organization design

1Harris is the Chicago Board of Trade Professor of Finance and Business Economics, GraduateSchool of Business, University of Chicago. Raviv is the Alan E. Peterson Distinguished Professor ofFinance, Kellogg Graduate School of Management, Northwestern University. We thank David Besankoand Ed Zajac for helpful comments. Address correspondence to Professor Milton Harris, GraduateSchool of Business, 1101 East 58th Street, Chicago, IL 60637; telephone: 773-702-2549; E-mail:[email protected]; Web: http://gsbwww.uchicago.edu/fac/milton.harris/more/.

First Draft: July 7, 1999Current Draft: July 29, 1999

ORGANIZATION DESIGN

by

Milton Harris

and

Artur Raviv1

ABSTRACT

This paper attempts to explain organization structure based on optimal coordination of interactionsamong activities. The main idea is that each manager is capable of detecting and coordinatinginteractions only within his limited area of expertise. Only the CEO can coordinate company-wideinteractions. The optimal design of the organization trades off the costs and benefits of variousconfigurations of managers. Our results consist of classifying the characteristics of activities andmanagerial costs that lead to the matrix organization, the functional hierarchy, the divisional hierarchy, ora flat hierarchy. We also investigate the effect of changing the fixed and variable costs of managers onthe nature of the optimal organization.

D:\Userdata\Research\Hierarchies\Organization Design.wpd 1 July 29, 1999 (3:36PM)

Organization Design

by Milton Harris and Artur Raviv

Organizations are observed to exist with various structures. Many organizations are designed as

hierarchies with each manager reporting to one and only one manager at the next higher level. Other

organizations employ a matrix structure in which each low level manager reports to two or more

superiors.

Within the hierarchical structure, there is considerable variation in the number of levels and in

the set of activities grouped together. The two main groupings are divisional and functional. In a

divisional hierarchy, all the activities pertaining to a single product (or perhaps set of products) are

grouped together into a division. For example, until the late 1980s Procter and Gamble (P & G) had a

relatively flat hierarchical structure with only two levels. At the lower level were the brand managers.

Each brand such as Tide, Crisco, Head and Shoulders and Scope had its own brand manager who

was singularly accountable for his or her brands performance. [Robbins (1990, p. 295)]. The only

other layer was headquarters. P & G then introduced a layer in between the brand managers and

headquarters. This layer contains division managers for product categories, such as laundry detergents,

each responsible for advertising, sales, production, research, etc. for the brands in his or her category. In

a functional hierarchy, by contrast, activities pertaining to a particular function are organized into

departments. For example, at Maytag, these departments include R & D, manufacturing, marketing,

corporate planning, personnel, finance, labor relations, and legal [see Robbins (1990, pp. 286-87)]. In a

functional hierarchy, the marketing department would, for example, coordinate marketing activities for

all products.

Matrix structure, which involves dual-authority relations [Jennergren (1981, p. 43)] can

combine divisional and functional structures. For example, the manager in charge of design for project A

2See the survey by Jennergren (1981) for a summary of this literature.


reports both to the Project A division manager and to the head of the design engineering group. In

another example of the matrix form, the president of a unit producing power transformers in Norway for

Asea Brown Boveri (ABB) reports to the president of ABB Norway and to the head of the Power

Transformer Business Area [see Baron and Besanko (1997, p. 2)].

An interesting topic in the theory of the firm, relatively under explored in the economics

literature, is what determines whether an organization adopts a matrix or hierarchical structure, how

many levels are involved and how activities are grouped. Several authors in the organization behavior

literature have argued that the choice between divisional and functional structures is driven by the

relative importance of coordination of functional activities within a product line and economies of scale

from combining similar functions across product lines.2 The advantage of a divisional structure is that it

allows better coordination among the various functions, such as manufacturing, product design,

personnel, and marketing, required to produce and sell a product. Segregating these functions by product

divisions, however, results in the failure to exploit economies of scale available if, for example,

marketing for all products is handled by a central marketing department. Trading off these advantages, it

is argued, determines whether one adopts a divisional or a functional hierarchy.

