Rational Inattention and Organizational Focus
Wouter Dessein
Andrea Galeotti
Tano Santos
Columbia University
University of Essex
Columbia University
June 4, 2014
Abstract
We examine the allocation of attention in team production. Coordinated adaptation
to task-speci…c shocks requires that agents pay attention to each other, but attention
is scarce. At the optimum, all attention is evenly allocated to a select number of
“leaders”. The organization excels in a small number of focal tasks at the expense of
all others. Our results shed light on the importance of leadership, strategy and “core
competencies,”as well as new trends in organization design. We also derive implications
for the optimal size of organizations. Surprisingly, improvements in communication
technology may result in smaller but more adaptive organizations.
We thank Ricardo Alonso, Arthur Campbell, Roland Benabou, Luis Garicano, Navin Kartik, Jin Li,
and Tim Van Zandt for helpful comments, as well as seminar and conference audiences at Brown, Columbia
GSB, Essex, EUI, Harvard Business School, Northwestern Kellogg, MIT Sloan, Paris School of Economics,
Pompeu Fabre, Princeton, Queens, Rochester, Toronto, Tsinghua University, UCL, ULB, Warwick, the
CEPR Conference on Incentive, Management and Organizations (Milan), ESEM Malaga, ESSET Gerzensee,
and NYU Stern IO Day. Andrea Galeotti is grateful to the European Research Council for support through
ERC-starting grant (award no. 283454) and to The Leverhulme Trust for support through the Philip
Leverhulme Prize.
Economics, according to Lionel Robbins’s famous dictum, is “the science which studies
human behavior as a relationship between ends and scarce means which have alternative
uses.”As emphasized by Herbert Simon, attention may well be the ultimate scarce resource
in the economy: “a wealth of information creates a poverty of attention and a need to allocate
that attention e¢ ciently among the overabundance of information sources,” (Simon 1971,
pp. 40–41). This paper studies the optimal allocation of (scarce) attention in organizations.
Despite a fully symmetric environment and decreasing marginal returns to attention, we
show that when attention is scarce, organizational focus and leadership naturally arise as
a response to organizational trade-o¤s between coordination and adaptation. As such, our
results provide micro-foundations for a central idea in the management literature that …rms
should focus on a limited set of “core competencies”(Prahalad and Hamel 1990)1 and …rms
that aim to be “all things to all people,” will be “caught in the middle” and fail (Porter
1985;1996). We develop comparative statics as to when organizational focus is more or less
important, and shed light on new trends in organizational design including more shared
leadership, an increase in horizontal communication linkages, and a reduction in the overall
size of the organization (downsizing and downscoping).2
Our starting point is that organizations have a limited communication capacity and must
use this resource judiciously in order to coordinate their production processes.3 We propose
a model of team production in which a number of complementary tasks, such as engineering,
purchasing, manufacturing, marketing and selling must be implemented in a coordinated
fashion. Alternatively, di¤erent tasks may correspond to di¤erent products or locations (e.g.
a multi-product …rm which exploits economies of scope). Each agent is in charge of one
task and must adapt this task to local information or “shocks”. Such adaptation, however,
results in coordination failures with other tasks unless agents communicate e¤ectively. Organizational focus, then, takes the form of allocating more scarce organizational attention to
one task –or one agent –than another task.
1
According to Prahalad and Hamel, such core competences represent the “collective learning in the
organization, especially how to coordinate diverse production skills” (p. 82)
2
See Whittington et al. (1999), Guadalupe et al. (2012), and Roberts and Saloner (2013).
3
As Arrow (1974, p.53) stated “The information has to be coordinated if it is to be of any use to the
organization. More formally stated, communication channels have to be created within the organization.”
1
If attention is abundant there is no need for organizational focus in our model. All tasks
can then be very adaptive and well coordinated, and it is optimal to distribute attention
evenly. In contrast, if attention is scarce and coordination is important, it is optimal to treat
tasks asymmetrically. A few agents should then be allowed to be very responsive to their local
information, and all attention should be focused on those agents and their tasks in order
to avoid coordination failures. In contrast, coordination with all other tasks is achieved
by limiting their adaptiveness. All tasks are then well coordinated, but only a few tasks
are adaptive.4 Leadership, where a few agents monopolize scarce attention and take most
of the initiative, arises endogenously. In contrast, a “balanced”organization that spreads
attention evenly across tasks is “stuck in the middle”: tasks are neither very adaptive nor
are they very well coordinated. Importantly, the above result obtains even when all tasks
and agents are ex ante identical and there are decreasing marginal returns to attention: as
in most standard models of information acquisition, we assume that every additional unit
of attention allocated to an agent reduces the uncertainty about this agent’s actions by an
ever smaller amount.
The mechanism underlying the above result is a fundamental complementarity between
the attention devoted to an agent, and the initiative taken by this agent. Agents take
initiative by adapting their task to local information. But agents who are ignored by others
are forced to also largely ignore their own private information, as taking initiative would
then result in substantial coordination failures. Conversely, it is a waste of resources to
allocate scarce attention to an agent who takes little or no initiative. Following the same
logic, the more attention an agent receives, the more initiative this agent can take, and the
more important it is to devote scarce attention to this agent in order to ensure coordination.
Because of the above complementarities, members in an organization either communicate
intensively about a particular task, or they ignore it. As we show, an optimal organization
then equally divides all attention among a select number of tasks or agents, which we refer
4
This correspond to the two general ways in which organizations can be coordinated according to March
and Simon (1958): “The type of coordination used in the organization is a function of the extent to which the
situation is standardized. (...) We may label coordination based on pre-established schedules coordination
by plan, and coordination that involves transmission of new information coordination by feedback. The more
stable and predictable the situation, the greater the reliance on coordination by plan.” (p182)
2
to as ‘leaders’. The scarcer is attention, the smaller is the number of tasks on which the
organization focuses. Interestingly, those chosen tasks then often receive much more attention
– and are much more adaptive – than if attention were to be abundant.
Our results support the notion that firms need to have a clear strategy – they must choose
a set of performance dimensions or tasks in the value-chain to focus on.5 By the same token,
we provide insights as to how focused firms should be. Over the last decades, there have
been enormous technological innovations in communication and coordination technologies
(E-mail, wireless communication and computing, intra networks, etc.). Our results suggest
that having a narrow focus becomes less important as information technology relaxes the
communication and attention constraints of organizations. The resulting organization is
often less well coordinated, less cohesive, but has a broader focus – it pays attention to the
task-specific information of a larger number of agents. This is consistent with new trends in
organizational design towards more network-like organizations where communication flows
are horizontal rather than vertical, and decision-making and influence is broadly shared
in the organization. Such novel organizations have been documented in both case studies
(for example “Proctor & Gamble Organization 2005,” HBS case 9-707-519)6 and large scale
empirical studies (Whittington et al. 1999, Guadalupe et al. 2012).7 Conversely, in fastmoving environments where speedy decisions are important – a competitive market place
where reacting quickly to a competitor’s move is of the essence, or a platoon in the field of
battle – there is often little time for extensive communication.8 Our model predicts that
5
6
See Van den Steen (2013) for a different view and formalization of “what is strategy”.
In this case study, Piskorski and Spadini document how P&G has moved towards a novel organizational
structure in which a separate product organization (responsible for global marketing and product development), a sales organization (responsible for delivery and customization to local markets), and a business
services organization are interdependent units who are giving equal weight in decision-making processes, and
achieve coordination through social networks and horizontal communication, rather than vertical authority
relationships. In the past, geographically organized sales organizations dominated P&G, slowing down the
development and roll-out of new products.
7
Guadalupe et al. document how, in recent decades, C-level executive teams in Fortune 500 firms have
almost doubled in size, mainly because of the inclusion of more functional managers.
8
As argued by Roberts and Saloner (2013), increased competition, through globalization of markets and
industries, “requires firms to be able to change more quickly and respond faster to market developments.”
3
leadership and organizational focus is more important in such environments compared to
settings where extensive communication is feasible prior to taking action.
In our basic model, the size or “scope” of the organization is fixed. In multi-product
firms, however, different tasks may correspond to different types of products or services,
and it is natural to think of the number of tasks as being endogenous. In Section 4, we
therefore endogenize the number of tasks by introducing economies of scale or scope: certain
fixed costs can be shared among tasks (e.g. production facilities or a distribution network),
yielding benefits to size. The size of organizations, however, is limited by the need for coordination and limited organizational attention. When the number of tasks is endogenous, we
uncover an important trade-off between organizational size and organizational focus. As we
show, smaller organizations have more leaders and, hence, the information of more agents
is reflected in decision-making. As an organization grows larger, leadership becomes more
concentrated as there is more need for coordination. This is consistent with the experience of many entrepreneurial firms, whose culture of joint-decision-making and open lateral
communication often disappears as they grow larger and more hierarchical.9
Two empirically relevant drivers of organizational size are the volatility of the environment and changes in information and communication technologies. Consistent with recent
trends in “de-scoping”(Whittington et al. 1999), we show that the optimal scope of organizations decreases as the environment becomes more volatile and adaptation becomes more
important.10 Intuitively, by reducing the number of tasks that it undertakes, the organization
reduces its coordination needs, hence allowing for better adaptation. At the same time, the
number of tasks that receive attention increases. Hence, as the environment becomes more
volatile, there is a move from large, focussed organizations that maximize scope economies
to smaller, but more adaptive and balanced organizations. Improvements in information
(p822)
9
In a classic management article, Greiner (1972) discusses how young, creative organizations often face
a crisis of leadership as they grow bigger. According to Greiner, an initial phase of creativity must give way
to a phase of direction, where companies that continue to grow, do so under “able, directive leadership”.
10
As noted by Siggelkow and Rivkin (2005), ”rapid technological change, deregulation, and globalization
have intensified competition and increased the turbulence that managers face, forcing them to adopt new,
more responsive organizational forms.” (p101) See also footnote 8.
4
technology might be conjectured to always increase the size of organizations, as they allow
for better coordination. Interestingly, we show that information technology has a decidedly
ambiguous impact on firm scope. Intuitively, information technology makes it optimal for
organizations to shift towards a strategy that emphasizes adaptation to its environment, but
smaller firms with shared leadership are better configured to do so. Hence, while for low
levels of information technology, large, non-adaptive firms exploiting economies of scale are
optimal, for intermediate levels of information technology, smaller, more flexible firms are
often preferred.
Outline. After reviewing the related literature in Section 1, we describe our model in Section
2. Section 3 provides a characterization of the optimal allocation of attention. Section 4
endogenizes the number of tasks, with larger organizations exploiting economies of scope
but facing more daunting coordination problems. We conclude in Section 5 by discussing
the implications of our model for skill-heterogeneity and the entrenchment of leadership and
functional cultures in organizations, as well as some missing elements such as power struggles
and conflicts of interests. Proofs of Propositions, as well as model extensions, are relegated
to the Appendix.
1
Literature Review
Our paper is part of a larger literature on team theory (Marschak and Radner 1972, Radner
1993), which studies games where agents share the same objective, but have asymmetric
information. As Garicano and Van Zandt (2013) emphasize, team theory is a natural framework in which to analyze the questions raised in this paper, as these models study interrelated decision-making when information is dispersed and there are limits to communication,
a canonical organizational problem.11
Our paper further belongs to a recent literature in organizational economics, most prominently Dessein and Santos (2006) (DS hereafter), Alonso et al. (2008), Rantakari (2008),
and Bolton, Brunnermeir and Veldkamp (2013), that emphasizes coordination adaptation
11
See, e.g., Cremer (1980, 1993), Geanakoplos and Milgrom (1991), Prat (2002), Wernerfelt (2004), Hart
and Moore (2005), Cremer et al. (2007) and Alonso et al. (2013).
