optimal surplus at level ,. Q.E.D.
Proof of Proposition 7
For : :
0
, it follows from (22) that
i
2
) =
i
(·
c
(i)
, /
c,i
)
= ¹(1 ÷j (·
c
(i)
) (·
c
(i)
)
o
_
/
c,i
_
t1
÷
1(·
c
(i)
)
o
_
/
c,i
_
t
÷
1
2
¸·
c
(i)
_
(o
i1
)
2
+)
2
_
_
/
c,i
_
2
< ¹(1 ÷j (·
c
(i)
) (·
c
(i)
)
o
_
/
c,i
_
t1
÷
1(·
c
(i)
)
o
_
/
c,i
_
t
÷
1
2
¸·
c
(i)
_
(o
i1
)
2
+
_
:
0
_
2
_
_
/
c,i
_
2
< ¹(1 ÷j (·
c
0 (i)
) (·
c
0 (i)
)
o
_
/
c
0
,i
_
t1
÷
1(·
c
0 (i)
)
o
_
/
c
0
,i
_
t
÷
1
2
¸·
c
0 (i)
_
(o
i1
)
2
+
_
:
0
_
2
_
_
/
c
0
,i
_
2
=
i
(·
c
0 (i)
, /
c
0
,i
) =
i
02
)
In the above, we explicitly indicate the dependence of the optimal surplus, the optimal number of
managers, and the optimal pay-performance sensitivity on the intrinsic risk. The …rst inequality
follows from the fact that : :
0
, while the second follows from the de…nitions of the optimal
number of managers and pay-performance sensitivity at level i when the project’s intrinsic risk is
:
02
.
If o
0
o
0
0
, it follows from (6) that (o
i1
)
2
_
o
0
i1
_
2
. Expressing the dependence of the
optimal surplus, number of managers and their pay-performance sensitivity at level i on the initial
transient risk, we have
i
((o
0
)
2
) =
i
(·
o
0
(i)
, /
o
0
,i
)
= ¹(1 ÷j
_
·
o
0
(i)
_
(·
o
0
(i)
)
o
_
/
o
0
,i
_
t1
÷
1(·
o
0
(i)
)
o
_
/
o
0
,i
_
t
÷
1
2
¸·
o
0
(i)
_
(o
i1
)
2
+)
2
__
/
o
0
,i
_
2
< ¹(1 ÷j
_
·
o
0
(i)
_
(·
o
0
(i)
)
o
_
/
o
0
,i
_
t1
÷
1(·
o
0
(i)
)
o
_
/
o
0
,i
_
t
÷
1
2
¸·
o
0
(i)
_
_
o
0
i1
_
2
+)
2
__
/
o
0
,i
_
2
< ¹(1 ÷j
_
·
o
0
0
(i)
_
(·
o
0
0
(i)
)
o
_
/
o
0
0
,i
_
t1
÷
1(·
o
0
0
(i)
)
o
_
/
o
0
0
,i
_
t
÷
1
2
¸·
o
0
0
(i)
_
_
o
0
i1
_
2
+)
2
__
/
o
0
0
,i
_
2
=
i
(·
o
0
0
(i)
, /
o
0
0
,i
) =
i
(o
02
0
)
The …rst inequality follows from the fact that o
0
o
0
0
, while the second follows from the de…nitions
of the optimal number of managers and pay-performance sensitivity at level i when the project’s
intrinsic risk is o
02
0
. Q.E.D.
Proof of Proposition 8
33
By (28),
1
n
= arg max
1
_
:1 ÷
1
i=1
`
i
_
+
1
i=1
i
, (50)
where the subscript indicates the dependence of the number of levels on the mean project quality
:. It follows from (50) that
:1
n
+
1
r
i=1
i
÷`
i
1
n
+ +
1
r
+a
i=1
i
÷`
i
\: ¸ 7
+
(51)
It follows from the above that
1
r
+a
i=1
r
`
i
:: +
1
r
+a
i=1
r
i
(52)
By Proposition 2, the optimal surplus
i
at level i does not depend on the project’s mean
quality :. The …xed costs `
i
are deterministic and also independent of :. It then follows directly
from (52) that, if :
0
< :,
1
r
+a
i=1
r
`
i
:
0
: +
1
r
+a
i=1
r
i
so that
:
0
1
n
+
1
r
i=1
i
÷`
i
:
0
(1
n
+ +
1
r
+a
i=1
i
÷`
i
\: ¸ 7
+
(53)
But (53) implies that the optimal height 1
n
0
when the mean project quality is :
0
< : must satisfy
1
n
0
< 1
n
. Q.E.D.
Proof of Proposition 9
By (52), 1
c,o
0
(where the subscripts indicate the explicit dependence on the project’s intrinsic
and transient risks) solves (28) if and only if
1
s,¬
0
+a
i=1
s,¬
0
`
i
:: +
1
s,¬
0
+a
i=1
s,¬
0
c,o
0
,i
(54)
where the additional subscripts on
c,o
0
,i
indicate its dependence on :, o
0
. By Proposition 7,
c,o
0
,i
decreases with : and o
0
. Hence,
1
s,¬
0
+a
i=1
s,¬
0
c,o
0
,i
_
1
s,¬
0
+a
i=1
s,¬
0
c
0
,o
0
0
,i
(55)
for :
0
_ : and o
0
0
_ o
0
. (54) and (55) together imply that the optimal height 1
c
0
,o
0
0
when the
project’s intrinsic and transient risks are :
0
and o
0
0
, respectively, must satisfy 1
c
0
,o
0
0
< 1
c,o
0
. Q.E.D.
Proof of Proposition 10
We …rst show that
0b
.
0(S
2
)
< 0, where o
2
= :
2
+ (o
i1
)
2
. Since (o
i1
)
2
decreases with i by (6),
it follows that o
2
also decreases, which implies that /
i
increases with i. By the implicit function
34
theorem,
0/
i
0(o
2
)
= ÷
0
2
.
0b0(S)
2
0
2
.
0.
2
÷
0
2
.
0.0(S)
2
0
2
.
0b0.
0
2
.
0b
2
0
2
.
0.
2
÷
_
0
2
.
0b0.
_
2
(·(i)
, /
i
) (56)
The denominator in the fraction on the right hand side above is positive by the second order
conditions for the maximum of the function
i
. We need to show that
0
2
i
0/0 (o)
2
0
2
i
0·
2
÷
0
2
i
0·0 (o)
2
0
2
i
0/0·
(·(i)
, /
i
) 0,
or alternatively
(/
i
) (·(i)
)
2
(o)
2
_
0
2
i
0/0 (o)
2
0
2
i
0·
2
÷
0
2
i
0·0 (o)
2
0
2
i
0/0·
_
0 (57)
By (22) and (57), it su¢ces to establish that
÷(·(i)
)
2
0
2
i
0·
2
÷
1
2
(/
i
) (·(i)
)
0
2
i
0/0·
(58)
In the proof of Proposition 4, we have established (38). By the second order condition for the
maximum of the function
i
,
0
2
i
0/
2
0
2
i
0·
2
÷
_
0
2
i
0/0·
_
2
0 (59)
Since
÷/
i
·(i)
0
2
i
0/0·
÷
1
2
(/
i
)
2
0
2
i
0/
2
,
we must have
÷/
i
·(i)
0
2
i
0/0·
< ÷2 (·(i)
)
2
0
2
i
0/
2
, (60)
for (59) to hold. But (60) is equivalent to (58). It follows that
0b
.
0(S
2
)
< 0 so that managers at higher
hierarchical levels have greater pay-performance sensitivities. Q.E.D.
Proof of Proposition 11
By the proof of Proposition 10,
0b
.
0(S
2
)
< 0. Since o
2
increases with :
2
as well as with (o
0
)
2
, it
follows that /
i
decreases with :
2
and with (o
0
)
2
. Q.E.D.
Appendix B
In this Appendix, we endogenously derive the managers’ certainty equivalent reservation pay-
o¤s. We assume that a manager (i, ,) can expropriate a proportion of the output (c
1
i
, ..., c
.(i)
i
)
generated at level i, but incurs personal costs from doing so. Contracts can only be contingent
on the change in the implementation payo¤ net of rents expropriated by managers. Given a con-
tract (a
)
i
, /
)
i
), the manager’s certainty equivalent payo¤ if she expropriates a proportion r of the
35
maximum output at her level and (hypothetically) incurs no personal costs is
¹(r)
= 1
)
i1
_
_
a
)
i
+/
)
i
_
1(i) ÷1(i ÷1) ÷r(c
1
i
, ..., c
.(i)
i
)
_
+r(c
1
i
, ..., c
.(i)
i
)÷
1
2
¸
2
(/
)
i
)
2
_
(o
i1
)
2
+)
2
_
_
_
= 1
)
i1
_
a
)
i
+/
)
i
(1(i) ÷1(i ÷1))
_
÷
1
2
¸
2
(/
)
i
)
2
_
(o
i1
)
2
+)
2
_
+ (61)
Portion expropriated by manager
¸ .. ¸
(1 ÷/
)
i
)r(c
1
i
, ..., c
.(i)
i
)
From (61), the manager expropriates a proportion (1 ÷ /
)
i
)r of the maximum output at her level
if she is not caught. The manager incurs personal costs from expropriation that are a proportion
i(·(i))(1 ÷/
)
i
)r of her certainty equivalent payo¤ under the contract assuming that she does not
expropriate, that is,
Personal Costs = i(·(i))(1÷/
)
i
)r
_
1
)
i1
_
a
)
i
+/
)
i
(1(i) ÷1(i ÷1))
_
÷
1
2
¸
2
(/
)
i
)
2
_
(o
i1
)
2
+)
2
_
_
(62)
The manager’s personal costs, therefore, increase with the proportion of the output that she ex-
propriates. The factor i(·(i)) ¸ (0, 1] in (62) allows for the manager’s personal costs to depend
on the number of managers at her level.
By (61) and (62), the manager’s certainty equivalent payo¤ net of her personal costs is
¹
act
(r) = C + (1 ÷/
)
i
)r(c
1
i
, ..., c
.(i)
i
) ÷i(·(i))(1 ÷/
)
i
)rC, (63)
where
C = a
)
i
+/
)
i
(1(i) ÷1(i ÷1)) ÷
1
2
¸
2
(/
)
i
)
2
_
(o
i1
)
2
+)
2
_
is the manager’s certainty equivalent payo¤ if she does not expropriate. By (63),
C =
1
i(·(i))
(c
1
i
, ..., c
.(i)
i
) = ¸(·(i))(c
1
i
, ..., c
.(i)
i
) (64)
is the minimum certainty equivalent payo¤ that the manager must be guaranteed by the contract
to make her indi¤erent to expropriation. It is reasonable to assume i(.) is decreasing, which
implies that the manager’s proportional personal costs of expropriation decline with the number
of managers at her level because monitoring the managers becomes more di¢cult. By (64), ¸(.) is
increasing, and j(·(i)) = ·(i)¸(·(i)) is also increasing (see Assumption 2).
