Divide and Conquer? Decentralized Firm Structure May
Promote Cooperation
Job Market Paper
Michal Goldberg∗
December 12, 2013
Abstract
I consider a model in which an entrepreneur’s objective is to maximize cooperation in a
multidivisional organization. In order to do so, the entrepreneur can choose the number of
production units in a profit center. The profit centers may rely on their long-term relationships with one another for cooperation. I use a repeated games setting that endogenizes
the cooperation decisions. I show that in certain cases, in contrast to the common wisdom,
decentralization may promote cooperation. In particular, I show that the stronger the interdependencies among units are, the smaller the optimal number of units in each profit
center. I further use this setting to examine efficiency gains of mergers and acquisitions.
Efficiency gains of a merger are below the value of synergies between the acquiring firm
and the target when the ties are not strong enough to sustain cooperation. However, the
potential synergy between an acquiring firm and a target may be an underestimate of the
efficiency gains of a merger when the ties are sufficiently strong to cause a firm to shift from
a noncooperative to a cooperative equilibrium. I characterize these cases and draw policy
implications.
∗ Carnegie
Mellon University, [email protected]. I thank Heski Bar-Isaac, Serguey Braguinsky, Boyan Jo-
vanovic, Alessandro Lizzeri, Yaron Yehezkel, and seminar participants at Carnegie Mellon University for valuable
comments. All errors are mine.
1
1
Introduction
The economics of organizations literature has long been concerned with the optimal structure of
firms (e.g., [Cremer, 1980], [Williamson, 1967], and [Hart and Moore, 2005]). However, while in
other fields dynamic considerations are a key in understanding economic forces, the models that
analyze the optimal structure of firms rely mostly on static considerations. Dynamic incentives
are particularly important when studying the relationship among different parts of a firm because
one of the attributes that distinguishes a large firm from a collection of firms is the long-term
relationships among its parts. This paper studies the optimal firm’s structure, and in particular,
the optimal level of decentralization, that would promote the highest level of cooperation among
different parts of a firm in a dynamic setting.
The common wisdom is that centralization enhances intra-firm cooperation while decentralization provides better incentives within a unit (e.g., [Roberts, 2007]). The optimal firm
structures then balances cooperation against incentives. In contrast, this paper shows that
decentralization may increase intra-firm cooperation.
In the model, intra-firm cooperation generates profits to the firm because potential synergies
can be realized. Some of these synergies can be realized efficiently by using a contract and
transfer pricing (such as supply of a good that does not require special adaptation), or by coordinating a move to a more efficient equilibrium (such as coordination on a standard adaptation).
However, some other synergies cannot be efficiently realized using transfer pricing or through
within-period coordination, but can be realized by an implicit contract, relying on the long-term
relationships among parties in a firm.
The structure of the firm affects the incentives of the units to cooperate with each other.
In this paper, I study the optimal firm structure that maximizes profits from cooperation. In
particular, I study the optimal level of centralization that maximizes cooperation. In the most
decentralized firm structure, each production unit forms an independent profit center, and all
decisions are made at the production unit level. In more centralized firm structures, a few
production units are grouped together to form a profit center. All units in a profit center
2
maximize the profits of the profit center, and all decisions are made at the profit center level. I
allow for a range of profit center sizes, to account for various levels of centralization. To avoid
the obvious case in which a perfect centralization achieves the highest degree of cooperation, i.e.
that all decisions are made by a single authority, I assume that a firm has to be divided into at
least two profit centers.1
When dynamic incentives are considered, decentralization may promote cooperation, as a
profit center that is small and can achieve little without cooperation may anticipate gaining
more from cooperation than a larger profit center. However, centralization may reduce the cost
associated with cooperation, as the larger the size of the profit center, the fewer units outside the
profit center, and consequently, a larger profit center is required to accommodate fewer needs.
The optimal level of centralization is determined by balancing this trade off.
More specifically, I initially assume a model in which the number of production units is
exogenous, but the number of units in each profit center is determined by the entrepreneur.
Cooperation is essential because each period each of the production units requires some need
that one or more of the other units may be able to accommodate. This could be a pricing or
output strategy in a market shared by more than one unit, adoption of some technology standard,
or information regarding some market that is of interest to multiple units. Accommodating that
need is costly, but the benefit to the requester exceeds the cost. That is, accommodating needs
is always beneficial for the firm, but in a static world if the units maximize only their own
profit, and in the absence of additional incentives, the needs of other profit centers are never
accommodated, as this is a prisoner’s dilemma game. However, I assume that the units are in
long-term relationships among themselves. The units may be able to utilize these relationships
and sustain cooperation.
Although in a static world the optimal firm structure is the most centralized possible in
this situation, in a dynamic setting, the optimal firm structure varies with the probability that
1 The
exact reason for why organizing the firm in one profit center is not possible is beyond the scope of this
paper. There could be many reasons for that, for example, monitoring may not be adequate when a large firm
is organized into a single profit center.
3
some unit is able to accommodate a need of another unit, (the interdependence degree). The
interdependence degree can be viewed as a measure of the interdependency of the units in a
firm with each other.
I first analyze the case in which the ability to accommodate needs is identically and independently distributed across production units. I show that when the interdependence degree
among the units is higher, the optimal firm structure is less centralized. The reason is that an
increase in the interdependence degree among production units increases the probability that a
large profit center can accommodate its needs on its own. This reduces the continuation payoff
of cooperation and hence reduces the incentive of a profit center to cooperate. Meanwhile, the
average cost of cooperation is lower as the profit center grows larger for two reasons. First, each
profit center has to take into account the cost of accommodating fewer needs in total. Second,
units that are in the same profit center are able to coordinate among themselves offerings to
accommodate certain needs, and thus are able to avoid considering the cost of accommodating
the same need. These effects are prominent when the interdependence among production units
is low, and hence the profit centers expect to gain little from cooperation, but are still required
to be willing to accommodate all needs they can.
In addition, I find that when the interdependence of units is identical, the optimal number
of units in a profit center does not change with the size of the firm. An increase in the number
of production units in a firm increases the cost of cooperation as well as the reward from
cooperation as each need is more likely to be accommodated. However, in the symmetric case,
these two effects offset each other. A by-product of this result is that the number of profit
centers increases with the size of a firm.
These results have a few implications. First, one should expect that when the units in a firm
are highly interdependent of each other, they will be organized in a decentralized way. This is
a counterintuitive result because in a static model, interdependent units should form a profit
center together. Another implication is that the number of profit centers should be higher in
larger firms.
4
Next, I consider alternative assumptions regarding the distribution of the degrees of interdependence. Because it is complicated to analyze the general case for this problem, I consider
extreme assumptions in order to understand the economic forces present in each case. I find that
in cases where the units are heterogeneous and one unit is very likely to be able to accommodate
all needs, the most decentralized structure achieves the best cooperation. The reason is that the
incentive of a profit center that includes such a unit is even lower than in the symmetrical case
because a large part of the profit center’s needs can be accommodated by the dominant unit.
Furthermore, in order to analyze the effect of correlation in the interdependence across units,
I analyze two extreme cases of perfect negative and perfect positive correlations. In the extreme
negative correlation case I assume that at most one unit is able to accommodate any given need.
