Organizational Design of Multi-Product Multi-Division Firms
Joaquı́n Coleff ∗
[Job Market Paper]
Abstract
We seek to understand how a multi-product multi-market firm (e.g., a multinational firm) designs
its optimal organizational structure. In this paper, we analyze how these two important characteristics of the firm determine the information flows within the firm, affecting its profitability and how, in
turn, the firm influences these flows by allocating decision rights among its managers’ centralizing
or decentralizing decision-making, and by designing the compensation schemes. We find that, being
multi-product (having to allocate a scarce resource between markets) negatively correlates organizational decisions in the different product markets. In the market with higher returns to differentiation,
decision-making is decentralized. In the market with lower returns to differentiation, albeit there is
a bias toward centralization, decisions are only centralized if the difference in returns of different
markets is sufficiently high. The negative correlation increases with the possibility of controlling
resource allocation. Our paper contributes to the literature on organizational design by analyzing the
case of multinational firms. Our results are robust to different generalizations.
Keywords: Multinational, Multi-product, Organization Design, Resource Scarcity, Cheap Talk
JEL Classification: D2, D8, L2, G34
∗ I am indebted to Susanna Esteban for her support and helpful guide. I also thank to Daniel Garcia, Carlos Ponce, Antonio
Cabrales, Marco Celentani, Marc Möller, Dolores de la Mata, Joel Lopez Real and Jessica Bordón. All remaining errors are my
own. Contact: jcoleff@eco.uc3m.es. Departamento de Economı́a, Universidad Carlos III de Madrid, Getafe (Madrid), Spain.
1
1
Introduction
In this paper, we seek to understand how a multi-product multi-market firm (e.g., a multinational firm)
designs its optimal organizational structure. We analyze how these two important characteristics of the
firm determine the information flows within the firm, affecting its profitability and how, in turn, the firm
influences these flows by allocating decision rights among its managers’ centralizing or decentralizing
decision-making and by designing the compensation schemes. Our paper contributes to the literature on
organizational design by analyzing the case of multinational firms.
In the model, a monopolist manufactures two products and sells them in two different country markets. The demands for the products are independent; they are coffee and candy bars. In each country,
the firm has a manager to whom it may allocate decision rights. The two country managers are under
the umbrella of a headquarters’ office (hereafter, HQQ) that has the same objective function as the firm.
Country demands are heterogeneous (e.g., have different demand elasticities) and thus country profits
are maximized when the product’s characteristics target the specific country. The country manager, who
obtains a signal of what the “ideal” product in his own country would be, only imperfectly observes his
own country’s demand heterogeneity. However, the manager can improve the quality (precision) of his
signal by devoting more time and effort to information acquisition.
Managerial time is a scarce resource and must be allocated to the two product markets that the country
manager oversees. Time and effort are complementary. Effort determines the quality of the signal, while
time lowers the cost of exerting it. That is, exerting effort over a longer period of time (being less
time constrained) is less costly to the manager. Nonetheless, devoting more time to one market implies
devoting less time to the other, which then has a higher cost of effort. In other words, the two markets
work as the two sides of the same coin.
The firm must decide whether, and how much, to customize the products to the countries, adding
hazelnut flavoring to its coffee or mixing crunchy rice puffs in its chocolate bars. This decision can be
delegated to the country’s manager, resulting in a decentralized structure, or be centralized in the HQQ’s
office. Country managers are self-interested and their pays are endogenous to the firm. If their pay is
simply a proportional share of the firm’s aggregate profits, we say their interests are aligned (with the
firm), while, otherwise, if they are an unequal average of the country and the aggregate profits, we say
their incentives are misaligned.
After observing the private signal, the manager acts upon his information. With centralized decisionmaking, the manager sends his reports to the HQQ’s office. It is this office that, after receiving the
reports of the two managers, decides the specifics of the products that are manufactured for each country.
Instead, with decentralized decision-making, reports are exchanged between country managers, who then
unilaterally determine the characteristics of the country’s products. Obviously, when sending reports, the
manager may be strategic, aiming to bias the characteristics of the products that are chosen by the HQQ
or by the other country manager.
Managers may misreport their observed signals to strategically bias product decisions. If the chosen
products are identical across country markets, there are economies of scale in production and thus, cost
savings. If, instead, the products are heterogeneous, the firm raises revenues as it implements thirddegree price discrimination. Together, these imply that in his misreporting, the manager seeks to make
the two countries sell the same identical product and in turn, make this product be his home country’s
2
ideal. From the firm’s point of view, having the manager misreport information has a cost since it lowers
the accuracy of the information transmitted, upon which other agents (HQQ or other country) make
product decisions. To align the manager’s incentives, the firm can modify the manager’s compensation,
making it more or less aligned with the firm’s profit, and/or decentralize decision-making.
To understand the workings of the model and what results are obtained from the described environment, it is useful to start with the existing literature. As our model builds upon this literature, it shares
intuitions. Alonso, Dessein, and Matouschek (2008)1 analyze the problem of a multi-market (two countries) single-product firm when the manager of each country perfectly, but privately observes information
about true demand characteristics. As in our paper, there are economies of scale in homogenizing products and there are also gains from price discrimination when customizing the products to the respective
country markets. Different from our model, however, is that information is exogenous and does not arise
from exerting effort or allocating a scarce resource.
Alonso, Dessein, and Matouschek (2008) show that, when the compensation scheme aligns incentives, the firm prefers to decentralize decisions, while, when it misaligns them, the firm prefers centralization. To see this, consider the case of perfect alignment (country managers and HQQ have the same
objective function). Since there is perfect alignment, the maximization problems of all agents are the
same and thus, regardless of who decides, the same decisions are made. As the misalignment increases,
increasing the weight on the own country profit, the manager starts having incentives to misrepresent
information. In his report, the manager aims to implement product characteristics that are close to his
ideal product, as this increases the country’s profit. If the misalignment is small, the policy the country
manager chooses with decentralization is similar to the one the HQQ would choose, albeit the manager
makes his choice with better information, reducing the mismatch with the country’s true ideal product.
The manager bases his choice on his own observed information and the report that the other manager
sends. Since the two managers have similar objective functions (the misalignment is small), there is little
incentive to lie in the reports to each other and thus there is only a small bias in their choices. As the
misalignment of incentives increases, however, communication under decentralization becomes too poor
as they seek to effectively bias the other country’s product implementation. As a result, the HQQ prefers
to centralize decisions and pursue lower costs through product standardization. Although the managers
also misreport information in their communication with the HQQ the bias is smaller than it would be
if the information were sent to the other manager. The HQQ’s objective function, being the sum of the
two country profits, is closer to the country’s objective function than the objective function of the other
country and thus there are less incentives to lie.
Rantakari (2010) can be viewed as adding a new trade-off to this single-product two-country environment. Now the manager does not perfectly observe the country’s characteristics, instead, as in our paper,
he obtains a signal that can be made more precise by exerting costly effort. As in Alonso, Dessein, and
Matouschek (2008), if the objective functions are aligned, the manager has no incentive to lie when reporting the market’s characteristics. However, because the cost of effort is only incurred by the manager
and thus in his aligned objective function carries more weight than the country’s profit, he exerts less effort than optimal for the firm. With less effort, the manager obtains lower quality information, choosing
a product that does not coincide with the country’s ideal and this lowers profits. To give incentives to the
1 See
Rantakari (2008) for the case of asymmetric organizational design in the environment of two country single-product
firm with private (perfectly observed) information.
3
manager to exert more effort and obtain higher quality information, the firm must compensate him by
misaligning the objective function, but, then, the manager has incentives to misreport the signal, which,
in turn, lowers profits. When the misalignment needed to induce effort is too large, the communication
between countries is too imperfect, as their diverging objective functions leads them to strongly bias the
reports, and this makes the firm prefer centralized decision-making.
In our paper, we extend this environment to make the firm multi-product and multi-market. The
firm sells two products, with independent demands, in two different country markets. The economic
interaction in being multi-market arises because the manager has a limited resource (managerial time)
to devote to information acquisition in the two markets. The product markets differ in their profitability,
making the quality of information more valuable in one market than in the other.
Being multi-product modifies the existing predictions. If the time allocation between product markets
is exogenous, the problem of the firm is equivalent to the problem of two multi-market single-product
firms and thus the results in Rantakari (2010) apply to each firm–product–separately. If, instead, the
allocation of time is endogenous to the manager, the results in Rantakari (2010) do not generalize. Since
the firm wants to shift resources (more time and effort) toward the profitable market, the firm misaligns
incentives and decentralizes decision-making in this market, and aligns incentives and centralizes decisions in the unprofitable one. That is, the firm modifies the return to effort and time in the two markets
to induce the correct shift of resources.
Nonetheless, for the unprofitable market, the previous allocation of decision rights may not be optimal
as it misses out on one effect. If there are small differences in the country’s profitabilities, the cost
of shifting resources away from the unprofitable market and thus, obtaining bad quality information
in this market, is large. Low quality information means that product decisions with centralization are
“incorrect” and yield low profitability. Recall, however, that this market could be almost as profitable as
the other if product differentiation were implemented. Accordingly, the firm modifies its decision to also
decentralize the unprofitable market so that more resources are devoted to it. That is, if the differences in
profitability between markets are small, both markets are decentralized. If they are large, the profitable
market is decentralized while the unprofitable one is centralized.
It is worth noting that the finding that the firm may prefer to decentralize the two markets depends
crucially on the manager’s time allocation being non-verifiable. If the HQQ were to monitor the allocation of time, the firm would not need to use decentralization in the unprofitable market as a way to
provide incentives. The firm could simply force the manager to devote more time to it. Then, with more
time being devoted, the manager would obtain, and transmit, better information and this would, in turn,
raise the profitability of product standardization, which would increase the returns from centralizing the
unprofitable market.
Lastly, there are other strains of literature that our work relates to. Athey and Roberts (2001), Friebel
and Raith (2010), and Dessein, Garicano, and Gertner (forthcoming) explore other environments but
share with Rantakari (2010) and our paper the trade-off between the choice of effort and the quality of
decision making. Our paper also relates to the team theory literature, which is summarized in Mookherjee
(2006). This literature analyzes the trade-offs between performance and incentives (and not the allocation
of decision rights) by assuming that contracts are complete, which implies that the revelation principle
applies. In our framework, and in the one of Alonso, Dessein, and Matouschek (2008) and Rantakari
(2010), imposing completeness of the contracts would imply that decision rights are always centralized,
4
making the framework unsuitable to understand the problem of organizational design. Contract incompleteness (i.e., pays contingent on realized profits) and decision-making allocation are inherent to the
same problem.
The paper is organized as follows. In Section 2, we describe the model and solve it in Section 3. In
our framework, the benefits of the economies of scale are given by the technology to the firm. In Section
4 we discuss the extent to which it is possible to endogenieze the decision of choosing the technology,
which is still an open debate in the literature. In Section 5 we extend the model to consider externalities
in information acquisition, showing that our results are robust. In Section 6, we conclude.
2
Setup of the Model
A multi division firm produces two products, A and B, that sells in two different regional markets, 1 and 2.
Each regional division is controlled by a local division manager who is in charged to obtain information
about market characteristic. We denote by θ ji the market characteristic of product j in region i.2 Product
demands are unrelated both by region and by product, which implies that market characteristics are
independently distributed. We assume θ ji is uniformly distributed over the interval [−θ j , θ j ], where θ A
and θ B are bounded and, without loss of generality, θ A ≥ θ B . We define a firm’s product strategy3 for
market A as a pair of actions (aA1 , aA2 ). For a realization of market characteristic θA1 for product A in
region 1 this product strategy for market A generates the following profits for local division 1,
ΠA1 = K(β ) − (aA1 − θ1A )2 − β (aA1 − aA2 )2 .
The term (aA1 − θ1A )2 is associated the fit between product strategy and local market characteristic in
region 1. The better the adaptation to local market conditions, the higher the profit of local division 1 from
product A. The term (aA1 − aA2 )2 represents the ability of the firm to coordinate strategies within product
A among different regions. The more standardized the product strategy, the lower the (aA1 − aA2 )2 and
the better the coordination between different regions. The term K(β ) captures the maximum profits that
the firm can obtain which are increasing in the level of divisional integration β .4 Similarly, we define the
profit for each regional division in each product, i.e. ΠB1 , ΠA2 and ΠB2 .
The payoff of local managers, however, depends on the compensation scheme designed by the headquarters. We denote by s j the share of the division 2’s profit in market j that is awarded to division 1 and
(1 − s j ) the share of division 1’s profits in market j that remains in division 1. The value of s j (∈ [0, 0.5])
defines how narrow or broad incentives are in each division respect to market j. In extreme cases, when
s j = 0 a local division cares only about his own profits about market j, and when s j = 0.5 a local division
cares only about half of firm’s total profits. Considering the effort cost (which we describes below), the
2 The parameter θ represents demand characteristic that can be used by the firm to increase benefits. For example, suppose
a market with horizontal product differentiation and installed capacity, where preferences are single-peaked at θ . The firm’s
profits increase as her product strategy is closer to the bliss point θ .
3 In this paper we use the concept of strategy to represent two different ideas. ”Firm’s strategy” is intended to represent the set
of decisions that the firm as a whole takes and the objectives it pursues. ”Players’ strategies”, in the sense of the game-theoretic
literature, will refer to the mappings from histories into actions that every agent is entitled to choose.