We address the issues relating to organization design mentioned above using a model based on

coordinating interactions across various activities. In our view, coordinating interactions requires costly

expertise embodied in managers. The optimal organizational structure trades off the benefits of

coordination against the cost of the necessary expertise. In this sense it is similar to the arguments of

organization theorists summarized in the previous paragraph. We provide a formal model that extends

the ideas beyond the functional vs. divisional choice. The model endogenizes the choice of organization

structure allowing us to make predictions regarding the use of matrix vs. hierarchical structures. If a

hierarchy is used, we rationalize the choice of functional vs. divisional grouping and vertical span.


We model a firm as consisting of activities such as producing products or components, designing

products, marketing products, etc. Each activity originates with a project manager who is assumed to

be essential to generating the activity and to have no function other than generating and possibly

managing his activity. If a set of activities interacts in a given period, there are benefits to coordinating

these activities. Reaping these benefits requires coordination by a manager with the correct expertise

(project managers have no coordination expertise). The territory between the project managers and the

CEO, may be populated by various middle managers. Each middle manager is capable of coordinating

a specific pair of activities. In addition to the benefits of coordinating pairs of activities, there are

incremental benefits to coordinating a set of activities on a company-wide basis. Only the CEO is

capable of this company-wide coordination (the CEO can also coordinate pairs of activities and may

engage in other unmodeled activities).

These managers may have both fixed and variable costs. The fixed cost of a manager must be

paid as long as the manager is available to coordinate activities for the firm, regardless of whether that

manager is actually used. A managers variable cost is paid only if the managers expertise is actually

used to coordinate activities. This is an opportunity cost of the managers time that is related to his value

in other activities not modeled here. For example, the variable cost of a CEO might be related to her

value in strategic planning.

The organization design problem consists of two components, a long run problem and a series of

short run problems. The long run problem is to choose which coordination capabilities to have available.

We view this as a once-and-for-all decision that trades off the fixed cost of various managers against the

potential benefits of having them available to coordinate specific activities. The short run problem is to

decide, given that a particular set of managers is available, how these should be used. This is a periodic

decision that, given the characteristics of current activities, trades off the likely benefits of coordinating

those activities that can be coordinated against the variable costs of the managers who are present.


Solving the long and short run problems results in an optimal organization structure. Thus an

organization structure consists of a set of middle managers who are available for coordinating activities

and a set of instructions for using these managers, the project managers and the CEO, given the

characteristics of the current set of activities.

For example, consider ABB and suppose the types of activities that will be available each period

are designing power transformers for use in the U.S., designing transformers for use in Norway,

marketing transformers in the U.S., and marketing transformers in Norway. The specific types of

transformers in demand may change from period to period, but suppose that there are sometimes benefits

from planning a marketing campaign for a given country tied closely to the transformer design for that

country. Also suppose that there may be significant production economies from coordinating transformer

designs for the two countries and that there may be benefits from coordinating marketing efforts across

countries so as to project a global corporate image. At any given time, the size of these benefits will

depend on which transformers are in demand at that time. Coordinating all these activities would require

four middle managers, two country presidents to coordinate design and marketing, one for each of the

two countries, one head of transformer design to coordinate design and one chief of marketing to

coordinate marketing across the two countries. In addition, there may be benefits of having the CEO

coordinate all four activities. If all four of the middle managers are available, it may be optimal always

to use all four or not coordinate activities or allow the CEO to coordinate activities. This will depend on

the variable costs of the middle managers and the CEO. Moreover, given that the four middle managers,

if present, will often be used, it may be optimal to have all four middle managers available. This will

depend on the fixed costs of these managers. Thus, ABBs observed matrix structure in which the

president of each countrys design group reports both to the head of the transformer design business area

and to the country president (and similarly for the marketing heads in each country) could be rationalized

3Actually, things are a bit more complicated than this simple rationalization. If, for example, theCEOs variable cost is low relative to that of middle managers, then all projects may be referred directlyto the CEO, bypassing the middle managers. Below we assume conditions under which a matrixorganization is optimal.


by low fixed and variable costs relative to the expected benefits from attempting to exploit interactions.3