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38
doc_613124431.pdf
Managerial economics as defined by Edwin Mansfield is "concerned with application of the economic concepts and economic analysis to the problems of formulating rational managerial decision
Project Characteristics, Organizational Structure, and
Managerial Incentives
+
Ajay Subramanian
Dept. of Risk Management and Insurance
J. Mack Robinson College of Business
Georgia State University
Email: [email protected]
Anand Venkateswaran
Dept. of Finance and Insurance
College of Business Administration
Northeastern University
Email: [email protected]
Richard Fu
Dept. of Accounting and Finance
College of Management
San Jose State University
Email: [email protected]
June 8, 2010
We thank Paul Bolster, Robert Gertner, Emir Kamenica, Canice Prendergast, Luis Rayo, Haresh Sapra, Lars
Stole, and seminar participants at the University of Chicago Booth School of Business, Georgia State University, and
Northeastern University for valuable comments. Subramanian gratefully acknowledges …nancial support from the
Research Program Council at the Robinson College of Business. The usual disclaimers apply.
Project Characteristics, Organizational Structure, and
Managerial Incentives
Abstract
We develop a model to show how agency con‡icts between shareholders and managers,
manager synergies, and the threat of expropriation by managers interact to a¤ect a …rm’s
internal organizational structure and the incentives of its managers. Our theory provides a
novel explanation based on agency considerations for two empirical regularities; …rms typically
have pyramidal organizational forms; and the pay-performance sensitivities of managers increase
with their hierarchical level. We also derive a number of implications that relate characteristics
of a …rm’s pool of projects—their risk and pro…tability— to its internal organization and the
incentives of its managers. (i) The optimal breadth of a …rm’s organization increases with the
risk of the …rm’s projects. (ii) The optimal height of the organization declines with risk and
increases with pro…tability. (iii) The pay-performance sensitivities of top managers increase
with pro…tability. Our results explain recent empirical evidence that …rms “‡atten” over time.
Broadly, our study contributes to the literature on the theory of the …rm by providing insights
into sources of unobserved heterogeneity among …rms, the organization of managers within …rms,
and the design of their incentives.
Key Words: Organizational Structure, Hierarchy, Incentives, Uncertainty, Synergies, Bargain-
ing Power
1 Introduction
A large and growing body of literature is devoted to the investigation of the determinants of the
size and internal organization of …rms. Bolton and Dewatripont (2005) identify the core issues at
the heart of the problem. “Why do hierarchies exist? How are e¢cient hierarchical organizations
designed? What determines the number of layers (or tiers) in a hierarchy, the pay structure, and
employees’ incentives along the hierarchical ladder?” A number of studies examine these issues
from di¤erent perspectives: (i) the e¤ects of uncertainty and learning about the quality of a …rm’s
projects (e.g. Sah and Stiglitz, 1986); (ii) information processing and communication costs (e.g.
Radner, 1992, Bolton and Dewatripont, 1994, Garicano, 2000); (iii) the link between organizational
structure and incentives (e.g. Qian, 1994, Maskin et al, 2000, Mookherjee and Reichelstein, 2001);
and (iv) the e¤ects of bargaining and expropriation (Stole and Zwiebel, 1996, Rajan and Zingales,
2001).
We develop a framework that integrates features of models analyzed in these relatively inde-
pendent strands of the literature to show how characteristics of a …rm’s external environment—the
risk and pro…tability of its projects—interact with its internal characteristics—agency con‡icts,
manager synergies and expropriation/bargaining power—to a¤ect its organizational structure and
the incentives of its managers. Our theory o¤ers a novel explanation based on manager–shareholder
agency con‡icts for two empirical regularities: …rms usually have pyramidal organizational forms;
and higher-level managers receive higher-powered incentive compensation than lower-level ones.
We also derive a number of testable predictions that relate project characteristics to organizational
structure—the number of hierarchical levels, the number of managers at each level, and the varia-
tion of managerial incentives across levels. Our results highlight the e¤ects of latent variables such
as manager synergies and expropriation/bargaining power on a …rm’s internal organization and,
thereby, executive compensation. We, therefore, shed light on some of the sources of unobserved
heterogeneity among …rms, and the channels through which they in‡uence executive compensation.
To motivate the model, consider the following illustrative examples. Fidelity wants to select
a portfolio in which to invest capital provided by investors. Asset managers in the lowest tier of
the organization gather information on potential investments. Higher tiers …lter this information
1
to narrow down the set of potential investments and determine the appropriate capital allocations.
Finally, the highest tiers of portfolio managers select the portfolio. The case of Walmart seeking
to open outlets in a new country o¤ers a similar example. Lower level managers conduct initial
feasibility studies to determine market conditions such as demand and the extent of competition,
and submit reports to their superiors. Higher tier managers conduct further analyses to identify
potential locations and outlet sizes. Finally, top-level managers select the locations and the scale
of operations.
Although the above examples describe …rms in di¤erent industries, they share certain key fea-
tures. The preliminary tasks in the development of the …rm’s “project” are delegated to agents at
the lowest level of the organization. At this stage, there is a high degree of uncertainty about the
“quality" of the project that determines its payo¤. For example, in the case of Walmart, when
agents in the lowest tier conduct initial feasibility studies, there is signi…cant uncertainty about
the scale and pro…tability of Walmart’s foreign operations. The agents in the lowest tier exert
e¤ort to develop the project, and then pass it on to the next tier in the organization that, in turn,
further develops the project before passing it on to the higher level. As the project …lters up the
organization, the uncertainty about its quality diminishes. For instance, in the case of Fidelity,
the uncertainty about the composition of its portfolio is resolved as information about potential
investments moves up the organization. The project is implemented after it passes through the
highest tier.
We attempt to capture the essence of the above examples in a model of the internal organi-
zational structure of a …rm. Speci…cally, we consider an organization with multiple hierarchical
levels and multiple decision–makers or managers at each level. Our focus is on the development
of a project rather than its …nal implementation. The total number of managers, the number of
levels in the organization, the number of managers at each level, and their incentives are endoge-
nously determined by the characteristics of the …rm’s pool of projects (hereafter, the project). As
illustrated by the above examples, we analyze the development of a project by successive tiers of
managers.
We model the implementation payo¤ of the project at any level, which is the value of future
2
earnings from the project if it is hypothetically immediately implemented without further develop-
ment by higher-level managers. The implementation payo¤ evolves as a Gaussian process and is
observable. The change in the implementation payo¤ of the project at each hierarchical level has
three components—the project’s intrinsic quality, the output generated by the managers’ e¤ort,
and the project’s intrinsic risk. The project’s intrinsic quality represents the component of the
expected change in the implementation payo¤ that is independent of the managers’ e¤ort. There is
imperfect, but symmetric, information about the project’s intrinsic quality. Agents have normally
distributed prior assessments of the intrinsic quality and update their assessments based on ob-
servations of the project’s implementation payo¤s. The variance of agents’ posterior assessments
of the project’s intrinsic quality at any date is its transient risk. The transient risk is resolved
over time as successive levels of managers develop the project and generate information about its
intrinsic quality. The variance of the project’s implementation payo¤, which is constant over time,
is the intrinsic risk of the project. We incorporate positive synergies among managers at each
level so that multiple managers complement each other. Manager synergies cause the marginal
productivity of each manager to be enhanced by the e¤ort exerted by other managers at the same
level. The output generated by the managers is increasing and concave in their e¤ort.
Managers can expropriate a proportion of the output generated at each level. Only the imple-
mentation payo¤ net of rents expropriated by managers is veri…able (by a third party such as a
court of law) and, therefore, contractible. Manager incur personal costs from expropriation. Man-
agers are risk-averse with preferences that display constant absolute risk aversion (CARA), and
receive explicit contracts from the …rm’s owners (shareholders) that are publicly observable.
The risk-neutral shareholders of the …rm choose its organizational structure (the number of
levels and the number of managers at each level) at the initial date and o¤er each manager a
contract conditional on the current information about project quality. We show that it is optimal
for shareholders to prevent manager expropriation by designing each manager’s contract so that it
guarantees her a “certainty equivalent” reservation payo¤ that is proportional to the output gen-
erated by all managers at her level. Hence, the threat of expropriation by managers endogenously
provides them with signi…cant bargaining power vis-a-vis shareholders. Further, the total propor-
3
tion of the output generated at any level that the managers receive as compensation increases with
the number of managers. The managers’ e¤ort choices are determined in Nash equilibrium of the
game in which each manager rationally anticipates the e¤ort exerted by other managers.
1
In the
symmetric equilibrium, managers at the same level receive identical contracts and exert the same
e¤ort.
Organizational Structure; The Key Tradeo¤s: The synergies among managers at each
level create an incentive for the shareholders to increase the number of managers. However, the
managers’ compensation as a proportion of the output they generate also increases with the number
of managers. The optimal number of managers at each level re‡ects the tradeo¤ between the bene…ts
of synergies among managers, and the costs of compensating them.
Because successive tiers of managers enhance the project’s payo¤ through their e¤ort choices,
increasing the number of hierarchical levels has a positive e¤ect on the project’s payo¤. Further, as
illustrated by our motivating examples, managers at higher levels have superior information about
the project’s …nal payo¤ and, therefore, face lower transient risk. As a result, shareholders can
exploit the bene…cial e¤ects of better risk-sharing with higher-level managers. The project’s payo¤
is, however, lowered by …xed costs that increase with the number of tiers in the organization. The
shareholders choose the number of levels in the organization to trade o¤ the bene…ts of further
project development against the increased costs associated with a larger number of hierarchical
levels.
Pyramidal Organizational Form: The …rm’s optimal organizational structure has the
commonly observed pyramidal form, that is, the number of managers declines with the hierarchical
level. Because managers at higher levels face lower transient risk, they receive more powerful
incentives and generate greater output. The total proportion of the output generated by managers
that they receive as compensation, however, also increases with the number of managers. The
output the managers generate is, however, concave in their e¤ort so that the marginal e¤ect of an
additional manager declines with the number of managers. Consequently, the relative impact of
managerial compensation costs in determining the number of managers is greater at higher levels.
1
Because managers have CARA preferences and payo¤s are normally distributed, we follow Holmstrom and Mil-
grom (1991) in considering contracts that are a¢ne in the increments of the implementation payo¤.
4
Hence, the number of managers at higher levels is smaller than the number at lower levels, but
each manager is more productive.
If there is perfect information about projects (there is no transient risk), the number of managers
does not vary across levels so that the organizational structure has a matrix form. Uncertainty and
learning, therefore, play key roles in generating the commonly observed pyramidal form. Our theory
o¤ers a novel explanation for the pyramidal organizational form based on manager-shareholder
agency con‡icts. In this respect, we complement earlier studies that explain the pyramidal form
using information processing costs (Bolton and Dewatripont, 1994) or variations in the di¢culty
of tasks across managerial tiers (Garicano, 2000).
Project Characteristics and Organizational Structure: The number of managers at each
hierarchical level increases with the project’s intrinsic and transient risks. The output generated
by managers at each hierarchical level declines with the project’s (intrinsic and transient) risks
because it is costlier to provide incentives to the risk-averse managers so that they exert lower
e¤ort in equilibrium. Since the total managerial compensation at each level is proportional to the
output generated, its relative importance in determining the number of managers at that level
declines with risk. The optimal number of managers at each hierarchical level in the organization,
therefore, increases with the project’s intrinsic and transient risk.