The extreme positive correlation case is the opposite, and if one unit is able to accommodate
some need, all other units are able to accommodate it as well. In both of these cases, the most
centralized structure possible achieves the best cooperation. I conjecture from this that in order
for decentralization to promote cooperation, there must be a chance that more than one unit
would be able to accommodate at least some of the needs, but the units cannot be perfect
substitutes for each other.
A next natural step is to treat the interdependence among units as an endogenous choice.
To do so, I characterize the maximum cost an entrepreneur may be willing to bear to increase
the level of interdependence among the units in a firm. At any level of interdependence among
the units, it is optimal to pay a positive cost to increase the level of interdependence among
the units. However, the maximum cost that an entrepreneur will be willing to pay to increase
the level of interdependence varies. In particular, this maximal cost increases at the level of
interdependence around the critical level of interdependence, above which full cooperation can
be sustained.
One way to increase interdependence among units is to extend the boundaries of the firm
by mergers and acquisitions. Synergies are often cited as a reason for mergers and acquisitions,
and are sometimes used to justify the approval of mergers and acquisitions ([U.S. Department
5
of Justice and the Federal Trade Commission, 2010]). However, the effect of a potential synergy
on the performance of the merged firm varies. For example, Moeller, Schlingemann, and Stulz
[2004] find that the larger the firm size the lower the gains of the shareholders of the acquiring
firm.
This paper provides a new way to evaluate the increased economic efficiency in mergers and
acquisitions. The value of the synergy between the target firm and the acquiring firm may not
correctly represent the added economic efficiency of the merger. Although there may be the
potential for a synergy, it may not be realized because the target and acquiring firm cannot
sustain cooperation. In particular, I show that when cooperation cannot be sustained in the
acquiring firm prior to the acquisition, and the target firm has the same potential synergies with
the acquiring firm as the rest of the units in the firm, cooperation will not be sustainable in the
merged firm, and the synergies will not be realized. In that case, the added economic efficiency
of mergers can be below the value of the potential synergy.
However, there are cases in which the added economic efficiency of mergers exceeds the
potential synergy between the acquiring firm and the target. That can happen when the target
has very strong potential synergies with the acquiring firm. In that case, the acquisition can
lead the entire firm to shift from a noncooperative to a cooperative equilibrium. I show that
when all units in the acquiring firm are symmetrical in their interdependence of each other,
and the potential synergies with the target are stronger, shifting to a cooperative equilibrium is
more likely when the size of the acquiring firm is smaller. This is consistent with Moeller et al.
[2004].
The rest of this paper proceeds as follows. Section 2 discusses the related literature. Section
3 describes an example of interdependencies among units in a firm. Section 4 describes the
formal model and discusses the assumptions made. Section 5 analyzes the equilibrium when
the interdependence degrees among units are independent. Section 6 analyzes the equilibrium
under a few alternative assumptions regarding the correlation of interdependencies. Section 7
relaxes the assumption that the interdependencies are exogenous, and discusses the value of
6
an increase in interdependencies among units in a firm. Section 8 analyzes one way in which
interdependencies could be increased – namely by mergers and acquisitions, characterizes the
cases in which an acquisition could increase firm value by more than the synergies between the
acquiring firm and the target, and draws policy implications. Section 9 concludes. Finally, all
proofs are in the appendix.
2
Related Literature
There is a large body of literature on organizational design (see [Harris and Raviv, 2002] for a
survey). However, the existing literature primarily uses static considerations, whereas in this
paper dynamic incentives are in the basis of the analyses. A major focus of the literature
on organizational design is information transmission and aggregation within an organization
(e.g., [Bolton and Dewatripont, 1994], [Harris and Raviv, 2002], [Maskin et al., 2000], [Baccara
and Bar-Isaac, 2008], [Stein, 2002], [Qian et al., 2006], and [Alonso et al., 2008]. My model
can be applied to information transmission within an organization, if the need that can be
accommodated is information. However, the model is not specific for information and may also
be useful in analyzing other attributes.
A part of the literature on organizational design is concerned with the optimal functions of
each unit – that is, a unitary form (”U form”) versus a multidivisional form (”M form”). Another
part of the literature (e.g., [Alonso et al., 2008], [Stein, 2002]) essentially takes the optimal
function of each unit as given, and is concerned with the optimal level of decentralization. This
paper adds to the latter. I take the function of each unit, and the degree of interdependence
among them as given and analyze the optimal level of decentralization.
Alonso et al. [2008] is similar to this paper in the sense that in some cases, better cooperation
can be achieved through a less centralized firm structure. However, their result was driven by
how organizations aggregate dispersed information and relied on the divisions’ interests being
sufficiently aligned to each other. My model does not assume any correlation in the interests of
the unit managers, but achieves cooperation through long term relationships among the units.
7
The literature that uses repeated games in order to analyze the optimal structure of a firm
includes Baker, Gibbons, and Murphy [2002]. Baker et al. analyzed the optimal ownership
structure of a firm when a downstream party uses a good produced by an upstream party. The
ownership structure affects the outside options of the contracting parties, and as a result affects
the level of cooperation among them. The difference between Baker et al. and my model is that
my model analyzes horizontal relationships, while in Baker et al. the units are related vertically.
My model relies on the fact that the units depend mutually on each other.
3
An Example of Interdependence Among Units
This section will describe an example of interdependence among units in a firm that will apply
to this model. Consider, for example, four of the subsidiaries of Amazon.com: Quidsi, Zappos,
Woot, and A9.com. Quidsi is an online retailer for household items; Zappos is an online retailer
that specializes in shoes; Woot is an online daily deals retailer; A9.com is an IT company that
specializes in search technology, used by Amazon.com, some of its subsidiaries, and other online
retailers, and in online advertisements. There are many potential sources for synergy among
these subsidiaries. The three online retailers could coordinate prices, share shipping facilities
and shipping contracts, advertise on each other’s websites, and adopt user interface technology
developed by one of the other companies. A9.com could adapt their software for the use of one or
more of the retailers. It could also benefit from obtaining sales data from the retailers in order to
improve its advertising. Part of these synergies can be easily realized using explicit agreements
and transfer pricing. Other potential synergies are a result of a coordination problem, in which
cooperation benefits all parties within the period, such as in taking advantage of economies of
scale in order to reduce shipping costs. Finally, some synergies cannot be realized using the
methods above, but could be realized by using the long-term relationships among the units.
One of these synergies is user interface technology. For example, suppose that Quidsi is
seeking to improve its search engine. An efficient way to do so may involve software developed
by A9. One could argue that Quidsi should simply buy the software from A9. However, as
8
pointed out by Holmstrom and Tirole [1991], if the subsidiaries rely solely on transfer pricing
for this transaction, some inefficiencies could occur. For example, A9 might not invest enough
in relationship-specific capital, to increase the value of its product to the outside market, which
would increase the price it could charge Quidsi. Contracting on the exact features of the product
in all contingencies could be costly, which eliminates a detailed contract as a solution to the
problem. In terms of the model, the accommodation required by Quidsi is to have Quidsi invest
in a search engine that suits specifically the needs of Quidsi.