Our results consist of classifying the characteristics of activities and managerial costs that lead to

the matrix organization, the functional hierarchy, the divisional hierarchy, or a flat hierarchy. If we

define the net benefits of company-wide coordination as the incremental benefits of such coordination

net of the variable cost of using the CEO, then we expect to observe a flat structure when the net benefits

of company-wide coordination are low and the fixed costs of middle managers are high. On the other

hand, the matrix structure is optimal when the fixed cost of middle managers is low. How low depends

on the variable cost of the CEO; the largest fixed cost for which the matrix structure is optimal is higher

when the variable cost of the CEO is higher, holding constant the benefits of company-wide

coordination. Finally, hierarchies are best when the fixed cost of middle managers is moderate and the

CEOs variable cost is low or medium, provided that the net benefits of company-wide coordination are

not too low.

To understand the intuition for these results, it is helpful first to realize that the middle managers

have two functions in our model. One is to coordinate pairs of projects when they interact. The other is

to generate information that allows more efficient use of the CEO. In particular, middle managers allow

a more accurate assessment of whether a company-wide interaction is present. This information enables

the firm to reap the benefits of company-wide coordination in some situations when it would otherwise

be suboptimal. When the fixed costs of middle managers are high, the net benefit of middle managers in

pairwise coordination is low. Therefore, it is worthwhile to employ them only if their other contribution

is high. The value of the information generated by the middle managers is high when the net benefits of

company-wide coordination are high. Consequently, when fixed costs are high and the net benefits of


company-wide coordination are low a flat structure, in which projects are left to the project managers, is

called for. Otherwise, it is best to use at least some middle managers, i.e., adopt a matrix or hierarchical

structure. Recall that the matrix structure involves using many middle managers, since each project

manager reports to (at least) two. The hierarchical form uses fewer middle managers than the matrix.

Thus high fixed costs of middle managers favors the hierarchical form relative to the matrix. On the

other hand, relative to the hierarchy, the extra middle managers in the matrix form allow the firm to more

accurately avoid use of the CEO when the company-wide interaction is absent. This is more valuable

when the CEOs variable cost is larger, i.e., larger opportunity cost of the CEO favors the matrix over the

hierarchy.

A number of empirical implications follow from these results. Under certain additional

assumptions, we show that organization structure will exhibit a sort of life cycle as the organization

grows in complexity and size. In particular, we show that the structure will progress from a divisional

hierarchy to a functional hierarchy to a matrix structure and, finally, to a flat, highly decentralized

structure. We also show that conglomerates that are organized as hierarchies may be expected to exhibit

divisional, as opposed to functional, hierarchies. Finally, we show that firms that do not face tight

resource constraints, highly regulated firms, and firms in stable environments will tend to have flat

organizational structures.

The remainder of the paper is organized as follows. A brief review of the literature is contained

in the next section. The model is presented formally in Section 2. We then solve the short run problem

for four, two and no middle managers in Sections 3, 4 and 5. The long run problem is analyzed in

Section 6. Empirical implications are considered in Section 7 and conclusions are presented in Section 8.

1. Literature Review

The economics literature on organization design is, as mentioned above, somewhat sparse. One


approach, adopted by Radner (1993), is to assume that processing information takes valuable time. To

reduce the delay, one can use parallel processing involving several people processing part of the

information at the same time. Delay reduction can be traded off against the cost of more processors.

Generally, this does not result in the types of organization structures we usually observe. Bolton and

Dewatripont (1994) have a similar approach but emphasize the tradeoff between specialization and

communication. They show that in most cases, the optimal organization structure combines a hierarchy

with a conveyer belt type of structure. Sah and Stiglitz (1986) also focus on sequential vs. parallel

processing of information but investigate the tradeoff between type I and type II errors to determine when

sequential processing is better than parallel processing and vice versa.

Garicano (1997) provides an elegant theory of hierarchies based on expertise that is similar is

some respects to ours. He postulates the presence of experts who can be ranked according the difficulty

of the problems they can solve. Experts in a given cohort can solve all problems that can be solved by

those in lower cohorts plus some more difficult problems. Experts who can solve more problems are

correspondingly more expensive. More difficult problems occur less frequently than less difficult ones,

however. This results in a pyramidical hierarchy with more workers at lower levels and fewer at higher

levels. In analyzing hierarchies, we more-or-less assume a pyramidical structure but allow contingent

referral of projects. We also consider experts with non-nested expertise allowing for the optimality of

matrix forms.