Next, we show that the number of hierarchical levels declines with the project’s intrinsic and
transient risks. As discussed earlier, the optimal number of hierarchical levels trades o¤ the positive
e¤ects of further project development against the increased …xed costs associated with a larger
number of levels. As the output generated by each hierarchical level decreases with project risk,
the relative impact of the …xed costs in determining the number of levels increases. Hence, the
optimal number of hierarchical levels decreases with project risk.
The e¤ects of the project’s intrinsic and transient risks on the height and breadth of the …rm’s
organization are novel predictions of our analysis that arise from the interplay among costs of
risk-sharing, manager synergies and compensation costs. These implications are not obtained in
earlier models such as Qian (1994) in which all payo¤s are deterministic, or in studies that focus
on the e¤ects of information processing and communication costs abstracting away from incentive
5
considerations (e.g. Bolton and Dewatripont, 1994, Garicano, 2000).
We show that the number of levels increases with the project’s initial mean intrinsic quality
or pro…tability. Because the project’s expected payo¤ increases with its pro…tability, the marginal
e¤ect of …xed costs on the optimal number of levels declines. The number of hierarchical levels,
therefore, increases with pro…tability.
Our results imply that riskier …rms have ‡atter organizational structures, and more pro…table
…rms have taller organizational structures. Insofar as pro…tability is expected to decline, and risk
levels expected to increase, as industries mature and become more competitive, our results are
consistent with anecdotal evidence that …rms are “‡attening” over time.
Project Characteristics and Manager Incentives: The managers’ incentive contracts
are simultaneously and endogenously determined along with the …rm’s organizational structure.
The managers’ pay-performance sensitivities—the sensitivities of managers’ compensation to the
change in the implementation payo¤—increase with their hierarchical level, and decrease with the
project’s intrinsic and transient risks.
2
Pay-performance sensitivities increase with level because
costs of risk-sharing decline, which is consistent with evidence that higher-level managers receive
relatively greater equity-based compensation than lower level managers.
Because the number of levels increases with the project’s pro…tability, the pay-performance
sensitivities of top-level managers increase. In contrast, in traditional principal-single agent mod-
els, the pay-performance sensitivity of the agent is only a¤ected by risk (e.g. Holmstrom and
Milgrom, 1987). The positive e¤ect of the pro…tability of the …rm’s projects on top managers’ pay-
performance sensitivities is, therefore, a key distinguishing prediction of our theory, which arises
from the modeling of multiple managers in di¤erent hierarchical levels.
2 Related Literature
Previous literature examines the determinants of …rms’ internal organizational structures from
di¤erent perspectives. Sah and Stiglitz (1986) analyze the link between the NPV of a …rm’s projects
2
Empirically, the pay-performance sensitivity is typically measured as the fraction of stock and option holdings
by an employee, to the total market value of equity of the …rm. The higher the fraction, the greater is the pay-
performance sensitivity.
6
and its internal organization using a model in which there is imperfect information about the quality
of a …rm’s pool of projects, and there are two managers who have exogenously speci…ed probabilities
of accepting bad projects or rejecting good ones. Building on Calvo and Wellisz (1978), Qian
(1994) develops a more general model of a …rm’s organization with deterministic payo¤s in which
managers are imperfectly monitored and their e¤ort choices are endogenous. We complement these
studies by developing a model of the hierarchical organization of a …rm in which there is imperfect
information and learning about the quality of projects as well as risk in their payo¤s. We also
incorporate synergies among managers’ e¤ort choices (e.g. Athey and Roberts, 2001, Dessein et al,
2007). These distinguishing features of our model lead to implications for the e¤ects of project risk
and quality uncertainty on organizational structure, as well as the variation of managers’ incentives
(as distinct from their compensation levels) across the tiers of the …rm. Further, we endogenously
derive the commonly observed pyramidal organizational structure.
Another important stream of the literature investigates the e¤ects of information processing and
communication costs on …rms’ organizations (Keren and Levhari, 1983, Radner, 1992, 1993, Bolton
and Dewatripont, 1994, Van Zandt, 1999, Garicano, 2000, Dessein, 2002, Dessein and Santos, 2003,
Vayanos, 2003). We complement this literature by focusing on the link between organizational
structure and incentives. Bolton and Dewatripont (1994) and Garicano (2000) also endogenously
derive a pyramidal organizational form. In Bolton and Dewatripont (1994), a pyramidal structure
arises from the tradeo¤ between the returns to specialization and information processing costs. In
Garicano (2000), it arises from variations in the frequency and di¢culty of tasks handled by agents
across tiers. Because their focus is on the e¤ects of information processing and communication
costs on organizational structure, these studies abstract away from incentive considerations. In
our model, a pyramidal structure arises from the interactions among costs of risk-sharing, manager
synergies, and bargaining power. Incentive considerations, therefore, play a key role in generating
a pyramidal form in our analysis.
An independent strand of the literature investigates the e¤ects of bargaining and expropriation
on organizational structure (e.g. Stole and Zwiebel, 1996). Harris and Raviv (2002) develop a model
in which organizational structure re‡ects the tradeo¤ between the coordination of the expertise of
7
managers against the costs of compensating them. We complement these studies by considering
a general hierarchical organization with multiple levels and multiple managers at each level. In
our framework, a …rm’s choice of organizational structure depends on the presence of uncertainty
about project quality, and the resulting process of learning induced by managers’ e¤ort. Further,
managers are compensated through explicit contracts. These distinguishing features of our model
lead to new implications for the impact of risk and pro…tability on managers’ incentive structures
and their organization.
An independent strand of the literature investigates the e¤ects of bargaining and expropriation
on organizational structure (e.g. Stole and Zwiebel, 1996 a, b, Rajan and Zingales, 2001). We
complement these studies by considering a general hierarchical organization with multiple levels
and multiple managers at each level. In our framework, a …rm’s choice of organizational structure
depends on the presence of uncertainty about project quality, and the resulting process of learning
induced by managers’ e¤ort. Further, managers are compensated through explicit contracts. These
distinguishing features of our model lead to new implications for the impact of risk and pro…tability
on managers’ incentive structures and their organization.
3
3 The Model
We develop a model of the internal organization of a …rm in which agents are organized in multiple
hierarchical levels. Our model could also be viewed as that of a division within a large …rm. As in
Sah and Stiglitz (1986), Geanakoplos and Milgrom (1991) and Radner (1992), we focus on higher-
level decision-making activities such as project development. We hereafter refer to the agents
in the organization as “managers.” The total number of managers, the number of levels in the
organization, the number of managers at each level, and their incentive structures are determined
endogenously by the characteristics of the …rm’s pool of projects or investment opportunity set.
Our focus is on the development of a project rather than the generation of the initial “idea”
for the project or its eventual implementation. For instance, in the example of Walmart in the
3
A body of literature analyzes hierarchical organizations as a general “mechanism design” problem (see Martimort,
2005, Mookherjee, 2006 for recent surveys). Another strand of the literature compares several aspects of unitary (U-
form) and divisional (M-form) organizations (e.g. Aghion and Tirole, 1995, Maskin et al, 2000, Qian et al, 2006).
8
introduction, top managers (or, indeed any other agent) might conceive (or suggest) the initial
idea of opening outlets in a new country. The development of this “project” (the establishment of
Walmart’s operations in the country), however, requires e¤ort by successive tiers of managers in
Walmart’s organizational structure. After top management makes the …nal decisions on the scale of
Walmart’s operations, the actual implementation of the project (the construction of outlet stores,
warehouses, etc.) is carried out. Similarly, the composition of Fidelity’s …nal investment portfolio
requires the e¤ort of successive levels of portfolio managers who gather and process information on
several potential investments and recommend appropriate capital allocations. After top portfolio
managers act on the recommendations and decide on the …nal composition of portfolio, the tasks
of actually allocating capital to the various investments are carried out. The illustrative examples
highlight the following aspects of the model: (i) the focus on project development rather than
idea conception or project implementation, and (ii) project development involves actions and the
transmission of information by successive tiers of managers upwards through an organization (see
Milgrom and Roberts (1991, Chapter 4)).
3.1 The Structure of the Organization
There are · managers, 1 levels in the organization, and ·(i) managers at a level i, where “1” is
the lowest level and “1” the highest. ·, 1, ·(i) are later determined endogenously. Managers are
identical ex ante, but are di¤erentiated ex post by their positions in the organization. This allows
us to focus on the incentive e¤ects of organizational design without obfuscating the implications
with those arising due to manager heterogeneity. (Our results are unaltered if we assume that
managers at the same hierarchical level have the same ability, but ability increases with hierarchical
level.) Consistent with the preceding discussion, and as in studies such as Sah and Stiglitz (1986),
Geanakoplos and Milgrom (1991) and Radner (1992), managers at the lowest level or tier of the
organization exert e¤ort to develop the project before passing it to the next tier in the hierarchy that,
in turn, develops the project before passing it on further, and so on. The project is implemented
and its payo¤ is realized after it passes through all levels in the organization. Hence, with 1 levels
in the organization, projects’ payo¤s are realized after they have passed through all 1 levels. The
9
…nal implementation of the project generally involves more routine tasks that are carried out by
lower-level employees such as production workers. Because our focus is on the development of the
project rather than its implementation, these additional agents are not explicitly modeled as in Sah
and Stiglitz (1986), Geanakoplos and Milgrom (1991) and Radner (1992).
3.2 The Implementation Payo¤ Process
We focus on a representative project that requires an initial capital investment that is hereafter
normalized to 1. There are multiple periods with equally spaced dates ¦0, 1, 2, ...¦. Managers
at hierarchical level i work on the project during period [i ÷ 1, i]. The key state variable is the
implementation payo¤ 1(.) of the project at any date. The implementation payo¤ 1(i) at date i is
the total payo¤ (present value of future earnings) from the project if it is hypothetically implemented
after passing through the …rst i levels of the organization without further development by managers
at higher levels. The implementation payo¤ process 1(.) is observable. As we discuss in more detail
later, the implementation payo¤ is non-veri…able because managers can expropriate a portion of
the output they generate. Consequently, only the implementation payo¤ net of rents expropriated
by managers is veri…able and, therefore, contractible.
At each level of the organization, managers exert unobservable e¤ort to develop the project.
Their e¤ort increases the project’s implementation payo¤. There are positive externalities created
by the possibility of synergies among managers at each level. Synergies among managers could,
for example, arise from the fact that managers have limited information processing capacities so
that di¤erent managers could carry out complementary tasks involved in the development of a
project. In reality, synergies could exist within a hierarchical level as well as across di¤erent levels.
We restrict consideration to synergies within each hierarchical level to simplify the model. The
project’s …nal payo¤ is, however, generated by the e¤ort of successive levels of managers. Further,
as we discuss later, the output generated by lower-level managers lowers the uncertainty faced by
higher-level managers. Consequently, they are more productive as they can be provided with more
powerful incentives. In these respects, we do capture at least some aspects of “cross level” synergies.
Suppose manager , at level i exerts e¤ort -
)
i
. We use subscripts to denote the hierarchical level
10
and superscripts to denote the particular manager. The change in the project’s implementation
payo¤ over period [i ÷1, i] is
1
¦.
¡
.