It is important for this model that the relationship between A9 and the rest of the subsidiaries
is not purely vertical, in the sense that some of the needs of A9 could be accommodated by
other subsidiaries. An example of this would be sales data the retailers have that A9 could use
to advance its online advertising business. Sales data could be useful for A9 to, for example,
predict which consumers would be interested in a particular product. Using a contract and
transfer pricing might not be possible in this case because of legal issues. The accommodation
required here by A9 is to receive sales data from one or more of the retailers. This is costly to
the retailers, but the benefit to A9 might exceed this cost. It should be noted here, that in this
example, unlike with search engine improvement, it is likely that some of the subsidiaries would
be substitutes for each other because if, for example, A9 is interested in kids’ shoes sales data,
it could get them from either of the subsidiaries considered above.
4
The Model
An entrepreneur seeks an optimal way to organize a firm. The firm consists of n production
units. These are the most basic units of production, and cannot be further divided. However,
the entrepreneur can choose whether the decisions will be made at the production unit level
(decentralization), or at a higher level centralization. In the decentralized structure, each production unit is an independent division, and makes decisions on its own. In the centralized
structure, each k production units are grouped together to form a profit center, and decisions
are made at the profit center level.
9
The horizon is infinite. Time is discrete and is discounted by a common discount factor
of δ. To focus the attention on cooperation within the firm, I assume that the value of the
firm is driven solely by cooperation among the production units. Each period each of the
production units develops some need. Unit i is able to accommodate a need of division j in
a given period with probability pij , the interdependence degree between production units i and
j.
2
While accommodating this need is costly, the benefit to the requesting production unit
exceeds this cost. The ability to accommodate a need is a public knowledge. However, the units
cannot transfer money among them, so that market for these needs (i.e. transfer pricing) is
unavailable. Also, the headquarters cannot directly motivate unit managers to accommodate
these needs (e.g., the compensation contract of a unit manager cannot depend on whether the
unit he manages had accommodated a certain need).
3
More precisely, I assume that period t profit of production unit i is:
n
X
(atj,i b − ati,j c)
(1)
j=1
where ati,j = 1 if production unit i accommodates a need of production unit j at period t.
While it is possible that more than one production unit is able to accommodate some need, the
benefit materializes at most once. That is,
Pn
j=1
ati,j ≤ 1. b and c are constants that satisfy
b > c > 0, and stand for the benefit from accommodating a need and its cost respectively.
Accordingly, the total profit of the firm at period t is:
n
n X
X
(ati,j b − ati,j c).
(2)
i=1 j=1
The profit of a profit center is the sum of the profits of its production units. I assume
each production unit in a profit center maximizes the profit of the profit center it belongs to.
2 Different
assumptions regarding the relations between the distribution of pij across production units will
be made in the following sections.
3 The
setting is adopted from the trading favors literature ([Mobius, 2001], [Hopenhayn and Hauser, 2004]).
However, this literature is mostly concerned with the case the opportunity to provide a favor is not a public
knowledge, and consequently it is impossible to sustain a simple Nash reversion equilibrium. In this paper the
opportunity to provide a favor is a public knowledge, but the trading favors setting is used to analyze cooperation
in different firm structures.
10
In that I assume away free rider problems that bring about decentralization, but rather, any
decentralization in this model can emerge only if it achieves better cooperation.
4.1
Discussion of Assumptions
An important assumption is that profit centers cannot use transfer pricing to trade among
themselves. One might argue that an intra-firm market would mitigate the cooperation problem,
and will insure that the allocation is efficient. However, informational frictions may make the
transaction costs too high, which in turn will prevent accommodations of some of the needs.
Holmstrom and Tirole [1991] analyze a model in which the existence of an intra-firm market
results in an inefficient allocation. Incentive problems arise in their model because of unobserved
managerial investment in cost reduction. Their analysis distinguishes between the case in which
profit centers are allowed to trade in outside markets and the case in which profit centers are
not allowed to trade in outside markets. If profit centers are allowed to trade in outside markets,
the market may monitor the quality of the product, but managers might invest in quality that
is comparable to the market instead of relationship-specific investments. In addition, managers
might invest in activities that enhance their outside options that may not be efficient otherwise.
If the profit centers are not allowed to trade in outside markets, a profit center might not have
sufficient incentive to invest in quality and cost reducing technology.
Although the assumption in this model is that the profit centers cannot use pricing mechanisms, the model can be applied only for the needs, or the part of them, that cannot be
accommodated otherwise, due to the reasons described above or due to any other inefficiency.
That is, if it is assumed that the profit centers can use transfer pricing, there will be needs that
are not accommodated in equilibrium because of different inefficiencies. These needs may still
be accommodated if the profit centers utilize the long term relationships among themselves, as
discussed in this paper.
It has been also assumed that a firm cannot consist of a single profit center; a firm has to
be divided into at least two profit centers. An analogous assumption is that the compensation
11
of profit center managers cannot be tied only to the performance of the entire firm; it must, at
least partially, be tied to the performance of a smaller unit. It should be noted that without this
assumption in place, full cooperation is always achieved, regardless of organizational structure.
This assumption would be justified if in case all employees are compensated according to
the performance of the entire firm, a free rider problem occurs. As a result, the profit center
managers do not maximize the value of their firm. Although this is not specifically modeled, the
underlying assumption is that there is a bound on the size of a profit center such that only if a
profit center is larger than this bound a free rider problem occurs. One can think of a situation
in which a manager is able to monitor the actions of a group of a particular size, but is unable
to monitor the actions of a larger group.
This assumption is most appropriate for large firms. If a firm is small enough such that tying
the compensation of all employees to the performance of the firm ensures perfect cooperation,
the model is not relevant. The model is more relevant, however, for larger firms in which free
riding might exist.
The fact that firms compensate their employees according to performance measures of units
that are smaller than the entire firm supports this assumption. Firms use profit centers, cost
centers, expense centers, and investment centers in order to evaluate the performance of its
units. Many firms use these measures in order to compensate their employees. For example,
Briggs & Stratton adopted in 1990 a system in which 40% of the bonuses of managers are based
on a measure of divisional performance, 40% on a measure of corporate performance, and 20%
on a measure of personal performance.
The paper analyzes different assumptions regarding the interdependence across units. Since
the general case is complicated to analyze, specific assumptions are made. However, in order for
the framework to be valid, it is sufficient that the interdependence among production units is
bilateral in the sense that each profit center may require a need that some other unit may be able
to accommodate. This does not require that the same two units will be able to accommodate
each other’s needs, but that each division can be both the recipient and the sender of some
12
needs. The framework does not apply in case that the relationships among units are purely
vertical, as in a case in which a unit is able to accommodate another unit needs but cannot
benefit from cooperation with other units.