4 The level of integration represents the losses for not having a good coordinating between product strategies. Some papers
require that this level of divisional integration is a firm’s choice variable and some papers do not. For a discussion about it see
Section 5.
5
payoff of division 1 in market j is
U j1 = (1 − s j )Π j1 + s j Π j2 −C(e,t),
j ∈ {A, B}.
The headquarters maximizes the expected sum of divisions’ payoff choosing the structure of the firm,
as defined by the allocation of decision rights about product strategy, denoted by g and the compensation
scheme s for each market. Decision rights can be centralized or decentralized. Under centralization
of decision rights, each manager sends reports about market characteristics to the headquarters before
she makes a decision over product strategy. Under decentralization, local managers may communicate
between themselves before they take a decision concerning the product to be sold in their own region.
This means that, under decentralization, firm’s product strategy results from the addition of two separated
decisions.5 Let g represents the allocation of decision rights, with g ∈ {C; D} (or {cent; dec}).
The problem would be simple if there were no agency problems. First, as communication is soft
and non-verifiable, local managers act strategically to exaggerate their local information in their own
interest. Following the literature starting from Alonso et al (2008), we model informal communication
as a one-round cheap talk model (Crawford and Sobel, 1982).6
Moreover, information is not perfectly observed. Local managers observe an imperfect signal of
√
market characteristic. The realization of the signal θ̂ equals the true value θ with probability e and a
√
random draw from the same distribution of θ with probability 1 − e. The quality of this signal reflects
the effort and resources local managers apply to market learning, which is observable but non-verifiable
√
to the organizational participants. The cost of acquiring a signal of quality e exerting a level e of effort
and allocating t resources is C(e,t) ≡ µ(t)C(e)σ 2 . Each division has a budget of 1 of resources, i.e.
0 ≤ t ≤ 1 and tA + tB ≤ 1 per division, and effort is a free choice in e ∈ [0, 1]. We assume C(e) > 0,
C0 (e) > 0, C00 (e) > 0, µ(t) > 0, µ 0 (t) < 0, µ 00 (t) < 0, and limt→0 µ(t) = +∞. We normalize the cost
function to be proportional to market variability, σ 2 . The function C(e) captures the evolution of the
effort cost in learning about market characteristic, while the function µ(t) scale the marginal cost of that
effort. We assume that the allocation of the limited amount of managerial time affects µ(t) directly.
Exerting effort and allocating resources in collecting information in one market are complementary.
Effort determines the quality of the signal (its precise), while time reduces the cost of this effort. It is less
costly for the manager to exert the same amount of effort if this is exerted over a longer period of time.7
We assume that resource allocation is delegated to local divisions. The headquarters may have calculated which is the optimal amount of resources in each local division for operational functioning. However, we believe that within each division, local managers administer how to distribute these resources
5 Our framework is symmetric and we follow the literature to concentrate in symmetric structures for each market which
implies centralizing or decentralizing decision making, i.e., s j1 = s j2 = s j and g j1 = g j2 = g j . See Alonso et al (2008) and
Rantakari (2008).
6 Also see Geanakoplos and Milgrom (1991) for a reflection over information transmission within organizations: “we assume
that only the surface content of a message like “produce 100 widgets” can be grasped costlessly; the subtler content, which
depends on drawing an inference from the message using knowledge of the sender’s decision rule, can be inferred only at a
cost.”
7 Time and effort in one activity are complementary. Then, effort in collecting information about different products are substitutes. Our model applies the multi-tasking model of Holmstron and Milgrom (1991) for substitute effort in the environment
of organizational design.
6
for learning about market characteristics.8 However, we can think that delegating resource allocation is
based on positive externalities in market learning. We adapt our model to this case in an extension.
Finally, the timing of the model (see Figure 1) is as follows: first, the headquarters chooses the firm’s
structure (decision making and incentives alignment) for each product; second, the private information
θi j about each product in each market is realized although not yet observed; third, local managers simultaneously and independently choose how much resources and effort devote to collect local information
for each product. This allocation determines the signal accuracy; forth, signals θ̂i j about the true value
of θi j are delivered; fifth, strategic communication takes place; sixth, actions are chosen and, finally,
payoffs are delivered.
Signals are
delivered
Private information realices,
but not observed
Structure is decided: decision
making, integration and
compensation scheme
Actions are
chosen
Communication
takes place
Divisions allocate resources
and choose effort
Figure 1: Timing.
B1
t1B
e1B
θˆ1B
Manager
Region 1
t1 A
e1 A
HQ
m2 B
m1 A
m2 A
θˆ1 A a1 A
a2 A
A1
t1B
t2 B
θˆ2 B e2 B
a1B a2 B
m1B
B1
B2
A2
e1B
θˆ2 A
t1 A
e1 A
t2 A
e2 A
(a) Mixed centralization and decentralization
t2 B
θˆ1B a1B
a2 B
m2 B
m1B
m1 A
m2 A
Manager
Region 1
Manager
Region 2
B2
θˆ1 A a1 A
A1
θˆ2 B e2 B
Manager
Region 2
a2 A
A2
θˆ2 A
t2 A
e2 A
(b) Decentralization in both products’ markets
Figure 2: Organizational Structure depending on decision making.
We derive the equilibria that maximizes total expected profits by backward induction. Given signals
and communication outcome, we find the best response functions. Then, anticipating these best response
functions, local managers engage in optimal strategic communication. Anticipating the optimal strategic
8 See Geanakoplos and Milgrom (1991): “To take advantage of the information processing potential of a group of managers,
it is necessary to have the managers attend to different things. But these differences are themselves the major cause of failure
of coordination among the several managers.” In their model, “a chief executive allocates production targets, capital and other
resources to division managers who in turn reallocate the budgeted items to their subordinates, etc. until the resources and
targets reach the shops where production takes place.”
7
communication and actions, each local manager allocates resources and exerts effort to collect accurate
local information. Finally, given optimal behavior, the headquarters chooses the optimal design.
3
Equilibrium
To find the optimal structure we calculate the incentive’s alignment and division integration that maximizes total expected profits of the organization for each possible combination of decision making, taking
as given the best response of local managers in collecting, transmitting and using information. Then, we
compared which structure provides higher expected profits.
In Section 3.1 we analyze how information is transmitted and used. In Section 3.2 we analyze how
local managers acquire local information. In Section 3.3 we study the resource allocation problem.
Finally, in section 3.4 we describe which organization structure maximizes total profits depending on
parameters values.
Respect to the equilibrium concept, we focus on Pareto Efficient and Perfect Bayesian Nash equilibriums. In some stages, there are multiple equilibria that can be ranked from a pareto optimality perspective,
therefore we concentrate on those equilibria that leave the agents with the maximum expected payoffs.
We assume that agents can coordinate over those equilibriums when it is mutually beneficial.
3.1
Actions and communication: transmitting and using information
Assume the organization structure defined by g ∈ {C; D} (or {cent; dec}), β and s. Each division i
has private information about market characteristic in his own region, θi . We solve how information is
transmitted and use for one product. The solution is similar for in both products. The solution follows
from Alonso, Dessein and Matouscheck (2008).
Under centralization, the headquarters communicates with each local division and forms beliefs about
local market characteristics, i.e. Eθ1 and Eθ2 , and chooses the firm’s product strategy solving
max U1 +U2 = π1 + π2 = K(β ) − (a1 − Eθ1 )2 − (a2 − Eθ2 )2 − 2β (a1 − a2 )2 .
a1 ,a2
Under decentralization, local divisions communicate between themselves, forms beliefs about the other
division action, i.e. division 1 forms beliefs Ea2 , and choose their action solving
max U1 = (1 − s)π1 + sπ2 = K(β ) − (1 − s)(a1 − θ1 )2 − sE(a2 − θ2 )2 − β (a1 − Ea2 )2 .
a1
The following proposition characterizes the optimal actions for both cases.
Proposition 1. 1.a Conditioned on beliefs defined by E2 [θ1 ] ≡ E2 [θ1 /m1 ] and E1 [θ2 ] ≡ E1 [θ2 /m2 ],
optimal actions under decentralization are
(1 − s)
β
β
1−s+β
D
a1 (m1 , m2 , θ1 ) =
θ1 +
E2 [θ1 ] +
E1 [θ2 ] , and
(1 − s + β )
(1 − s + β ) 1 − s + 2β
1 − s + 2β
(1 − s)
β
β
1−s+β
D
a2 (m1 , m2 , θ2 ) =
θ2 +
E1 [θ2 ] +
E2 [θ1 ] .
(1 − s + β )
(1 − s + β ) 1 − s + 2β
1 − s + 2β
8
1.b Conditioned on beliefs, optimal actions under centralization are
aC1 (m1 , m2 ) =
aC2 (m1 , m2 ) =
2β
1 + 2β
E[θ1 /m1 ] +
E[θ2 /m2 ], and
1 + 4β
1 + 4β
2β
1 + 2β
E[θ1 /m1 ] +
E[θ2 /m2 ].
1 + 4β
1 + 4β
The proof of Proposition 1 is based on Alonso et al (2008) and Rantakari (2008, 2009).
Optimal decision making reveals that managers can not truthfully transmit the information acquired.
Each manager has incentives to lie in order to improve the benefits of his own division. The intrinsic
)
incentives to lie under decentralization are E2 [θ1 /m1 ] − θ1 = (1−2s)(1−s+β
θ1 ≡ ωD θ1 and under cens(1−s)+β
tralization are m1 − θ1 = (1−2s)β
1−s+β θ1 ≡ ωC θ1 . Given these incentives to lie the only incentive compatible
communication for the sender of the message is, as described in Crawford and Sobel (1982), a partition
of the state space. Recall that we have assumed that the market characteristic of product j is uniformly
distributed in [−θ j , θ j ]. Then, we characterized the truthfully revealing partitions and communication
equilibria.
Proposition 2. Fix two positive integer N1 and N2 , there exists at least one perfect bayesian equilibrium
characterized by functions (υ1 (m1 /θ1 ), υ2 (m2 /θ2 ), a1 (m2 , m1 , θ1 ), a2 (m1 , m2 , θ2 ), g1 (θ2 /m2 ), g2 (θ1 /m1 )).
The communication rule υ j (m j /θ j ), decision rule a j (mi , m j , θ j ), and beliefs g j (θi /mi ) satisfies:
2 .a υ j (m j /θ j ) is uniform, with support on [d j,i−1 , d j,i ] if θ j ∈ [d j,i−1 , d j,i ].
2 .b g j (θ j /m j ) is uniform, with support on [d j,i−1 , d j,i ] if m j ∈ [d j,i−1 , d j,i ].
)
2 .c The boundaries are defined by: i) d j,i+1 −d j,i = d j,i −d j,i−1 +4 (1−2s)(1−s+β
d j,i for i = 1, ..., N j −1
s(1−s)+β
under decentralization; and ii) d j,i+1 − d j,i = d j,i − d j,i−1 + 4 (1−2s)β
1−s+β d j,i for i = 1, ..., N j − 1 under
centralization.
2 .d The decision of each worker are defined by Proposition 1.a under decentralization and Proposition
1.b under centralization.
Taking the boundaries d0 = −θ and dN = θ of the space, the solution is defined by
di = θ
with x = (1 + 2ω) +
xi (1 + yN ) − yi (1 + xN )
x N − yN
0 ≤ i ≤ N,
p
p
(1 + 2ω)2 − 1 and y = (1 + 2ω) − (1 + 2ω)2 − 1, and ω = ωD ≡
under decentralization and ω = ωC ≡
(1−2s)β
1−s+β
(1−2s)(1−s+β )
s(1−s)+β
under centralization. Note that xy = 1, x > 1 and y < 1.
The proof of Proposition 2 is based on Alonso et al (2008) and Rantakari (2008) and (2010). Proposition 2 describes the communication equilibriums with N-partitions of the space [−θ j , θ j ]. Lets define
mi as the posterior of the receiver of message mi about the private information of local manager i. After
communication takes place, the successful rate of communication is E[m2i ] = Vi σ 2j , where Vi represents
2
the proportion of local information variance (σ 2j ≡
θj
3 )
9
that is communicated. The rate Vi is increasing in
the number of partitions N. The equilibriums with +∞-partitions are the ones that achieve the maximum
expected payoffs for both managers. Onwards, we concentrate on these +∞-partitions equilibriums,
3+3ω
which deliver the rate of transmission equal to V j = 3+4ω
, with ω ∈ {ωC , ωD }.
After information is acquired there are two costs. One cost related with how well information is used
characterized by Λ. Under centralization the headquarters achieves the minimum cost, given that she
maximizes total expected profits with information available. Under decentralization, however, each local
manager has a narrow focus and does not internalize the externality that decision making has on the other
division.
The second cost is associated with information transmission and is characterized by Γ(1 − V ). The
factor V represents how well information is communicated between the ones who have the information
and the ones who makes decisions, i.e. how accurate is communication. The factor Γ represents how important is that communication, i.e. the value of communication accuracy. Comparing centralization and
decentralization, under centralization there is more accurate communication because the conflict between
each division and the headquarter is lower, i.e. there is less incentives to exaggerate private information.
However, the value of that communication is also higher under centralization. The information that is
not communicated under centralization is lost, in the sense that nobody can use that information for making decisions. Under decentralization, however, local managers use, at least, their own information for
making their own decisions.