5
trade-offs as a key mechanism determining organizational form. Following Marshak and Radner (1972), all these papers study how organizations are optimally designed to coordinate
production and assume a similar quadratic pay-off structure as our paper.
Closest to us is DS, which studies optimal task specialization in organizations, but restricts
communication flows to be symmetric; instead the present paper takes task-specialization
as given and endogenizes communication patterns.12 Calvo-Armagenol, de Marti and Prat
(2011) also endogenizes communication patterns in a framework similar to that of DS. Their
focus, however, is on how asymmetries in pay-off externalities between pairs of agents result
in asymmetric communication flows and differential influence for agents.13
Alonso et al. (2008) and Rantakari (2008) depart from the team theory formulation
in DS by introducing incentive conflicts and by modelling communication as cheap talk.14
Cheap talk communication is not suitable to study the allocation of attention, however, as
the quality of communication only depends on incentive conflicts between agents, and not
on time or information-processing constraints. The focus of these papers is on the optimal
allocation of decision-making authority whereas the present paper is concerned with the
optimal allocation of organizational attention.15
A number of other papers study the optimal characteristics of leaders in adaptationcoordination models where economic actors must respond to a common “global” shock, but
also care about choosing similar actions as other agents in the organization or economy.16
12
Autor (2009) and Bolton and Dewatripont (2013) offer textbook presentations of the DS model as well
as a description of its main characteristics.
13
In a symmetric set-up, there are no asymmetric communication patterns in Calvo-Armagnol et al. : each
agent is equally influential and there are no leaders. In contrast, we show how leadership and asymmetric
information flows arise naturally in symmetric settings.The main difference is that Calvo-Armagenal et al.
posit a communication technology with strong decreasing marginal returns, always enough to overwhelm
the convexities induced by the coordination-adaptation trade-offs. Other differences are that agents are
self-interested and invest in both active and passive communication.
14
See also Hagenback and Koessler 2010, and Galeotti et al. 2013, which study strategic communication
in networks.
15
Also Dewatripont (2006) introduces incentives in DS by modeling communication as a moral hazard in
teams problem, drawing upon Dewatripont and Tirole (2005). Again, implications for the optimal allocation
of decision-making authority are derived.
16
Our model differs from such beauty-contest models (Morris and Shin 2002, Hellwig and Veldkamp 2009)
6
Closest to us, Bolton, Brunnermeir and Veldkamp (2013) highlight the benefits of leader
resoluteness in achieving coordination.17 Similarly, Van den Steen (2005) shows how having
leaders with strong beliefs may be desirable as they give direction to the firm by affecting
the employee’s choice of project. The present paper does not speak to the characteristics of
successful leaders. Instead, we show how and when leadership arises endogenously in a team
of ex ante identical agents.
Finally, the arguments for organizational focus introduced in the present paper are reminiscent of at least two other literatures in organizational economics. In multitask incentive
theory (Holmstrom and Milgrom, 1991 and 1994), a narrow task-assignment may allow a
principal to provide higher-powered incentives to an agent. In multitask career concerns
models, Dewatripont et al. (1999) show how incentives are impaired by an agent pursuing
multiple objectives. The key insight in this literature is that it is easier to provide incentives to specialized agents. The above theories thus offer rationales for specialization at
the individual level but are silent on the issue of organizational focus, which is the topic of
this paper. More related to this paper, the literature on ‘narrow business strategies’ and
‘vision’ (Rotemberg and Saloner 1994, 2000) has argued that the commitment by a principal
or leader to select a certain type of projects provides strong incentives for agents to exert
effort related to such projects. In the same vein, Rantakari (2013b) shows how a principal
optimally favors one agent over another (competing) agent in order to improve both idea
generation and strategic communication. As in the multitask models above, ‘focus’ is thus
again a tool to improve incentives.
2
The model
We posit a team-theoretic model, based on Dessein and Santos (2006), Alonso et al. (2008)
and Rantakari (2008), in which production requires the combination of n tasks, each carried
out by a different agent. The implementation of a task is informed by the realization of a
in that agents must adapt their actions to local shocks, an assumption which is more suitable to study
information flows within organizations.
17
See also Dewan and Myatt (2008) and Morris and Shin (2010), who show how the ability of a leader to
convey her information clearly to followers is often more important than the precision of her information.
7
task-specific shock, only observed by the agent in charge of that task. Organizational tradeoffs arise because agents need to adapt to this privately observed shock while maintaining
coordination across different tasks. Communication allows agents to partially share their
local information with other members of the organization, but requires the use of scarce
organizational resources (e.g. time, attention). The paper studies the optimal allocation of
scarce communication capacity or ‘attention’.
2.1
Production
Production involves the implementation of n tasks, each performed by one agent i ∈ N =
{1, 2, ..., n}. The profits of the organization depend on (i) how well each task is adapted
to its organizational environment and (ii) how well each task is coordinated with the other
tasks. For this purpose, agent i must take a primary action, qii ∈ R, and a coordinating
action, qij ∈ R, for each task j ∈ N \ {i}.
In particular, agent i observes a piece of information θi , a shock with variance σθ2 and
mean 0, which is relevant for the proper implementation of the assigned task. We refer to θi
as the local information of agent i and assume its realization is independent across agents.
Let qi = [qi1 , qi2 ..., qii , ..., qin ] be the actions taken by agent i. Given a particular realization
of the local information, θ = [θ1 , θ2 , ..., θn ], and a choice of actions, q = [q1 , q2 , ..., qn ], the
realized profit of the organization is:
π (q|θ) =
X
πi (q|θ) ,
(1)
i∈N
where, for each i ∈ N ,


πi (q|θ) = − (qii − θi )2 + β
X
(qji − qii )2 
(2)
j∈N \{i}
The performance of task i, as measured by πi , then depends on how well adapted agent i0 s
primary action qii is to task i0 s local information θi and how well coordinated agent j 0 s
coordinating action qji is to agent i0 s primary action qii . The parameter β > 0 measures
task-interdependence and the importance of coordination relative to adaptation.
Communication frictions and adaptation-coordination trade-offs: If agent i could perfectly
communicate his primary action to agent j, there is no trade-off between adaptation and
8
coordination: agent i then optimally sets qii = θi and agent j ensures coordination by setting
qji = qii . In the presence of communication frictions, however, adaptation-coordination tradeoffs arise. Indeed, assume agent j receives no information about qii and therefore sets
qji = E(θi ) = 0. Then the more agent i adapts qii to θi the larger are the coordination
costs with task j. By ignoring his local information θi and setting qii = 0, however, agent i
can ensure perfect coordination with task j. Thus expression (1) captures the notion, going
back to at least March and Simon (1958), that it is adaptation to unpredictable contingencies,
combined with communication frictions, that create coordination problems.18
An alternative specification: In Appendix B we study an alternative specification which
imposes qii = qij = qi so that each agent chooses a single action, in line with Alonso et al.
(2008) and Rantakari (2008). In this case, even with perfect communication, agents face a
trade-off between adapting their action to local information and maintaining coordination
with the actions of other agents. While this makes the analysis less transparent, as each
action now optimally depends on all local shocks, Appendix B shows that qualitatively
identical results obtain.
2.2
Attention as a scarce resource and communication frictions
To improve coordination between tasks, agents can communicate their local information.19
Such communication will often be imperfect, but the more time or attention an agent j
devotes to understanding agent i0 s primary action qii the more likely it is he understands
which corresponding coordinating action qji is required to ensure coordination.20
18
“(D)ifficulties arise only if program execution rests on contingencies that cannot be predicted perfectly
in advance. In this case, coordinating activity is required (...) to provide information to each subprogram
unit about the relevant activities of the others.” (p. 180). March and Simon also emphasize the role of
‘complementary actions’ in achieving coordination: “To the extent that contingencies arise, not anticipated
in the schedule, coordination requires communication to give notice of deviations from planned or predicted
conditions, or to give instructions for changes in activity to adjust to these deviations.” (p182).
19
Whether what is communicated is the realization of the local information or the choice of the primary
action is not key because, as we shall show, there is a one-to-one correspondence between them.
20
Note that in our model, agents have no incentive to hide information, as in Campbell et al. (2013)
and Blanes i Vidal and Moller (2013), or exaggerate information, as in Alonso et al. (2008) and Rantakari
9
We assume that communication occurs in public meetings so that each agent j 6= i
devotes the same amount of time or attention to any given agent i. An attention allocation
t = [t1 , t2 , ..., tn ], where ti ≥ 0 for all i ∈ N , then represents the “air-time” or “attention”
any agent i receives. Alternatively, ti is the meeting time devoted to discussing task ti ,
regardless of whom speaks. Organizational attention is scarce, which is captured by the
following attention constraint:
X
ti ≤ τ,
(3)
i∈N
where τ < ∞. For example, τ could be the length of time agents spend in meetings as
opposed to production. The assumption that τ is exogenous simplifies the analysis but it
is not crucial for the results.21 In Appendix E we consider alternative models of communication such as bilateral communication (e.g. agent 3 may devote more attention to agent
1 than agent 2 does) and individual attention constraints (so that the time agents 3 and 4
communicate does not affect the attention constraints of agents 1 and 2). We show that they
result in attention allocations that are equivalent to the ones that are optimal under public
communication.
We adopt the same binary communication technology as in Dessein and Santos (2006).
Each agent j receives a message mji concerning θi . With a probability 1 − r(ti ), mji is pure
noise and the choice of the relevant coordinating action qji cannot be made contingent on it.
In contrast, with a probability r(ti ) agent j perfectly understands agent i, that is mji = θi .
We posit the following attention function:
1 − r(ti ) = (1 − p)ti ,
(4)
where ti can be interpreted as the number of signals an agent j 6= i receives about θi and
where each of those signals correctly reveal θi with an independent probability p. If ti is
continuous rather than discrete, communication technology (4) can be viewed as a Poisson
process where λ = ln(1 − p)−1 is the constant hazard rate that the receiver correctly learns
(2008). As is common in the team-theory literature, we focus on non-strategic (technological) limits to
communication.
21
We have derived a number of results for when τ is endogenous which show that our findings are robust to
P
an endogenous communication capacity. Specifically, we allow the designer to choose t at a cost g( i∈N ti ),
where g(.) is a convex function. The analysis is presented in Appendix F.
10
the local information of the sender and, hence, 1 − r(ti ) = e−λti . We further assume that
agent i never knows whether his communication with agent j 6= i was successful or not.
Thus, when deciding on a primary action qii he takes into account that with a probability
r(ti ) agent j will be informed about his choice.
Note that the value of communicating an additional signal about θi – in term of increasing
the probability r(ti ) of successful communication – is decreasing in the number of signals ti
that have already been communicated. Indeed, the more signals that have been sent about
θi the more likely it is that an agent j 6= i has already learned the realization of θi . A natural
feature of communication technology (4) is thus that it implies decreasing marginal returns
to communicating about a particular task-specific shock.