Vayanos stresses the interaction of information, i.e., the idea that the best project in a subset may

depend on the nature of projects outside that subset. This feature is absent from other models in the

economics literature, e.g., Radner (1993), Bolton and Dewatripont (1994), and Garicano (1997) but is

one we emphasize. The application Vayanos models is portfolio selection. He assumes a hierarchy in

which managers at each level examine a set of portfolios suggested by subordinates and an exogenously

determined set of assets (except the lowest level managers who examines only assets). Each manager

4Calvo and Wellisz (1979) also assume a hierarchical structure. Their main focus is explainingwage differentials across levels of the hierarchy.

5We will use the shorthand project A, project B, etc. to mean the project from activity A, theproject from activity B, etc.


chooses weights for combining these portfolios and assets into a larger portfolio (without changing the

weights of the items in the submitted portfolios). Managers ignore assets outside their purvue when

choosing weights. The main result is that each agent in the hierarchy (except the lowest) has exactly one

subordinate and also examines some assets directly.4

The organization behavior literature has investigated the empirical relationships between

decentralization of decisions and such variables as size (measured by employment) and vertical

integration. Blau and Schoenherr (1971), Child (1973), Donaldson and Warner (1974), Hinings and Lee

(1971), Kandwalla (1974) and Pugh et al. (1969) all find a positive relationship between size and extent

of decentralization. Kandwalla (1974) also documents a positive relationship between vertical

integration and decentralization. Child (1973) finds that the vertical span (number of levels) of hierarchy

is positively related to size.

2. Model

We model a firm that, for tractability, is assumed to engage in only four activities, labeled A, B,

C, and D. Each period, a project is drawn randomly from each activity. All projects are completed

within the period. If, for example, activity A is producing Chevrolets, the current project might be design

of headlights. Various subsets of these four projects may or may not interact. We denote an interaction

between two projects by juxtaposing their activity labels, e.g., AB denotes an interaction between

projects A and B.5 The set of such pairwise interactions is denoted by = {AB,CD,AC,BD}. Which of

these interactions occurs is given by an elementary event e . That is, e is interpreted as the event that

exactly those pairs of projects M e interact and no others. Thus, for example, the event e = {AB,CD}

6It should be pointed out that we abstract from all private information and incentive problems. While these may also be important determinants of organization structure [see Qian (1994), Singh(1985)], we limit the scope of the current paper to a consideration of coordination issues.


p([]) p if M P,

r if M R.

p(e) M e

p([]) M e

[1p([])]. (1)

indicates that projects A and B interact, projects C and D interact, and no other projects interact. The

event e = indicates that all four possible interactions occurred. We refer to this event as a company-

wide interaction. The event e = indicates that no interaction has occurred. The set of elementary

events is denoted by E = 2 (the set of all subsets of ).

The set of projects realized in a given period is defined by the probabilities p(e) of the various

feasible interactions e M E. These probabilities are not known prior to the realization of the projects but

are observed by everyone in the firm once the projects are realized.6 To simplify the analysis and to

capture the notion that some interactions tend to be similar to each other, we divide the set of feasible

interactions, , into two groups, P = {AB,CD} and R = {AC,BD}. The interpretation is that interactions

in a given group are similar to each other. For example, suppose A is production of Tide, B is marketing

of Tide, C is production of Cheer, and D is marketing of Cheer. Then the above grouping reflects the

assumption that interactions within a product line are similar to each other. We take similarity to the

extreme by assuming the interactions in a given group are identical in terms of probability of occurrence.