We identify these costs in each local division under both centralization and decentralization. The
expected profits for each division in each product are described in the following Lemma.
Lemma 1. Under both structures the expected profit function for each is characterized by



E[Πi ] = K(β ) − 

 2
Λi + Γii (1 −Vi ) + Λi + Γ ji (1 −V j ) 
σ .
|
{z
}
{z
}  θ
|
due to i information due to j information
with the following expressions for centralization VC =
ΛiC , and Γi jC = −ΛiC . For decentralization the values
β [(1−s)2 +β ]
,
(1−s+2β )2
ΓiiD =
β [(1−s)2 +β ]
(1−s+β )2
− ΛiD , and Γi jD = β
3(1−s)(1+2β )
β
(1−2s)β +3(1−s)(1+2β ) , ΛiC = 1+4β , ΓiiC
3(1−s)(1−s+2β )
are VD = (1−2s)(1−s+β
)+3(1−s)(1−s+2β ) ,
(1−s)2
(1−s+β )2
(1)
= 1−
ΛiD =
− ΛiD .
Lemma 1 describes the situation under perfect information, all the algebra for constructing this expression is in Appendix 8. For this expression we omit the cost µ(t)C(e)σθ2 of acquiring information that
is introduced in the next section.
Summarizing the results of this section, we have characterized the value of the information under both
centralization and decentralization. After information is acquired, it is transmitted and used through local
managers to decision makers. Under both centralization and decentralization, the value of information
is increasing in incentive alignment, s, and decreasing in local division integration, β . Misaligning
incentives (reducing s) reduces the value of information under both structures. This effect is higher
under decentralization if s ≤ s(β ), with s(β ) increasing in β . Similarly, increasing divisions integration
(increasing β ) reduces the value of information under both structures but more under decentralization.
Hence, the value of information is higher under centralization if incentives are sufficiently misaligned
10
and divisions are sufficiently integrated, i.e. s low and β high.
3.2
Acquiring Information
Let us assume now that information is imperfect and costly. Each manager invests an effort e in acquiring an imperfect signal θ̂ of the true value θ . The realization of the signal is the true value with
√
√
probability e but a random draw from state distribution with probability 1 − e. The higher the effort
the more accurate the signal. Following Rantakari (2010) we use e as a measure of the quality of primary
information which has a cost µ(t)C(e)σ 2 with e ∈ [0, 1] and C(e) = −(e + log(1 − e)).9
Acquiring information of quality ei by division i and information of quality e j by division j plus the
cost of acquiring information means that now Equation (1) becomes, (for details go to Appendix 8)


 2

E[Πi ] = K(β ) − 
− Λi − Γii (1 −Vi )] +e j [Λi + Γ ji (1 −V j )] +µC(ei )
 σθ .
1 − ei [1
|
{z
}
{z
}
|
due to i information
due to j information
(2)
To understand the inefficiency in information acquisition, we distinguish between the profit of each
division, Π, and the value generated by each division, π. Aggregated profits equals aggregated value
generated, i.e. ∑k=i, j πk = ∑k=i, j Πk . However, local divisions do not internalize the externality that their
own information generates on the other division, then πk Q Πk . The profits of a division is represented
in Equation (2), however, the value generated by a division is



 2
E[πi ] = K(β ) − 
(Γii + Γi j )(1 −Vi )] +µC(ei )
1 − ei |[1 − (Λi + Λ j ) − {z
 σθ .
}
=ψii due to i information
(3)
A local manager acquires information considering the effect that his own information has on his own
payoff and not on the value that the information generates. Given g, β and s the expected utility of each
manager over a project is E[Ui ] = (1 − s)E[Πi ] + sE[Π j ] or,
E[Ui ] = K(β ) + − 1 + ei ψ̃ii + e j ψ̃ ji − µC(ei ) σθ2 ,
(4)
with ψ̃ii ≡ (1 − s)[1 − Λi − Γii (1 −Vi )] − s[Λ j + Γi j (1 −Vi )], ψ̃ ji ≡ s[1 − Λ j − Γ j j (1 −V j )] − (1 − s)[Λi +
Γ ji (1 −V j )], ψ̃ j j ≡ (1 − s)[1 − Λ j − Γ j j (1 −V j )] − s[Λi + Γ ji (1 −V j )] and ψ̃i j ≡ s[1 − Λi − Γii (1 −Vi )] −
(1 − s)[Λ j + Γi j (1 − Vi )]. It is crucial to remark that the value of the information for local manager,
ψ̃(β , s, g), is decreasing in β and s under both centralization and decentralization. However, the value of
information ψ(β , s, g) is decreasing in β but increasing in s (as mentioned above) under both centralization and decentralization. Notice that as ψi 6= ψ̃ii the information acquired is not optimal.
The following Lemma describes the effort choice
9 The
result follows for more general expressions of C(e) but this expression captures all the general properties and contributes with simplicity.
11
Lemma 2. Under both decentralization and centralization the effort choice is characterized by,
2.a The optimal level of effort is given by ψ = µC0 (e∗ ), however
2.b Each local division chooses effort according to ψ̃ = µC0 (ê).
The comparative static implies that
∂e
∂µ
0
C (e)
= − µC
00 (e) < 0,
∂e
∂β
∗
< 0, ∂∂es > 0 and
ization and decentralization. For the function C(e) = −(e + log(1 − e)), ê =
∂ ê
∂ s < 0 under both centralψ̃
ψ
∗
µ+ψ̃ and e = µ+ψ .
The proof of Lemma 2 consists in solving the first order condition respect to e of manager’s problem.
The objective function is defined by Equation 3 for Lemma 2.a and by Equation 4 for Lemma 2.b.
The information acquired by local managers ê is decreasing in local managers incentives alignment s
because it depends on perceived value of that information, ψ̃, while the optimal amount of information
is increasing in incentive alignment s, because it depends on the real value of information, ψ̃.
Up to now, we have described the incentives to exert effort for information acquisition, i.e. e, and
how much value that information generates ψ(β , s). Under both structures, centralization and decentralization, local managers face similar trade-offs. Under decentralization, however, the perceived value of
information tend to be much bigger than under centralization, i.e ψ̃(β , s) bigger under decentralization
except when β is excessively high and s excessively low. The advantage of decentralization is that local
divisions use their own information to adapt production strategy to their local markets, so that the value
of information is higher for them. They value more the information under decentralization excepts in
extreme circumstances where standardization is very important and compensation schemes are narrowed
to local divisions’ profits.
Finally, one may wonder if managerial efforts in different divisions are strategic complements or
strategic substitutes. Intuitively, one manager may prefer to exert a lower costly effort when the other
manager has acquired a lot of information about his own country market meaning that efforts are strategic
substitutes. The opposite it could also be true in which case efforts would be strategic complements. In
this paper efforts are neither strategic complement nor strategic substitutes. The effects cancel out due
to 1) independence of θ j1 and θ j2 (for j ∈ {A, B}), 2) independence m1 and m2 in the communication
game, and 3) to the functional form of the profit function.10 Relaxing these assumptions to capture the
strategic interaction of the efforts appears a promising avenue for future research.
3.3
Resource allocation
Local managers has a budget of resources equal to 1 that they allocate to different markets for acquiring
information as much as possible. The allocation of these resources determines the marginal cost of
information acquisition about market characteristics, i.e. µA (tA ) and µB (tB ), with the constraint tA + tB ≤
1. In fact, we show that information acquisition and resources allocation are complementary. There are
incentives to allocate more resources in markets where there are more information acquisition, higher e,
and more information is acquired for markets that receives more resources.
We analyze two situations. As a benchmark case we analyze the optimal resource allocation, i.e. how
the headquarters would allocate those resources within each division. In the second case we analyze how
10 Go
to Appendix 8 for the details in the expected profit function under both structures.
12
local managers effectively allocate resources. The objective function of the headquarters is
o n
o
n
E[πiA ] + E[πiB ] = KA − [1 − eiA ψA + µAC(eiA )] σA2 + KB − [1 − eiB ψB + µBC(eiB )] σB2 .
(5)
Note that we refer only to the sum E[πiA ] + E[πiB ] in the objective function of the headquarters because
this is the part involved in the decision of tA and tB within each local division. The objective function of
division i is E[UiA ] + E[UiB ]
{K − σA2 [1 − eiA ψ̃iiA − e jA ψ̃ jiA + µAC(eiA )]} + {K − σB2 [1 − eiB ψ̃iiB − e jB ψ̃ jiB + µBC(eiB )]}.
(6)
For the optimal case, the headquarters chooses tA (constraint is always binding, tB = 1 −tA ) of division
i maximizing the profit in Equation (5) subject to optimal effort described in Lemma 2.b. In the second
case, each manager chooses tA and tB maximizing his expected payoff defined in Equation (6).
Lemma 3. The resource allocation tA among division is determined as follows:
3.a The headquarters chooses tA according to
−
∂e
∂ µB 2
∂ eB
∂ µA 2
σA C(eA ) − A [ψA − µAC0 (eA )] = −
σB C(eB ) −
[ψB − µBC0 (eB )]
∂tA
∂ µA
∂tB
∂ µB
(7)
3.b The local manager chooses tA according to11
−
∂ µB
∂ µA
C(eiA )σA2 = −
C(eiB )σB2 ,
∂tA
∂tB
(8)
We assume that µ(t) is sufficiently concave for an interior solution to exist.12 The comparative static
shows that a product market receives more time when more information is acquired, i.e. higher e, and
when the market is more variable or disperse, i.e. higher σ 2 .
As noted in the last Section, local managers information acquisition effort is not efficient, and thus
their resource allocation will also be distorted. Indeed, if the information would be acquired efficiently,
the term ∂∂µe [ψ − µC0 (e)] disappears due to Lemma 2.a and the resource allocation is always characterized by Lemma 3.b. As local managers omit this externality when acquiring information, whenever the
headquarters are able to influence the allocation of resources, they would try to correct this allocation as
described in Lemma 3.a.
However, whenever the headquarters are unable to influence this allocation, the local managers will
choose tA considering the opportunity cost of those resources on their own profits. This is consistent
with Geanakoplos and Milgrom (1991) who concludes “that managers at each level optimally focus
attention only on those variables that determine the marginal productivity of resources and the marginal
costs of production in the units under their command”. As described above, the choice of tA is only
affected by market variability, σ 2 and the function C(e). Hence, to reallocate resources in favor of
market A, headquarters must promote higher effort in market A, increasing C(eA ), lower effort in market
B, reducing C(eB ), or both.
11 Note
12 The
that this condition is derivated using envelope theorem.
convexity of resource allocation outweighs the effort convexity problem and incentive alignment convexity problem.
13
If σA2 = σB2 there will be a symmetric allocation of resources and identical effort when µ is sufficiently
convex. Note that from Lemma 3.b, the allocation of resources is symmetric only if eA = eB , and from
Lemma 2.a we can have identical effort if resource allocation is symmetric.
e
. For example, the function
The cost function µC(e) = −µ(e + log(1 − e)) has slope C0 (e) = µ 1−e
b
0.5
13
µ = µ t , with b ∈ (0, 1) and µ ∈ ℜ++ . For this function the resource allocation choice is tA =
1
with
1
1+H 1+b
H≡
C(eiB ) σB2
C(eiA ) σA2
and
HH ≡
C(eiB ) − ∂∂ µeBB [ψB − µBC0 (eB )] σ 2
C(eiA ) −
B
2
∂ eA
0
σ
∂ µA [ψA − µAC (eA )] A
with H when resource allocation is decided by each local manager and HH when resource allocation
b
1
is chosen by the headquarters. The cost of acquiring information is µA (tA ) = µ0.5b 1 + H 1+b . The
following Lemma is crucial for latter results.
b
and
Lemma 4. Define t ∗ as the value that maximizes [−µ(t) − µ(1 − t)H]. Assume µ(t, b) = µ 0.5
t
∗
0 < H ≤ 1. Given b0 ∈ [0, 1], the set of all functions with b1 ∈ [b0 , 1] satisfy the property that µ(t0 , b0 ) ≥
µ(t1∗ , b1 ).
The proof of the Lemma 4 is in Section 7.1. This Lemma describes that if resource allocation is more
important, then the marginal cost of acquiring information is lower in the market with higher variance.
In other words, a higher b implies lower µA and higher µB if σA2 > σB2 . Consequently, b represents the
importance of resource allocation.
3.4
Optimal structure
To find the optimal structure of the firm, we must evaluate the best response of local managers for each
possible structure and compare the total expected payoff obtained under each possible combination of
decision rights, i.e. centralization and decentralization in each product market. The problem is as follows
max
g,s
∑
KA − [1 − eA ψA + µAC(eA )] σA2 + KB − [1 − eB ψB + µBC(eB )] σB2 ,
(9)
k=1,2
subject to the optimal effort choice described in Lemma 2.b and resource allocation choice described in
Lemma 3.b. We gain in simplicity to expose our results in terms of the ratio of market variances denoted
σ2
with σAB ≡ σA2 , which satisfies σAB ≥ 1 since θ A ≥ θ B . The absolute value of market variances matters if
B
we must define also the optimal β . For robustness, we extend the results for the case that the headquarters
can also choose β . The first order conditions for the Propositions of this section are in Appendix 7.3.