An alternative specification: From (4), agent j 0 s expected residual variance about θi equals:
RV (ti ) ≡ E [V ar(θi |ti signals about θi )] = σθ2 (1 − r(ti )) = σθ2 e−λti
(5)
As we formally discuss in Appendix D, an identical residual variance as in (5) obtains if
messages mji and local information θi are normally distributed and the attention constraint
(3) is modeled as a constraint on the total reduction in entropy, as in Information Theory
(Cover and Thomas 1991) and the literature on Rational Inattention (Sims 2003). Since
expected profits for a given attention allocation t can be written as a function of the residual
variance, identical results obtain with this alternative communication technology.
2.3
Timing of the game
The timing of our model goes as follows:
1. Organizational design: Optimal attention allocation t is chosen.
2. Local information: Agent i ∈ N observes local information θi .
3. Communication: Agents allocate attention according to t.
4. Adaptation and Coordination: Agent i ∈ N chooses primary action qii and complementary actions qij , for all j ∈ N \ {i}.
11
We model the resulting agents’ action in stage four as the equilibrium outcome of a
Bayesian Game with common payoffs.22 Equivalently, one can think of the designer specifying
optimal decision rules in the organizational design stage (see Garicano and Van Zandt 2013).
The type of agent i, denoted by Ii , describes the realization of his local information θi and
the information mij he has received about the local shock of each agent j 6= i.
3
Organizational Focus
We now first provide a characterization of the optimal allocation of attention when there are
two agents and two tasks. This is enough to convey many intuitions of the model. Section
3.2 then generalizes the result for the case of n tasks.
3.1
Organizational focus with two agents
Without loss of generality, we focus on characterizing equilibria in linear strategies, where
qii = aii θi
and
qji = E [qii |Ij ] = aii E [θj |Ii ] .
(6)
It is further useful to think of the organization designer choosing both decision rules qii = aii θi
and communication qualities (r1 , r2 ), with 1 − ri = e−λti , rather than simply an allocation
of attention (t1 , t2 ). The attention constraint t1 + t2 ≤ τ is then equivalent to
(1 − r1 ) (1 − r2 ) ≥ e−λτ .
(7)
The communication constraint (7) will be binding at the optimum and characterizes by
how much the communication quality about task i must be reduced in order to improve
communication about task j 6= i. Note that the sum of communication qualities r1 + r2 is
maximized by setting r1 = r2 . The organization design problem is then equivalent to the
organization designer choosing (r1 , r2 , a11 , a22 ) in order to maximize the expected profits
E [π (q|θ)] = −(1 − a11 )2 σθ2 − (1 − a22 )2 σθ2 − β(1 − r1 )a211 σθ2 − β(1 − r2 )a222 σθ2 ,
22
(8)
More generally, for our analysis, it does not matter if agent i maximizes firm peformance π (q|θ) (common
pay-offs) as in Dessein and Santos (2006) or a mix of firm performance and task performance, that is
(1 − ω)π (q|θ) + ωπi (q|θ) with ω ∈ (0, 1) , as in Alonso et al. (2008) and Rantakari (2008).
12
subject to constraint (7).
Inspecting (8), it is immediate that aii and ri are complementary choices: the larger is
aii , the larger should be ri and vice versa. Intuitively, the more adaptive is an agent’s task
(the larger is aii ), the more important it is to communicate effectively regarding this task
in order to ensure coordination. In return, when communication about a task is better, it
is optimal for the agent-in-charge to become more adaptive to task-specific shocks, as the
other agent is more likely to take the appropriate coordinating action. Indeed, optimizing
a11 and a22 given r1 and r2 yields
a11 =
1
1 + β(1 − r1 )
and
a22 =
1
.
1 + β(1 − r2 )
(9)
Hence, the larger is ri , the more adaptive is task i to its local shock θi . In the absence
of decreasing marginal returns to attention, the above complementarity creates a powerful
force towards organizational focus where all attention is devoted to one task.
Substituting (9) and (7) into (8), and taking the derivative of expected profits with
respect to r1 yields
where a11 and a22
dE [π(q|θ)]
dr2
2 2
2 2
= βa11 σθ + βa22 σθ
,
(10)
dr1
dr1
are given by (9). As we discuss next, for r1 > r2 , the sign of (10) depends
on how important is coordination (β) and how scarce is attention (τ ).
As a benchmark, consider the case where dr2 /dr1 = −1, so that any increase in the
effectiveness of communication about task 1 is offset by an equal decrease in communication
quality about task 2. This would have been the case, for example, if the communication
constraint were given by r1 + r2 ≤ τ rather than (7). Since a11 > a22 whenever r1 > r2 ,
we have that dE [π (θ)] /dr1 > 0 whenever r1 > r2 . It follows that the organization designer
chooses either (r1 , r2 ) = (r(τ ), 0) or (r1 , r2 ) = (0, r(τ )) as long as r(τ ) < 1. In other words,
with constant returns to attention (as reflected in dr2 /dr1 = −1) , it is optimal to only
communicate about one task (say task 1) and let this task be very adaptive (a11 large).
Coordination about the other task (say task 2) about which there is no communication
(r2 = 0), is then achieved by restricting its adaptiveness (setting a22 low).
Given constraint (7), however, we have that for r1 > r2 :
dr2
1 − r2
=−
< −1.
dr1
1 − r1
13
(11)
Whenever r1 > r2 , any increase in the effectiveness of communication about task 1 is then
offset by a more than equal decrease in the communication quality about task 2. Constraint
(7) therefore reflects decreasing marginal returns to attention, providing a counter-veiling
force against organizational focus.
We distinguish between high and low task-interdependence. When task-interdependence
is low, that is β < 1, one can verify that (10) is negative whenever r1 > r2 . The organization
designer then optimally sets r1 = r2 and a11 = a22 .
Proposition 1 If β < 1 then organizational balance is optimal: (t∗1 , t∗2 ) = ( τ2 , τ2 ).
Proposition 1 states that if coordination is not much of a concern, then it is always
optimal to split attention evenly among both task. Intuitively, given the limited need for
coordination, the adaptiveness of any given task is only marginally affected by improvements
in communication quality. The fundamental complementarity at the heart of our model –
where a higher communication quality results in more adaptive tasks, making it optimal to
further improve communication – is then relatively weak. In contrast, decreasing marginal
return to attention (as reflected in dr2 /dr1 ) are independent of β. For β sufficiently small,
the organization designer then optimally sets r1 = r2 .
Whenever β > 1, however, complementarities induced by the need for coordination may
result in organizational focus. Consider the maximal communication quality r1 = r(τ ) that
can be achieved by allocating all attention to task 1. From (10), even when there is no
communication about task 2 and r1 = r(τ ), we have that dE(π)/dr1 > 0 provided r(τ ) is
small enough. Hence, when attention is sufficiently scarce (τ small) organizational focus is
locally optimal. In contrast, when attention is not scarce (τ is large), then (r1 , r2 ) = (r(τ ), 0)
cannot be optimal as −dr2 /dr1 in (10) grows without bound. The following proposition
characterizes the optimal allocation of attention for β > 1 :
Proposition 2 Suppose β > 1. There exists > (β) such that:
(i) Organizational focus is optimal, t∗1 ∈ {0, τ } and t∗2 = τ − t∗1 , if and only if τ ≤ > (β)
(ii) Organizational balance is optimal, (t∗1 , t∗2 ) = ( τ2 , τ2 ), if τ > > (β)
(iii) >(β) is increasing in the importance of coordination, β.
14
Figure 1 summarizes Proposition 1 and Proposition 2. Organizations which are “ somewhat” focused are never optimal. Indeed, if full focus is not optimal, the organization divides
its attention equally among both tasks. Intuitively, given the complementarities between the
adaptiveness of a task and the attention devoted to a task, the organization either completely
ignores a task, or it devotes a substantial amount of attention to it. At the threshold > (β) ,
the organization makes this shift from no attention to one task, to an equal amount of
attention to both tasks.
Ex-ante and ex-post coordination. A useful way to understand the above results is through
the notion that there are two ways to maintain coordination in an organization. One way is
for the organization to devote substantial attention to a task (setting ri high). The agent in
charge of this task can then be very responsive to his local information as the other agents in
the organization will likely be aware of his actions, by means of communication, and take the
appropriate coordinating actions. In Dessein and Santos (2006), this was referred to as expost coordination. An alternative way is for the agent to simply ignore his private information
and always implement his task in the same manner (setting aii low). Other agents can then
maintain coordination with this task without having to devote any attention to it. This can
be seen as ex-ante coordination. The notions of ex-ante and ex-post coordination correspond
to the two general ways in which organizations can be coordinated according to March and
Simon (1958): coordination by plan, which requires that tasks are executed in a more or
less standardized way (e.g. following standard operating procedures), and coordination by
feedback, which involves the transmission of new information. As March and Simon note ”it
is possible to reduce the volume of communication required from day-to-day by substituting
coordination by plan for coordination by feedback” (p.183).
While in Dessein and Santos (2006) all tasks were treated symmetrically by assumption,
the insight of Proposition 2 is that when attention is scarce (that is τ < > (β)), it is optimal
to coordinate ex-ante on one of the tasks and coordinate ex-post on the other task. The
first task is then very rigid and insensitive to its local information, so that the organization
can afford to ignore this task and fully allocate its attention to the second task, allowing
it to be flexible and adaptive. Despite a limited attention capacity, both tasks are then
well coordinated, but only one task is very sensitive to its environment. In contrast, when
15
Figure 1: Focused and balanced organizations in the two-agent case.
τ
6
> (β)
Balanced organizations
t = [ τ2 , τ2 ]
Focused organizations
t = [τ, 0] or t = [0, τ ]
-
β
βb
attention is plentiful, it is optimal for both tasks to be very adaptive, as they both can be
coordinated ex-post through communication.23
Comparative statics with respect to communication technology. Proposition 2 yields an interesting comparative static result with respect to exogenous changes in the communication
capacity τ. Improvements in the communication technology (E-mail, wireless communication devices, intranets, ...) can be interpreted as an exogenous increase in τ . An implication
of Proposition 2, therefore, is that such technological improvements result in a shift from
focused organizations which are centered around one task and excel on that task at the
expense of others, towards more balanced organizations which aim to perform equally well
on all tasks, but excel in none.
The above insight also has implications for the importance of leadership in teams. At
23
An alternative explanation for rigid, bureaucratic decision-making is provided by the influence-cost
literature (Milgrom and Roberts 1988). See Powell (2013) for a model in which the organization designer
reduces the adaptiveness of decisions to new information in order to limit influence activities.
16
the threshold > (β) the organization changes from having a single agent who monopolizes
all information flows (the leader) to a structure with shared leadership. Hence, an increased
communication capacity may come at the expense of the original leader in an organization,
who may face a discrete loss of power and influence in the organization. As a result, his task
is less adapted to its environment and, typically, other tasks are less well coordinated with
it. From having a complete monopoly on attention in the organization, this leader now must
share it equally with the other agent engaged in team production.
As discussed in the introduction, the above predictions are consistent with new trends in
organizational design towards more network-like organizations where communication flows
are horizontal rather than vertical, and decision-making and influence is broadly shared
in the organization, as documented in both case studies (for example “Proctor & Gamble
Organization 2005,” HBS case 9-707-519) and large scale empirical studies (Whittington et
al. 1999, Guadalupe et al. 2012).24
Task asymmetries and strategic direction. In our model, tasks are ex ante symmetric and it
does not matter which task the organization focuses on. In reality, of course, tasks are likely
to differ from each other. The question, then, is not only how focused to be, but which tasks
to focus on. A first asymmetry concerns the expected magnitude of the shocks faced by
each task. If shock θi has a variance σi2 with σ12 > σ22 , then it is easy to show that whenever
τ ≤ > (β) , the organization optimally focusses on task 1, that is t = (τ, 0). Therefore, the
result that the organization is indifferent as to which task to focus on is knife-edge. As there
may be uncertainty about the variance of each task-specific shock, this creates a role for
management in setting the “strategy” of the firm by identifying which task the organization
should focus on. As we discuss in Section 5, in a dynamic version of our model, the focus of
an organization may be difficult to change over time because of complementary organization
design choices such as hiring decisions and investments in human capital. Firms which focus
on the “right” task may then outperform their competitors over time. Firms who refuse to
make a choice and split attention, however, are likely to be worst off.