¦
(i) ÷1(i ÷1) =
_
¸
¸
¸
¸
¸
_
Intrinsic Quality
¸..¸
j +
Managers’ Output
¸ .. ¸
.(i)
)=1
_
_
)
_
_
.(i)
I=1
-
I
i
_
_
q(-
)
i
)
_
_
+
Intrinsic Risk
¸..¸
c
i
_
¸
¸
¸
¸
¸
_
. (1)
In (1), 1(i ÷ 1) is the project’s implementation payo¤ after it has passed through i ÷ 1 levels.
1
¦.
¡
.
¦
(i) is the project’s implementation payo¤ after it has passed through the i
th
level; the subscript
explicitly indicates its dependence on the e¤ort of the managers at level i.
Manager Output
From (1), the contribution of manager , at level i to the change in the project’s implementation
payo¤ is )
_
.(i)
I=1
-
I
i
_
q(-
)
i
). We assume that )(.) and q(.) are both strictly increasing so that each
manager’s marginal productivity is enhanced by the aggregate e¤ort exerted by all managers at the
same level.
4
De…ne
_
-
1
i
, -
2
i
, ..., -
.(i)
i
_
=
.(i)
)=1
_
_
)
_
_
.(i)
I=1
-
I
i
_
_
q(-
)
i
)
_
_
, (2)
which is the output generated by the managers at level i.
Assumption 1
_
-
1
i
, -
2
i
, ..., -
.(i)
i
_
is strictly increasing in each of its arguments, strictly concave
in
_
-
1
i
, -
2
i
, ..., -
.(i)
i
_
, and twice continuously di¤erentiable.
The above assumption captures the fact that, although there are positive synergies among
managers arising from the bene…ts of the coordination of activities across multiple managers, there
are decreasing returns to scale in such coordination.
The Project’s Intrinsic Risk
In (1), c
i
is a normal random variable with mean 0 and variance :
2
. The random variables
¦c
i
; i = 1, ..., 1¦ are independently and identically distributed. The variance :
2
of the noise variables
¦c
i
¦ is the project’s intrinsic risk.
4
We could alter the model to allow for the marginal productivity of each manager to be enhanced by the e¤ort
exerted by the other managers at her level (that is, excluding her e¤ort). This modi…cation complicates the notation
and analysis without altering any of our implications.
11
The Project’s Intrinsic Quality and Transient Risk
The parameter j in (1) is the intrinsic quality of the project, which determines the true expected
change (the expected change from the standpoint of the hypothetical omniscient agent) in the
project’s implementation payo¤ independent of the managers’ e¤ort. All agents have imperfect,
but symmetric information about j . The common initial prior distribution of j is normal with
mean : and variance (o
0
)
2
, that is,
j ~ ·
, (o0
)
2
) (3)
Hereafter, we refer to the parameter : as the project’s initial mean quality.
All agents rationally update their assessments of the project’s intrinsic quality in a Bayesian
manner based on their observations of the project’s implementation payo¤s, which serve as signals
of the project’s intrinsic quality. It follows that managers at higher levels in the …rm have more
accurate information about the project’s intrinsic quality than those at lower levels. The variance
of the posterior distribution of j at any date is the project’s transient risk. Because the uncertainty
about project quality is resolved over time due to Bayesian learning by agents, the project’s transient
risk declines over time.
In the context of the Fidelity example in the introduction, the uncertainty about the composition
of the …nal investment portfolio declines as successive tiers of portfolio managers gather information
on potential investments. In the case of Walmart, the uncertainty about the scale and pro…tability
of Walmart’s operations in the foreign country declines as one moves up the organization. As
another example, consider the case of Bank of America that wishes to screen a large initial pool of
applicants for loans. Loan managers at each level screen loan applications. The uncertainty about
the average quality of the pool of loans declines as successive tiers of loan managers screen the
applications until the …nal loans are sanctioned by top management.
12
3.3 Preferences, Manager Expropriation and Contracting
Managers have identical CARA preferences with multiplicative disutilities of e¤ort described by
the Von Neumann-Morgenstern utility function
l(r, c) = ÷exp
_
÷¸(r ÷cc
c
)
_
, c 2 (4)
where r is the manager’s state-dependent payo¤, and c is her e¤ort. The risk aversion coe¢cient
¸ and the e¤ort disutility function cc
c
are identical across managers. The …rm’s shareholders are
risk-neutral. The discount rates of the shareholders and managers are normalized to zero.
Managers receive explicit contracts from shareholders. By (1), the managers at level i directly
in‡uence the change in the implementation payo¤ over period [i ÷1, i]. Without loss of generality,
we assume that managers at a particular level i receive explicit contracts at date i ÷1 before they
work on the project, that is, managers at successive levels are recursively o¤ered contracts as the
project moves up the organization.
5
Managers at each level can expropriate a proportion of the output they generate, but incur
personal costs from expropriation. Only the implementation payo¤ net of rents expropriated by
managers is veri…able by a third party (e.g. a court of law) so that contracts can only be contin-
gent on the implementation payo¤ net of expropriated rents. In Appendix B, we explicitly model
manager expropriation and show that it is optimal for the shareholders to prevent manager expro-
priation by o¤ering each manager a contract that guarantees her a “certainty equivalent” payo¤
that is proportional to the output generated at her level (we describe this more precisely later). To
avoid complicating the notation and exposition, we assume this result in the following discussion
and refer the reader to Appendix B for the endogenous derivation of managers’ certainty equivalent
payo¤s.
5
Suppose instead that all managers receive contracts at date 0: Because information about the project’s intrinsic
quality is revealed over time, optimal contracts for managers at any level i would be contingent on the information
revealed at date i 1, and would be identical to the contracts we derive in our analysis.
13
3.4 The Information Structure and Bayesian Learning
The information …ltration ¦1
i
¦ of the underlying probability space (, 1, 1) , which is shared by
all agents in the economy, is generated by the implementation payo¤ process 1(.). To simplify
notation, we drop the subscripts indicating the dependence of the implementation payo¤ on the
managers’ e¤ort whenever there is no danger of confusion.
Suppose the equilibrium e¤ort exerted by manager , at level i is c
)
i
. In equilibrium, the e¤ort
exerted by each manager is correctly anticipated by all other agents. From standard Bayesian
updating formulae (Oksendal, 2001), the posterior assessment of the project’s intrinsic quality
after it has passed through levels 1, 2, ..., 1 (which is shared by all agents in equilibrium) is normal
with mean :
1
and variance (o
1
)
2
where :
1
, (o
1
)
2
satisfy
:
1
=
)2
:
11
+ (o
11
)
2
_
1(1) ÷1(1 ÷1) ÷
_
c
1
1
, ..., c
.(1)
1
__
(o
11
)
2
+
)2
, :
0
= :, (5)
(o
1
)
2
=
(o
11
)
2
)2
(o
11
)
2
+
)2
. (6)
In (5), the function (.) is de…ned in (2). We note that (o
)
)
2
< (o
)1
)
2
, that is, the project’s
transient risk declines as it moves up the organizational hierarchy. Note that a manager’s informa-
tion di¤ers from that of other agents “o¤ equilibrium” because it includes knowledge of her own
e¤ort. In equilibrium, however, because managers’ e¤ort choices are correctly anticipated by all
agents, all agents’ information …ltrations coincide.
3.5 Contract Structure
The implementation payo¤ process is normally distributed, the managers have CARA preferences,
and the shareholders are risk-neutral. Accordingly, we follow studies such as Gibbons and Murphy
(1992) by restricting consideration to renegotiation-proof contracts that are a¢ne functions of the
change in the implementation payo¤ (see also Section 4.2 of Bolton and Dewatripont, 2005). More
14
precisely, a manager , at level i is o¤ered a contract that has a payo¤
Q
)
i
= a
)
i
+/
)
i
[1(i) ÷1(i ÷1)] . (7)
The above incorporates the results of the analysis in Appendix B that it is optimal for share-
holders to prevent manager expropriation by o¤ering each manager a contract that guarantees her
a certainty equivalent payo¤ (to be described shortly) that is a proportion of the output generated
by all managers at her level. Each manager’s contract is, therefore, directly contingent on the
change in the implementation payo¤.
In (7), a manager’s contract is contingent on the output generated at her level. If the contract
were also contingent on the output generated at higher levels, then it would impose additional risk
on the manager even though the manager does not contribute to the output generated at higher
levels. Such a contract could, therefore, be altered by an ex post Pareto improving renegotiation
that eliminates the manager’s exposure to the output generated at higher levels, that is, the contract
would not be renegotiation-proof. By restricting consideration to renegotiation-proof contracts, we
can therefore assume that the contract has the form (7).
Given the observable contracts o¤ered to all managers, their e¤ort choices are determined in
Nash equilibrium of the game in which each manager rationally anticipates the e¤ort exerted by
all other managers in response to their incentives. By (1), each manager only a¤ects the output at
her level. Therefore, it is not di¢cult to see that the manager’s e¤ort choice only depends on her
contract and her inferences about the e¤ort choices of other managers at her level.
More precisely, suppose that a particular manager , in hierarchical level i (hereafter denoted
as manager (i, ,)) anticipates that the e¤ort choices of other managers at her level are c
)
i
=
¦c
c
i
; : ,= ,¦ . By (4), given her contract parameters a
)
i
, /
)
i
, manager (i, ,) exerts e¤ort c
_
a
)
i
, /
)
i
, c
)
i
_
that solves
c
_
a
)
i
, /
)
i
, c
)
i
_
(8)
= arg max
c0
÷¸cc
c
+¸1
)
i1
_
a
)
i
+/
)
i
[1(i) ÷1(i ÷1)]
_
÷
1
2
¸
2
\
)
i1
_
a
)
i
+/
)
i
[1(i) ÷1(i ÷1)]
_
.
15
In (8), 1
)
i1
is the mean and \
)
i1
is the variance conditional on the information 1
)
i1
of manager
(i, ,), which di¤ers from 1
i1
"o¤ equilibrium" as it includes knowledge of her own e¤ort.
From (1) and (2),
c
_
a
)
i
, /
)
i
, c
)
i
_
= arg max
c0
÷¸cc
c
+¸1
)
i1
_
a
)
i
+/
)
i
_
j +
_
c, c
)
i
_
+c
i
__
(9)
÷
1
2
¸
2
\
)
i1
_
a
)
i
+/
)
i
_
j +
_
c, c
)
i
_
+c
i
__
= arg max
c0
÷¸cc
c
+¸1
)
i1
_
a
)
i
+/
)
i
_
j +
_
c, c
)
i
_
+c
i
__
÷
1
2
¸
2
(/
)
i
)
2
_
(o
i1
)
2
+
)2
_
.
The second equality above follows from (5) and (6), as well as the fact that c
i
is normally dis-
tributed, and manager (i, ,)’s posterior assessment of j is also normal. It follows from (9) that
the optimal e¤ort exerted by the manager depends only on her pay-performance sensitivity /
)
i
and
the anticipated e¤ort of other managers at the same hierarchical level. Henceforth, we drop the
argument a
)
i
of the optimal e¤ort. It follows that, in Nash equilibrium of the game for a given
set of observable contracts o¤ered to all managers, each manager’s e¤ort only depends on the
pay-performance sensitivities of all managers at the same hierarchical level.