We first analyze the situation when resource allocation is not important, i.e. b → 0, as a benchmark
case. For this case, the optimal design for market A is independent of the optimal design for market B.
In the following proposition we summarize the main result.
13 With
the properties µ 0 (t) = −bµ
0.5b
t 1+b
< 0, µ 00 (t) = b(1 + b)µ
14
0.5b
t 2+b
> 0,
∂ µ 0 (t)
∂b
< 0, and
∂ µ 00 (t)
∂b
> 0.
b
Proposition 3. Assume µ = µ 0.5
with b → 0. There exists an increasing threshold µ̃(β ) above which
t
centralization outperforms decentralization and the choice of decision rights is independent for market
of product A and market of product B.
In Appendix 7.2 we develop an sketch of the formal proof that you may find in Rantakari (2010).
We provide here some useful intuition for our next results. As explained in Section 3.1, centralization
outperforms decentralization only if standardization is quite important and compensation schemes of
local managers are narrowed to their own profits, i.e. β sufficiently high and s sufficiently low. For
a given β , centralization performs better than decentralization if incentives of each local manager and
the headquarters are sufficiently misaligned. The main trade off is clearly observed in the first order
condition respect to s,
∂e
∂ψ
[ψ − ψ̃] + e
∂s
∂s
= 0.
(10)
When the manager’s payoff depends more on total firm’s profit, i.e., s is higher, there is an increase
in the value of the information acquired, e ∂∂ψs , but also an increment in the value of acquiring further
information ∂∂ es [ψ − ψ̃], because ψ increases in s while ψ̃ decreases in s. When the cost of information
is low, there is a lot of information acquisition, and the headquarters aligns managers incentives with the
firm’s profits to increase the value of that information. In this case decentralization performs better than
centralization. When the cost of information is high, the headquarters prioritizes information acquisition,
narrowing local managers incentives to their own division profit, and, eventually, the firm performs better
under centralization.
The structure of the firm balances the moral hazard problem of suboptimal information acquisition
with the suboptimal value generated by this information in decision making. When informational cost
is high, the headquarters follows a product standardization strategy through centralization, but when
informational cost is low, she follows a product differentiation strategy through decentralization.
Although it is not directly stated, we can infer by proposition 3 that when centralization performs as
good as decentralization, the compensation scheme is different under both structure. Assume that the
informational cost is just the threshold µ̃, and the headquarters is indifferent between centralizing of
decentralizing decision making. This indifference between centralization and decentralization requires
that s ≤ s(β ). As explained in Section 3.1 the value of information decreases in s more under decentralization than under centralization when incentives are misaligned, i.e. s ≤ s(β ). Hence, at the same
informational cost, the optimal profit sharing s under centralization must be lower than the profit sharing
under decentralization, i.e. scent < sdec . Once the headquarters must provide incentives for information
acquisition, centralization can handle better at the margin the trade-off between acquiring and transmitting information, allowing for a lower s.
Proposition 3 also asserts that the threshold µ̃(β ) decreases in β . If economies of scale are sufficiently important, it is more likely that standardization characterizes the firm’s product strategy. With
high needs for coordination, the headquarters prefers centralization unless local managers have sufficient
information to compensate the efficiency loss in production costs. Hence, there is a lower informational
cost threshold for decentralizing decision making.
When b → 0, resource allocation is not important, and the marginal cost of information is given by
15
µ in each market. The headquarters decentralize decision making if both markets if µ ≤ µ̃, and centralizes them otherwise. Lets now see how the optimal structure changes as resource allocation becomes
important.
b
Proposition 4. Assume µ = µ 0.5
with b ∈ (0, 1) and µ ∈ ℜ++ . If resources are allocated efficiently,
t
there exists a threshold in the ratio of market volatility σ̃AB (µ̃) above which the optimal design requires
a mix structure with centralization in market of product B and decentralization in market of product A.
Proof. The headquarters chooses tA and tB according to condition (7) in Lemma 3. We guarantee an
interior solution of resource allocation when the function µ(t) is sufficiently convex. The allocation
of tA is increasing in the ratio σAB , implying that µB is increasing in σAB and µA is decreasing in σAB .
Given resource allocation, µA and µB are defined and Proposition 3 applies in each market: if µ j > µ̃
centralization outperforms decentralization, but if µ j < µ̃ decentralization outperforms centralization in
market of product j.
Proposition 4 has several implications for the optimal internal design of the firm and, consequently,
the optimal firm’s product strategy. The headquarters recognizes that the value of information is higher in
markets with higher variability than in markets with lower variability. The main reason is straightforward:
the expected losses of a wrong product strategy are higher when consumers’ tastes are more uncertain.
Hence, the optimal resource allocation will concentrate more resources in markets with higher variability.
This is shown in Lemma 3.a: the higher the ratio σAB the more resources allocated to learning about
market of product A and the less resources allocated to learning about market of product B. Thus,
there is higher informational cost and less information acquisition in market of product B, and lower
informational cost and more information acquisition in market of product A.
On the other hand, the headquarters recognizes that the information acquired by local managers is
suboptimal. However, the suboptimal effort is more serious in the market with high variability. Thus, the
headquarters would encourage more learning in market A to balance this suboptimality effort. Resource
allocation helps to lessen the problem of moral hazard in the market with higher variability.
As the ratio of market variability becomes higher, the allocation of resources to market A increases
and to market B decreases. The headquarters will, eventually, find it optimal to follow a differentiation
product strategy in market A and standardization product strategy in market B. To implement this choice
the headquarters will decentralize decision making in market A providing a compensation scheme that
aligns local managers’ incentives with the firm’s profits. In market B, however, since the aim is to pursue
standardization, it is better to centralize decision making with compensation scheme that narrows local
managers incentives to division profits.
In Figure 4 we show the relation between informational cost µ for each market when resources are
allocated efficiently for the particular case that µ < µ̃, i.e. the headquarters has control over resources.
If µ < µ̃, then the headquarters would decentralize decision making in both markets if σAB < σ̃AB .
In the following Proposition we point out how the loss of control affects the firm’s structure. When
the headquarters has no control over decentralized resources, she modifies the strategy of the firm to
capture as much benefit as possible from resource allocation. We denoted with ϕ the measure of the
inefficiency in resource allocation.
16
b
Proposition 5. Assume µ = µ 0.5
with b ∈ (0, 1) and µ ∈ ℜ++ . When local managers allocate
t
resources, there exists a threshold in the ratio of market volatility σ̂AB (µ̃) above which the optimal design
requires a mixed structure with centralization in market of product B and decentralization in market of
product A. Also µ̃(ϕ) is increasing in ϕ.
Lets assume that µ < µ̃, which means that decentralization is optimal if markets have similar volatility, i.e. if ratio σAB ∼ 1. Compared with Proposition 4 there are two effects that encourages decentralization in this case. First, the headquarters would allocate more resources in the market with high
dispersion that local managers in fact do. Second, the headquarters would change the design of the internal structure to correct this misallocation of resources. Compared with Proposition 4 both effects favors
decentralization in the market with low dispersion, i.e. favors product differentiation strategy in market
B.
On the one hand, local managers allocate resources according to condition in Lemma 3.b. The cost
of acquiring information changes with the relative market dispersion, i.e. tA and µB increases with ratio
σAB , and µA decreases with σAB . However, as local managers do not internalize the effect that resources
would have in reducing the inefficiency in information acquisition, they allocate more in markets with
low variability. This favors decentralization in low volatility markets.
On the other hand, due to the suboptimal allocation of resources, the headquarters uses the structure
to motivate a reallocation of those resources in favor of markets with high dispersion. However, the only
mechanisms available are to promote more information acquisition in those markets or less information
acquisition in markets with low dispersion, or both. Hence, there is a change in firm’s product strategy to
promote higher information acquisition in high disperse markets: aligning incentives of local managers
with firm’s profits in low volatility markets and narrowing incentives of local managers to their profits in
high volatility markets.
Hence, these two channels increase the likelihood that the headquarters chooses decentralization in
market B, as a way to overcome the inefficiencies in the allocation of resources. For centralization to be
optimal the dispersion ratio σAB must be higher.
Summarizing, the externalities in information sharing generates suboptimal information acquisition.
The headquarters alleviates this inefficiency allocating more resources to the markets where the problem
is more serious, i.e. the markets with high dispersion. However, when lacking control over resources,
she uses the internal design of the firm, allocating decisions rights and modifying the compensation
scheme to substitute the effect of resource allocation. Hence, resource allocation and decision rights are
imperfect substitutes in the sense that controlling resources lead to a mix strategy of centralization and
decentralization, but having a lack of control derives in a decentralized organization.
In the first order condition respect to s we observe the main changes in the headquarters incentives.
Lets focus on market B that has lower dispersion,
∂ eB
∂ ψB
[ψB − ψ̃B ] + eB
∂ sB
∂ sB
= −
∂tA
ϕ.
∂ sB
(11)
This equation equals Equation (10) plus a term − ∂∂tsAB ϕ that represents the marginal effect of modifying
the incentives alignment in market B over the resource allocation tA alleviating the inefficiency ϕ (you
find the expression of ϕ in the Appendix 7.3). When the headquarters allocates resources the inefficiency
17
is zero and she decides to centralize decision making in market B when the cost of acquiring information
in that market is sufficiently high, i.e. µB > µ̃. However, when lacking control over resources, she uses
the organizational design to reduce the inefficiency ϕ, through decentralizing decision rights and aligning
incentives in market of product B.
The inefficiency ϕ increases the threshold µ̃ for market B but reduces it for market A ( ∂∂µ̃ϕB > 0). With
this inefficiency, there is an extra benefit to align incentives in market B, i.e. sB is higher at the same
informational cost µ. The informational cost that motivates a change of strategy from decentralization
to centralization must be higher. For market A we find the opposite. Now, the inefficiency appears like
an extra cost of increasing sA , then the threshold µ̃A decreases with the inefficiency ϕ.
By symmetry, we can analyze the implications of Proposition 5 for the case with µ > µ̃, in which
case the optimal organizational design when the ratio σAB is small requires centralizing decision making
and a mix structure arises when the volatility ratio σAB is high.
In the following proposition we state our main results that compares the threshold ratios of Propositions 5 and 4, and how they change with the importance of resources b. (proof in Appendix 7.4)
b
with b ∈ (0, 1).
Proposition 6. Assume µ = µ 0.5
t
6.a σ̃AB (µ̃) < σ̂AB (µ̃).
6.b
∂ σ̃AB
∂b
<
∂ σ̂AB
∂b
< 0.
First, Proposition 6 says that some firms are following product differentiation strategies over their
products due to the lack of control over resources. If local market’s characteristic for one product are
quite volatile -a market in which information is very imprecisely collected- the headquarters prefer to
follow a product differentiation strategy over that market, even when it implies to follow product standardization strategy over other markets. However, when headquarters lacks control over resources she
modifies the firm strategy to product differentiation in all the markets.
Moreover, Proposition 6 emphasizes that the more important resource allocation for market learning,
the more the firm will follow product differentiation.14 When the effect of resource allocation on informational cost is higher, the headquarters would tend to decentralize more decision making due to the
lack of control over resources.
In Figure 3 we run a numerical example15 showing how the optimal contract depends on the ratio of
variances. The aligning of incentives is higher in the market with higher variance. There are more returns
to differentiation than in the other market. The difference increases as the ratio of variances increases.
There is a threshold above which the organizational design changes, centralizing decisions about the
product with lower variance. This change also affects the contract. The objectives functions depend
more on the local division profits to motivate more information acquisition in both markets, however it
misaligns more in the centralized product.
The results of this section can be empirically identify in at least two ways. First, comparing multiproduct multi-market firms with single-product multi-market firms and observing that there may be differences in their organizational design. There are differences in organizational structure of firms operating in the same market that can not be explained neither by demand differences nor by supply conditions
14 Eventually,
15 For
the firm may stop providing some products to concentrate resources in the more volatile market.
this example β = 4, µ = 0.65, b = 0.5.
18
or retailers environment. In this paper we offer a framework for differences that are born within internal
organization.
A second way to identify the result in this paper is comparing multi-product multi-market firms along
time. If there are some shocks affecting the returns to product differentiation of different product, the
firm will likely modify its organizational design to increase its profits. However, it may be difficult to
observe this case because firms might modify the organizational design informally when one product has
higher or lower returns to differentiation.16
A
s Dec
s
A
s Dec
B
Dec
B
sCent
Decentralize A
Decentralize B
Decentralize A
Centralize B
σˆ A2
σ A2
Figure 3: Optimal contract in both products as a function of σA2 , assuming σB2 ≡ 1.
4
Discussion
There is an ongoing discussion on the literature about how β is determined. Some authors argue that
the headquarters is free to decide the degree of integration between the two different units.17 Some
other authors, however, believe that the need for coordination is an exogenous constraint given by the
technology (adopted and available), legal environment, culture, etc.18
In any case, most authors agree that not all elements of the structure can be modified with the same
ease and speed. An organization may revise the compensation scheme or its allocation of decision rights
quite often, i.e. every one or two years. However, changing its degree of integration requires updating
the equipment, logistic and information technologies operated and the firm must devote more time and
resources to do so.
16 For instance, suppose that a formal procedure to introduce a new product requires approval from Headquarters in a centralized organization; then if the headquarters realizes that the firm can profit more in one segment through customization (higher
returns to differentiation) it can communicate informally to country managers that approval requirements are relaxed. This can
be understood as decentralizing the decision of introducing new products in that particular segments.