A second asymmetry is one where one task imposes larger coordination costs (delays, low
product quality) should the other task not take the appropriate coordinating action. For
24
See also Roberts and Saloner (2013), Section 3.5.
17
example, in designing a car, important changes made to how the engine works, may have
important consequences for the remainder of the design. Should attention be focused on
those highly interdependent tasks? Appendix C extends the two-task model to allow tasks
to have different degrees of interdependence. We show that, perhaps counter-intuitively,
when some tasks are more interdependent than others and attention is relatively scarce, it
is optimal not to focus attention on highly interdependent tasks, but instead restrict their
adaptiveness.
3.2
Organizational focus with many agents
We now extend our analysis to allow for an arbitrary number of agents in the team. We
first show that the optimal organizational form is the `−leader organization, which features
a number ` ≤ n of equally adaptive agents (leaders) to whom the organization devotes
an equal amount of attention, whereas no attention is devoted to any agent who is not a
leader. We then provide comparative statics to highlight how the optimal organizational
form adapts to change in communication capacity, task-interdependence and the complexity
of the production process.
Definition: The `−leader organization. In an `−leader organization, the set of agents
can be partitioned in a set of leaders L (t) and followers F (t) such that
1. The number of leaders is ` ≤ n.
2. For each follower i ∈ F (t), ti = 0.
3. For each leader j ∈ L (t), tj = τ` .
An `−leader organization has the property that there is a number of agents `, which we
call leaders, to whom all agents (including other leaders) pay equal attention, and a second
class of agents to whom no other agent in the organization pays attention.
Proposition 3 The optimal organization is an `−leader organization with ` ∈ {1, 2, · · · , n}.
The proof of Proposition 3 follows from the observation that in an optimal organization,
all agents who receive some positive attention must receive the same attention. To see
18
the intuition behind this result, let i and j be two tasks and assume that ti > tj > 0, in
violation of the claim. In the case of two tasks, it was shown that either t∗1 ∈ {0, τ } , or
t∗1 = t∗2 = τ /2. Following the same logic, one can equally show that, keeping the attention
allocated to all other tasks k ∈
/ {i, j} fixed, profits can always be strictly increased by either
setting t0i = ti + tj and t0j = 0 or, alternatively, equalizing attention across tasks i and j,
that is setting ti = tj = (ti + tj )/2. As in the two tasks case, it is optimal to either allocate
a substantial amount of attention to any given task, allowing it to become very adaptive
and coordinate this task ex post, or force a task to largely ignore its local information and
coordinate this task with others ex ante, which does not require any attention.
When the communication network takes the form of an `−leader organization, the expected profit function can be written as:
E [π (q|θ)] = −` (1 − al )2 + (1 − rl )a2l β(n − 1) σθ2
− (n − `) (1 − af )2 + a2f β(n − 1) σθ2
(12)
where 1 − rl = e−λτ /` , and
al =
1
1 + (1 − rl )β(n − 1)
and
af =
1
1 + β(n − 1)
The optimal number of leaders, then, is given by
`∗ = argmax`∈{1,2,··· ,n} E [π (θ)] .
(13)
Armed with (13) we are able to offer a sharp characterization of the `−leader organization
as a function of the organization’s communication capacity τ and the task-interdependence
or coordination parameter β.
Proposition 4 There exists 0 < β (n) < ... < β (` + 1) < β (`) < ...< β (2) such that
1. `∗ is decreasing in β, i.e., `∗ = n if β < β (n) , `∗ = ` ∈ {2, · · · , n − 1} if β ∈
(β (` + 1) , β (`)), and `∗ = 1 if β > β (2)
2. For all ` ∈ {1, ..., n}, β (`) is increasing in τ and limτ →∞ β (`) = ∞.
19
Figure 2: Optimal number of leaders and adaptation as a function of β
Example: n = 20, σθ2 = 1, τ = 50/19, and β ∈ [0, 1]. Panel A: Optimal number of leaders, `∗ , as a function
of the importance of coordination β. Panel B: Leader adaptation Ωl ≡ al .
Panel B: Leader adaptation
1
18
0.95
16
0.9
14
0.85
12
0.8
Ωl
l*
Panel A: Optimal number of leaders
20
10
0.75
8
0.7
6
0.65
4
0.6
2
0.55
0
0
0.2
0.4
0.6
0.8
0.5
1
β
0
0.2
0.4
0.6
0.8
1
β
Figure 2 illustrates the results of Proposition 4 for a specific numerical example (n = 20,
σθ2 = 1 and τ = 50). Start with Panel A, which plots the optimal number of leaders `∗
as a function of the importance of coordination, β ∈ (0, 1). A balanced organization is
optimal when coordination is sufficiently unimportant. In this specific example, whenever
β < β (20) ≈ 0.06 the organization is fully balanced, that is, `∗ = n = 20. As coordination
becomes more important, the communication becomes more focused around fewer leaders.
Finally, when tasks are sufficiently interdependent, when β > β (1) ≈ 0.73, the organization
has a single leader, `∗ = 1.
Panel B of Figure 2 shows how al , the adaptiveness of each leader j ∈ L (t) to his
local shock, changes as tasks become more interdependent as measured by β. Interestingly,
leaders tend to be much more adaptive when coordination costs are higher, as they then
share influence with fewer other leaders. For example, when β = 0.8, there is only one
leader, but this leader is, roughly, 50% more adaptive to his local information then when
20
β = 0.2 and the number of leaders equals `∗ = 3.
Proposition 4 also shows how an exogenous change in attention capacity τ, increases the
number of leaders and makes the organization more balanced. Again, this implies that as
communication technology improves, organizations become less focused and leadership is
more broadly shared.
We conclude this section by providing a result on how the optimal number of leaders
depends on the complexity (or organizational size) of the production process, as captured
by the number n of complementary tasks. This result will also be useful when analyzing
endogenous organizational size (Section 4). Let `∗ (n) be the optimal number of leaders given
that the production requires the implementation of n tasks.
Proposition 5 Larger organizations are more focussed than smaller organizations: if `∗ (n+
1) < n + 1, then `∗ (n + 1) ≤ `∗ (n).
As an organization grows larger and its production becomes more complex, coordinationadaptation trade-offs become more pronounced, forcing the organization to become more
focused. Despite the organization having more members, fewer of them receive attention.
This result is consistent with the life cycle theory of organizations (Greiner 1972) according
to which small entrepreneurial firms, as they grow bigger, move from an initial phase of
creativity, where almost all employees are involved in decision-making to a phase of direction,
where “able, directive leadership” is installed, and where “[t]he new manager and his or her
key supervisors assume most of the responsibility for instituting direction.” Consistent with
our model, Greiner also describes how firms become less adaptive and responsive to their
environment as they grow larger.
4
Organizational Size
One interpretation of our model is that each task corresponds to a different type of product
or service that is produced or delivered by a multi-product firm. Under this interpretation,
it is natural for the organization designer to optimally chose the size of the organization.
By engaging in multiple tasks, firms can spread out fixed costs and realize scope economies
21
(Panzar and Willig, 1981). Doing so, however, increases coordination costs as now more tasks
need to be coordinated. We therefore modify our model to include fixed production costs,
such as production facilities or a distribution network. We investigate how the optimal size
(or scope) of an organization interacts with organizational focus, and how organizational size
and focus are jointly optimized in response to more volatile environments and improvements
in communication technology.25
4.1
A model of endogenous organizational size
We posit that the realized profit of an organization which performs n tasks (or produces n
products) equals
π(n) = nP − F + π(q|θ),
(14)
where P > 0 represents the revenues that can be obtained per task, F > 0 are the fixed
costs which are shared by all tasks/products, and the last term in (14) is the profits of the n
tasks model (expression 1). We further assume that regardless of the size of the organization,
there is a fixed communication capacity τ which can be spent communicating about the n
product-lines. Intuitively, as the organization grows, meetings become larger as more agents
need to be coordinated. But each agent can still only spend time τ in such a meeting, hence
the length of the public meeting remains τ .
As shown in Proposition 3, for a given organizational size n, the optimal organization is
a `−leader organization. If `∗ (n) is the optimal number of leaders given n tasks, then the
expected profits of an organization of size n are given by
E [π(n)] = nP − F − nσθ2 + `∗ (n)
+ (n − `∗ (n))
σθ2
1 + (n − 1)βe−2τ /`∗ (n)
σθ2
.
1 + (n − 1)β
We assume that organizational size is chosen to maximize profits per product-line:
1
n∗ = arg max E[π(n)].
n n
25
We endogenize organizational size, not firm size. See Friebel and Raith (2010) and Dessein (2014) (and
references therein) for models of multi-product firms with endogenous firm boundaries.
22
Our underlying assumption is that firms, whenever profitable, have the option to operate a
set of product lines independently as a separate organization.26
The following proposition shows the optimal organizational response to an increase in
the volatility of the environment σθ2 .
Proposition 6 The optimal organization size n∗ is decreasing in σθ2 . Furthermore, organizational balance, `∗ (n∗ )/n∗ , is increasing in σθ2 . If `∗ (n∗ ) < n∗ , the number of leaders is
increasing in σθ2 .
Management scholars have cited many reasons for the rise of new organizational forms,
but one line of explanation which is especially prominent is the “increased turbulence” that
managers face because of rapid technological changes, deregulation, and globalization (Rivkin
and Siggelkow, 2005; Roberts and Saloner, 2013).27 As σθ2 increases, so do the incentives
to adapt, which in turn bring coordination costs. By narrowing firm scope (reducing n∗ ),
organizations partially reduce these coordination costs, allowing for a better adaptation. Put
differently, organizations trade-off economics of scale and scope with adaptation to a changing
environment. Proposition 6 therefore reflects the common idea that smaller organizations
are more “nimble” and “flexible.”
Since for a given organizational size n, the number of leaders `∗ is independent of σθ2 , and,
from Propositions 5, `∗ weakly decreases with n, we obtain that a more volatile environment
not only results in smaller but also more balanced organizations.2829
4.2
Information technology and organizational size
Given that limited attention and coordination costs restrict organizational size in our model,
one may (naively) expect organizational size to be increasing in the communication capacity
26
Organizational boundaries are then based on who communicates with whom: Agents belong to the same
organization if they communicate with each other.
27
The other prominent line of explanation, information technology, is addressed in the next section.
28
Comparative statics with respect to an increase in the shareable fixed costs F are less interesting: n∗ is
increasing in F , `∗ (n∗ )/n∗ , is decreasing in F , and if `∗ (n∗ ) < n∗ , the number of leaders is decreasing in F.
29
Rantakari (2013a) obtains a related result in a different setting. He shows how firms operating in more
volatile environments decentralize decision-making and reduce task-interdependence, whereas in our model,
firms become more balanced and reduce firm scope.