Formally, for M , let [] denote the event that at least the interaction represented by occurred, i.e.,

[] = {e M EG M e}. Assume

We also assume that interactions are independent, i.e,

Let be the support of , where and are the random variables whose realizations, p and r, are the(p,r) p r

interaction probabilities for interactions in groups P and R, respectively. Also, denote the means by =p


E and = E . We assume {(p,r) M [0,1]2 Gp > r}, i.e, interactions between A and B and between Cp r r

and D are always the more likely interactions, while interactions between A and C and B and D are less

likely. In terms of the above example, the assumption is that interactions within a product line are more

likely than those across product lines. If, on the other hand, the economies of scale from combining

production activities and those from combining marketing activities are more likely than interactions

across functions within product lines, then we would simply relabel the activities so that A is production

of Tide, B is production of Cheer, C is marketing of Tide, and D is marketing of Cheer.

There are three types of potential managers, project managers, middle managers, and a CEO.

There is one project manager for each activity. These project managers are required to generate the

projects and participate in managing them but cannot coordinate interactions between projects.

If a set of projects does interact, there are benefits to coordinating them. Discovering and

reaping these benefits requires investigation and coordination by a middle manager with the correct

expertise. For each interaction M , a middle manager, denoted m, may be hired who can discover

and coordinate this interaction. The set of potentially available middle managers is denoted by M =

{mG M }. We can think of the middle managers in M as product division managers, managers of

functional areas (e.g., marketing manager), country managers, etc., depending on the interpretation of the

activities A, B, C, and D. Note, we omit managers who can exploit two pairs or three pairs, but not

company-wide interactions, on the grounds that anyone with expertise that spans more than one

interaction must have company-wide expertise. This is the role of the CEO, denoted m*. Let MP (MR)

denote the set of middle managers who can discover and coordinate interactions in P (respectively, R),

i.e.,

MP = {mG M P} and MR = {mG M R}.

As mentioned, manager m can exploit the interaction represented by (and only that

interaction). The CEO is assumed to be able to discover and exploit any interaction, but only the CEO


(m,e) 1 if M e,

0 otherwise.(2)

B(,e) m M

(m,e).

(,e) B(M,e)B(,e)s1(e), (3)

can exploit the company-wide interaction which is present if e = . Without loss of generality, we

assume that, if no interactions are exploited, revenue of the firm is zero. Incremental benefits

corresponding to various interactions are normalized to one, except for the company-wide interaction.

Thus, the probabilities p([]) also play the role of expected benefits of the potential interactions.

Incremental benefits for the company-wide interaction are given by s. Formally, for the middle

managers, we have, for M ,

For the CEO, incremental benefits depend on which benefits, if any, middle managers have already

exploited. In particular, m* generates benefits of 1 for each interaction present but not exploited by a

middle manager plus s if all four pairwise interactions are present. Thus s represents the additional

benefit from company-wide coordination of interactions. Formally, for any subset of middle managers

M, let

Then, incremental benefit for m* if projects have already been referred to the middle managers in M

is given by

where 1(e) = 1 if e = and 0 otherwise.

This formalism admits many possible interpretations. For example, A and B can be innovations

in product 1, C & D innovations in product 2, say, A and C are electrical devices in Chevys and

Cadillacs, respectively, and B and D are chassis of Chevys and Cadillacs, respectively. If project A turns

out to be headlight improvement and B crash resistance, then A and B are likely to interact in that both

improve safety. If they do interact, exploiting this interaction through a coordinated marketing effort


emphasizing safety will produce incremental benefits. If, however, B is roominess, then A and B are

unlikely to interact. One can interpret mAB (mCD) as the manager of the Chevy (Cadillac) Division and

mAC (mBD) as the head of electronics (chassis).

An alternative interpretation is that A and B are innovations in products 1 and 2, respectively,

while C and D are new marketing techniques for the two products, respectively. The innovations may

have common components calling for coordinated production. The new marketing techniques may call

for a common ad campaign for the two products. One can interpret mAB (mCD) as the manager of the

Production (Marketing) Division and mAC (mBD) as the manager of the Product 1 (2) Division.

As mentioned in the Introduction, managers have both fixed and variable costs. The fixed cost is

a cost associated with employing the manager whether that manager is actually used to coordinate any

given periods projects. That is, we assume that employment is a long term commitment that spans

several periods. Once a manager is employed, the firm must pay the managers fixed cost for the term of

the commitment. Empirically, we identify the managers fixed cost with his salary. For simplicity, we

assume that all middle managers have the same fixed cost, denoted F. A managers variable cost is an

opportunity cost of the managers time that is related to his value in other activities not modeled here.