5
The Optimal Firm Structure in Case the Interdependence Degrees Are Independent
This section analyzes the optimal structure of a firm in the case in which the interdependence
level is identically and independently distributed across production units. The firms are identical ex ante in their relations to each other, and the ability of one division to accommodate
needs of other units does not increase nor decrease the probability any other division is able to
accommodate that need as well. While this assumption is extreme, its purpose is to illustrate
the optimal structure of a firm when units are interchangeable with each other in some of their
capacities. That could be the case if the firm is organized in an M-form and units produce
goods that compete in the same market. For example, suppose that ”Opel” subsidiary of GM
would like to enter the North American market, and requires some accommodation from a car
manufacturer. This could be, for example, withholding of production for some time in order
to allow for the entry. In this case, ”Buick” and ”Chevrolet” could have similar prospects in
accommodating the requests of ”Opel”.
I consider a range of feasible firm structures. The firm structures are characterized by the
number of production units in each profit center, which reflects the level of centralization:
Definition 1. Firm structure A is called more centralized than firm structure B if the number of
production units in each profit center in firm structure A is larger than the number of production
units in each profit center in firm structure B.
In the most decentralized firm structure, each production unit forms a separate profit center,
and accordingly maximizes only its own profits. In this case, no needs are accommodated unless
cooperation among the profit centers is sustained. As a result, if cooperation among the profit
13
centers cannot be sustained, the value of the firm is zero. In more centralized firm structures,
more than one production unit forms a profit center. All decisions are made at the profit center
level. It is assume that the profit center manager is able to enforce costlessly any action that is
taken within his profit center. In the centralized firm structure, even if there is no cooperation
among profit centers, each profit center may accommodate the needs of its own production units.
This is due to the fact that the profit center manager maximizes the sum of the profits of its
production units. If a production unit is able to accommodate a need of another member of
the profit center, accommodating it increases the profit of the profit center directly. However,
the profit centers may be able sustain cooperation among themselves, and accommodate each
other’s needs. It is assumed that there is a priority in accommodating needs within the profit
center, such that a profit center may ask another profit center to accommodate a need only if it
cannot accommodate it on its own.
I analyze a grim trigger strategy for this game. In the cooperative phase, profit centers
request the other profit centers to accommodate needs that cannot be accommodated within
the profit center. A profit center accommodates all requested needs. Play remains in the
cooperative phase as long as none of the profit centers has deviated. If a profit center has
deviated, play switches to a punishment phase forever. In the punishment phase, none of the
profit centers offer to accommodate needs of other profit centers.
5.1
Profit Centers of Equal Size
I first restrict the analysis to profit centers of equal size. The number of production units in the
firm, together with other restrictions, then determine the feasible number of production units
in a firm. First, by assumption, a profit center cannot consist of more than
n
2
production units.
The set of feasible number of production units in a profit center is further restricted to be a
divisor of the number of production units in the firm. Finally, the analysis allows for additional
restrictions on the number of production units in a firm.
The following definitions will be useful in order to characterize the optimal organizational
14
design:
Definition 2. K ≡ {k1 , k2 , ..., ks } where k1 < k2 < ... < ks is a sequence that specifies the set
of all feasible number of production units in a profit center.
Definition 3. Denote by d∼K xe = {ki |ki−1 < x, ki ≥ x}. Similarly, b∼K xc = {ki |ki ≤
x, ki+1 > x}.
The following proposition characterizes the optimal organizational design.
Proposition 1. Assume the discount factor δ is large enough such that it is possible to sustain full cooperation in some organizational design.4 The interdependence degree determines the
optimal number of production units forming a profit center. The optimal number of production
nj
k l
mo
ln 12
ln 12
units in a profit center is given by k ∗ (p) ≡ max ∼K ln(1−p)
, ∼K ln(1−p)
. For large enough
increases in p,
5
the optimal number of units in a profit center k ∗ (p) decreases in p, the interde-
pendence degree. The optimal number of production units in each profit center is independent of
the number of production units in a firm. The optimal number of profit centers increases with
the number of production units in a firm.
The proof appears in appendix A.
The proposition states the optimal firm structure for cooperation. The criteria used to judge
among the possible structures is the lowest discount factor for a fixed set of parameters that
is required in order to sustain cooperation. The minimal discount factor that is required in
order to sustain cooperation depends on the cost of accommodating a need, the reward, the
interdependence level, and finally the number of units in a profit center. The optimal number of
units that encompasses a profit center is determined by choosing the number of units in a profit
4 The
discount factor should satisfy δ > arg maxk∈K
c
.
(1−p)k−1 (1−(1−p)k )(b−c)+c
If the discount factor is
below this value, the optimal organizational design is the most centralized possible, that is, each profit center
encompasses of ks production units. In this structure, the highest degree of partial cooperation can be sustained.
1
1
ln 2
ln 2
5 A sufficient condition that guarantees that if p > p k ∗ (p ) > k ∗ (p ), is
2
1
2
1
∼K ln(1−p1 ) 6= ∼K ln(1−p2 ) .
1
ln 1
ln 2
2
For a small difference between p1 , p2 it could be that p1 > p2 and k∗ (p1 ) = ln(1−p)
and k∗ (p2 ) = ln(1−p)
,
and accordingly k∗ (p1 ) > k∗ (p2 ).
15
center that minimizes the discount factor for a fixed cost of accommodating a need, reward
of accommodating a need, and interdependence level. The choice of lowest discount factor
that sustains cooperation pertains to a situation where the entrepreneur faces some uncertainty
regarding the discount factor at the point where he chooses the firm structure. In this case he
will prefer the structure that requires the lowest discount factor possible6,7 .
If the discount factor is too low it is impossible to sustain full cooperation in any organizational design. In that case, the optimal firm structure is that which yields the highest level
of cooperation in a static model. The highest level of cooperation in a static model, as shown
by Chandler [1977] and others, can be obtained in the most centralized design possible, namely
when there are two profit centers of equal size. In this case, the production units in each profit
center cooperate among themselves, but there is no cooperation between the profit centers. The
loss in welfare relative to the case of full cooperation is that there may be needs that cannot be
accommodated within a profit center, but can be accommodated by one of the units in a profit
center.
Once the discount factor is large enough such that it is possible to sustain cooperation in
some organizational design, the optimal organizational design depends on the interdependence
degree p. The higher is the independence level, the fewer units form a profit center, and the
organizational structure is more decentralized. The optimal number of production units in a
profit center does not depend on the total number of production units in a firm.
Empirical evidence about the relation between interdependence and organizational structure is rare because of both the challenge in measuring interdependence and the scarcity of
data on organizational structure. One paper that examines this relationship is Aiken and Hage
[1968].Aiken and Hage find in a sample of health and welfare organizations from the 60’s that
interdependence, approximated by the number of joint programs, is associated with decentralization.
6 Halonen
7 Identical
[2002] uses a similar criterion to judge among possible ownership structures.
results are obtained if the criterion is minimal highest cost of accommodating a need that can
support cooperation, or maximal lowest reward of an accommodated need.
16
In order to understand the reason for the differences in the optimal firm structure that are
generated by the interdependence degree, consider the two extremes: when the interdependence
degree is very high and when the interdependence degree is very low. When comparing the costs
and rewards of conforming to the cooperation strategy for a profit center, it is convenient to
compare the average cost and average reward per production unit. Although the profit center
compares the total reward and total cost, comparing the average cost and reward per production
unit is an equivalent and more intuitive way to compare the two.