23
τ, and to be decreasing in the interdependence of tasks, β. What truly restricts organizational
size, however, is the adaptiveness of the organization. In our model, there are no constraints
on organizational size as long as agents do not adapt tasks to local shocks. When communication capacity is limited, organizations may therefore pursue two distinctive organizational
strategies: (1) Adaptation: (Largely) give up on scope economies, and have a small, but
adaptive organization, or (2) Economies of Scope: (Largely) give up on adapting to local
shocks, and instead leverage economies of scale and scope.
When scope economies (F ) are large, but communication capacity (τ ) is very limited, organizations optimally choose to minimize average fixed costs at the expense of adaptation to
local shocks. Organization-wise, this strategy consists of having a large, rigid organization,
and focussing all attention on one or a few leaders. Large, non-adaptive organizations with
one or a few leaders are then optimal. As τ becomes larger, however, the organization may
gradually want to use the extra communication capacity to become more adaptive. Doing so
without incurring substantial coordination costs, however, requires reducing organizational
size, often substantially. At the same time, a smaller size allows the organization to pay
attention to a larger number of tasks or leaders. It is only when the communication capacity is sufficiently large that organizations can pursue both objectives, scale economies and
adaptation, in which case organizational size is increasing again in τ .
Figures 3 illustrates the above organizational strategies in response to changes in τ . For
simplicity, it is assumed that n∗ is constrained to n∗ ≤ n = 18. For τ very small, the
organization size is set at the maximum, n∗ = n. As communication capacity τ increases,
the organization is transformed from a “large, rigid bureaucracy” with eighteen tasks but
only one leader who monopolizes all attention into a “nimble, adaptive democracy” with
only six tasks who share the attention evenly. For larger values of τ, organizational size
slowly increases again with τ and attention remains evenly distributed.
To the extent improvements in information and communication technology (ICT) correspond to an increase in the communication capacity, our model predicts that improvements
in ICT may result in a shift from large inflexible organizations emphasizing economies of
scale and scope, towards smaller, more balanced organizations, which are focused on be-
24
Figure 3: Endogenous organizational scope and focus as a function of τ
Example: maximum number of tasks n = 18, σθ2 = 1, β = .25 and F = 3. Optimal organizational size, n∗
(continuous line), and of leaders, `∗ (dashed line), as a function of τ .
18
16
14
12
l* n*
10
8
6
4
2
0
0
1
2
3
4
5
τ
6
7
8
9
10
ing adaptive to external shocks and emphasize horizontal communication linkages.30 This
is consistent with recent trends in organization design, as described by Whittington et al.
(1999) and Roberts and Saloner (2013). According to our model, only organizations that are
already very adaptive, respond to ICT improvements by increasing organizational scope. Alternatively, observed trends toward de-sizing and de-scoping may have been a response to an
increased variability in the environment (Proposition 6), for example because of globalization
and increased competition (Rivkin and Siggelkow, 2005; Roberts and Saloner, 2013).
While we have emphasized non-monotone comparative statics with respect to τ, similar
intuitions apply when comparative statics with respect to β are considered. When tasks are
sufficiently interdependent (β large), organizations may give up on adaptation and maximize
scale and scope economies by increasing organizational size.
30
This prediction stands in contrast with those of obtained in recent team-theory models that model or-
ganizations as information-processing (Bolton and Dewatripont 1994) or problem-solving institutions (Garicano, 2000; Garicano and Rossi-Hansberg, 2006). While these papers also characterize optimal information
flows in organizations, improvements in communication technology unambiguously result in larger and more
centralized organizations.
25
5
Concluding remarks
In this paper, we have studied the optimal allocation of scarce attention in organizations that
face competing objectives of coordination, adaptation and, in the latter part of our paper,
economies of scale and scope. We believe the proposed framework sheds light not only on
the importance of organizational focus, but also on complex and often contradictory trends
in organizational design, often attributed to improvements in information technology and an
increasingly volatile business environment. To conclude, we expand on three elements which
are absent from our model: skill-heterogeneity among employees, conflicts of interests and
coordination through hierarchies.
Skill-heterogeneity, functional cultures and the permanence of leadership. In
order to isolate the role of leadership and focus in organizations, we considered a model in
which all tasks and agents are ex ante symmetric. Indeed, when attention is not scarce,
all tasks are treated symmetrically and the organization is a “team of leaders.” It is informational constraints, not technological constraints, that induce the need for leadership and
organizational focus.31 Therefore, which tasks the organization focusses on, or who becomes
a leader, is irrelevant in our model. The importance of organizational focus, however, does
have implications for what type of employees an organization may want to hire. It may
therefore cause asymmetries in the skill-level of employees. Higher-skilled employees may
understand better how to adapt to a changing environment. If higher-skilled employees command higher wages, organizations that are focused on, say, engineering, will then optimally
recruit higher-skilled engineers than organizations which are focussed on, say, marketing and
sales. If communication will mainly pertain to coordinating engineering initiatives, it may
further be optimal to hire employees with an engineering background even for marketing
positions in order to foster communication. Over time “functional cultures” may develop,
where certain parts of the organization become “backwaters,” whereas other parts dominate
not only decision-making but also the talent pool of the organization. Technology companies
are often reputed for their ”functional cultures”. Google, for example, proclaims “[it] is and
31
Appendix B shows how our model can accommodate technological trade-offs between adaptation and
coordination, and yield similar insights.
26
always will be an engineering company,” and almost all employees have advanced engineering
degrees even in non-engineering tasks. Walter Isaacson’s biography of Steve Jobs, on the
other hand, paints a picture of Apple as a company run by designers, where engineers were
very often ignored. To the extent that employee turn-over is limited and hiring is costly and
incremental, an initial focus on a particular set of tasks may therefore have long-term consequences. While our model is static, this suggests it may be neither feasible nor desirable
for an organization to “rotate” its focus or leadership.
Conflict, power struggles and organizational change. Incentives play no role in our
theory, but if agents care mainly about the performance of “their task” (as modeled, for
example, in Alonso et al. 2008), power struggles and conflict may arise as to whom becomes the focus of the organization. A dynamic version of our model where the allocation
of attention is decentralized, rather than optimally allocated by an organization designer,
may investigate how agents act strategically to become the center of attention. To the extent that leadership is an endogenous and self-enforcing phenomenon, organizations may get
stuck in inefficient equilibria. Indeed, agents pay attention to whomever they believe is very
adaptive, and agents are adaptive only when they are the the center of attention. Hence,
multiple leadership equilibria may exist. Leadership may further become entrenched, with
current leaders (agents who are currently the focus of the organization) being unwilling to
give up their central position in the network. Our model further suggests that an increase
in communication capacity can be “traumatic” for the existing leadership of the organization. As communication capacity increases, overall adaptiveness increases, but so can be the
number of leaders. Given that attention is shared equally among the leaders of the organization, this often implies that each individual leader receives less attention. This effect can
be particularly pronounced when there is a move away from a one-leader organization to a
setting with multiple leaders.
Coordination and Hierarchies. In our model, production is carried out by a team
of agents and decision-making is de facto decentralized. Informal leadership then arises
endogenously to ensure coordination. In practice, organizations often achieve coordination by
27
centralizing decisions and/or introducing layers of hierarchy. In the context of our model, the
organization could centralize production in the hands of a headquarter manager, who takes
all production decisions after communicating with agents. Centralized production has the
virtue of ensuring perfectly coordinated decisions. Centralized organizations, however, are
likely to be less adaptive to shocks as they must rely on (limited) communication with agents.
In ongoing work, we show that whenever attention is scarce, decentralized team production
is then preferred over centralized production. Hence, centralized production is unlikely to
be optimal in fast-moving environments where τ is small. As discussed above, empirical
evidence indicates that many organizations have gone through a process of delayering in
recent decades, suggesting they have come to rely less on hierarchies for coordination. To
shed light on this trend, it would be interesting to introduce multi-layered hierarchies in our
model, where such layers provide an alternative way of coordinating production.
28
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APPENDIX
Appendix A: Proofs of the propositions
Proof of Proposition 1 and 2. Substituting (9) into (8) we can write expected pro…ts as
2
X
:
(qj ) = n 2 +
1 + (1 r(ti ))
i=1;2
Let t1 = t and t2 =
(15)
t; we consider, without loss of generality, that t 2 [ =2; ]. Taking the derivative of
(15) with respect to t we obtain
e (
[1 + e
e t
[1 + e t ]2
@E [ (qj )]
=
@t
t)
(
:
t) ]2
(16)
After some plain algebra it follows that
@E [ (qj )]
> 0 () 1
@t
Note that if
Consider
then 1
2
2
1 then 1
e
> 0 for all
e
< 0:
e
0; hence, optimality implies that t = =2.
2
> 1; de…ne T( ) so that 1
2
e
T( )
= 0. Note that T( ) is increasing in . If
2
> T( ) then 1
< 0 and therefore optimality implies that t 2 f0; g. If
e
< T( )
> 0 and
therefore optimality implies that t = =2. This completes the proof of Proposition 1 and 2.
Proof of Proposition 3. Proposition 3 follows as a consequence of the following Lemma 7.
Lemma 7 In an optimal organization if ti > 0 and tj > 0 then ti = tj .
Proof of Lemma 7. Using the same method as in Section 3.1, we obtain that expected pro…ts equal
E [ (qj )] =
n
X
(1
aii )2
2
+ (n
1)(1
r(ti ))a2ii
2
;
i=1
where aii = 1=(1 + (n
r(ti )): Substituting aii and re-arranging terms, we obtain
n h
i
X
2
E( (q; tj )) = n 2 +
1+ (n 1)(1 r(ti ) :
1)(1
i=1
Suppose, for a contradiction, that ti > tj > 0. Consider now two alternative organizations. One organization, denoted by t0 , is the same as organization t, but t0i = ti
and t0j = tj + . The second organization,
denoted by ^t, is the same as organization t, but t^i = ti + and t^j = tj
. These constructions are derived
for some small and positive . Since the three organizations only di¤er in the way attention is distributed
for task i and task j, each other task l 6= i; j performs equally across the three organizations. We can write
E [ (q; tj )]
= C+
2
E [ (q; t0 j )]
= C+
2
= C+
2
E
q; ^tj
1
1 + (n
1 + (n
1 + (n
1)e
1
1)e
1
1)e
34
ti
(ti
+
1
1 + (n
)
(ti + )
+
+
1)e
1 + (n
1 + (n
tj
1
1)e
1
1)e
;
(tj + )
(tj
)
;
:
Since t is optimal, we must have that
E [ (q; tj )] > E [ (q; t0 j )] :
This is is equivalent to
h
e
and, since ti > tj , for small
2
1)2 e
(n
(ti +tj )
tj
(ti
e
)
ih
2
tj
we have that e
1)2 e
(n
(ti
e
)
i
1 > 0;
(ti +tj )
> 0 and therefore optimality of t requires that
1 > 0.
Similarly, since t is optimal, we must have that
q; ^tj
E [ (q; tj )] > E
This is equivalent to
h
(tj
e
(tj
and, since ti > tj , we have that e
1)2 e
(ti +tj )
)
ti
e
)
ih
ti
e
2
1)2 e
(n
:
i
1 > 0;
(ti +tj )
> 0, and therefore optimality of t requires that
2
(n
1 < 0. We have then reached a contradiction. This completes the proof of Lemma 7.