17 See Rantakari (2010).
18 See Alonso, et al (2008), Dessein, Garicano and Gertner (2010).
19
For instance, Eccles and Holland (1989) describe the case of Suchard when European Union merges
most west european countries as a unique market. It took several years and lots of resources to adapt the
company to the new situation. Thomas (2011) also mentions how the market structure modifies the needs
for standardization in the market of west europe. Procter & Gamble and Unilever reorganized production
after the pass of European Regulations in 1992. These firms launched programs to reduce the number of
products and to centralize production in fewer plants,19 which have required a lot of time and resources.
Our results however, do not depend on whether the coordination needs are exogenous or endogenous.
For this reason, we have analyzed first our results leaving β fixed and then described that our results
remain in the same direction when β is an additional firm’s choice.
5
5.1
Extensions
Choosing β
In the previous section we have discussed that some papers endogenize β and some others leave β as
exogenous. Our results are independent of this decision. We extend the results of Propositions 4, 5 and
6 for the case that headquarters chooses β . Results are shown in Figure 5.a for Proposition 3, Figure 5.b
for Proposition 4, Figure 6.a are for Propositions 5 and 6.a, and Figure 6.b for Proposition 6.b.
The decision of β does not modify qualitatively our previous results. However, it does have an
important implication for the strategy followed by the firm. If the headquarters decides the optimal β
for the structure of the organization, there is a negative relation between β and σ 2 . There is a high risk
of requiring high levels of coordination between local managers, as long as there is high uncertainty
about market characteristics. Then, for high values of σ 2 the headquarters prefers to provide autonomy
to local managers about the products to be offered in their markets and to follow a product differentiation
strategy.
Analogously, the headquarters follows an standardize product strategy in those markets with low
variability. The effect of lossing control over resources would also affect the decision of β . Let introduce
the formal Proposition.
b
Proposition 7. Assume µ = µ 0.5
with b ∈ (0, 1). There exist a cutoff σ̃ 2 (µ) and increasing relations
t
2 (σ 2 , µ), σ̃ 2 (σ 2 , µ), σ̃ 2 (σ 2 , µ) and σ̃ 2 (σ 2 , µ) such that:
σ̃AM
B
BM A
B A
A B
7.a If resources are allocated efficiently (headquarters), the decision rights over market j is centralized
if σ 2j < σ̃ 2j (σ−2 j , µ), for j ∈ {A, B}.
7.b If resources are allocated by local managers, the decision rights over market j is centralized if
σ 2j < σ̃ 2jM (σ−2 j , µ), for j ∈ {A, B}.
2 (σ 2 , µ) and σ̃ 2 (σ 2 , µ) < σ̃ 2 (σ 2 , µ).
7.c σ̃A2 (σB2 , µ) < σ̃AM
B
B A
BM A
7.d Thresholds are increasing in b.
19 These programs are “path to growth” (Unilever in 2000), Unilever 2010 (Unilever in 2004) and “Organization 2005”
(Proctor & Gamble in 1999). The program of Unilever included “a more streamlined brand portfolio, moving from 1600
brands in 1999 to a target number of 400 by the end of 2004”. “P&G aimed to improve supply chain management of the
proliferation of product, pricing, labeling, and packaging variations”.
20
Proposition 7.a, is based on Proposition 5 of Rantakari (2010) and determines that centralization
is chosen as an optimal structure if the market volatility, σ 2 , is sufficiently low, which is more likely
for higher informational cost µ. This relation is shown in Figure 5.a. Proposition 7.b remarks the
effect of resource allocation on the headquarters decision over centralization and decentralization. When
allocating resources becomes important, the headquarters promotes a lower cost in markets with high
variability. Not only the ratio of variances matters, but also the level of environmental variability in all
markets to determine whether centralizing or decentralizing decision making.
For instance, a lower β fits better in a market with higher volatility, which also makes decentralization
more attractive. On high volatility markets, the ratio that makes the headquarters prefer to centralize
decision making in some markets increases. These thresholds are represented by increasing relations in
Figure 5.b. As in our baseline analysis, these threshold are higher when the headquarters lacks control
over resources, which is shown in Figure 6.a. Then, the relations are increasing in the importance of
resource allocation as shown in Figure 6.b.
µ B (x)
µ
µ MB (x)
µ~ (ϕ )
µ~
Centralization
Decentralization
µ
µ MA (x)
µ A (x)
1
σ~AB
σ̂AB
σ AB =
σ A2
σ B2
Figure 4: Optimal decision making as a function of volatility ratio and informational cost.
5.2
Delegation
We assume that the headquarters delegates resource allocation on local division managers. We can
extend our model to see that this delegation is optimal if there are externalities when learning about
regional demands. For example, if learning about the demand of product B in region 1 can provide some
additional information when learning about the demand of product A in region 1, the firm could prefer to
merge to subdivisions in region 1, like many international firms do. Lets call subdivisions A1 and B1 for
the rest of this section.
Assume that the realization of the signal θ̂A1 that managers 1 observes about θA1 is the true value
√
with probability eA1 + αeB1 and a random draw from the state distribution of θA1 with probability 1 −
√
1
eA1 + αeB1 , with 0 < α < 1, and e ∈ [0, 1+α
] Introducing this setup does not modify the communication
problem. It generates an externality in information acquisition that is:
21
µ
A
σ B2
substitution between
tA and tB
Centralization
2
σ~BM
(σ A2 )
µ
45º line with σ A2 = σ B2
DA, DB
α (µ )
σ (µ )
2
B
Decentralization
B ≡ σ~B2 (σ A2 )
CA, DB
CA, CB
DA, CB
α (µ )
σ 2 ( µ ) 2 2 σ A2
σ~B (σ A )
σ A2 ( µ )
σ2
(a) Comparing µ and σ 2
σ A2
(b) Comparing σA2 and σB2
Figure 5: Optimal structure depending on marginal informational cost, µ, and market variability,
σ 2.
1- if subdivisions A1 and B1 are separated, local manager in division A1 collects information according to: ψ̃A (β , s, g) = µAC0 (êA ).
2- if subdivisions A1 and B1 are integrated, local manager in division A1 collects information accordσ2
ing to: ψ̃A (β , s, g) + σB2 α ψ̃B (β , s, g) = µAC0 (êA ).
A
3- If effort were contractible, the headquarters would ask the local manager to collect information
σ2
according to: ψA + σB2 αψB = µAC0 (e∗A ).
A
The net effect of this externality in learning about market characteristics is intuitive. For a given value
of α, integration is profitable if b is sufficiently low. There will be a cutoff on b such that delegation is
optimal as long as b is below that value. As α increases, learning from products A and B becomes
more complementary, which means that products are less rival. The inefficiency in resource allocation
is reduced. Although there are some new insights, the main implications of this paper over the firm’s
optimal structure remains. To see a comparison of this problem with our previous model see Appendix
7.9.
6
Conclusion
In this paper, we analyze the internal design of a multi-product multi-division firm. We model the
economic interactions of being multi-product that born within the organization. We focus on the effect
that the opportunity costs of resources have on decision making and compensation schemes.
Implementing a product differentiation policy on some products is more profitable than on other
products. This relative profitability generate opportunity cost of allocating scarce resources in each
market. We show that the firm may prefer to centralize decision making in markets with lower returns to
differentiation and decentralize decision making in markets with high returns to differentiation.
Empirically the result can be observed comparing the organizational design of multi-product multimarket firms with single-product multi-market firms. It can also be observed through dynamic changes
22
2
(σ B2 , ϕ )
σ B2 AM ≡ σ~AM
A ≡ σ~A2 (σ B2 )
A0
σ B2
Lemma 5
substitution between
tA and tB
B ≡ σ~B2 (σ A2 )
DA, DB
2
BM ≡ σ~BM
(σ A2 , ϕ )
σ B2 ( µ )
A1
σ B2 ( µ )
B0
CA, DB
DA, CB
CA, CB
σ A2 ( µ )
σ A2
σ A2 ( µ )
(a) Firm’s and Managers’ cutoff
B1
σ A2
(b) Change in resource allocation importance
Figure 6: Optimal structure for change on the importance of resource allocation
of multi-product multi-market firms who modify their organizational design when face shocks affecting
the relative returns to product differentiation in their product lines.
In this paper we have made a first move to extend the economic literature of organizational design
into the analysis of multi-product firms. Much of this literature focuses on internal economic problems
that frequently appear in international multi-product firms. Henceforth, an analysis of the multi-product
multi-division firm is not only important but also necessary. We show that the allocation of decision
making and the choice of the firm’s strategy for one product are not independent on how headquarters
office design the decision making and the strategy for markets of other products. This is in line with
Roberts (2004) who points out that “The structure [of the firm] does not follow strategy any more than
strategy follows structure”.
We have focused on the impact of the opportunity cost of resources on the internal organizational
design. There is still further necessary work on analyzing other economic interactions that multinational
firms face for being multi-product, e.g. interactions born in the demand side.
References
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7
Appendix
7.1
Proof of Lemma 4
Proof of Lemma 4 in page 14
b
Proof. Without loss of generality, assume µ = 1, µ(t) = 0.5
. Looking for t ∗ that maximizes [−µ(t) −
t
h
ib
1
1
and µ(t ∗ ) = 0.5b 1 + (H) 1+b . Since 0 < H ≤ 1 then t ∗ ≥ 0.5.
µ(1 − t)H] we find that t ∗ =
1
1+(H) 1+b
We need to prove that µ(t ∗ ) is decreasing in b for 0 < b <h0.5. Take logarithm
in both sides of µ(t ∗ )
i
1
and take derivatives respect to b: log µ(t ∗ ) = b log 0.5 + b log 1 + (H) 1+b , and
1
h
i
1
d log µ(t ∗ )
(H) 1+b
b
i
= log 0.5 + log 1 + (H) 1+b − h
log (H)
1
2
db
1 + (H) 1+b (1 + b)
in last term, notice that 1 − t ∗ ≡
1
1+b
(H)
1
1+(H) 1+b
becomes log 0.5 + log [H2], with H2 ≡
20 Recall
1
H
(12)
(1 − t ∗ < 0.5). working with the last two terms, Equation 13
(1−t ∗ )b2
(1+b)
1+bt ∗
+ (H) (1+b)2 . All I need to prove is that H2 < 2.20 For
that log 0.5 + log 2 = log 1 = 0.
24
1
3
the extreme case that t ∗ = 0.5 and b = 1 we have that H1 8 + (H) 8 ≤ 2 holds if 0.008 < H ≤ 1. Notice
that the expression decreases as b decreases or t ∗ increases. For completing the proof, lets check these
derivatives respect to b and t ∗ ,
∂ H2
=
∂b
1
H
(1−t ∗ )b2
(1+b)
log(
1+bt ∗
1
1−b
2 − t ∗ + bt ∗
(1+b)2 > 0.
)(1 − t ∗ )
−
log(H)
(H)
H
(1 + b)3
(1 + b)3
Notice that log(H) < 0 and log( H1 ) > 0 since H ≤ 1 . And respect to t ∗ ,
(1−t ∗ )b2
1+bt ∗
∂ H2
1 (1+b)
1
b
b
(1+b)2 < 0.
=
−
log(
)
+
log(H)
(H)
∂t ∗
H
H (1 + b)2
(1 + b)2
h
ib−1
∗)
−b
1
b
b 1 + (H) 1+b
Finally, we check that the derivative ∂ µ(t
=
(H) 1+b > 0 which implies that if
0.5
1+b
∂H
H decreases, then the cost µ(t ∗ ) decreases and the cost µ(1 − t ∗ ) increases, even more than we expect
due to this Lemma. The proof is complete.
d 2 log µ(t ∗ )
dbdH
7.2
−1
1+b
=


(H)
b log H 
1
bH
1
h
i 1 −
h
i >0
−
1
1
2
1 + b 1 + (H) 1+b
(1 + b) (1 + b) 1 + (H) 1+b
(13)
Proof of Proposition 3
Sketching the proof of Proposition 3 in page 14. This analysis is based on proposition 5 of Rantakari
(2010).
Sketch of the proof. Consider the optimal structure design for market A. First, the first order condition
respect to s shows a balance between less but more valuable information or more but less valuable information: eA ∂∂ψsAA + ∂∂ esAA [ψA − ψ̃A ] = 0. As local managers’ efforts or information collected is suboptimal,
modifying the compensation scheme helps to correct this inefficiency. A comparative static shows that
incentives are narrowed to local division payoff when information is more expensive ∂∂µs < 0. Balancing
the value and the amount of information, ψ and e, the headquarters must foster information acquisition
when it is expensive, even when this generates a reduction in the value of that information.
Evaluated their optimal compensation scheme, decentralization outperforms centralization if the value
generated is better, i.e. when eA ψA + µAC(eA ) is higher under decentralization than under centralization.