For future reference, let
c
)
i
= c
)
i
(/
1
i
, ..., /
.(i)
i
), (10)
denote the equilibrium e¤ort exerted by manager (i, ,) given the pay-performance sensitivities,
(/
1
i
, ..., /
.(i)
i
), of all managers at level i. By (2), the corresponding output generated by the managers
at level i is (c
1
i
, ..., c
.(i)
i
).
As a consequence of the analysis in Appendix B, there is no expropriation by managers if and
only if each manager’s contract guarantees her a “certainty equivalent” payo¤ that is a proportion
of the maximum output generated at her level. More precisely, a manager (i, ,)’s contract is feasible
if and only if it guarantees her a “certainty equivalent” payo¤ that is a proportion ¸(·(i)) ¸ (0, 1)
of the maximum output, (c
1
i
, ..., c
.(i)
i
), generated by all managers at level i (given their observable
16
contracts). For future reference, de…ne
j(·(i)) = ·(i)¸(·(i)), (11)
which is the total proportion of the output guaranteed to all managers at level i. In Appendix
B, we show that, if a manager’s personal costs from expropriation increase with the number of
managers at her level, which is consistent with the notion that monitoring managers at each level
becomes more di¢cult as their number increases, then j(.) is strictly increasing. That is, the total
proportion of the output generated by the managers at any level that they receive as compensation
increases with the number of managers. We therefore assume the following.
Assumption 2 j(.) is strictly increasing.
3.6 The Payo¤ and Objectives of Shareholders
As in studies such as Keren and Levhari (1983) and Qian (1994), there are …xed costs associated
with each tier in the organization that reduce the shareholders’ payo¤. These costs could arise from
operating costs, depreciation, information processing costs, etc. The level i in the organization is
associated with a …xed cost `
i
.
Assumption 3 `
i
is deterministic, strictly increasing, and strictly convex in i.
The payo¤ to the …rm’s shareholders net of the compensation paid out to managers and the
…xed costs ¦`
i
; i = 1, ..., 1¦ is
1
)
= 1(1) ÷
1
i=1
.(i)
)=1
_
a
)
i
+/
)
i
[1(i) ÷1(i ÷1)]
_
(12)
=
_
j1 +
1
i=1
_
c
1
i
, ..., c
.(i)
i
_
÷
1
i=1
`
i
+
1
i=1
c
i
_
÷
1
i=1
.(i)
)=1
_
a
)
i
+/
)
i
[1(i) ÷1(i ÷1)]
_
.
The managers’ e¤ort choices are given by (9) and the second equality above follows from (1).
In equilibrium, the risk-neutral shareholders choose the number of managers, the organizational
structure, and a feasible set of contracts for the managers, to maximize their expected payo¤
17
rationally anticipating the managers’ ex post e¤ort choices in response to their incentives. The
organizational structure parameters 1, ¦·(i)¦ (the number of hierarchical levels and the number
of managers at each level) are chosen at the initial date, whereas the managers’ contracts are
dynamically determined based on the current information about the project’s quality.
For a given organizational structure (1. ¦·(i)¦) , the optimal contractual parameters solve
_
a
)
i
, /
)
i
_
= arg max
¦o
¡
.
,b
¡
.
¦
1
i1
_
_
_
_
_
j1 ÷
1
a=1
`
a
+
1
a=1
_
c
1
a
, ..., c
.(a)
a
_
+
1
a=1
c
a
_
÷
1
a=1
.(a)
j=1
[a
j
a
+/
j
a
[1(r) ÷1(r ÷1)]]
_
_
_
_
, (13)
subject to the constraints that each manager (i, ,)
0
’s certainty equivalent payo¤ is a proportion
¸(·(i)) of the output generated at level i, that is,
1
i1
_
a
)
i
+/
)
i
(1(i) ÷1(i ÷1))
_
÷
1
2
¸\
i1
_
a
)
i
+/
)
i
(1(i) ÷1(i ÷1))
_
÷c
_
c
)
i
_
c
(14)
_ ¸(·(i))
_
c
1
i
, ..., c
.(i)
i
_
, , = 1, ..., ·(i)
In (13) and (14), 1(i) is the implementation payo¤ after the project has passed through i levels.
Using (5) and (6), we can rewrite the constraints (14) as
1
i1
_
a
)
i
+/
)
i
(1(i) ÷1(i ÷1))
_
÷
1
2
¸
_
/
)
i
_
2
_
(o
i1
)
2
+
)2
_
÷c
_
c
)
i
_
c
_ ¸(·(i))
_
c
1
i
, ..., c
.(i)
i
_
.
(15)
If \ (1, ¦·(i)¦ denotes the shareholders’ expected payo¤ as a function of the number of levels
1 and the managers at each level ¦·(i)¦, then they choose these at date 0 to solve
(1
, ¦·(i)¦
) = arg max
(1,f.(i)g)
\ (1, ¦·(i)¦) (16)
4 The Equilibrium
Since managers at each hierarchical level are identical, we focus on the symmetric equilibrium in
which all managers at the same level receive identical contracts and exert the same e¤ort. Fix the
organizational structure (1, ¦·(i)¦. Suppose that managers at level i receive the contract ¦a
i
, /
i
¦.
18
The e¤ort c
i
exerted by each manager at level i must satisfy the following …rst order conditions:
÷cc (c
i
)
c1
+/
i
0
0c
)
i
[
c
1
.
=c
2
.
=...=c
^(.)
.
= c
.
= 0; , = 1, ..., ·(i). (17)
From (13) and (15), the shareholders optimally choose the pay-performance sensitivities of
the managers, and adjust the …xed components of their compensation so that their participation
constraints are binding. Substituting (15) (with equality) in (13), and using (17), the shareholders’
optimal choice of the pay-performance sensitivity of the managers at level i solves
/
i
= arg max
b
.
(1 ÷j(·(i)))(c
i
(/
i
), ..., c
i
(/
i
)) ÷·(i)
_
c (c
i
(/
i
))
c
+
1
2
¸
_
(o
i1
)
2
+
)2
_
(/
i
)
2
_
,
(18)
where c
i
(/
i
) solves (17) (the argument explicitly indicates its dependence on the pay-performance
sensitivity /
i
), and j(.) is de…ned in (11).
To facilitate our subsequent analysis, we consider a class of tractable functional forms for the
functions )(.) and q(.) that determine the e¤ects of the managers’ e¤ort on output in (1).
Assumption 4
)(r) = .r
c
; q(j) = j
o
; . 0, c +, < 1. (19)
The following proposition explicitly characterizes the optimal e¤ort choices of managers for a given
organizational structure and incentive contracts.
Proposition 1 (E¤ort Choices for Given Organizational Structure and Incentives)
Suppose the organizational structure is (1, ¦·(i)¦), and managers at level i receive a contract
(a
i
, /
i
).
(i) The e¤ort exerted by managers at level i is
c
i
(1, ¦·(i)¦ , /
i
) =
_
.(c +,)(·(i))
c
cc
_ 1
¿oc
(/
i
)
1
¿oc
, (20)
(ii) The optimal number of managers at hierarchical level i, ·(i)
, and their pay-performance
19
sensitivities, /
i
, solve
(·(i)
, /
i
) = arg max
.(i),b
.
i
(·(i), /
i
), where (21)
i
(·(i), /
i
) = ¹(1 ÷j (·(i))) (·(i))
o
(/
i
)
t1
÷1(·(i))
o
(/
i
)
t
÷
1
2
¸·(i)
_
(o
i1
)
2
+
)2
_
(/
i
)
2
(22)
¹ =
_
.
¿
o+c
(c +,)
cc
_
o+c
¿oc
, 1 =
_
.(c +,)
c
o+c
¿
c
_ 1
¿oc
(23)
o =
cc +c ÷c ÷,
c ÷c ÷,
, t =
c
c ÷c ÷,
(24)
By (22), the number of managers at level i and their optimal pay-performance sensitivities do
not depend on the number of hierarchical levels 1 in the organization. In (21) and (22), the function
i
is the output generated by managers at level i net of their compensation. Since this accrues to
shareholders, we refer to it as the shareholder surplus at level i.
We now characterize the optimal number of managers at each hierarchical level and their pay-
performance sensitivities, which solve (21). To ease the subsequent notation and exposition, we
allow for the number of managers, ·(i), at level i to take non-integer values. We assume that
the …rst order conditions of the optimization program (21) are necessary and su¢cient to identify
(·(i)
, /
i
) .
Proposition 2 (The Optimal Number of Managers at each Level and their Incentives)
The equilibrium number of managers at level i, ·(i)
, and their optimal pay-performance sensitiv-
ities, /
i
, solve the following equations.
0 = ¹(t ÷1)(1 ÷j (·(i)
))(·(i)
)
o
(/
i
)
t2
÷1t(·(i)
)
o
(/
i
)
t1
(25)
÷¸·(i)
_
(o
i1
)
2
+
)2
_
(/
i
) ,
0 = ¹o(1 ÷j (·(i)
))(·(i)
)
o1
(/
i
)
t1
÷1o(·(i)
)
o1
(/
i
)
t
(26)
÷
1
2
¸
_
(o
i1
)
2
+
)2
_
(/
i
)
2
÷oj
0
(·(i)
) (·(i)
)
o
(/
i
)
t1
.
We now determine the shareholders’ optimal choice of the number of hierarchical levels in the
20
organization. Let
i
=
i
(·(i)
, /
i
) (27)
The following proposition characterizes the optimal number of hierarchical levels in the organization.
Proposition 3 (The Optimal Number of Hierarchical Levels) The optimal number of hier-
archical levels, 1
, is …nite and solves
1
= arg max
1
_
:
0
1 ÷
1
i=1
`
i
_
+
1
i=1
i
(28)
It follows from Proposition 7 in the next section that the surplus generated at each hierarchical
level
i
is uniformly bounded. Since the …xed cost `
i
is increasing and strictly convex in i by
Assumption 3, the optimal number of hierarchical levels is …nite.
5 Organizational Structure and Incentives
In this section, we derive properties of the equilibrium. We analyze the e¤ects of project charac-
teristics on the …rm’s organizational structure and the managers’ incentives.
5.1 Organizational Form
We …rst examine the variation of the number of managers across hierarchical levels. The following
proposition shows that the optimal number of managers at a given level decreases as we move up
the organization, that is, the organization has the traditional pyramidal form.
Proposition 4 (Pyramidal Organizational Form) The optimal number of managers, ·(i)
,
at hierarchical level i decreases with i.
The above result arises from the subtle interplay between various forces. First, managers at
higher levels in the organization have better information about project quality and face lower
transient risk. This leads to lower costs of risk-sharing (that is, these managers can be provided with
more powerful incentives), which implies that, ceteris paribus, the output generated by managers
21
is higher as one moves up the organization. However, the total compensation of managers, as a
proportion of the output they generate, also increases with the number of managers. Since output
is concave in managers’ e¤ort, the marginal positive impact of the e¤ort of an additional manager
declines with the number of managers. Consequently, the marginal negative impact of the costs
of compensating managers on the optimal choice of the number of managers is greater at higher
hierarchical levels. As the hierarchical level increases, therefore, it is optimal to have fewer managers
who are more productive.