Proof of Proposition 4. Expected payo¤s of an `-leader organization can be rewritten as
E( (q; tj )) =
n
2
2
+`
1 + (n
2
1)(1
+ (n
r( =`))
`)
1 + (n
1)
:
Hence, we obtain that
dE [ (q; tj )]
=
d`
where
h
(`; ; ; n) = ` 1
and that
2
(1 + (n
e
`
ih
1 + (n
2
d2 E [ (q; tj )]
=
d`d`
1))` 1 + (n
(n
dE[ (q;tj )]
d`
Denote by ~ the solution to 1
Since 1
(n
1)e
`
`
1)e
`
1)e
is decreasing in
n
i
3
( (n
h
1
(`; ; ; n);
`
(n
1) + 1)e
1)e
(`; ; ; n) is decreasing in
is the same as the sign of
~ (n
`
1) 2 e
`3 1 + (n
Observation 1. By direct veri…cation, the function
that the sign of
1)e
1)e
`
i
`
;
:
for all `; ; n. Note also
(`; ; ; n).
= 0. Also, denote by ^ the solution to 1
^ (n
1)e
and decreasing in `, the following observation follows:
35
= 0.
Observation 2. (2a) ~ < ^ for all ; n; (2b) If
d2 E[ (q;tj )]
d`d`
d2 E[ (q;tj )]
d`d`
< ~ then
< 0 for all `; (2c) If
> ^ then
> 0 for all `.
We now show that there exists a
( ; n) > 0 such that for all
<
optimal organization is ` = n. Denote by ( ; n) the solution to
n 1
e
(n; ( ; n); x; n) = 0. Explicitly,
e
n
n
~:
( ; n) =
n 1
e
( ; n) the number of leaders in the
n
Observation 3. Direct veri…cation implies (3a) ( ; n) < ~ for all ; n; (3b) ( ; n) is increasing in .
Observation 3a together with observation 2b imply that
for all
< ( ; n), the lower value of
dE [ (q; tj )]
j`=n =
d`
because, by observation 1,
Hence, for all
dE[ (q;tj )]
d`
dE[ (q;tj )]
d`
is declining in ` for all
is obtained when ` = n, and, at ` = n we have
2
(1 + (n
(n; ; ; n) >
< ( ; n). So,
1))n 1 + (n
1)e
(n; ; ; n) > 0;
n
(n; ( ; n); ; n), and, by de…nition,
(n; ( ; n); ; n) = 0.
< ( ; n) the expected returns of an `-leader organization are increasing in the number of
leaders, which implies that the optimal organization has ` = n leaders.
Next, observation 3b together with the observation that lim
!0
( ; n) = 1, imply that for all
< 1, the
optimal organization has ` = n leaders, regardless of the level of .
We now show that there exists a ( ; n) > ( ; n) such that for all
the number of leaders is ` = 1. Denote by ( ; n) the solution to
( ; n) =
1
e
e
1+e
> ( ; n) in the optimal organization
(1; ( ; n); ; n) = 0. Explicitly
^:
Observation 4. Direct veri…cation shows: 4a. ~ < ( ; n) < ^ , for all
and n; 4b.
( ; n) is increasing in
(1; ; ; n) < 0 for all
> ( ; n). Similarly,
.
Observation 1 together with
(1; ( ; n); ; n) = 0 imply that
observation 1 together with
(n; ( ; n); ; n) = 0 and observation 4a, imply that
> ( ; n). So,
dE[ (q;tj )]
d`
(n; ; ; n) < 0 for all
is negative at ` = 1 and at ` = n. Observation 4a and observation 2b implies
36
that
dE[ (q;tj )]
d`
is either …rst decreasing in ` and then increasing in ` (when
2 [ ( ; n); ^ ]) or it is always
> ^ ]). Hence, the pro…ts of the organization are decreasing in ` for all
increasing in ` (when
> ( ) and
therefore the optimal organization has ` = 1 leader.
We now conclude by considering the case where
2 ( ( ; n); ( ; n)). From the analysis above we infer that
the marginal expected pro…ts to ` of the organization around ` = 1 are positive, because (1; ; ; n) > 0, and
that the marginal expected pro…ts of the organization around ` = n are negative, because
Furthermore, observation 2b implies that, for all
organization,
dE[ (q;tj )]
,
d`
dE[ (q;tj )]
j`=`
d`
2 ( ( ; n); ( ; n)), the marginal expected pro…ts of the
are either always decreasing in ` (when
in ` and then increasing in ` (when
(n; ; ; n) < 0.
2 [ ( ; n); ~ ]) or they are …rst decreasing
2 [ ~ ; ( ; n)]). Hence, there exists a unique ` 2 [1; n] such that
= 0; such value of ` is the solution to
(` ; ; x; n) = 0 and, ` maximizes the expected
pro…t of the organization. Finally, by applying the implicit function theorem, d` =d
< 0 if and only if
d (` ; ; ; n)=d` < 0. Note that this last inequality holds because the fact that there exists a unique `
in which
(` ; ; ; n) = 0 and the fact that
2 ( ( ; n); ( ; n)) the function
(1; ; ; n) > 0 and
(n; ; ; n) < 0, assure that for all
(`; ; ; n) is decreasing around ` .
We have therefore shown that for every ` 2 f1; :::; n
1g there exists a
= (`+1) the optimal organization has ` = `+1 leaders; b. if
has either ` = ` leaders or ` = ` + 1 leaders, and c. if
(` + 1) <
(`) such that: a. if
2 ( (`+1); (`)) the optimal organization
= (`) the optimal organization has ` = ` leaders.
We now show that the optimal number of leaders ` is increasing in , which, in view of the above analysis,
amounts in showing that, for every ` 2 f1; :::; n
`,
such that at
=
`
1g there exists a unique value of
2 ( (` + 1); (`)), say
the expected pro…t of the `-leader organization is the same as the expected pro…t of
the ` + 1-leader organization. This is what we show next.
For brevity de…ne G(x) = e
x
and denote by
(`; ) the di¤erence between the expected pro…t generated
by the ` + 1-leader organization and the expected pro…t generated by the `-leader organization. We obtain
(`; ) =
2
1 + (n
`+1
1)G(` + 1)
`
1 + (n
Taking the minimum common denominator, we have that
(1 + (n
1)) [(` + 1)(1 + (n
1)G(`))
[1 + (n
37
1)G(`)
1
1 + (n
1)
:
(`; ) = 0 if, and only if,
`(1 + (n
1)G(` + 1))]
1)G(`)][1 + (n
1)G(` + 1)]
=
0:
This is a quadratic equation in
to check that
and therefore there are only two solutions of . Moreover, it is immediate
= 0 is one of the solution. Hence, there is only one non-zero solution. We have therefore
completed the proof of the …rst part of proposition 4.
To complete the proof of the proposition, we show that, for every ` 2 f1; :::; n
increasing in . De…ne t =
; then the cut o¤
(1+ (n 1)) (` + 1)(1 + (n
1)e
t
`
)
1g, the cut o¤
t
`+1
1)e
)
1 + (n
1)e
t
`+1
1 + (n
which, after some algebra, is
`+1
=
"
n
t
`(`+1)
t
e `+1 + `e
t
`
1 `+e
(1 + `)
t
`(`+1):
(1 + `)e
#
Note that nominator is increasing in t because
d `e
t
`(`+1)
t
+ e `+1
=
dt
t
1
e `+1
`+1
t
`2 +`
e
< 0;
whereas the denominator is decreasing in t because
t
`
d e
t
`(`+1)
(1 + `)e
1
e
`
=
dt
t
`
t
`2 +`
e
< 0:
It follows that
d
`+1
> 0:
d
Note further that
lim
= lim
`+1
!1
!1
t
1 `+1
e
= +1
`
This concludes the proof of Proposition 4.
Proof of Proposition 5: Recall that in an organization of size n and ` leaders the pro…ts are
E [ (n; `)] = nP
F
n
2
2
2
+`
1 + (n
=`
1) e
+ (n
`)
1 + (n
1)
+ (n + 1
`)
:
Similarly, in an organization of size n + 1 and ` leaders the pro…ts are
E [ (n + 1; `)]
=
=
(n + 1)P
(n + 1)P
F
F
(n + 1)
(n + 1)
2
2
2
+`
2
=`
1+n e
2
+
1 + (n
1) ^
2
+`
1 + (n
2
1) ^ e
=`
+ (n
where
^=
is
is the (non-zero) solution of
`+1
`(1 + (n
1
`+1
n
(n
38
1)
`)
1 + (n
1) ^
1+n
1)e
t
`
= 0;
Given the above derivation we have that
2
2
`n ( ) = arg max
`
l=1:::n
and
`n+1 ( ^ ) = arg max
l=1:::n+1
Suppose …rst that `n+1 ( ^ )
1 + (n
"
+ (n
=`
1) e
`)
1 + (n
1)
2
2
`
1 + (n
1) ^ e
=`
+ (n
`)
1) ^
1 + (n
n. It the follows from Proposition 4 that since ^ >
#
then `n+1 ( ^ )
`n ( );
which concludes the proof of the …rst part of the proposition.
Suppose now that `n+1 ( ^ ) = n + 1; From Proposition 4 we know that: A. `n+1 ( ^ ) = n + 1 if, and only if,
^
( ; n + 1) (where ( ; n + 1) is derived in Proposition 4) and B.
in ` = 1:::n + 1 for all
0
( ; n + 1). Since, as the derivation above show,
0
evaluated at the same value of
evaluated at ^
dE[ (n+1;`)]
d`
, we have that
dE[ (n;`)]
d`
is positive and decreasing
dE[ (n+1;`)]
d`
=
dE[ (n;`)]
d`
when
is positive and decreasing for all ` = 1:::n when
( ; n + 1). But then, since B holds and since
<^
( ; n + 1), it follows
dE[ (n;`)]
d`
is
positive and decreasing for all ` = 1:::n when evaluated at . Hence `n ( ) = n. This completes the proof.
2
Proof of Proposition 6: We prove that the optimal organization size is decreasing in
. Recall that
`n+1 is the optimal number of leaders given n + 1 tasks and `n is the optimal number of leaders given ntasks.
Then
E [ (n; `n )]
=P
n
whereas
E
(n + 1; `n+1 )
=P
n+1
2
1
n
1 + (n
1
1) e
"
1
F=(n+1)+
n + 1 1 + (n
`n+1
1) ^ e
2
F=n +
`n
=`n
+ (n
`n )
1
1 + (n 1)
2
(17)
(n
=`n+1
`n+1 )
1
+
+
^
1 + (n 1)
1 + (n 1) ^
#
2
;
(18)
where ^ =
n
(n 1)
> :
Suppose …rst that `n+1
n. Then, Proposition 5 implies that `n
`n+1 . To prove the proposition is
then su¢ cient to show that
E
is decreasing in
`n
1 + (n
Since `n
2
1
1) e
: Since ^ > ^ e
=`n
+ (n
`n )
=`
(n + 1; `n+1 )
E [ (n; `n )]
n+1
n
; a su¢ cient condition for to be decreasing in
1
1 + (n 1)
> `n+1
1 + (n
1
1) ^ e
=`n+1
+ (n
2
`n+1 )
is that
1
1 + (n
1) ^
` and ^ > ; this is indeed satis…ed.
Next, assume that `n+1 = n + 1; Proposition 5 then implies that `n = n. Hence
"
#
1
1
2
=
+ F=n F=(n + 1):
=n
=(n+1)
1 + (n 1) e
1 + (n 1) ^ e
Since ^ > ; it follows that
is decreasing in
2
. The second part of the proposition follows from this result
and proposition 5.
39
For Online Publication. Appendix B: Technological trade-o¤s between adaptation and coordination.