The, comparison is simple when considering that effort positively relates with the managers’ perceived
value of the information acquired, i.e. ψ̃. The value and the perceived value of information are greater
under decentralization with the only exception that consists on a β sufficiently high and a s sufficiently
low, i.e. β > β (s) (decreasing). This results follows directly from Section 3.1 (based on Alonso et al,
2008) and from Section 3.2
25
7.3
Analysis of Propositions 5 and 4
First oder conditions for Proposition in page 16. The first order conditions are
∂ eA
∂ µB 2
∂ µA 2
∂ eB
0
0
σ C(eA ) −
[ψA − µAC (eA )] = −
−
σ C(eB ) −
[ψB − µBC (eB )]
∂tA A
∂ µA
∂tB B
∂ µB
∂ ψA ∂ eA
0
+
[ψA − µAC (eA )] = 0
eA
∂ sA ∂ sA
∂ ψB ∂ eB
2
0
σB eB
+
[ψB − µBC (eB )] = 0
∂ sB ∂ sB
σA2
∂ ψ A ∂ eA
0
K
eA
+
[ψA − µAC (eA ) = 0,
∂ βA ∂ βA
∂ ψ B ∂ eB
0
2
0
K (βB ) − σB eB
+
[ψB − µBC (eB ) = 0.
∂ βB ∂ βB
0
(βA ) − σA2
(14)
(15)
(16)
(17)
(18)
First, we could replace the optimal effort choice µAC0 (eA ) = ψ̃ into all first order conditions. Once tA
is chosen, the decisions of (s, β , g) are described by Rantakari (2010). Given the convexity of µ(t), the
decision of tA depends directly on who makes effort choice. In either case there exists increasing relations
σ̃B2 (σA2 ) and σ̃A2 (σB2 ) such that product A is centralized if σA2 < σ̃A2 (σB2 ) and decentralized if σA2 ≥ σ̃A2 (σB2 ),
and product B is centralized if σB2 < σ̃B2 (σA2 ) and decentralized if σB2 ≥ σ̃B2 (σA2 ).
If local divisions control resources, the first order conditions are
∂tA
∂ ψA ∂ eA
0
+
[ψA − µAC (eA )]
= −
ϕ,
eA
∂ sA
∂ sA
∂ sA
∂tA
∂ ψB ∂ eB
σB2 eB
+
[ψB − µBC0 (eB )]
= −
ϕ.
∂ sB
∂ sB
∂ sB
σA2
∂ ψA ∂ eA
∂tA
K 0 (βA ) − σA2 eA
+
[ψA − µAC0 (eA )
= −
ϕ,
∂ βA ∂ βA
∂ βA
∂tA
∂ ψB ∂ eB
K 0 (βB ) − σB2 eB
+
[ψB − µBC0 (eB )
= −
ϕ,
∂ βB ∂ βB
∂ βB
(19)
(20)
(21)
(22)
with
ϕ
≡ σA2
∂ µA ∂ eA ∂ µB ∂ eB ψA − µAC0 (eA ) − σB2
ψB − µBC0 (eB ) .
∂tA ∂ µA
∂tB ∂ µB
(23)
We can replace the optimal allocation of time chosen by managers into ϕ. There exists increasing rela2 (σ 2 ) and σ̃ 2 (σ 2 ) such that decision making about product A is centralized if σ 2 < σ̃ 2 (σ 2 )
tions σ̃BM
A
AM B
A
AM B
2 (σ 2 ), and decision making about product B is centralized if σ 2 < σ̃ 2 (σ 2 )
and decentralized if σA2 ≥ σ̃AM
B
B
BM A
2 (σ 2 ).
and decentralized if σB2 ≥ σ̃BM
A
When σA2 > σB2 then ϕ > 0, and thus the firm prefers more resources in product A than already
implemented by the manager, i.e. tA < tA∗ . The RHS of Equations (31) and (33) are positive and the
26
RHS of Equations (32) and (34) are negative. The headquarters uses incentive alignment, s, and division
integration, β to affect the resource allocation choice of local managers. Then, when σA2 > σB2 , the
headquarters reduces incentive alignment of product A, i.e. ∇sA , increases incentive alignment of product
B, i.e. ∆sB , reduces integration of product A, i.e ∇βA , and increases integration of product B, i.e. ∆βB .
In other words, local managers put too much resources in product B, because they underestimate the
opportunity cost of resources. Then, the headquarters finds not so important to provide incentives for
information acquisition in this product and then decentralizing it appears more profitable.
7.4
Proof of Proposition 6
Proof of Proposition 6 in page 18.
Proof. Lets assume that σAB = σ̃AB . If this is the case and local managers allocate resources, the optimal
design of the organization requires decentralization decision making in both markets. Lest analyze the
case where µ < µ̃.
First, since local managers ignores the inefficiency ϕ, they allocate more resources to market B than
the headquarters would. Given tA∗ > tˆA , we have that µA (tA∗ ) < µA (tˆA ) and µB (1 − tA∗ ) > µB (1 − tˆA ). As
information is cheaper in market B, the headquarters would provide higher sB on that market, which
favors decentralization of decisions rights.
However, the headquarters recognizes the inefficient allocation of resources that generates ϕ. Changing the organizational design implies increasing sB which also favors decentralization of decision rights.
Given, this extra effect of correcting the inefficiency ϕ implies that the informational cost threshold µ̃
for centralization to be the optimal must be higher.
For the second part of the proof we need to realize that REMARK: It is cumbersome and time consuming to show that
b −1
∂ 2 µB (1 − t ∗ )
∂2
b
1+b
µ0.5 1 + H
< 0.
=
∂ b∂ H
∂ b∂ H
(24)
Now, notice that µB increases in b, but increases much more when the headquarters allocates resources, due to HH < H. Then, the value that the Headquarters given to the inefficiency increases with
b. The benefit of increasing sB is higher and then the threshold µ̃(ϕ) increases, which implies that there
are more decentralization due to loss of control.
7.5
Proposition
From the resource allocation problem we know that the informational cost in market B is µB = µ0.5b [1 +
−1
H 1+b ]b with H being different in the case that local managers allocate resources or headquarters allocate
resources. In both cases the slope respect to the variability ratio σAB is positive. However, the slope is
higher when the headquarters allocates resources as shown in the Figure 4 in page 21. The function is
27
concave respect to the ratio σAB .21
Lets see that µ̃(ϕ) > µ̃. From the problem without resource allocation we have that there exists a
cutoff µ̃ such that centralization outgains decentralization if µ > µ̃. Lets says that for exactly µ = µ̃ there
exists scent < sdec that are equally preferable. We show that in market B with externalities in resources
allocation the optimal values of s increase under both centralization and decentralization. Assume that
scent
and sdec
B
B are the optimal and equally preferable, then at those values,
ψ̃ 2 ∂∂ψs
ψ̃ 2 ∂∂ψs
h
i = µ̃ < h
i
− ψ̃ ∂∂ψs + ∂∂ψ̃s [ψ − ψ̃]
− ψ̃ ∂∂ψs + ∂∂ψ̃s [ψ − ψ̃ + ϕ ∗ Ξ]
with ϕ =
∂ µB
∂tB
h
∂ eA
∂ µA
i
C(eB )
∂ eB
−
[ψA − ψ̃A ] C(e
[ψ
−
ψ̃
]
and Ξ ≡
B
B
∂ µB
A)
∂ µB ∂ eB 2 0
∂tB ∂ sB σB C (eB )
∂ 2 µA 2
∂ 2 µB 2
2 σA C(eA )+ ∂t 2 σB C(eB )
∂tA
B
< 0, and ϕ ∗ Ξ < 0.
This implies first that s increases under both centralization and decentralization, and also that decentralization becomes more likely to be chosen. Then, for being indifferent between decentralization and
centralization the value µ̃(ϕ) must be higher, µ̃(ϕ) > µ̃.
7.6
Proof of Proposition 7.a
Proof of Proposition 7 in page 20. What we need to prove is the following:
2 (σ̃ 2 ) = σ̃ 2 (σ̃ 2 ) = σ̃ 2 (σ̃ 2 ) = σ̃ 2 (σ̃ 2 ).
i - The relations intersect at σ̃ 2 (µ) = σ̃BM
B
AM
A
2 (σ 2 ) ≤ σ̃ 2 (σ 2 ), and for σ 2 > σ̃ 2 (µ) we have that
ii - For σA2 > σ̃ 2 (µ) we have that σ̃ 2 (µ) < σ̃BM
B A
B
A
2
2
2
2
2
σ̃ (µ) < σ̃AM (σB ) ≤ σ̃A (σB ).
2 (σ 2 ) and for σ 2 < σ̃ 2 (µ) it is true that
iii - Similarly, for σB2 < σ̃ 2 (µ) it holds that σ̃A2 (σB2 ) < σ̃AM
B
A
2 (σ 2 ).
σ̃B2 (σA2 ) < σ̃BM
A
Proof. The first part is direct. If the variance in both markets are the same, the equilibria is completely
symmetric and the resource allocation is optimal. If variances are equal to σ̃ 2 (µ), they are in point that
shows indifference between centralizing and decentralizing decision rights in both markets.
2 (σ 2 ) < σ̃ 2 (σ 2 ). Assume, without loss of generality, that
For σA2 > σ̃ 2 (µ) we have that σ̃ 2 (µ) < σ̃BM
B A
A
σA2 > σB2 > σ̃ 2 (µ).
Given that σA2 > σB2 > σ 2 (µ), it is optimal for the manager to reallocated time from B to A, then
2 (σ 2 )
µB > µ. The manager strictly prefer to centralize project B if σB2 = σ 2 (µ), and thus σ̃ 2 (µ) < σ̃BM
A
holds.
2 (σ 2 ) ≤ σ̃ 2 (σ 2 ) let notice that at σ 2 = σ̃ 2 (σ 2 ) the manager would never strictly
For proving that σ̃BM
B A
B
B A
A
prefer to centralize project B. If the firm is indifferent between centralizing and decentralizing at σ̃B2 (σA2 ),
the optimal time choice tA∗ is given by condition (26) in page 30. The manager chooses time tA according
2
21 Given ∂ µB
∂ H2
> 0 and
∂ µB
∂H
< 0,
∂ 2 µB
∂ 2 µB
=
2
∂ H2
∂ σAB
∂H
∂ σAB
2
+
2
H
∂ µB
∂ µB
∂ µB ∂ 2 H
=
H
+
2
< 0.
2
∂ H ∂ σAB
∂H
(σAB )2 ∂ H 2
28
to equation (8) in page 13, then tA < tA∗ . If the manager reduces time to project A, then ∆µA and ∇µB
leading to ∆sB , ∇sA , ∆βB , and ∇βA . As explained by Rantakari (2010), a reduction in effort cost ∇µB
increases the effort for more information ∆eB so it is more damaging for the firm not to use that information accurately (ψ), and the firm increases sB , i.e. ∆sB increases the value of using that information
(∆ψ).
On the other hand, the fact that tA < tA∗ also raises the term ϕ which represents the externality omitted
by managers: then ∆sB , ∇sA , ∆βB , and ∇βA .
Summarizing, at σB2 = σ̃B2 (σA2 ) there are incentives to decentralize project B: a firm with lower cost µB
and higher incentive alignment sB performs better under decentralization. To find the threshold for being
indifferent between centralizing and decentralizing the variance of project B must decrease. This implies
2 (σ 2 ) < σ̃ 2 (σ 2 ). Doing a similar reasoning for σ 2 > σ 2 > σ̃ 2 (µ)) implying σ̃ 2 (σ 2 ) < σ̃ 2 (σ 2 ),
that σ̃BM
B A
B
A
A
AM B
A B
2 (σ 2 ) and σ 2 < σ 2 < σ̃ 2 (µ) implying σ̃ 2 (σ 2 ) < σ̃ 2 (σ 2 ),
2
σB < σA2 < σ̃ 2 (µ) implying σ̃A2 (σB2 ) < σ̃AM
B
B
B A
BM A
A
the proof is complete.
7.7
Proof of Proposition 7.b
Proof of Proposition 7 in page 20
Proof. By Lemma 4 we have that ∂∂µbA < 0 and ∂∂µbB > 0. Recall that H and HH was defined in Section
∂ µA
∂ µB
∂ µA ∂ H
∂ µA
dµB
∂ µB ∂ H
∂H
A
3.3. Then dµ
db = ∂ b + ∂ H ∂ b < 0, because ∂ H > 0 and ∂ b < 0. Similarly, db = ∂ b + ∂ H ∂ b > 0,
because ∂∂µHB < 0 and ∂∂Hb < 0. Then, µA decreases and µB increases.
Without loss of generality we focus on the case σA2 > σ̃ 2 (µ) and we analyze the increment in σ̃B2 (σA2 ).
Assume (σA2 , σB2 ) are such that the headquarters is indifferent between centralizing and decentralizing
project B, i.e. σB2 = σ̃B2 (σA2 ). If µB increases due to a greater b, now the headquarters reduces sB to
compensate the reduction in information acquisition. However, a reduction in sB affects more the value
of information under decentralization than under centralization. Now, the headquarters strictly prefers to
centralize project B and the cutoff σ̃B2 (σA2 ) must to be higher.
Finally, notice that the firm that was indifferent between centralizing and decentralizing at σ̃B2 (σA2 )
now strictly prefer to centralize project B. As µB increases, there is another cutoff σ̂B2 (σA2 ) (> σ̃B2 (σA2 ))
that is indifferent between centralizing and decentralizing. Notice that at higher cost µB the firm reduces
sB improving the relative performance of centralization respect to decentralization at σ̃B2 (σA2 ).
2 (σ 2 ) inThe threshold when resource allocation is delegated to managers also increases, i.e. σ̃BM
A
creases. The inefficiency ϕ that the headquarters want to correct with structure design also increases
but the threshold increases. First, because σ̃B2 (σA2 ) increases and, second, because the effect on higher
informational cost favors centralization of decision rights in market B.