We assume that the middle managers have no function other than discovering and coordinating

interactions between projects. Therefore, the variable cost of the middle managers is zero. The CEO,

however, is assumed to have other duties such as strategic planning. Consequently the CEOs variable

cost is positive and is denoted by Q. We further simplify the problem (and eliminate some uninteresting

cases) by assuming that the value added by the CEO if there is a company-wide interaction exceeds her

variable cost, even if all four middle managers are used, ie.,

Assumption 1. s > Q.

Since project managers are assumed to be essential to generating projects and to have no function

other than generating and possibly managing projects, the cost (both fixed and variable) of project


V(matrixGp,r) 2(pr) p 2r 2(sQ) 4F. (4)

managers can be ignored. The CEOs fixed cost can also be ignored, since she is required exogenously.

3. The Short-Run Problem with All Four Middle Managers: The MatrixOrganization

Recall that to find an optimal organization design, one must solve both the long-run and short-run

problems. This must be done by first solving the short-run problem for each possible subset of available

middle managers. The long-run problem can then be solved by comparing the expected benefits

obtainable from optimally using various subsets of middle managers (the value of the short-run solution)

net of the fixed costs of the given middle managers. In this section, we set up and analyze the short-run

problem, assuming that all middle managers are employed by the firm. Later, we will consider the

problem when only a subset of middle managers is available and analyze the long-run problem.

When all middle managers are employed, the short-run problem is to decide, given the current

project characteristics, p and r, which managers should be used to check for and coordinate possible

interactions and in what order and in which contingencies they should be used. The problem is vastly

simplified, however, by the assumption that the middle managers have no variable costs. This implies

that, given that all four middle managers are present, it is optimal to have them investigate the four

possible interactions first, before referring any decisions to the CEO. This strategy allows the firm to

reap any benefits from interactions that are present and involve the CEO only if it is known that a

company-wide interaction requiring her special expertise exists. The strategy corresponds to the matrix

organization described in the Introduction. That is, each project manager refers his project to two upper

level managers: project A is referred unconditionally both to mAB and mAC, project B is referred

unconditionally both to mAB and mBD, etc.

Using Assumption 1, the value of the matrix organization net of fixed costs, given interaction

probabilities p and r, can easily be computed as

7There is one other possibility, namely employing one manager from each of MP and MR. Thisstructure does not, however, resemble a hierarchy (since one project will be referred to two middlemanagers) or any other commonly observed organizational forms. Consequently, we rule out thispossibility.


The intuition for this result is as follows. Given that all four middle managers will be used and that, if

the company-wide interaction occurs, projects are referred to the CEO, the expected benefit is the

expected benefit from each single interaction, p+p+r+r, plus the expected value added of the CEO net of

her variable cost, p2r2(s>Q). The expected cost is the fixed cost of the four middle managers, 4F. The

difference between the expected benefit and the expected cost gives the value of the matrix form in (4).

Whether the matrix organization is better than, say, a divisional hierarchy involves comparing the

expected value of the matrix form with that of a divisional hierarchy. To make such comparisons, we

must develop the net value of hierarchies. We start, in the next section, by analyzing the short run

problem with only two middle managers available.

4. The Short-Run Problem with Two Middle Managers: Hierarchies

In this section we consider the short run problem assuming that only two middle managers are

available. Since interactions are identical within a group, either P or R, there are only two relevant cases:

only managers in MR are available or only those in MP are available.7 We consider, for each case, the

optimal use of the given two managers.

When only managers from MR or only those from MP are available, the structure resembles a

hierarchy in which each project is referred (at most) to one and only one manager at the next level. For

example, if only managers from MP are available, projects A and B are referred to mAB and projects C

and D are referred to mCD. Consequently, we refer to these two situations as the R-hierarchy and the P-

hierarchy, respectively. Before proceeding, we simplify the analysis by making the following

assumption.

Assumption 2. For all (p,r) M , Q < 2r+r2s.

8Any arbitrarily small positive variable cost of the middle managers would break the tie in favorof direct referral to the CEO.