We show that our insights hold in a model of coordination a la Alonso, Dessein, Matouschek
(2008), Rantakari (2008) and Calvo-Armengol et al (2011). We consider the case for two agents,
but everything can be generalized to n agents. In these class of models, instead of having the
distinction between primary action and complementary action, each agent chooses one single action.
We posit that agent i chooses qi . Given a particular realization of the string of local information,
= [ 1;
2 ],
and a choice of actions, q = [q1 ; q2 ], the realized pro…t of the organization is:
(qj ) = K
where
2
1)
(q1
(q2
2
2)
(q1
q2 )2 ;
(19)
is some positive constant. As in the model developed in our paper, agent i has information
set Ii that contains the local shock
i
and a message mj about local shock
j.
The communication
technology follows the description in our basic model.
Standard computation allows us to derive agents’ best replies, for a given network t = (t;
t).
We obtain:
1
1+
1
1+
q1 =
q2 =
[
1
+ E [q2 jI1 ]]
(20)
[
2
+ E [q1 jI2 ]]
(21)
Without loss of generality we focus on characterizing equilibria in linear strategies; we write (20)
and (21) as
q1 = a11 (t1 )
1
+ a12 (t2 )E[ 2 jI1 ]
(22)
q2 = a22 (t2 )
1
+ a21 (t1 )E[ 1 jI2 ]
(23)
Substituting the guess (22) and (23) into (20) and (21), we …nd that the equilibrium actions are
q1 =
q2 =
1+
1+2 +
1+
1+2 +
2
e
t1 1
+
2
e
t2 2
+
1+2 +
2
e
t2
E[ 2 jI1 ]
(24)
1+2 +
2
e
t1
E[ 1 jI2 ]
(25)
Finally substituting (24) and (25) into (19) and taking unconditional expectations we …nd that
the problem
max E (qj ) s:t:t1 + t2 =
t
40
is equivalent to
1
]1+2 +
max
t2[0;
where t = t1 and t2 =
2
e
t
+
1
2
1+2 +
e
(
t)
t.
Similarly to the proof of proposition 1 and proposition 2, it is easy to verify that
@E (qj )
> 0 () (1 + 2 )2
@t
4
e
> 0:
We then obtain a result that is qualitatively the same as the one stated in Proposition 3. For every
there exists a ( ) > 0, so that for all
for every
< ( ) the optimal organization has t = =2, whereas
> ( ) the optimal organization has t = f0; g. Furthermore, ( ) is increasing in .
For Online Publication: Appendix C: Task Asymmetries.
We have shown that organizational focus may be optimal even when tasks (and agents) are
ex-ante identical. In reality, tasks are of course likely to di¤er from each other. The question,
then, is not only how focussed to be, but which tasks to focus on. An interesting asymmetry is one
where some tasks impose larger coordination costs (delays, low product quality) should other tasks
not take the appropriate coordinating actions. For example, in designing a car, important changes
made to how the engine works, may have important consequences for the remainder of the design.
Should attention be focused on those highly interdependent tasks? In this section we show that
this is not necessarily the case. For conciseness of the argument, we consider the two-task case.
p
Let the coordination parameters be 1 and 2 for task 1 and 2, respectively.De…ne =
1 2,
the geometric mean of
1
and
1
The parameter
increase in
=
2
and consider situations where
(1 + )
and
2
=
(1 + )
1
:
thus determines the “spread” between the coordination costs across tasks: An
> 0 increases the coordination costs associated with task 1 and decreases that of task
2, leaving the geometric average, a su¢ cient statistic for how costly lack of coordination is to the
organization, unchanged. When
Proposition 8 Assume
1. If
1
>
= 0 the case collapses to the one considered in Section 2.
2
1, then:
< > ( ), the optimal organization is focused on task 2, i.e., (t1 ; t2 ) = (0; ).
41
2. If
> ( ), let b be the solution to (1 + b)2 e
(a) If < b then
(b) If
2
=1:
> t1 > t2 > 0:
b, then (t1 ; t2 ) = ( ; 0) :
If attention is limited,
< > ( ), then all attention is focused on the task which is least
interdependent: Task 2. The reason is that allocating limited attention to task 1 is essentially
not worth it as it would translate into limited adaptation given the large coordination costs the
organization would bear. Instead, it is better to provide all attention to task 2 and let task 2 be
adaptive. Task 1 is then coordinated by restricting its adaptiveness.
Instead when the attention capacity is larger and the asymmetry
is not too large, both tasks
receive attention but task 1 receives more than task 2. Intuitively, if both tasks are allowed to
be adaptive, more attention needs to be devoted to that task that is more interdependent. If
asymmetries between both tasks are su¢ ciently large, task 2 may even receive no attention for
> > ( ) : At the threshold ^ = > ( ) ; the organization then switches from being fully focussed
on task 2 to being fully focussed on task 1:
Proof of Proposition 8. Replicating the analysis for the model with two-tasks, by allowing for
asymmetries, we obtain that equilibrium actions are
qii = aii
i
and qji = aii E[ j jIi ] and aii =
1
ie
1+
t
:
Furthermore, we can express expected pro…t for a given t as
E[ (qj ] =
(1
a11 )2
2
(1
a22 )2
2
1 (1
r1 )a211
2
2 (1
r2 )a222
2
;
(26)
Hence, the organizational problem is to choose t1 = t 2 [0; ] to maximize expression (26). Repeating the arguments developed for the symmetric case, we obtain that the pro…ts of the organization
are decreasing in t, if, and only if,
[1
1 2e
][
1e
t
2e
(
t)
] > 0:
It is convenient to divide the analysis in two cases. Recall that we are assuming that
(which is equivalent of assuming
2
> ^ = 1).
42
(27)
> 1+
Case 1. Assume that
1e
2
> 0, or > b. This assumption implies that
1e
t
2e
(
t)
>
0 for all t 2 [0; ]. This in turn implies that the objective function is decreasing in t if, and only if,
1
which is always satis…ed because
1 2e
< 0 ()
> 1 + . So, if
< ln
< ln
> b, it is optimal to set t = 0 and
and
there is focus on task 2.
Case 2. Assume now that
2e
(
t)
1e
2
< b. Since
< 0, or
declines in t, it follows that there exists a t so that
such t solves
1= 2
=e
t ) =e
(
t
and since
1
>
2
and e
2e
1
t
1e
t
> 0 and since
2e
(
t )
1e
t
= 0. Indeed,
is decreasing in t, it follows that
t > =2. The next two observations complete the proof:
First, if 1
1 2e
> 0, then the objective function is increasing in t for t
in t for all t > t . Hence, in the optimal organization t = t . Second, if 1
objective function is decreasing in t for all t
t and it is decreasing
1 2e
t and increasing in t for all t
< 0, then the
t . Hence, there
are two candidates for the minimum: either t = 0 or t = . Comparing the two organizations it
reveals that since 1
task 2. Note also that 1
if,
1 2e
< 0 the optimal organization has t = 0, and so there is focus on
1 2e
> 0 and
1e
2
< 0, are mutually compatible, if and only
> 1 + , which holds by assumption. This concludes the proof of Proposition 8.
For Online Publication. Appendix D: Information Theory.
We now show that an identical residual variance as in (5) obtains if messages mji and local
information
i
are normally distributed and the attention constraint (3) is modelled as a constraint
on the total reduction in entropy, as in Information Theory (Cover and Thomas 1991) and the
literature on Rational Inattention (Sims 2003). Since, in an equilibrium with linear strategies,
the expected pro…ts for a given attention allocation t can be written as a function of the residual
variance, identical results obtain with this alternative communication technology.32
32
It is easy to show that, for a given t, the expected pro…ts in an equilibrium with linear strategies can be
written as:
E[ (tj )]
n
2
+
n
X
i=1
Details are available upon request to the authors.
43
2
2
+ (n
1)RV(ti )
:
For simplicity, we focus on the two-task case. Let mi be a message about
i
and let m =
(m1 ; m2 ). The mutual information between m and ; denoted by I( ; m); equals the average
amount by which the observation of m reduces uncertainty about the state ; where the ex ante
uncertainty is measured by the (di¤erential) entropy of ,
Z
H( ) =
f ( ) log f ( )d ;
and the uncertainty after observing m is measured by the corresponding entropy
Z
H( jm) =
f ( jm) log f ( jm)d :
Denoting by
the (Shannon) capacity of the communication channel, the constraint on information
conveyed by m about
is given by
33
I( ; m) = H( )
H( jm)
:
(28)
Following Sims (2003) and the subsequent literature on rational inattention, we assume that
and
2
1
are (independently) normally distributed, and communicated through a Gaussian commu-
nication channel which contaminates its inputs with independent normally distributed noise, e.g.,
mi =
i
+ i , where
i
is normally distributed. As a result, also m1 and m2 and the conditional
distributions F ( 1 jm1 ) and F ( 2 jm2 ) are independently normally distributed. As noted by Sims,
Gaussian communication channels minimize the variance of F ( i jmi ) given the constraint (28) on
the mutual information between
i:
34
Given that
1
and
i
and mi : Hence, they maximize the correlation between mi with
are independently distributed, we have
2
I( ; m) = I( 1 ; m1 ) + I( 2 ; m2 );
where I( i ; mi ) = H( i )
2
is given by
1
2
ln(2 e
H( i jmi ): Moreover, since the entropy of a normal variable with variance
2 );
we obtain
I( i ; mi ) =
33
(29)
1
ln
2
2
ln Var( i jmi ) :
(30)
The capacity of a channel is a measure of the maximum data rate that can be reliably transmitted over
the channel. Shannon capacity has proven to be an appropriate concept for studying information ‡ows in a
variety of disciplines: probability theory, communication theory, computer science, mathematics, statistics,
as well as in both portfolio theory and macroeconomics.
34
This follows from a well known result in information theory that among all distributions with the same
level of entropy, the normal distribution minimizes the variance.
44
It follows that the constraint (28) on the mutual information between
ln
2
2
ln Var( 1 jm1 ) + ln
ln Var( 2 jm2 )
We can now re-interpret the mutual information between mi and
the organization to task i: Denoting t1
and m can be rewritten as
I( 1 ; m1 ) and t2
2 :
i
(31)
as the attention devoted by
I( 2 ; m2 ); the constraint on mutual
information (28) imposed by the Shannon capacity becomes equivalent to our attention constraint
t1 + t2
:
Using the above formalization, we obtain a tractable expression for RV(ti )
deed, from (30) and ti
V ar( i jmi ). In-
I( i ; mi ), we have
2
ln RV (ti ) = ln
2ti ;
i = 1; 2:
(32)
or still
RV (ti ) =
where t1 + t2
2
e
2ti
;
i = 1; 2;
(33)
:
For Online Publication. Appendix E: Alternative communication models.
E.1. Bilateral communication with aggregate organizational constraints.
We now consider that communication is bilateral and that the constraint is at the organizational
level. Formally, the allocation of attention is t = ftji gji2N , where tji denotes the amount of communication between agent i and agent j about local information
i.
Let
be the total communication
capacity of the organization. Then, we require that t satis…es
XX
i
j
We maintain the assumption that r(tij ) = 1
e
tij
tij .
:
The following equivalent result obtains:
Result 1. In an optimal organization under bilateral communication and constraint
allocation of attention t = ftji g satis…es
tji = tPi for all i; j 2 N ;
45
, the
where tP = ftP1 ; :::; tPn g is the allocation of attention in an optimal organization under public communication and constraint
P
= =(n
1).