In the appendix is proven that if µA decreases and µ 0 (t) increases, then ϕ becomes higher. However,
also derivatives ∂tsAA and ∂tsBA are affected (probably they are smaller now). Managers’ cutoffs also go up
2 (σ 2 ) > σ̃ 2 (σ 2 ).
σ̂BM
BM A
A
29
7.8
Moral Hazard
Introduce here an assumption over C(e) that guarantees that tˆ < t ∗ < tFB . We place an assumption that
guarantees that our comparison in this model is well defined. From Lemma 2 we can replace ψ =
C0 (e)
µC0 (e∗ ) and ∂∂µe = − µC
00 (e) in the bracket in the left hand side of Equation 7, which becomes C(e) +
C0 (e)
0 ∗
0
µC00 (e) [µC (e ) − µC (e)].
We analyze the problem for the case that moral hazard problem is sufficiently
important which guarantees that tˆ < t ∗ < tFB . The function C(e) = −(e + log(1 − e)) provides enough
convexity for the problem to be non-trivial.
Ass 1 For C(e) we assume that parameters satisfies the following inequality:
C0 (ê)C000 (ê)
C00 (e∗ ) ∂ e∗
0 ∗
0
0
0
[C
(e
)
−C
(
ê)]
+C
(
ê)
.
(25)
2C (ê) < 1 −
[C00 (ê)]2
C00 (ê) ∂ ê
h
i
e∗
(1−e)2
e
e
For C(e) = −(e + log(1 − e)) we require that 1−e
2 − (1−e
< (1 − 2e) 1−e
∗ − 1−e . Implying
∗ )2
that the condition holds if ê < e(e∗ ), where e(e∗ ) (defined by Equation (25)) is decreasing in e∗ .
This assumption requires that the informational cost parameter µ is not extremely high, i.e. that
the moral hazard problem is sufficiently important to be solved. if µ is too high, there is almost no
effort, neither ê nor e∗ is sufficiently high.
7.8.1
Derivatives
From manager’s time allocation foc − ∂∂tµAA σA2C(eA ) = − ∂∂tµBB σB2C(eB ) we calculate the derivatives:
∂ µA ∂ eA 2 0
∂tA
∂tA
∂tA ∂ sA σA C (eA )
< 0 and
= − ∂ 2µ
=
∂ 2 µB 2
A
2
∂ sA
∂
sB
2 σ C(eA ) +
2 σ C(eB )
∂tA
A
∂tB
B
∂ µB ∂ eB 2 0
∂tB ∂ sB σB C (eB )
2
2
∂ µA 2
σ C(eA ) + ∂∂tµ2B σB2C(eB )
∂t 2 A
B
>0
A
These derivatives are sufficiently small in absolute value since µ 00 (t) >> 0 and also u000 (t) << 0.
Moreover | ∂∂tsAB | > | ∂∂tsAA |, implying that sB increases more than the reduction in sA when local managers
decides how specialized to be in each project.
µA
From manager’s effort allocation ∂∂ esAA = ∂∂ψ̃sAA (ψ̃ +µ
.
)2
A
7.9
A
First order conditions of extension in Section 5
The headquarters objective function is:
n
o n
o
E[πiA ] + E[πiB ] = KA − [1 − (eA + αeB )ψA + µAC(eiA )] σA2 + KB − [1 − (eB + αeA )ψB + µBC(eiB )] σB2 .
When headquarters control resources, the first order conditions are
"
#
"
#
σA2
σB2
∂ eA
∂ µB 2
∂ eB
∂ µA 2
0
0
σ C(eA ) −
[ψA + 2 αψB − µAC (eA )] = −
σ C(eB ) −
[ψB + 2 αψA − µBC (eB )]
−
∂tA A
∂ µA
∂tB B
∂ µB
σA
σB
30
(26)
∂ ψA ∂ eA
σB2
0
(eA + αeB )
+
[ψA + 2 αψB − µAC (eA )] = 0
∂ sA ∂ sA
σA
σA2
∂ ψB ∂ eB
2
0
+
[ψB + 2 αψA − µBC (eB )] = 0
σB (eB + αeA )
∂ sB ∂ sB
σB
σA2
(27)
(28)
σB2
∂ ψA ∂ eA
0
+
[ψA + 2 αψB − µAC (eA )
= 0,
K
(eA + αeB )
∂ βA ∂ βA
σA
σA2
∂ ψB ∂ eB
0
2
0
K (βB ) − σB (eB + αeA )
+
[ψB + 2 αψA − µBC (eB )
= 0.
∂ βB ∂ βB
σB
0
(βA ) − σA2
(29)
(30)
Once tA is chosen, the decisions of (s, β , g) are described by Rantakari (2010). Given the convexity of
σ2
µ(t), the decision of tA depends directly on the relative variability σB2 , and on who makes effort choice.
A
In either case there exists increasing relations σ̃B2 (σA2 ) and σ̃A2 (σB2 ) such that product A is centralized
if σA2 < σ̃A2 (σB2 ) and decentralized if σA2 ≥ σ̃A2 (σB2 ), and product B is centralized if σB2 < σ̃B2 (σA2 ) and
decentralized if σB2 ≥ σ̃B2 (σA2 ).
If local divisions control resources, the first order conditions are
∂tA
σ2
∂ ψA ∂ eA
= −
+
[ψA + B2 αψB − µAC0 (eA )]
ϕ,
σA2 (eA + αeB )
∂ sA
∂ sA
∂ sA
σA
σA2
∂ ψB ∂ eB
∂tA
2
0
σB (eB + αeA )
+
[ψB + 2 αψA − µBC (eB )]
= −
ϕ.
∂ sB
∂ sB
∂ sB
σB
σB2
∂ ψA ∂ eA
0
+
[ψA + 2 αψB − µAC (eA )
K
(eA + αeB )
∂ βA ∂ βA
σA
σ2
∂ ψB ∂ eB
K 0 (βB ) − σB2 (eB + αeA )
+
[ψB + A2 αψA − µBC0 (eB )
∂ βB ∂ βB
σB
0
(βA ) − σA2
(31)
(32)
= −
∂tA
ϕ,
∂ βA
(33)
= −
∂tA
ϕ,
∂ βB
(34)
with
ϕ
≡ σA2
σ2
∂ µA ∂ eA
σ2
∂ µB ∂ eB
ψA + B2 αψB − µAC0 (eA ) − σB2
ψB + A2 αψA − µBC0 (eB ) .
∂tA ∂ µA
∂tB ∂ µB
σA
σB
(35)
We can replace the optimal allocation of time chosen by managers into ϕ. There exists increasing
2 (σ 2 ) and σ̃ 2 (σ 2 ) such that decision making in product A is centralized if σ 2 < σ̃ 2 (σ 2 )
relations σ̃BM
A
AM B
A
AM B
2 (σ 2 ), and decision making in product B is centralized if σ 2 < σ̃ 2 (σ 2 )
and decentralized if σA2 ≥ σ̃AM
B
B
BM A
2 (σ 2 ).
and decentralized if σB2 ≥ σ̃BM
A
When σA2 > σB2 then ϕ > 0, and thus the headquarters prefers more specialization in product A than
already implemented by the manager, i.e. tA < tA∗ . The RHS of Equations (31) and (33) are positive
and the RHS of Equations (32) and (34) are negative. The headquarters uses incentive alignment, s, and
division integration, β to affect the specialization choice of local managers. Then, when σA2 > σB2 , the
headquarters reduces incentive alignment of product A, i.e. ∇sA , increases incentive alignment of product
31
B, i.e. ∆sB , reduces integration of product A, i.e ∇βA , and increases integration of project B, i.e. ∆βB .
In other words, local managers put too much specialization in product B, because they underestimate the
benefit of specializing in product A. Then, the headquarters finds not so important to provide incentives
for information acquisition in this product and then decentralizing it appears more profitable.
8
Appendix 2: derivating all expressions
8.1
Close forms
In this appendix we construct the close form solutions of the expressions of ψ and ψ̃ under centralization
and decentralization that are the following:
(3(1 − s))(1 + 2β )2
,
(1 + 4β )(β (1 − 2s) + (3(1 − s))(2β + 1))
5s3 (1 − 2β ) + s2 (5β + 14(β 2 − 1)) + s(16β + 2β 2 + 13) − (4 + 6β 3 + 11β + 12β 2 )
,
ψd = −(1 − s)
((1 − s) + 2β )(1 − s + β )((1 − s + β )(1 − 2s) + 3(1 − s)(1 − s + 2β ))
(1 + 2β )((1 − s) + β (3 − 4s)
, and
ψ̃c = (3(1 − s))
((1 + 4β )(β (1 − 2s) + (3(1 − s))(2β + 1))
9β 2 + 11β + 13s2 β + 4 − 13s + 14s2 − 5s3 − 12sβ (2 + β )
ψ̃d = (1 − s)
.
((1 − s) + 2β )((1 − s + β )(1 − 2s) + 3(1 − s)(1 − s + 2β )))
ψc =
And its derivatives are under centralization:
3β (2β + 1)2
∂ ψc
=
≥ 0,
∂s
(1 + 4β )(8β (1 − s) − β + 3(1 − s))2
∂ ψ̃ c 3(2β + 1)(3(1 − s)2 + β 2 + 2(1 − s)β (10(1 − s) − 1) + 8β 2 (1 − s)(4(1 − s) − 1))
=
< 0.
∂s
(1 + 4β )2 (8β (1 − s) − β + 3(1 − s))2
(36)
And under decentralization are:
∂ ψd
β 2 ((1 − s)3 H1 + β (1 − s)2 H2 + β 2 sH3 + β 3 H4 + 12β 4 )
,≥ 0
=
∂s
((1 − s) + 2β )2 (1 − s + β )2 ((1 − s + β )(1 − 2s) + 3(1 − s)(1 − s + 2β ))2
(37)
with H1 ≡ 160(1 − s)3 − 120(1 − s)2 + 27(1 − s) − 2, H2 ≡ 680(1 − s)3 − 516(1 − s)2 + 1266(1 − s) + 10,
H3 ≡ 956(1 − s)3 − 744(1 − s)2 + 204(1 − s) − 16 and H4 ≡ 448(1 − s)3 − 360(1 − s)2 + 102(1 − s) − 8.
∂ ψ̃d
5(1 − s)5 (5(1 − s) − 2) + 6β (1 − s)4 (30(1 − s) − 11) + (1 − s)4 + 6β 2 (1 − s)3 (69(1 − s) − 20)
=
+
∂s
((1 − s) + 2β )2 (5(1 − s)2 − (1 − s) + 8β (1 − s) − β )2
6(1 − s)3 β + 4β 3 (1 − s)2 (104(1 − s) − 23) + 9(1 − s)2 β 2 + 8(1 − s)β 3 + 6β 4 (32(1 − s)2 − 8(1 − s) + 1)
< 0.
((1 − s) + 2β )2 (5(1 − s)2 − (1 − s) + 8β (1 − s) − β )2
32
8.2
Decision Making: using information
In this section we build the expression in equation (1) for Centralization and Decentralization before
communication outcome is calculated. To have the same terms we must replace the expression E[m2 ] =
[1 − (1 −V )]E[θ 2 ] and check all the terms.
Building the indirect function for E[π/m] given the equilibrium beliefs for m ≡ E[θ /m] and E[θ 2 /m] ≡
m2 . Remember that ΠA1 = K(β ) − [(aA1 − θ1A )2 + β (aA1 − aA2 )2 ] .
Given the optimal policy for decision making and the information transmission process the objective
function of each section can be expressed as a function of σ12 , σ22 , E[θ12 /m] and E[θ22 /m]. Let see how
we arrive to the optimal expression.
8.2.1
Centralization
The optimal decision making under centralization are:
1 + 2β
2β
E[θ1 /m1 ] +
E[θ2 /m2 ], and
1 + 4β
1 + 4β
2β
1 + 2β
E[θ1 /m1 ] +
E[θ2 /m2 ].
1 + 4β
1 + 4β
aC1 (m1 , m2 ) =
aC2 (m1 , m2 ) =
Then, replacing it in the expected profit of division 1 we get the following terms:
(aC1 − θ1 )2
2β (E[θ2 /m2 ] − E[θ1 /m1 ]) 2
=
E[θ1 /m1 ] − θ1 +
1 + 4β
= (m1 − θ1 )2 +
(aC1 − aC2 )2
4β 2 (m2 − m1 )2
2
+ 2β (m1 − θ1 )
(m2 − m1 )
(1 + 4β )
(1 + 4β )
4β 2 m22 + m21 − 2m2 m1
(m2 − m1 )
+ 2β (m1 − θ1 )
= m21 − 2m1 θ1 + θ12 +
2
(1 + 4β )
(1 + 4β )
2
E[θ1 /m1 , m2 ] − E[θ2 /m1 , m2 ]
m2 − 2m1 m2 + m22
=
= 1
1 + 4β
(1 + 4β )2
(38)
(39)
taking expectations we get that
E(aC1 − θ1 )2
=
E[θ12 ] − E[m21 ]
E[(aC1 − aC2 )2 ] =
E[m21 ] + E[m22 ]
(1 + 4β )2
+
4β 2 E[m22 ] + E[m21 ]
(1 + 4β )2
(40)
(41)
Now, lets build the expected profits E[Π] which add both terms weighted by 1 and β respectively. Notice
that taking ex-ante expectation the following terms are null E[m1 m2 ] = 0, E[(m1 − θ1 ) (m2 − m1 )] = 0.