An important implication of the intuition above is that, in the presence of perfect information
about project quality (that is, there is no transient risk), there are an equal number of managers
at each level of the organization because there is no variation in the costs of risk-sharing across
levels. Hence, the organizational structure has a matrix rather than pyramidal form. The presence
of uncertainty about the quality of a …rm’s investment opportunities, and the resulting process
of learning induced by the e¤ort exerted by managers to develop the …rm’s projects, distort the
…rm’s organizational structure to the commonly observed pyramidal form. Earlier studies have
highlighted the e¤ects of imperfect monitoring (for example, Qian 1994), expropriation risk (Rajan
and Zingales 2001), and communication costs (Radner 1992, 1993, Van Zandt 1999, Garicano 2001)
on …rms’ organizational structure. Our framework suggests that the presence of agency costs of risk-
sharing, imperfect information, and the process of learning about project quality are also important
determinants of organizational design.
Bolton and Dewatripont (1994) and Garicano (2000) also endogously derive a pyramidal organi-
zational structure in their frameworks. In Bolton and Dewatripont (1994), a pyramidal form arises
from the tradeo¤ between the gains to specialization in processing certain types of information
and the costs of communicating information. In Garicano (2000), it arises from the fact that it is
optimal to assign agents at lower levels to easier, frequent tasks and those at higher levels to harder,
less frequent tasks. Because their focus is on the e¤ects of information processing and communi-
cation costs on organizational structure, these studies abstract away from incentive considerations.
In contrast, in our model, a pyramidal organizational form arises from the interactive e¤ects of
costs of risk-sharing, which decline as one moves up the organization, manager synergies, and the
22
possibility of expropriation. Incentive considerations, therefore, play a key role in generating the
pyramidal form in our analysis.
5.2 Breadth of the Organization
We now analyze the e¤ects of project characteristics on the number of managers at each hierarchical
level or the breadth of the organization.
Proposition 5 (Risk, Uncertainty and Organizational Breadth) The optimal number of man-
agers, ·(i)
, at any level i increases with the project’s intrinsic risk :
2
as well as its initial transient
risk o
2
0
.
The number of managers at each level is determined by trading o¤ the bene…ts of positive
synergies among managers and the costs of compensating them. The output generated by managers
declines with risk due to the higher agency costs of risk-sharing with the …rm’s shareholders. Hence,
the total compensation received by managers at each level also declines with the project’s intrinsic
and transient risk. The relative (or marginal) e¤ect of managers’ compensation in determining the
optimal number of managers at any level, therefore, declines with risk so that the …rm can support
a larger number of managers at the optimum. Proposition 5 leads to the following implication.
Implication 1 High-risk …rms have a larger number of managers at each hierarchical level than
low-risk …rms.
The intuition for Proposition 5 suggests that the predicted e¤ects of risk and uncertainty on
organizational breadth arise from the interactions among costs of risk-sharing, manager synergies
and bargaining power. These are novel implications of our model that are not obtained in the
frameworks of studies such as Qian (1994) in which all payo¤s are deterministic, or in studies that
focus on the e¤ects of information processing costs, but abstract away from incentive considerations
(e.g. Bolton and Dewatripont, 1994, Garicano, 2000).
5.3 The Surplus Generated by Each Hierarchical Level
We now investigate the surplus generated by managers at each hierarchical level.
23
Proposition 6 (The Surplus Generated by Each Hierarchical Level) The surplus,
i
(see
27) generated at each level i increases with i.
This result supports the intuition described earlier for Proposition 4. Managers at higher levels
face lower transient risk leading to better risk-sharing with the …rm’s shareholders. As a result,
managers at higher levels can be provided with more powerful incentives implying that the output
they generate and the surplus increases with hierarchical level.
Because the number of managers declines with hierarchical level by Proposition 4, the average
surplus per manager increases disproportionately with hierarchical level. Note that managers are
ex ante identical in our framework so that the positive relation between average surplus and level
arises purely due to the e¤ects of learning and the resolution of uncertainty.
The following result shows that the surplus generated at any given level decreases with the
project’s intrinsic risk and initial transient risk. The result follows directly from the fact that, as
discussed earlier, higher risk levels leads to greater costs of risk-sharing between the risk-neutral
shareholders and risk-averse managers.
Proposition 7 (Risk, Uncertainty, and the Surplus Generated by Each Level) The sur-
plus,
i
(see 27) generated at each level i decreases with the project’s intrinsic risk :
2
as well as
its initial transient risk o
2
0
.
Propositions 6 and 7 lead to the following implications.
Implication 2 The surplus generated at each hierarchical level increases (disproportionately) with
the level and declines with the project’s intrinsic and transient risks.
5.4 Height of the Organization
We now examine the e¤ects of project characteristics on the number of hierarchical levels or height
of the organization.
Proposition 8 (Pro…tability and Organizational Height) The equilibrium number of levels,
1
, increases with the project’s initial mean quality or pro…tability :.
24
As we examine the e¤ect of the project’s mean quality keeping its intrinsic risk and initial
transient risk …xed, it follows from Proposition 2 and (27) that the surplus generated at any given
level is …xed. By (28), as the project’s mean quality increases, the expected change in the project’s
implementation payo¤ increases. As a result, the relative (or marginal) impact of the …xed costs
¦`
i
¦ as well as the costs of risk-sharing with managers on the determination of the optimal number
of hierarchical levels declines. Therefore, the equilibrium number of levels increases with the mean
quality. Since the total expected payo¤ increases with the number of levels, the number of levels
in the organization and the expected payo¤ to the …rm’s shareholders are also positively related.
The above proposition leads to the following implication.
Implication 3 More pro…table …rms are taller than less pro…table …rms.
The following proposition show that the optimal number of levels in the organization decreases
with the project’s intrinsic risk and the initial transient risk .
Proposition 9 (Risk, Uncertainty and Organizational Height) The optimal number of lev-
els, 1
, decreases with the project’s intrinsic risk :
2
as well as its initial transient risk o
2
0
.
The intuition for this result follows from the intuition for Proposition 7. An increase in the
intrinsic or transient risk of the project implies that the surplus generated at each hierarchical level
declines. Therefore, the marginal e¤ect of the depreciation costs ¦`
i
¦ in determining the optimal
number of levels increases. The optimal number of hierarchical levels, therefore, declines with the
project’s intrinsic and initial transient risk. Proposition 9 leads to the following implication.
Implication 4 High-risk …rms have a smaller number of hierarchical levels than low-risk …rms.
Implications 1, 3, and 4 together suggest that less pro…table and riskier …rms have a smaller
number of hierarchical levels and a larger number of managers at each level, that is, they are
“‡atter” than more pro…table and less risky …rms. Insofar as pro…tability is expected to decline,
and risk levels expected to increase as industries mature and become more competitive, these
predictions suggest that …rms “‡atten” over time, which appears to be consistent with anecdotal
evidence.
25
5.5 Manager Incentives
We now analyze the e¤ects of project characteristics on the incentive structures of managers. The
following result describes the variation of managers’ incentives across hierarchical levels.
Proposition 10 (Variation of Manager Incentives Across Levels) The pay-performance sen-
sitivity /
i
, of managers at level i increases with i.
The above result further supports the intuition that we described for our earlier results. Because
managers at higher levels face lower transient risk, their incentives can be better aligned with the
interests of the …rm’s shareholders, that is, the costs of risk-sharing are lower. It is important
to note, however, that the pay-performance sensitivity of a manager at any level is determined
endogenously along with the organizational structure; more speci…cally, on the equilibrium number
of managers at that level. Hence, it is not a priori obvious that pay-performance sensitivities increase
with the hierarchical level. Formally, we have the following implication, which is consistent with
anecdotal evidence that CEOs receive relatively greater equity-based compensation than lower level
managers.
Implication 5 The pay-performance sensitivity of a manager increases with her level in the or-
ganization.
We now show that the pay-performance sensitivity of a manager at a given level in the organi-
zation decreases with the project’s intrinsic and initial transient risk.
Proposition 11 (Risk, Uncertainty and Manager Incentives) The pay-performance sensi-
tivity /
i
, of managers at level i decreases with the project’s intrinsic risk :
2
as well as its initial
transient risk o
2
0
.
In contrast with traditional principal-single agent models such as Holmstrom and Milgrom
(1987), the organizational structure and the incentive structures of managers are simultaneously
and endogenously determined by the characteristics of the project. Proposition 11, therefore, does
not directly follow from the results of these models.
26
Implication 6 The pay-performance sensitivity of a manager decreases with the project’s (intrin-
sic and transient) risk.
The above implication is consistent with empirical evidence of a negative relationship between a
CEO’s pay-performance sensitivity and …rm risk (see Aggarwal and Samwick, 1999, Prendergast,
1999).
The following result, which follows immediately from Propositions 8, 9, 10, and 11, describes the
e¤ect of the project’s mean intrinsic quality or pro…tability on the incentives of top-level managers.
Proposition 12 (Incentives of Top-Level Managers) The pay-performance sensitivities of top-
level managers increase with the project’s initial mean intrinsic quality or pro…tability :.
The pay-performance sensitivities of top-level managers are simultaneously and endogenously
determined along with the number of levels in the organization. Hence, the modeling of the actions
of multiple managers in di¤erent hierarchical levels plays a key role in generating the positive
relation between top managers’ pay-performance sensitivities and the project’s initial mean quality
or pro…tability. In contrast, in traditional principal-single agent models with normally distributed
payo¤s, the pay-performance sensitivity of the agent is only a¤ected by risk (see Section 5 and
equation (32) in Holmstrom and Milgrom, 1987, Aggarwal and Samwick, 1999). We have the
following implication.
Implication 7 The pay-performance sensitivities of top-level managers increase with the prof-
itability of …rms’ investment opportunities.
The above implication suggests that there is a causal link between the pro…tability of the
…rm’s initial pool of projects and the incentives of its top-level managers, that is, pro…tability is
a determinant of top-manager incentives. The …nal payo¤s of the …rm’s projects are, however,
a¤ected by the e¤ort choices of managers at each hierarchical level that are, in turn, a¤ected by
their incentives. Consequently, managerial incentives and …rm performance are both endogenously
determined by characteristics of the …rm’s initial pool of projects that are di¢cult to directly
observe. The lack of a direct causal link between managerial incentives and …rm performance
complicates empirical analyses of their relationship (see Himmelberg et al, 1999).
27
6 Conclusions
We develop a theory of the impact of the characteristics of a …rm’s projects on the organization
of its managers and their incentives. By integrating aspects of models developed in relatively
independent strands of the literature, our analysis shows the interactive e¤ects of project risk,
quality uncertainty, information processing, manager synergies, and bargaining power on …rms’
organizational structures and the incentives of their managers.
The commonly observed pyramidal organizational form arises from the interplay among (i)
project quality uncertainty; (ii) information processing or learning; (iii) positive complementarities
among managers’ e¤ort choices; and (iv) managers’ bargaining power. The number of hierarchical
levels in a …rm increases with the pro…tability of its projects, and decreases with their risk. The
number of managers at each hierarchical level of the organization increases with risk. Insofar as
pro…tability is expected to decline, and risk levels expected to rise, as industries mature and become
more competitive, our theory provides an alternative explanation for recent empirical evidence that
…rms ‡atten over time.