First consider the case the interdependence among production units is almost perfect, that
is the probability a production unit is able to accommodate a need of some other production
unit approaches 1. The expected cost of conforming to the cooperation strategy for a profit
center that is able to accommodate the needs of all other production units equals to ck if the
structure is such that each profit center consists of k production units. The reason is that each
of the other profit centers are very likely to be able to accommodate that need as well. Since
there are a total of
n−k
k
profit centers, and for each given need the probability that each of the
profit centers is assigned to it is equal across profit centers, the probability a particular profit
center is selected to accommodate a given need is
k
n−k .
When considering the average cost per
production unit it is c independently of the size of a profit center. Notice here that although the
probability a profit center is requested to accommodate needs of other profit centers approaches
0, in order to sustain cooperation a profit center must be willing to bear this cost once requested.
The reward from cooperation, however, varies greatly with the organizational structure. In
any organizational structure in which a profit center consists of more than one production units,
the reward from cooperation approaches zero, since all needs are more likely accommodated
within the profit center. It further approaches zero faster the more centralized is the organizational structure. In the most decentralized structure, i.e. that in which each profit center
consists of a single production unit, the reward of each production unit approaches b , since the
probability a requested need is accommodated approaches 1. Clearly, when the interdependence
degree is very high, the higher the centralization, the harder is sustaining cooperation.
17
Next consider the case there is a very low level of interdependence among production units,
that is the probability a production unit is able to accommodate a need of some other production
unit approaches 0. The average reward per unit in a profit center from cooperation approaches
0 in all firm structures. The cost of conforming to the cooperation strategy for a production
unit that is able to accommodate the maximal number of needs if there are k production units
in each profit center approaches (n − k)c. This is because if it offers to accommodate all needs
it is likely to accommodate all of these, since the probability other production units are able to
accommodate them as well is low. This cost is an average of
(n−k)c
k
per production unit. As the
number of production units in a profit center k increases, the average cost per production unit
decreases. That is, the average reward from cooperation does not change with the organizational
structure, but the average cost decreases in the number of production units in a profit center.
As a result, the more centralized is the organizational structure, the easier it is to sustain
cooperation when the interdependence degree approaches 0.
Another feature of the optimal firm structure is that the number of production units in a
profit center is independent of the total number of production units in a firm n. The number
of production units in a firm may restrict the number of feasible number of production units in
a firm. However, holding the number of feasible number of production units in a firm constant,
the number of production units in a profit center does not change with the number of units in
a firm. An increase in the number of production units has two opposing effects on the incentive
to cooperate. First, it increases the continuation value of cooperation. The higher the number
of production units, the higher probability a unit’s needs are accommodated in the cooperation
phase, and accordingly the continuation value of cooperation increases. Second, an increase in
the number of production units in a firm increases the expected number of needs the binding
profit center is required to accommodate in equilibrium. These two numbers are proportional to
the probability that at least one of the profit centers can accommodate a given need. That is, an
increase in the number of units in a firm affects both the cost and the reward from cooperation
similarly, and as a result cooperation incentives do not change with the number of production
18
units in a firm.
A consequence of the independence of the number of production units in a profit center in
the total number of production units in a firm n is that, holding everything else constant, the
number of profit centers in a firm increases in the number of production units in a firm. The
reason is that the number of profit centers under the assumption of profit centers of equal size
is the number of production units divided by the size of a profit center. Since the size of a profit
center does not change with the number of production units in a firm, the number of profit
centers must increase in the number of production units in a firm.
This proposition has a few implications. First, the model predicts that the higher the
interdependence among production units in a firm the lower the decentralization in the sense
of a lower number of production units in each firm. Second, the model predicts that the size of
a profit center does not depend on the size of the firm, however, holding the interdependence
among production units constant, the number of profit centers increases in the size of the firm.
The following corollary is useful in case the interdependence is not observable.
Corollary 1. Decentralization is associated with higher profits.
Since a higher degree of interdependence increases profits, and interdependence is associated
with decentralization according to the proposition, decentralization is associated with higher
profits.
5.2
Profit Centers of Variable Size
I now extend the analysis to profit centers of variable size. Profit centers are not restricted to
be of the same size, but can rather take different sizes. Since this extension makes the problem
cumbersome to solve for the general case, it will be solved for an example.
In this example, there are four production units. The entrepreneur can choose among the
following partitions: in the ”symmetrical centralized” structure each two units form a profit
center, in the ”symmetrical decentralized” structure each unit forms a profit center, and in the
”asymmetrical” structure there are two profit centers, one that encompasses of one unit, and
19
the other encompasses of three units.
The following proposition characterizes the firm structure in that case:
Proposition 2. In the case there are four production units, an asymmetrical assignment is
never optimal. As a result, allowing for profit centers of variable sizes does not change the
optimal firm structure.
6
Alternative Interdependence Relations
Section 5 has analyzed the case the degrees of interdependence are identical and independent of
each other. The purpose of this section is to analyze how changes in this assumption affect the
optimal firm structure. One can think of two ways in which the relationships among units could
be different from this assumption. One is that the interdependence degree is not symmetrical
across units, that is, some units are more likely to accommodate needs of all or part of the
other units. Another way in which the potential synergies could be different is that there is a
correlation in the interdependence degree, positive or negative. If the ability of one division to
accommodate some need of another division increases the chance other units will be able to do
so as well, there is a positive correlation in the interdependence degree among units. If, however,
the ability of one division to accommodate some need reduces the chance other units will be able
to accommodate the same need, there is a negative correlation in the interdependence degree
among units.
Each of the assumptions made in this section is extreme, and shows how the optimal firm
structure is affected by changes in these directions. Most of the interdependencies among units
in a firm, however, would likely follow a mix of all assumptions made. In the example of
Amazon.com’s subsidiaries described above, for instance, it is likely that only A9 would be able
to accommodate software needs of the retailers. As a result, A9 might be more likely than
the rest of the subsidiaries considered to be able to accommodate needs of the retailers. When
considering the sales data needs of A9, however, all retailers considered have similar abilities in
being able to provide sales data to A9, which could depend on the market A9 would be interested
20
in at a certain period. Since some markets are common to all or a subset of the retailers, and
some are not, it is not clear if the ability of one of the retailers to provide some data increases
the chance other retailers would be able to provide similar data or not.
Proposition 3 describes the optimal firm structure in the following cases. First, the ”dominant unit” case assumes there is one unit that can accommodate all needs with certainty, while
all other units can accommodate a need with a positive equal probability. This is an extreme
case of asymmetry among the units. Second, the ”perfect negative correlation” case assumes
that at most one unit can accommodate a certain need. This assumption would be appropriate
if each unit is very different from the other. It is assumed that the probability each unit is able
to accommodate some need is symmetric across units8 . Finally, in the ”perfect positive correlation” case, if one unit is able to accommodate some need all other units are able to accommodate
that need as well.