33
Also note that E[m1 θ1 ] = E[m21 ]
"
K(β ) −
E[Π1 ] =
E[θ12 ] − E[m21 ] +
4β 2 E[m22 ] + E[m21 ]
(1 + 4β )2
1 + 3β
β
E[m21 ] +
E[Π1 ] =
K(β ) − E[θ12 ] −
(1 + 4β )
(1 + 4β )
E[m21 ] + E[m22 ]
+β
(1 + 4β )2
2
E[m2 ]
#!
or
(42)
REMARK: Since E[m1 ] = 0 and E[θ1 ] = 0, then E[m21 ] = V E[θ12 ] = E[θ12 ] − (1 −V )E[θ12 ]
And the division payoff is E[U1 ] = (1 − s)Π1 + sΠ2 .
8.2.2
Decentralization
The optimal decision making under decentralization are:
aD
1 (m1 , m2 , θ1 ) =
aD
2 (m1 , m2 , θ2 ) =
β
(1 − s)
β
1−s+β
θ1 +
m1 +
m2 , and
(1 − s + β )
(1 − s + β ) 1 − s + 2β
1 − s + 2β
1−s+β
(1 − s)
β
β
m2 +
m1 .
θ2 +
(1 − s + β )
(1 − s + β ) 1 − s + 2β
1 − s + 2β
Then, replacing it in the expected profit of division 1 we get the following terms:
2
(aD
1 − θ1 )
=
=
D 2
(aD
1 − a2 )
2
β
β
β
θ1 +
m1 +
(m2 − m1 )
(1 − s + β )
(1 − s + β )
1 − s + 2β
β2
β2
2β 2 (m1 − θ1 )(m2 − m1 )
2
2
(m
(m
−
θ
)
+
−
m
)
+
1
1
2
1
(1 − s + β )2
(1 − s + 2β )2
(1 − s + β )(1 − s + 2β )
2
2
2
2
2
2
β m1 − 2m1 θ1 + θ1
β m2 + m1 − 2m2 m1
2β 2 (m1 − θ1 )(m2 − m1 )
+
+
(43)
(1 − s + β )2
(1 − s + 2β )2
(1 − s + β )(1 − s + 2β )
=
−
(1 − s)(m2 − m1 ) 2
(1 − s)(θ1 − θ2 )
β
=
+
(1 − s + β )
(1 − s + β )
1 − s + 2β
2
(1 − s)
(1 − s)2 (m22 + m21 − 2m1 m2 )
β2
2
2
=
(θ
+
θ
−
2θ
θ
)
+
1
2
1
2
(1 − s + β )2
(1 − s + β )2
(1 − s + 2β )2
2β
(1 − s)2
(44)
+
(m2 θ1 + m1 θ2 − m2 θ2 − m1 θ1 )
(1 − s + β )2 1 − s + 2β
Now, lets build the expected profits E[Π] which add both terms weighted by 1 and β respectively.
Notice that taking ex-ante expectation the following terms are null E[m1 m2 ] = 0,E[m1 θ2 ] = 0, E[m2 θ1 ] =
0,E[θ1 θ2 ] = 0, E[(m1 − θ1 ) (m2 − m1 )] = 0. Also note that E[m1 θ1 ] = E[m21 ]
34
by parts22
2
E[(aD
1 − θ1 ) ] =
=
D 2
E[(aD
1 − a2 ) ] =
β 2 E[θ12 ]
(1 − s + β )2 − (1 − s + 2β )2 2
β2
2
+
β
E[m
]
+
E[m22 ],
1
(1 − s + β )2
(1 − s + β )2 (1 − s + 2β )2
(1 − s + 2β )2
β 2 E[θ12 ]
2(1 − s) + 3β
β2
3
2
−
β
E[m
]
+
E[m22 ]. (45)
1
(1 − s + β )2 (1 − s + β )2 (1 − s + 2β )2
(1 − s + 2β )2
[2(1 − s) + 3β ] β (1 − s)2
(1 − s)2
2
2
(E[θ
]
+
E[θ
])
−
(E[m22 ] + E[m21 ]).
(46)
1
2
(1 − s + β )2
(1 − s + β )2 (1 − s + 2β )2
And expected profits E[Π1 ] are
2
β (1 − s)2
β + (1 − s)2
β + (1 − s)2
β (1 − s)2
2
2
2 [2(1 − s) + 3β ] β + (1 − s)
2
2
K(β ) −
E[θ
]
+
β
E[θ
]
−
β
E[m
]
+
β
−
E[m
]
. (47)
2
1
1
2
(1 − s + β )2
(1 − s + β )2
(1 − s + β )2 (1 − s + 2β )2
(1 − s + 2β )2 (1 − s + β )2
Since E[θ ] = 0, E[θ 2 ] = V (θ ) =
8.3
2
θ
3
= σ 2.
Imperfect signals
Given the information acquired, specified by the effort e, each manager have a posterior about about
the true value. That is, given the realization of the signal θ̂ (that coincide with the true value of θ with
√
√
probability e), the manager’s posterior is θ̃ = E[θ /θ̂ ] = eθ̂ .23 This posterior is the best guess that
managers have over the true value, and they are going to set a product which closest to the bliss point.
With the posterior, the headquarters have also inferences after she receives managers’ reports that are
E[θ̃12 /m] and E[θ̃22 /m] The optimal decision making under centralization are:
aC1 (m1 , m2 ) =
aC2 (m1 , m2 ) =
2β
1 + 2β
E[θ̃1 /m1 ] +
E[θ̃2 /m2 ], and
1 + 4β
1 + 4β
2β
1 + 2β
E[θ̃1 /m1 ] +
E[θ̃2 /m2 ].
1 + 4β
1 + 4β
(1−s+β )2 −(1−s+2β )2
2(1−s)+3β
2
D
D 2
the profit function we have E[(aD
1 − θ1 ) ] + β E[(a1 − a2 ) ]. Note that (1−s+β )2 (1−s+2β )2 = −β (1−s+β )2 (1−s+2β )2 .
23 Note that we care about the mean of the posterior and not the posterior distribution. The posterior mean distribution is
√ √
uniform in [− eθ , eθ ].
22 In
35
Then, replacing it in the expected profit of manager in country 1 we get the following terms:
!2
2β E[θ̃2 /m2 ] − E[θ̃1 /m1 ]
E[θ̃1 /m1 ] − θ1 +
,
1 + 4β
(aC1 − θ1 )2 =
2
= (m1 − θ1 ) +
= (m1 − θ1 )2 +
(aC1 − aC2 )2
=
4β 2 (m2 − m1 )2
+ 2β (m1 − θ1 )
2
(1 + 4β )
4β 2 m22 + m21 − 2m2 m1
2
(1 + 4β )
E[θ̃1 /m1 , m2 ] − E[θ̃2 /m1 , m2 ]
1 + 4β
2
=
(m2 − m1 )
,
(1 + 4β )
+ 2β (m1 − θ1 )
(m2 − m1 )
.
(1 + 4β )
m21 − 2m1 m2 + m22
.
(1 + 4β )2
(48)
(49)
Notice that taking ex-ante expectation the following terms are null E[m1 m2 ] = 0, E[(m1 − θ1 ) (m2 − m1 )] =
0. Also note that E[m1 θ1 ] = E[m21 ]. Also, note that m2 = E[θ̃1 ]2 = e1 θ̂12 and then E[m21 ] = e1 E[θ̂12 ] =
2
e1V1 θ31 . Taking expectations
E[(aC1 − θ1 )2 ]
2
= E[(m1 − θ1 ) ] +
E[(aC1 − aC2 )2 ] =
4β 2 E[m22 ] + E[m21 ]
(1 + 4β )2
E[m21 ] + E[m22 ]
.
(1 + 4β )2
Building the expected profits E[Π] which add both terms weighted by 1 and β respectively.
β
2
2
2
(E[m1 ] + E[m2 ])
E[Π1 ] =
K(β ) − E[(m1 − θ1 ) ] +
(1 + 4β )
(50)
(51)
(52)
where the term E[(m1 − θ1 )2 ] is the difference between the real realization of θ and what the HQ believes
of θ̃ after receiving the report.
2
√
√
E[(m − θ )2 ] = E[ m − eθ̂ + eθ̂ − θ ]
2
√
√
√ 2
= E[ eE[θ̂ /m] − eθ̂ ] + E[ eθ̂ − θ ]
2
√
√ 2
√
with E[ eE[θ̂ /m] − eθ̂ ] = (1 −V )eE[θ 2 ] and E[ eθ̂ − θ ] = (1 − e)E[θ 2 ].
√ √
REMARK: E[ m − eθ̂
eθ̂ − θ ] = 0.
proof
√ √
√
√
√
E[ m − eθ̂
eθ̂ − θ ] = E[m eθ̂ − θ ] − E[ eθ̂
eθ̂ − θ ]
√
√
√
= E[m eθ̂ − θ ] − E[ eθ̂
eθ̂ − θ ]
√
The first term is not problematic E[m eθ̂ − θ ] = 0, but the second term deserves a little more of
√
√
√
attention to notice that E[ eθ̂
eθ̂ − θ ] = E[eθ̂ 2 ] − E[ eθ̂ θ ] = 0. I prove that = E[eθ̂ 2 ] = eE[θ 2 ] =
36
√
E[ eθ̂ θ ]:
√
√ √
√ √
√ √
E[ eθ̂ θ ] = E[ e eθ θ + (1 − e) exθ ] = E[eθ 2 + (1 − e) exθ ] =
√ √
= eE[θ 2 ]
eE[θ 2 ] + (1 − e) e
E[xθ ]
| {z }
=0 independent
Now, we show the other part.
√
√
√
√
E[eθ̂ 2 ] = E[ eeθ 2 + (1 − e)ex2 ] = eeE[θ 2 ] + (1 − e)eE[x2 ] = eE[θ 2 ]
(53)
The proof is complete.
2
REMARK: Since E[m1 ] = 0 and E[θ1 ] = 0, then E[m21 ] = e1 E[θ̂12 ] = e1V1 θ31
2
!
β
1 + 3β
+ e2V2
1 − e1V1
(1 + 4β )
(1 + 4β )
θ
K(β ) −
3
E[Π1 ] =
And the division payoff is E[U1 ] = (1 − s)Π1 + sΠ2 . Adding up for both divisions
2
!
1 + 2β
1 + 2β
1 − e1V1
− e2V2
(1 + 4β )
(1 + 4β )
!
2
θ
2K(β ) −
3
E[Π1 ] + E[Π2 ] =
2K(β ) −
=
θ
[1 − e1 ψ1 − e2 ψ2 ]
3
(54)
A similarly analysis accounts for the case under decentralization. For further details you can see
Proposition 5 in Rantakari (2010).
8.4
8.4.1
Strategic Communication: transmission
Building E[m2 ] for any finite partition.
I must find the payoff for any finite partition. The general formula for a N j partition is:24
1
E[m ] =
2θ
2
By uniform distribution we get that
∑
Z dj d j + d j−1 2
R d j d j +d j−1 2
d j−1
2
j∈N j d j−1
2
dθ .
dθ = (d j − d j−1 )
(55)
d j +d j−1
2
2
.
(1−2s)(1−s+β )
under des(1−s)+β
i
N
i
N
x (1+y )−y (1+x )
0 ≤ i ≤ N with x =
xN −yN
From proposition 2 we have d j,i+1 − d j,i = d j,i − d j,i−1 + 4bd j,i with b ≡
centralization and b ≡ (1−2s)β
Also di = θ
1−s+β under centralization.
p
p
2
(1 + 2b) + (1 + 2b) − 1 and y = (1 + 2b) − (1 + 2b)2 − 1. Property xy = 1 and x > 1 applies to both
24 Note that m2
=
d j +d j−1
2
2
is expectation given a particular signal. Then, E[m2 ] is the ex-ante expectation is the expression.
37
cases and are used all along the algebra. Replacing it in the expression above we have (Equation 27 in
Alonso, Dessein and Matouscheck 2008).
2
2
E[m ] =
=
(x3N j − 1)(x − 1)2
(xN j + 1)2 (x + 1)(1 + y)
−
,
(xN j − 1)3 (x2 + x + 1)
xN j (xN j − yN j )2
2
(x3N j − 1)(x − 1)2
θ
xN j (x + 1)2
−
.
3 (xN j − 1)3 (x2 + x + 1) x(xN j − 1)2
θ
3
(56)
For the case with infinite partitions we have
2
θ (x + 1)2
2 1+b
=θ
= V σ 2 = [1 − (1 −V )]σ 2 .
lim E[m ] =
N j →+∞
4 (x2 + x + 1)
3 + 4b
2
(57)
if sender of message is truly believe, then he will report exaggerating. For example, if s = 0 they
β
have incentives to misreport θ R − θ = 1+β
θ under centralization and θ R − θ = 1+β
β θ under decentralization. Then reports are: θ R ≡
decentralization.
(1+4β )(1+2β )
θ
(1+2β )2 +β
=
1+2β
1+β θ
38
under centralization and θ R ≡
1+2β
β θ
under