Our model, in which organizational structure and manager incentives are simultaneously and
endogenously determined, also generates several implications for the design of executive compensa-
tion, and the link between executive compensation and organizational structure. First, managers’
pay-performance sensitivities (or “equity-based” compensation) increase with their level in the …rm.
Second, the pay-performance sensitivities of managers decline with the risk of the …rm’s projects
and the pay-performance sensitivities of top-level managers increase with the projects’ pro…tability.
Third, the height and breadth of the …rm’s organization decline with managers’ bargaining power.
Fourth, managers’ pay-performance sensitivities could increase with their bargaining power.
Our study contributes to the literature on the theory of the …rm by uncovering some of the
sources of unobserved heterogeneity among …rms, and showing the link between organizational
structure and executive compensation. We show how characteristics of a …rm’s external envi-
ronment – the risk and pro…tability of its projects – interact with its internal characteristics –
information processing costs, manager synergies, and bargaining power – to a¤ect the organization
and incentives of its managers.
28
Appendix A
Proof of Proposition 1
(i) The e¤ort c
i
exerted by each manager at level i (we drop the arguments indicating the depen-
dence on the organizational structure (1, ¦·(i)¦) and the pay-performance sensitivity /
i
to simplify
the notation) solves the …rst order conditions (17).
By (2) and (19),
(c
i
, ..., c
i
) = .·(i)
c+1
(c
i
)
c+o
(29)
Substituting (29) in (17) we have
/
i
.,(·(i))
c
(c
i
)
c+o1
+/
i
.c(·(i))
c
(c
i
)
c+o1
= cc (c
i
)
c1
. (30)
It follows directly from (30) that the e¤ort c
i
is given by (20).
(ii) By (16) and (18), the optimal number of managers at level i and their pay-performance
sensitivity /
i
solve
(·(i)
, /
i
) = arg max
.(i),b
.
(c
i
, ..., c
i
) ÷·(i)
_
c(c
i
)
c
+
1
2
¸
_
(o
i1
)
2
+
)2
_
(/
i
)
2
_
(31)
By (20) and (29), we have
(c
i
, ..., c
i
) = . (·(i))
c+1
(c
i
)
c+o
(32)
= . (·(i))
c+1
_
.(c +,)(·(i))
c
cc
_ o+c
¿oc
(/
i
)
o+c
¿oc
It follows from (31) and (32) that
(·(i)
, /
i
) = arg max
.(i),b
.
. (·(i))
c+1
_
.(c +,)(·(i))
c
cc
_ o+c
¿oc
(/
i
)
o+c
¿oc
÷·(i)c
_
.(c +,)(·(i))
c
cc
_ ¿
¿oc
(/
i
)
¿
¿oc
÷
1
2
¸·(i)
_
(o
i1
)
2
+
)2
_
(/
i
)
2
.
The above implies (21). Q.E.D.
Proof of Proposition 2
By (21) and Assumption ??, we have
0
i
(·(i)
, /
i
)
0·(i)
= 0;
0
i
(·(i)
, /
i
)
0/
i
= 0 (33)
(25) and (26) follow directly from (22) and (33). Q.E.D.
Proof of Proposition 3
By (16) and Proposition 2, the optimal number of hierarchical levels 1
solves
1
= arg max
1
\ (1, ¦·(i)
¦ .
29
By (1), (5), and (27),
\ (1, ¦·(i)
¦ = 1
0
[1(1) ÷1(0)] =
_
:
0
1 ÷
1
i=1
`
i
_
+
1
i=1
i
(28) follows directly from the above. By Proposition 7 that we prove later, the optimal surplus
generated at each hierarchical level
i
is bounded above by the surplus in the scenario where the
intrinsic and transient risks are both zero. It immediately follows from Assumption 3 that the
number of hierarchical levels is …nite. Q.E.D.
Proof of Proposition 4
The optimal number of managers at level i, ·(i)
and their pay-performance sensitivity solve (33).
De…ne
o
2
= (o
i1
)
2
+:
2
(34)
The proof proceeds by showing that
0.(i)
0(S
2
)
0. Since (o
i1
)
2
decreases with i by (6), it follows
that o
2
also decreases, which will then imply from the above that ·(i)
decreases with i. By the
implicit function theorem,
0·(i)
0(o
2
)
= ÷
0
2
.
0b0(S)
2
0
2
.
0b0.
÷
0
2
.
0.0(S)
2
0
2
.
0b
2
_
0
2
.
0b0.
_
2
÷
0
2
.
0b
2
0
2
.
0.
2
(·(i)
, /
i
) (35)
The denominator in the fraction on the right hand side above is negative by the second order
conditions for the maximum of the function
i
. To establish that
0.(i)
0(S
2
)
0, we need to show that
0
2
i
0/0 (o)
2
0
2
i
0/0·
÷
0
2
i
0·0 (o)
2
0
2
i
0/
2
(·(i)
, /
i
) 0, (36)
or alternatively
(/
i
)
2
·(i)
(o)
2
_
0
2
i
0/0 (o)
2
0
2
i
0/0·
÷
0
2
i
0·0 (o)
2
0
2
i
0/
2
_
0. (37)
By (22), to show (37) it su¢ces to establish that
÷/
i
·(i)
0
2
i
0/0·
÷
1
2
(/
i
)
2
0
2
i
0/
2
(38)
By (22) and (25), we have
/
i
0
i
0/
= ¹(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
÷1t(·(i)
)
o
(/
i
)
t
(39)
÷¸·(i)
_
o
2
_
(/
i
)
2
,
·(i)
0
i
0·
= ¹o(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
÷1o(·(i)
)
o
(/
i
)
t
(40)
÷
1
2
¸·(i)
_
o
2
_
(/
i
)
2
÷¹oj
0
(·(i)
)) (·(i)
)
o+1
(/
i
)
t1
.
30
Next, we note that
÷·(i)
/
i
0
2
i
0·0/
= ÷¹o(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
+1ot(·(i)
)
o
(/
i
)
t
(41)
+¸·(i)
_
o
2
_
(/
i
)
2
+¹o(t ÷1)j
0
(·(i)
)) (·(i)
)
o+1
(/
i
)
t1
By (33), /
i
0
.
0b
= 0 so that the right hand side of (39) is zero. Using this fact to simplify the right
hand side of (41), we obtain
÷·(i)
/
i
0
2
i
0·0/
= ¹o(t ÷1)j
0
(·(i)
)) (·(i)
)
o+1
(/
i
)
t1
(42)
÷(o ÷1)¸·(i)
_
o
2
_
(/
i
)
2
From (39), we have
÷(/
i
)
2
0
2
i
0/
2
= ÷¹(t ÷1)(t ÷2)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
(43)
+1t(t ÷1)(·(i)
)
o
(/
i
)
t
+¸·(i)
_
o
2
_
(/
i
)
2
= ÷¹(t ÷1)
2
(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
+¹(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
+1t(t ÷1)(·(i)
)
o
(/
i
)
t
+¸·(i)
_
o
2
_
(/
i
)
2
Since /
i
0
.
0b
= 0 by (33), it follows from (39) that
÷¹(t ÷1)
2
(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
+1t(t ÷1)(·(i)
)
o
(/
i
)
t
= ÷¸(t ÷1)·(i)
_
o
2
_
(/
i
)
2
Substituting the above in (43), we have
÷(/
i
)
2
0
2
i
0/
2
= ¹(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
÷¸(t ÷2)·(i)
_
o
2
_
(/
i
)
2
(44)
To establish (38), it follows from (42) and (44) that it su¢ces to establish that
¹o(t ÷1)j
0
(·(i)
)) (·(i)
)
o+1
(/
i
)
t1
÷(o ÷1)¸·(i)
_
o
2
_
(/
i
)
2
(45)
1
2
¹(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
÷
1
2
¸(t ÷2)·(i)
_
o
2
_
(/
i
)
2
By (33), ·(i)
0
.
0.
= 0. By (40),
¹o(t ÷1)j
0
(·(i)
)) (·(i)
)
o+1
(/
i
)
t1
= ¹o(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
(46)
÷1o(t ÷1)(·(i)
)
o
(/
i
)
t
÷
(t ÷1)
2
¸·(i)
_
o
2
_
(/
i
)
2
31
Substituting (46) and (45) and simplifying, we need to show that
¹o(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
÷1o(t ÷1)(·(i)
)
o
(/
i
)
t
(47)
÷(o ÷1 +
t ÷1
2
)¸·(i)
_
o
2
_
(/
i
)
2
1
2
¹(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
÷
1
2
¸(t ÷2)·(i)
_
o
2
_
(/
i
)
2
or
¹(o ÷
1
2
)(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
(48)
÷1o(t ÷1)(·(i)
)
o
(/
i
)
t
÷(o ÷
1
2
)¸·(i)
_
o
2
_
(/
i
)
2
0
Using the fact that the right hand side of (39) is zero by (33), we have
¹(o ÷
1
2
)(t ÷1)(1 ÷j (·(i)
)) (·(i)
)
o
(/
i
)
t1
÷(o ÷
1
2
)¸·(i)
_
o
2
_
(/
i
)
2
(49)
= 1(o ÷
1
2
)t(·(i)
)
o
(/
i
)
t
Substituting (49) in (48), it remains to show that
_
o ÷
t
2
_
1(·(i)
)
o
(/
i
)
t
0,
which will follow if o
t
2
. By (24), to show o
t
2
, it su¢ces to show that cc +
c
2
c +,,
which follows from the fact that c 2, 0 < c, , < 1, and c +, < 1. Q.E.D.
Proof of Proposition 5
By the result of Proposition 4,
0.(i)
0(S
2
)
0 for any i where o
2
= :
2
+(o
i1
)
2
. Since o
2
increases
with :
2
, it follows that
0.(i)
0(c
2
)
0. By (6), (o
i1
)
2
increases with the initial transient risk (o
0
)
2
. Hence,
0.(i)
0((o
0
)
2
)
0. Therefore, the number of managers at each hierarchical level increases with
the project’s intrinsic risk and initial transient risk. Q.E.D.
Proof of Proposition 6
Consider any two hierarchical levels i and , with i < ,. By (22), we have
i
=
i
(·(i)
, /
i
) = ¹(1 ÷j (·(i)
) (·(i)
)
o
(/
i
)
t1
÷
1(·(i)
)
o
(/
i
)
t
÷
1
2
¸·(i)
_
(o
i1
)
2
+
)2
_
(/
i
)
2
¹(1 ÷j (·(,)
) (·(,)
)
o
_
/
)
_
t1
÷
1(·(,)
)
o
_
/
)
_
t
÷
1
2
¸·(,)
_
(o
i1
)
2
+
)2
_
_
/
)
_
2
¹(1 ÷j (·(,)
) (·(,)
)
o
_
/
)
_
t1
÷
1(·(,)
)
o
_
/
)
_
t
÷
1
2
¸·(,)
_
(o
)1
)
2
+
)2
_
_
/
)
_
2
=
)
(·(,)
, /
)
) =
)
.