Proposition 3. Assume the discount factor δ is large enough such that it is possible to sustain
cooperation in some organizational design. The optimal firm structure is:
1. The most decentralized structure possible in the dominant unit case.
2. The most centralized structure possible in the perfect negative correlation case.
3. The centralized firm structure in the perfect positive correlation case.
The proof is in appendix B. In the dominant unit case, the more decentralized is the firm
structure, the better the cooperation that can be achieved. The reason is that the dominant
unit accommodates all needs of its profit center, other than its own need. The more units there
are in a profit center, the lower the probability the profit center that includes the dominant
unit requests that need, as it is more likely to be able to accommodate it on its own. At the
same time, the cost of cooperation increases in the number of units in a profit center because
there are fewer profit centers, and hence the probability it is selected to accommodate a need is
higher.
8 Since
at most one unit is able to accommodate a need, it has to be assumed that p ≤
21
1
.
n−1
In the perfect negative correlation case, the more centralized is the firm structure, the better
the cooperation that can be achieved. I conjecture from this result that in order for the decentralized structure to achieve better cooperation, at least some of the units must be substitutes
for each other in at least some of the cases.
In the perfect positive correlation case, partial and full cooperation achieve the same level
of cooperation. The reason is that once two units cooperate, adding more units to the alliance
cannot increase the number of needs that are accommodated, as the additional units can only
accommodate needs that are already accommodated. As a result, centralized firm structure is
at least weakly better than the decentralized firm structure. It should be noted here, however,
that in this case there is no advantage for more than two units to form a firm together, as the
addition of another unit does not increase the total profits of the firm even if full cooperation
can be achieved.
7
The Value of a Firm as a Function of the Interdependence Degree among its Production Units
The purpose of this section is to analyze the effect of the interdependence among the units in a
firm on its value. So far it has been shown that the level of interdependence among the units
in a firm affects the optimal structure of a firm. It is interesting, then, to analyze how does the
value of a firm responds to the level of interdependence among units. As expected, the value
of a firm increases in the interdependence degree, since then there is a better opportunity for
synergy among the units. However, the rate of increase in the value of the firm depends on the
level of interdependence.
Figure 1 illustrates the relations between the value of a firm and the level of interdependence
among its units in the case the ability to accommodate needs is identically and independently
distributed. When the interdependence degree is very low, full cooperation can not be sustained
in any of the structures considered. The reason is that a unit that is able to accommodate a need
22
of another unit, knows it is very unlikely that it would be rewarded in the future for cooperating,
since the probability another unit will be able to accommodate its needs is low. Consequently,
it prefers not to accommodate a need of another unit in the current period. Hence, for very low
levels of interdependence only partial cooperation can be sustained, and this can be achieved in
the centralized firm structure. Therefore, the value of the firm increases linearly in the level of
interdependence for low level of interdependence.
However, at some point there is a discontinuous ”jump” in the value of the firm as the
level of interdependence increases. This happens because there is a critical value of level of
interdependence, such that for any level of interdependence above it, full cooperation can be
sustained in some organizational structure. Therefore, around some level of interdependence, a
small increase in the level of interdependence among units causes a large increase in the value
of the firm. As the level of interdependence further increases, the rate of increase in the value
of the firm as the the level of interdependence increases depends on the correlation among
the interdependence levels. If the interdependence degrees are independently and identically
distributed, as discussed in section 5, the maximal cost an entrepreneur would be willing to pay
in order to increase the level of interdependence decreases in the level of interdependence within
this range.
The fact that there is a ”jump” in the value of the firm at some level of interdependence
can have implications if the entrepreneur is able to choose the level of interdependence in the
firm. The prediction is that for any level of interdependence among the units, the entrepreneur
would choose to bear some cost for increasing the interdependence among the units. However,
the model predicts that the maximal cost that the entrepreneur would be willing to bear in
order to increase the interdependence among units is non monotonic in the interdependence
level among the units. This cost is first constant, then increases and finally it may increase or
decrease in the level of interdependence among the units. The reason is that within the range in
which only partial cooperation can be sustained, increasing the level of interdependence would
only increase the value of the firm linearly, as each division cooperates with only one division.
23
Figure 1: The value of a firm as a function of the interdependence degree among units, depicted
for b = 1.5, c = 1, n = 4 and δ = 0.9.
However, if an increase in the level of interdependence among units can cause full cooperation
to be sustainable, the entrepreneur would be willing to bear a higher cost in order to increase
the level of interdependence.
8
An Implication to Mergers and Acquisitions
One way to increase the interdependence among units in a firm is by mergers and acquisitions.
Acquiring or merging with a firm with which the acquiring firm has synergies may induce not
only the acquiring firm to cooperate with the acquired firm, but also induce better cooperation
among the units in the acquiring firm. Synergies are often cited as the motive for mergers and
acquisitions. For example, Berkovitch and Narayanan [1993] show that synergy is the primary
motive in takeovers with positive total gains. Furthermore, the horizontal merger guidelines of
the U.S. Department of Justice and the Federal Trade Commission treats potential synergies as
a source of benefit to the public from mergers. For example: ”... merger-generated efficiencies
may enhance competition by permitting two ineffective competitors to form a more effective
competitor, e.g., by combining complementary assets.”(Section 10, U.S. Department of Justice
and the Federal Trade Commission [2010]). However, not all mergers result in an increase in
24
shareholder value. In particular, Moeller et al. [2004] show that shareholders of large companies
suffer a significant loss after an acquisition announcement. Moeller et al. further find that
shareholders of small firms gain on average following an acquisition announcement. The analysis
in this section shows that a small firm size is associated with increased cooperation within the
acquiring firm, and consequently a better shareholder value. This section further has policy
implications with regards to mergers and acquisitions.
Acquiring or merging with a firm with which the acquiring firm has a high potential of
synergies can increase the value of the acquiring firm both directly an indirectly. The direct
effect of a merger is that a potential synergy between the acquired firm and the acquiring firm
can be realized. However, an indirect effect can emerge if the addition of a new unit to the
firm can incentivize units of the acquiring firm to cooperate better among themselves. This can
happen if the cooperation gain resulting from cooperation in the merged firm is high enough such
that a profit center that found it optimal not to cooperate before the merger, finds it optimal
to cooperate when the additional unit is added to the firm. If this indirect effect is realized,
the gains from mergers could be significantly higher than gains from cooperation between the
acquired firm and the acquiring firm. In that case, one should expect a significant increase in
the value of the firm, as this may allow the firm to move from an equilibrium in which partial
cooperation can be achieved at best, to an equilibrium in which there is a full cooperation. This
translates to a ”jump” in the firm’s value in figure 1.
To study this situation in a simple way, the analysis in this section assumes the only possible
firm structure is such that in each profit center there is a single unit, that is K = {1}. I assume
further that the acquiring firm consists of n units. The assumptions regarding the acquiring
firm are otherwise identical to these in section 4. There is a proposed acquisition of a firm with
which the acquiring firm has a high potential for synergies. To study this in the simplest way
possible, I assume that the target firm can accommodate any need of any of the units with
certainty. Moreover, any of the units of the acquiring firm can accommodate the needs of the
target with certainty as well.
25
The following proposition characterizes the cases in which acquiring a firm in the above
setting can lead the acquiring firm to shift from a non-cooperative to a cooperative equilibrium.
Proposition 4. There exists a cutoff value n̂(b, c, p) > 1 such that whenever the size of the
acquiring firm n is smaller than n̂(b, c, p) an acquisition can lead a firm from a non-cooperative
to a cooperative equilibrium.