Proof of Result 1. The key step for this equivalence result is the proof of the following Lemma
Lemma 9 Consider bilateral communication and constraint . In an optimal organization all
agents devote the same attention to a particular agent, that is, for all i 2 N , tji = tki for all
j; k 2 N n fig.
Proof of Lemma 9. Suppose that t is optimal and, for a contradiction, assume that there exists
0. De…ne a new organization t0 , which is the same as t with the
some agent i such that tji > tki
exception that t0ji = tji
and t0ki = tki + , for some small and positive . Using the expression
for expected payo¤s, it is easy to verify that
E [ (q; tj )]
q; t0 j
E
0;
if, and only if,
e
Since t0ji = tji
+e
t0ki
e
tji
+e
tki
:
(34)
and t0ki = tki + , after some algebra we obtain that condition 34 is equivalent to
e
which, for
t0ji
tki
e
(tji
)
() tki
tji
;
su¢ ciently small, contradicts our initial hypothesis that tji > tki . This completes the
proof of Lemma 9.
Note that under bilateral communication and arbitrary capacity , Lemma 9 implies that the
optimal allocation of attention t satis…es tji = tli for all j; l 6= i. Hence, in the optimal organization
every agent j 6= i devotes the same attention to agent i, that is the restriction imposed by public
communication. It is immediate to see the relation between
and
P.
E.2. Individual Communication Constraints.
So far we have assumed that the communication constraint is determined at the organizational
level. Alternatively, each agent may have a limited communication capacity
I:
Formally, let each
agent have access to an individual communication channel, whose …nite capacity
46
I
can be used
to broadcast information to all other agents and/or to process information broadcasted by others.
Each agent i then optimally decides on a vector ti = [ti1 ; ti2 ; :::; tii ; :::; tin ] ; where
X
tij
I
8i 2 N ;
j2N
(35)
and where tii is the capacity devoted to broadcast information about
i,
and tij is the capacity
devoted to listen to the information broadcasted by agent j 6= i: We maintain the assumption that
1
r(tij ; tjj ) = e
maxftij ;tii g
We now proof the following equivalence result, which again implies
that the optimal organization is an `
leader organization with ` 2 f1; 2;
; ng leaders and that
the same comparative statics hold as in Proposition 4.
Result 2. Under individual communication and individual capacity constraint
I,
in the optimal
organization the allocation of attention t = fti jgi;j satis…es
tjj = tij = tbij 8i; j 2 N
where tb = ftbij gi6=j is the allocation of attention in the optimal organization under bilateral communication and capacity constraint
= (n
1) I .
Proof of Result 2. Consider the case of individual communication with individual capacity
constraint
a. tji
I.
Suppose that t is an optimal organization. It is immediate to see that t satis…es:
P
I for all ij 2 N . Now note that if tb is an optimal
tii for all i; j 2 N and b.
j tji =
organization under bilateral communication and constraint
= (n
1) I , then organization t
with tji = tii = tbji is a feasible organization under individual communication and satis…es property
a. and b. above. We now claim that t is optimal under individual communication and individual
capacity constraint
I.
Suppose there is another organization t that does strictly better than t .
First, note that the expected pro…t of an organization, for a given t, can be written in terms of
residual variances as follows
E[ (tj )] =
n
2
+
n
X
i=1
where RV(tji ; tii ) =
2 (1
2
2
+ (n
1)
Pn
j=1 RV(tji ; tii )
;
r(tji ; tii )). Second, t must satisfy property a and property b and therefore
minftji ; tii g = tji , and so the residual variance that agent j has about task i is RV(tji ). Since t
is strictly better than t is follows that the pro…le of residual variances fRV(tji )gji is better than
fRV(tji )gji . But then, construct ^
tb as follows: t^bji = tji . Note that ^
tb is feasible under bilateral
47
communication and capacity . Furthermore since the pro…le of residual variances fRV(tji )gji is
better than fRV(tji )gji , it must also be true that pro…le of residual variances fRV(t^bji )gji is better
than fRV(tbji )gji , and so ^
tb must be strictly better than tb , which contradicts our initial hypothesis
that tb is optimal.
For Online Publication. Appendix F: Endogenous Attention Capacity.
So far we have taken
to be a hard constraint in the amount of time agents can devote to com-
munication with each other. In practice this is another margin that organizations can use to
improve performance, by, for example, allowing more time for meetings and communication between teams. Equivalently, the organization can increase the e¤ective communication capacity ;
by cross-training and rotating employees, by hiring employee with higher cognitive abilities, or by
investing in communication technology. Assume thus that an organization can acquire a capacity
at a cost C ( ). C ( ) represents for example the costs of having team members engaged in
communications activities rather than in production. We assume that this cost has the following
properties:
C (0) = C 0 (0) = 0
C0 ( ) > 0
C 00 ( )
0 and C 000 ( )
0:
The problem of organizational design is now
max E (qj )
;t
Proposition 10 Assume that
> b , then
1. The optimal communication capacity
2. There exists
2
>
2
C( )
subject to
g if
2
(36)
2.
is increasing in
> 0 such that t1 2 f0;
(3) :
2
and t1 =
2
if
2
>
2.
Proof of Proposition 10. We prove each of the two parts of the proposition.
First part. We …rst show that the optimal capacity
is increasing in
2
in the focused organi-
zation and in the balanced organization. This, together with Proposition 2, implies the …rst part
of Proposition 10: the optimal capacity
is increasing in
48
2.
We consider the focused organization …rst. Recall that the expected pro…ts in the focused organization are
c
E[
Taking the derivative with respect to
)]
=
@
Since C 00 ( )
and, as
@E[
c (qj
@
)]
j
[1 + e
< 0. Moreover
"
2
2e
@ 2 E [ c (qj )]
=
@ @
[1 + e
0, C 000 ( )
increases, 1
negative for all
0 and 1
e
e
c (qj
@E[
)]
@
)]
c (qj
)]
2
is increasing in
=
c
of the optimality condition of
! 0 and that
h
3
]
1
c
j
=0
i
e
> 0, and that, since C 0 ( ) > 0,
#
+ C 00 ( ) :
(recall that
@ 2 E[ c (qj )]
@ @
> ^ = 1)
is either
and negative otherwise. Summarizing,
is (i) positive at
c
= 0, (ii) negative at
= 1 and (iii)
! 1 as
2
uniquely solves
c
2e
[1 + e
c
]
C 0 ( c ) = 0:
2
and since, from above,
the implicit function theorem implies that
2
)]
@
or it is …rst increasing and then decreasing in . As a consequence of
@
c (qj
@E[
becomes eventually positive, it follows that
@E [
@
c (qj
]
is negative for small value of
(i)-(iii) we obtain that the optimal capacity
@E[
C 0 ( ):
=1
we have shown that the function
Since
2
> 0, or it is positive for small value of
it is either decreasing in
C( ):
2e
We now observe that, since C 0 (0) = 0, it follows that
it follows that
e
1+ e
+
we have
c (qj
@E [
1
1+
2
(qj )] =
c
@ 2 E[ c (qj )]
j =
@ @
is an increasing function of
c
< 0, an application of
2.
From investigation
and the assumptions that C 0 (0) = 0, it follows that
c
! 0 as
! 1.
We now consider the case in which the organization is balanced. The expected pro…ts in the
balanced organization are
E
h
d
i
(qj ) =
Taking the derivative with respect to
@E
d (qj
@
2
2e
2
1+ e
we obtain
)
2e
=h
1+ e
49
C( ):
2
2
2
i2
C 0 ( ):
We can now proceed in the same fashion as in the case for the balanced organization to conclude
d
that the optimal capacity
uniquely solves
d (qj
@E
d
2
=h
1+ e
@
and that
d
2e
)
d
2,
is an increasing function of
C 0 ( d ) = 0;
i2
d
2
2
! 0 as
d
! 0 and
2
! 1 as
! 1.
Since the optimal capacity in the focused and balanced organization are both increasing in
2
and
since, by Proposition 2, the optimal organization is either focused or balanced, it follows that the
2.
optimal capacity of the optimal organization is increasing in
Second part. We now prove the second part of the proposition. First note that for a given
common
@E [
c (q;
@
d (q;
@E
j )]
@
j )
> 0;
if, and only if,
e
[1 + e
e
]2
2
[1 + e
2
]2
> 0;
and, after plain algebra, this condition is equivalent to
h
Since
2
3
e
c( 2)
e
e
2
is increasing in
ih
1
2
c (^ 2 )
2
2
e
3
2
i
>0
()
1
2
e
3
2
< 0:
ranging from 0 to 1, there exists a unique ^ 2 that solves 1
2
= 0. By construction, if
= ^ 2 , then
c (^ 2 )
=
d (^ 2 ).
The next observation is used
in the rest of the proof.
Observation 1.
d( 2)
<
c (^ 2 )
if, and only if,
2
< ^2.
3
c (^ 2 )
2
2
To see this note that since c is increasing in 2 , it follows that 1
e
< 0 for all 2 < ^ 2 .
@E [ d (qj )]
Hence,
j c ( 2 ) < 0, which implies that d ( 2 ) < c ( 2 ). Analogously, since is increasing
@
@E [ d (qj )]
c 2
2
in 2 , it follows that 1
e 3 ( ) > 0 for all 2 > ^ 2 . Hence,
j c ( 2 ) > 0, which implies
@
d( 2)
>
De…ne now
2
that
c ( 2 ).
as the solution to 1
2
e
d( 2)
= 0 and de…ne
50
2
be such that 1
2
e
c( 2)
= 0.
2
We now show that
> ^ 2 . By de…nition of ^ 2 and
1
which implies that
d( 2)
>
We now show that
2
2.
>
d( 2)
d( 2)
have that
c ( 2 ).
>
3
e
d (^ 2 ),
d (^ 2 )
=
2
c ( 2 ).
d
and since
2
e
d( 2)
Since
Hence, in order for
e
d( 2)
is increasing in
2
and
e
> ^ 2 and since
d( 2)
=
2
e
c( 2)
2
2,
then 1
2
e
c( 2)
0 and 1
2
2
2
e
it follows that
2
> ^2.
c( 2)
;
d( 2)
>
c( 2)
2
for all
to hold we must have that
< 0. From Proposition 2 we know that for all
optimal organization is focused. Hence, if
2
c( 2)
We now complete the proof of the second part of Proposition 10. If
and 1
;
we have that
2
=0=1
2
we have that
2
=0=1
2
By de…nition of
1
which implies that
2
2,
2
2,
then 1
such that 1
2
e
2
e
> ^ 2 , we
2
>
d( 2)
2.
0
0 the
the optimal organization is focused. Finally, if
d( 2)
> 0 and therefore, in view of Proposition 2,
it follows that the balanced organization is optimal.
From Part 1 of the Proposition, it pays to invest more in communication capacity when the
environment becomes more volatile. Intuitively, the cost of not being adapted is then larger and a
better communication capacity allows for better adaptation. From Part 2, a focused organization
is optimal in environments for which adaptation is not very important. Intuitively, a focused
organizations is optimal when the communication capacity is limited, and the organization does
not invest much in communication capacity when adaptation is not very important. Similarly,
balanced organizations are optimal when adaptation to the environment is very important, and the
organization invests heavily in communication capacity.
51
which adaptation is not very important. Intuitively, a focused organizations is optimal when the communication capacity is limited, and the organization does not invest much in communication capacity when
adaptation is not very important. Similarly, balanced organizations are optimal when adaptation to the
environment is very important, and the organization invests heavily in communication capacity.
52