The …rst two equalities above follow from the de…nition of the optimal surplus at level i. The …rst
inequality follows from the fact that (·(i)
, /
i
) maximize the surplus for managers at level i. The
32
second inequality follows from the fact that, by (6), (o
i1
)
2
< (o
)1
)
2
because i < ,. The last two
equalities follow from the de…nition of the optimal surplus at level ,. Q.E.D.
Proof of Proposition 7
For : :
0
, it follows from (22) that
i
2
) =
i
(·
c
(i)
, /
c,i
)
= ¹(1 ÷j (·
c
(i)
) (·
c
(i)
)
o
_
/
c,i
_
t1
÷
1(·
c
(i)
)
o
_
/
c,i
_
t
÷
1
2
¸·
c
(i)
_
(o
i1
)
2
+
)2
_
_
/
c,i
_
2
< ¹(1 ÷j (·
c
(i)
) (·
c
(i)
)
o
_
/
c,i
_
t1
÷
1(·
c
(i)
)
o
_
/
c,i
_
t
÷
1
2
¸·
c
(i)
_
(o
i1
)
2
+
_
:
0
_
2
_
_
/
c,i
_
2
< ¹(1 ÷j (·
c
0 (i)
) (·
c
0 (i)
)
o
_
/
c
0
,i
_
t1
÷
1(·
c
0 (i)
)
o
_
/
c
0
,i
_
t
÷
1
2
¸·
c
0 (i)
_
(o
i1
)
2
+
_
:
0
_
2
_
_
/
c
0
,i
_
2
=
i
(·
c
0 (i)
, /
c
0
,i
) =
i
02
)
In the above, we explicitly indicate the dependence of the optimal surplus, the optimal number of
managers, and the optimal pay-performance sensitivity on the intrinsic risk. The …rst inequality
follows from the fact that : :
0
, while the second follows from the de…nitions of the optimal
number of managers and pay-performance sensitivity at level i when the project’s intrinsic risk is
:
02
.
If o
0
o
0
0
, it follows from (6) that (o
i1
)
2
_
o
0
i1
_
2
. Expressing the dependence of the
optimal surplus, number of managers and their pay-performance sensitivity at level i on the initial
transient risk, we have
i
((o
0
)
2
) =
i
(·
o
0
(i)
, /
o
0
,i
)
= ¹(1 ÷j
_
·
o
0
(i)
_
(·
o
0
(i)
)
o
_
/
o
0
,i
_
t1
÷
1(·
o
0
(i)
)
o
_
/
o
0
,i
_
t
÷
1
2
¸·
o
0
(i)
_
(o
i1
)
2
+
)2
__
/
o
0
,i
_
2
< ¹(1 ÷j
_
·
o
0
(i)
_
(·
o
0
(i)
)
o
_
/
o
0
,i
_
t1
÷
1(·
o
0
(i)
)
o
_
/
o
0
,i
_
t
÷
1
2
¸·
o
0
(i)
_
_
o
0
i1
_
2
+
)2
__
/
o
0
,i
_
2
< ¹(1 ÷j
_
·
o
0
0
(i)
_
(·
o
0
0
(i)
)
o
_
/
o
0
0
,i
_
t1
÷
1(·
o
0
0
(i)
)
o
_
/
o
0
0
,i
_
t
÷
1
2
¸·
o
0
0
(i)
_
_
o
0
i1
_
2
+
)2
__
/
o
0
0
,i
_
2
=
i
(·
o
0
0
(i)
, /
o
0
0
,i
) =
i
(o
02
0
)
The …rst inequality follows from the fact that o
0
o
0
0
, while the second follows from the de…nitions
of the optimal number of managers and pay-performance sensitivity at level i when the project’s
intrinsic risk is o
02
0
. Q.E.D.
Proof of Proposition 8
33
By (28),
1
n
= arg max
1
_
:1 ÷
1
i=1
`
i
_
+
1
i=1
i
, (50)
where the subscript indicates the dependence of the number of levels on the mean project quality
:. It follows from (50) that
:1
n
+
1
r
i=1
i
÷`
i
1n
+
+1
r
+a
i=1
i
÷`
i
\: ¸ 7
+
(51)
It follows from the above that
1
r
+a
i=1
r
`
i
:: +
1
r
+a
i=1
r
i
(52)
By Proposition 2, the optimal surplus
i
at level i does not depend on the project’s mean
quality :. The …xed costs `
i
are deterministic and also independent of :. It then follows directly
from (52) that, if :
0
< :,
1
r
+a
i=1
r
`
i
:
0
: +
1
r
+a
i=1
r
i
so that
:
0
1
n
+
1
r
i=1
i
÷`
i
:
0
(1
n
+
+1
r
+a
i=1
i
÷`
i
\: ¸ 7
+
(53)
But (53) implies that the optimal height 1
n
0
when the mean project quality is :
0
< : must satisfy
1
n
0
< 1
n
. Q.E.D.
Proof of Proposition 9
By (52), 1
c,o
0
(where the subscripts indicate the explicit dependence on the project’s intrinsic
and transient risks) solves (28) if and only if
1
s,¬
0
+a
i=1
s,¬
0
`
i
:: +
1
s,¬
0
+a
i=1
s,¬
0
c,o
0
,i
(54)
where the additional subscripts on
c,o
0
,i
indicate its dependence on :, o
0
. By Proposition 7,
c,o
0
,i
decreases with : and o
0
. Hence,
1
s,¬
0
+a
i=1
s,¬
0
c,o
0
,i
_
1
s,¬
0
+a
i=1
s,¬
0
c
0
,o
0
0
,i
(55)
for :
0
_ : and o
0
0
_ o
0
. (54) and (55) together imply that the optimal height 1
c
0
,o
0
0
when the
project’s intrinsic and transient risks are :
0
and o
0
0
, respectively, must satisfy 1
c
0
,o
0
0
< 1
c,o
0
. Q.E.D.
Proof of Proposition 10
We …rst show that
0b
.
0(S
2
)
< 0, where o
2
= :
2
+ (o
i1
)
2
. Since (o
i1
)
2
decreases with i by (6),
it follows that o
2
also decreases, which implies that /
i
increases with i. By the implicit function
34
theorem,
0/
i
0(o
2
)
= ÷
0
2
.
0b0(S)
2
0
2
.
0.
2
÷
0
2
.
0.0(S)
2
0
2
.
0b0.
0
2
.
0b
2
0
2
.
0.
2
÷
_
0
2
.
0b0.
_
2
(·(i)
, /
i
) (56)
The denominator in the fraction on the right hand side above is positive by the second order
conditions for the maximum of the function
i
. We need to show that
0
2
i
0/0 (o)
2
0
2
i
0·
2
÷
0
2
i
0·0 (o)
2
0
2
i
0/0·
(·(i)
, /
i
) 0,
or alternatively
(/
i
) (·(i)
)
2
(o)
2
_
0
2
i
0/0 (o)
2
0
2
i
0·
2
÷
0
2
i
0·0 (o)
2
0
2
i
0/0·
_
0 (57)
By (22) and (57), it su¢ces to establish that
÷(·(i)
)
2
0
2
i
0·
2
÷
1
2
(/
i
) (·(i)
)
0
2
i
0/0·
(58)
In the proof of Proposition 4, we have established (38). By the second order condition for the
maximum of the function
i
,
0
2
i
0/
2
0
2
i
0·
2
÷
_
0
2
i
0/0·
_
2
0 (59)
Since
÷/
i
·(i)
0
2
i
0/0·
÷
1
2
(/
i
)
2
0
2
i
0/
2
,
we must have
÷/
i
·(i)
0
2
i
0/0·
< ÷2 (·(i)
)
2
0
2
i
0/
2
, (60)
for (59) to hold. But (60) is equivalent to (58). It follows that
0b
.
0(S
2
)
< 0 so that managers at higher
hierarchical levels have greater pay-performance sensitivities. Q.E.D.
Proof of Proposition 11
By the proof of Proposition 10,
0b
.
0(S
2
)
< 0. Since o
2
increases with :
2
as well as with (o
0
)
2
, it
follows that /
i
decreases with :
2
and with (o
0
)
2
. Q.E.D.
Appendix B
In this Appendix, we endogenously derive the managers’ certainty equivalent reservation pay-
o¤s. We assume that a manager (i, ,) can expropriate a proportion of the output (c
1
i
, ..., c
.(i)
i
)
generated at level i, but incurs personal costs from doing so. Contracts can only be contingent
on the change in the implementation payo¤ net of rents expropriated by managers. Given a con-
tract (a
)
i
, /
)
i
), the manager’s certainty equivalent payo¤ if she expropriates a proportion r of the
35
maximum output at her level and (hypothetically) incurs no personal costs is
¹(r)
= 1
)
i1
_
_
a
)
i
+/
)
i
_
1(i) ÷1(i ÷1) ÷r(c
1
i
, ..., c
.(i)
i
)
_
+r(c
1
i
, ..., c
.(i)
i
)÷
1
2
¸
2
(/
)
i
)
2
_
(o
i1
)
2
+
)2
_
_
_
= 1
)
i1
_
a
)
i
+/
)
i
(1(i) ÷1(i ÷1))
_
÷
1
2
¸
2
(/
)
i
)
2
_
(o
i1
)
2
+
)2
_
+ (61)
Portion expropriated by manager
¸ .. ¸
(1 ÷/
)
i
)r(c
1
i
, ..., c
.(i)
i
)
From (61), the manager expropriates a proportion (1 ÷ /
)
i
)r of the maximum output at her level
if she is not caught. The manager incurs personal costs from expropriation that are a proportion
i(·(i))(1 ÷/
)
i
)r of her certainty equivalent payo¤ under the contract assuming that she does not
expropriate, that is,
Personal Costs = i(·(i))(1÷/
)
i
)r
_
1
)
i1
_
a
)
i
+/
)
i
(1(i) ÷1(i ÷1))
_
÷
1
2
¸
2
(/
)
i
)
2
_
(o
i1
)
2
+
)2
_
_
(62)
The manager’s personal costs, therefore, increase with the proportion of the output that she ex-
propriates. The factor i(·(i)) ¸ (0, 1] in (62) allows for the manager’s personal costs to depend
on the number of managers at her level.
By (61) and (62), the manager’s certainty equivalent payo¤ net of her personal costs is
¹
act
(r) = C + (1 ÷/
)
i
)r(c
1
i
, ..., c
.(i)
i
) ÷i(·(i))(1 ÷/
)
i
)rC, (63)
where
C = a
)
i
+/
)
i
(1(i) ÷1(i ÷1)) ÷
1
2
¸
2
(/
)
i
)
2
_
(o
i1
)
2
+
)2
_
is the manager’s certainty equivalent payo¤ if she does not expropriate. By (63),
C =
1
i(·(i))
(c
1
i
, ..., c
.(i)
i
) = ¸(·(i))(c
1
i
, ..., c
.(i)
i
) (64)
is the minimum certainty equivalent payo¤ that the manager must be guaranteed by the contract
to make her indi¤erent to expropriation. It is reasonable to assume i(.) is decreasing, which
implies that the manager’s proportional personal costs of expropriation decline with the number
of managers at her level because monitoring the managers becomes more di¢cult. By (64), ¸(.) is
increasing, and j(·(i)) = ·(i)¸(·(i)) is also increasing (see Assumption 2).
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