The proof is in appendix C. According to the proposition, for every value of reward from
cooperation b, cost of accommodating a need c, and interdependence degree p there exists some
cutoff value of firm size n̂ such that if the size of the acquiring firm is smaller than n̂ it is easier
to sustain cooperation following a merger with a target firm with which all units in the acquiring
firm has strong interdependencies. When this is true, the minimal discount factor required in
order to sustain full cooperation is smaller in the newly formed firm relatively to the discount
factor before the acquisition. As a result, it could be then that the addition of the target to the
acquiring firm causes a firm to shift from a non cooperative to a cooperative equilibrium.
When the size of the acquiring firm is small enough an acquisition can result in a better
cooperation within the acquiring firm as the minimal discount factor required for cooperation
to be sustainable increases in the number of units in a merged firm in which one of the units
has a better synergies with the rest of the units than others, and is independent in the acquiring
firm, assuming the interdependence among all units is homogeneous. The reason is that the
minimal discount factor required for cooperation in the acquiring firm is independent of the
number of units in the firm (see Proposition 1), while the minimal discount factor required for
cooperation increases in the number of units in the merged firm. The reason for this is that,
looking at the condition for cooperation of the acquired unit, when it is certain that in the
cooperation phase all its needs are accommodated, and with certainty it can accommodate all
needs of other units, the minimal discount factor required for cooperation can depend on the
number of units in the firm only through the expected number of needs the acquired unit is
required to accommodate in the cooperative phase. Specifically, the minimal discount factor
required for cooperation increases in the expected cost of accommodating needs. The expected
26
cost of the target, in turn, decreases in the number of units in the acquiring firm.
The implication of this proposition on mergers and acquisitions is that the potential synergy
between the acquiring firm and the target may not be an accurate estimate of the benefit of a
merger. In particular, it could lead the firm to move from a non cooperative to a cooperative
equilibrium. In that case, the benefit of a merger includes also the potential synergy between
units in the firm. One of the cases in which that could happen, according to the proposition,
is when the size of the firm is small. This is consistent with Moeller et al. [2004], who find
that shareholders of small firms gain on average following an acquisition announcement. They
further show that shareholders of small firms gain significantly more than shareholders of large
firms.
The exact direction of the effect of firm size on the efficiency of acquisition may be specific
to the setting of this model, and in particular to the homogeneity of interdependence among the
acquiring firm’s units. However, this proposition is useful in order to show an explanation, in
addition to the existing explanations, to the association between firm’s size and gain following
an acquisition effect. The reason stems from the ability of the new acquisition to move from a
non cooperative equilibrium to a cooperative equilibrium.
This proposition also has policy implications. The increase in economic efficiency as a result
of a merger may be above the value of the synergy between the acquiring firm and the target.
Hence, in weighing the economics costs and benefits of a merger, a policy maker should consider
whether a merger is likely to result in an increased cooperation within the acquiring firm. For
example, the economic efficiency of a merger is predicted to be enhanced for small acquiring
firms.
9
Concluding Remarks
This paper has shown that dynamic incentives may affect the optimal structure of firms. Cooperation maximization alone results in reverse relations between profit center’s size and the
degree of interdependence among production units. Centralization, i.e. grouping a few units
27
together in each profit center, decreases the reward from cooperation since some of each unit
needs are accommodated by other units in the same profit center. Since cooperation within a
profit center is assumed to be perfect by other means, that reduces the incentive of a profit center to fully cooperate. The higher the interdependencies among units, an increase in the size of
a profit center increases its capability to accommodate its own needs by more, and consequently
the decrease in the incentive of a profit center is more prominent.
The paper further considers ways in which the interdependence degrees among units could
be endogenous. One way to do that is by mergers and acquisitions. The paper shows that by
merging with another firm with which the acquiring firm has a strong potential synergies, a
firm may shift from a non-cooperative to a cooperative equilibrium. In that case the potential
synergies between the acquiring firm and the target are an underestimation of the value of the
efficiency gains from the merger. This fact has a policy implications.
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A
Proof of proposition 1
Proof. In order to compare different firm structures, the incentives to cooperate in alternative
firm structures should be considered. Consider a firm structure in which each k production units
form a profit center. First, consider the payoff of a profit center in the cooperation phase. In the
cooperation phase, a profit center asks other profit centers to accommodate each of its k needs if
the fellow units in the profit center cannot accommodate it, that is with probability (1 − p)k−1 .
Then, each of the needs are accommodated if at least one other unit can accommodate it, that is
with probability 1 − (1 − p)n−k . By the symmetry of the game, the expected number of needs a
given profit center accommodates equal the expected number its own needs are accommodated
by other profit centers. Hence the expected payoff of cooperation for a given profit center is:
Vc ≡
k(1 − p)k−1 (1 − (1 − p)n−k )(b − c)
.
1−δ
(3)
To find the conditions under which cooperation can be sustained, the payoff of a profit center
that finds it the most costly to cooperate should be considered. The binding constraint would
come from a profit center that is able to accommodate needs of all n − k other profit centers.
Notice, however, that the expected number of needs that such a profit center is required to
accommodate is lower than n−k. The reason is that if any of the needs that the division is able to
accommodate can be accommodated by another division, there is a chance that another division
will be selected to accommodate that need, and the provider is decided randomly in this case.
30
The probability a profit center accommodates a need it can accommodate is calculated using
Bayes’ rule. To simplify the exposition, denote by A the event ”profit center i accommodates
need x, and denote by B the event ”profit center i is able to accommodate need x. Using this
notation, the probability a profit center accommodates a need it can is P (A|B). According to
Bayes rule,
n−k
1 × 1−(1−p)
n
P (B|A)P (A)
k −1
η ≡ P (A|B) =
.
=
P (B)
1 − (1 − p)k
(4)
The last equality follows since given a profit center is selected to accommodate a need the
probability it is able to accommodate is 1, the probability a profit center is selected to provide
a profit center ex ante is
1−(1−p)n−k
n
k −1
by symmetry, and finally the probability a profit center is
able to accommodate a given need is 1 − (1 − p)k .
Since the randomization for each need is independent, if a profit center is able to accommodate n − k needs, the expected number of needs that it accommodates is (n − k)η.
If a profit center deviates, its current needs may be accommodated by other profit centers,
but none of its future needs will be accommodated. Since the payoffs obtained from needs that
are accommodated in the current period are the same in case the profit center cooperates and
in case it does not, these payoffs can be ignored for a matter of comparing deviation against
cooperation payoff. Hence, a profit center chooses to cooperate in a given period if the future
payoff of cooperation is greater than the cost of cooperation in the current period. For a profit
center that is able to accommodate n − k needs, this condition is:
Vc
> (n − k)ηc.
1−δ
(5)
Substituting the values of Vc and η into the above inequality yields the following:
δk(1 − p)k−1
(n − k)k(1 − (1 − p)n−k )
(1 − (1 − p)n−k )(b − c) >
c.
1−δ
(n − k)(1 − (1 − p)k )
(6)
It is easy to see in the above inequality that all terms which depend on n cancel. Consequently,
the optimal number of units in a profit center is independent of n. Simplifying this condition a
31
lower bound for δ can be obtained:
δ>
(1 −
p)k−1 (1
c
.
− (1 − p)k )(b − c) + c
(7)
The optimal number of units in a profit center k ∗ can be obtained by minimizing the right
hand side of inequality (7) with respect to k:
min
k
(1 −
p)k−1 (1
c
.
− (1 − p)k )(b − c) + c
(8)
Since c and b are constants, b − c > c, and both the denominator and the nominator are positive,
the following function can be equivalently maximized:
max(1 − p)k (1 − (1 − p)k ).
(9)
ln(1 − p)(1 − p)k−1 (2(1 − p)k − 1) = 0.
(10)
k
The first order condition is:
Simplifying the above there is a single critical point at k̃(p) ≡
ln( 21 )
ln(1−p) .
Since the second order
condition is satisfied at k̃(p), k̃(p) maximizes equation (9). However, k̃(p) may not be in the set
of feasible profit center sizes K as defined in definition 2, and hence might not be the optimal
profit center size. Since the function in 9 is continuously differentiable and is maximized at k̃,
the maximal point that is restricted to belong to K must be either the one directly below or
k l
mo
nj
ln 12
ln 12
, ∼K ln(1−p)
.
the one directly above K̃(p), that is: k ∗ (p) ≡ max ∼K ln(1−p)
Taking a derivative of k̃(p) with respect to p reveals that k̃ decreases in p. There may
l
m
be, however, cases in which p1 > p2 but k ∗ (p1 ) > k ∗ (p2 ). This may happen if ∼K k̃(p1 ) =
l
m
j
k j
k
∼K
k̃(p2 ) and ∼K k̃(p1 ) = ∼K k̃(p2 ) . If the difference between p1 and p2 is large enough,
each corresponds to different values in the set K and K ∗ (p) decreases in p.
B
Proof of proposition 3
Proof.
1. Assume one of the units, the dominant unit, can accommodate any given need with
probability 1. The rest of the units can accommodate any given need with probability p,
32
and the IID assumption is maintained. Assume there are k units in each profit center, such
that the dominant unit is part of one of the profit centers, the strong profit center. The
strong profit center is clearly the one that exhibits the binding constraint. Each period,
the strong profit center requests one accommodation in case the units in the profit center
cannot accommodate the need of the dominant unit. Furthermore, the strong profit center
is always able to accommodate the needs of the other profit centers. Consequently the
value of cooperation for a strong profit center is:
Vc =
(1 − p)k−1 (1 − (1 − p)n−k )b − (n − k)ηc
,
1−δ
(11)
where η, the probability the strong profit center accommodates a certain need is given by:
n
k −2
η=
X
l=0
1
l+1
n
k
n
−2
(1 − (1 − p)k )l (1 − p)k( k −2−l) .
l
(12)
Cooperation can be sustained whenever:
δVc > (n − k)ηc.
(13)
Substituting the values of Vc and η into the above and simplifying, cooperation can be
sustained whenever:
δ(1 − p)k−1 b >
kc
.
1 − (1 − p)k
(14)
The left hand size of inequality (14) clearly decreases in k. To see that the right hand side
of (14) increases in k, consider its derivative with respect to k:
∂
kc
=c
∂k 1 − (1 − p)k
k(1 − p)k ln(1 − p) + 1 − (1 − p)k
((1 − p)k − 1)2
.
The nominator is clearly positive. Using the standard logarithm inequality, ln(1−p) ≥
(15)
−p
1−p .
Substituting this into the denominator:
k(1−p)k ln(1−p)+1−(1−p)k ≥ −pk(1−p)k−1 +1−(1−p)k = 1−(1−p)k−1 +(1−p)k−1 p(1+k) > 0.
(16)
This shows that the higher the number of units in a profit center, the larger is the discount
factor required in order to sustain cooperation. As a result, the optimal firm structure
33
is that with the lowest possible number of units in each profit center, that is the most
decentralized structure possible.
2. In the perfect negative correlation case at most one unit is able to accommodate a given
need. The value of cooperation if there are k units in a profit center is:
Vc =
k(1 − (k − 1)p)(n − k)p
k(n − k)p(b − c)
=
.
(1 − (k − 1)p)(1 − δ)
1−δ
(17)
Cooperation can be sustained whenever:
kδ(n − k)p(b − c)
> (n − k)c.
1−δ
(18)
Simplifying the above, cooperation can be sustained if the discount factor satisfies the
following condition:
δ>
c
.
kp(b − c) + c
(19)
It is easy to see that the larger the k the minimal discount factor required for cooperation
is lower.
3. The result follows since partial cooperation, that can always be sustained in the centralized
firm structure, achieves the same profits as full cooperation.
C
Proof of proposition 4
Proof. To compare the minimal discount factor required to sustain cooperation before and after
the merger, consider first a firm in the described setting before the merger. Substituting a profit
center size k of 1 in equation 7 yields a minimum discount factor of
c
.
p(b − c) + c
(20)
Next, consider the minimal discount factor required to sustain cooperation after the merger.
In this case, the constraint of the target firm will be binding, as it is assumed to be able to
34
accommodate needs of all other units. All needs of the target firm are accommodated in a
cooperative equilibrium. Hence, the value of cooperation per period of the target firm is:
Vc =
b − nηc
,
1−δ
(21)
where η is the expected number of needs the target firm is required to accommodate. Since each
of the other units can accommodate a given need with probability p, and the accommodating
unit is selected randomly out of the units that can accommodate a need, the expected number
of needs the target firm is required to accommodate each period is:
η=
n−1
X
l=0
1 − (1 − p)n
1 l
p (1 − p)n−1−l =
.
l+1
np
(22)
Cooperation can be sustained whenever δVc > ηc. Substituting equations (20) and (22) into the
above and simplifying, the minimal discount factor required for cooperation after the merger is:
δ>
c (1 − (1 − p)n )
.
bp
(23)
Comparing equations (20) and (23), cooperation is easier to sustain after the merger if:
c
c (1 − (1 − p)n )
>
.
p(b − c) + c
bp
(24)
To evaluate the effect of a change of a firm size on the consequences of a merger, first consider
the minimal discount factor required in each of the cases if n = 1. The minimal discount factor
before the merger is independent of n and is equal to
c
p(b−c)+c .
required for cooperation after the merger if n = 1 is
c
b.
The minimal discount factor
Since b > c the minimal discount
factor required for cooperation is smaller if n = 1 after the merger. Next, the derivative of the
minimal discount factor required for cooperation after the merger with respect to n is equal to
c
− bp
(1 − p)n ln(1 − p), which is positive for all n > 0. Finally, consider the minimal discount
factor as n approaches ∞. Before the merger, it is
c
p(b−c)+c ,
and after the merger it is
c
bp .
Since
b > c the minimal discount factor required for cooperation is larger after the merger than before.
Since both minimal discount factors are continuously differentiable in n, it follows that the two
functions intersect at some n̂(b, c, p) > 1, and the discount factor required for cooperation if
n < n̂(b, c, p) is lower after the merger, but the discount factor required for cooperation is lower
if n > n̂(b, c, p) before the merger.
35