This assumption implies that, when only two middle managers are available, if both interactions are

present, then all projects will be referred to the CEO.

First suppose only managers in MR are present. To calculate the value of an optimal strategy in

this case, we use backward induction. Suppose projects have been referred to the two managers in MR.

If both interactions are found (this happens with probability r2), one obtains an additional expected net

benefit of referring all projects to the CEO of 2p+p2s>Q. This expression is positive by Assumption 2

and the fact that p > r. Therefore, as mentioned, all projects will be referred. If at least one of the

interactions from the R group failed to occur (this happens with probability 1>r2), the additional net

benefit of referring all projects to the CEO is only 2p>Q. If this is positive, then the projects will be

referred but not otherwise. Therefore the value of an optimal continuation strategy in this situation is

max{2p>Q,0}. Thus the expected value of an optimal continuation strategy, given that both managers

from MR have been consulted, is r2(2p+p2s>Q)+(1>r2)max{2p>Q,0}. The expected benefit from

referring projects to the two middle managers is simply 2r. Note that if Q < 2p, then all projects will

eventually be referred to the CEO no matter what is discovered by the middle managers. This is

equivalent to simply referring all projects directly to the CEO, skipping the middle managers.

Consequently, we refer to this case as one in which all projects are referred directly to the CEO.8

The above discussion proves the following lemma.

Lemma 1. When only managers from MR are present, if Q > 2p, then projects A and C are referred to

mAC and B and D are referred to mBD simultaneously. If both interactions are present, then all projects are

referred to m*; otherwise, no further referrals are made. On the other hand, if Q @ 2p, then all projects

are referred directly to m*. The value of the optimal strategy (net of fixed costs), given interaction

probabilities p and r, is


V(R-hierarchyGp,r) 2F 2r r 2(2pp 2sQ), if Q > 2p,

2(pr) p 2r 2s Q, otherwise.(5)

V(P-hierarchyGp,r) 2F 2p p 2(2rr 2sQ), if Q > 2r,

2(pr) p 2r 2s Q, otherwise.(6)

V(flatGp,r) max S2(pr) p 2r 2s Q, 0[. (7)

By symmetry, we have the following analogous result for a P-hierarchy.

Lemma 2. When only managers from MP are present, if Q > 2r, then projects A and B are referred to

mAB and C and D are referred to mCD simultaneously. If both interactions are present, then all projects are

referred to m*; otherwise, no further referrals are made. On the other hand, if Q @ 2r, then all projects

are referred directly to m*. The value of the optimal strategy (net of fixed costs), given interaction

probabilities p and r, is

5. The Short Run Problem with No Middle Managers: Flat Structure

When no middle managers are employed, the short run problem is simply to decide whether to

refer all four projects to the CEO or to give up any coordination benefits and let the project managers run

the projects. If all projects are referred to the CEO, the expected net profit is 2(p+r)+p2r2s>Q. If no

projects are referred, the expected net profit is zero. Since there are no fixed costs in this case, the net

value of the flat structure, given interaction probabilities p and r, is

6. The Long Run Problem: Optimal Organization Design

Recall that, when solving the long run problem, the firm must choose which managers to employ

before it knows exactly what the project characteristics, i.e., interaction probabilities, will be. The first

step in solving the long run problem is to take expectations over of the net values of the various(p,r)

structures given (p,r) derived in the previous sections. The nature of these expectations for the cases of


V(R-hierarchy)V(P-hierarchy) ES(pr)[Q(pr)2(1pr)][. (8)

two middle managers and no middle managers depend on whether the interaction probabilities lie on one

branch of the value function with probability one, the other branch with probability one, or neither. In

particular, for the case of two middle managers, it is uncommon in practice for projects to be referred

directly to the CEO, bypassing existing levels of the hierarchy. Consequently, we assume there is no

probability that are such that projects are referred directly to the CEO when two middle managers(p,r)

are present. On the other hand, in flat organizations, projects are generally managed by the project

managers. Consequently, we assume that projects are never referred to the CEO in the flat organization.

This last assumption, which implies the previous one, is stated formally as follows.

Assumption 3. For all (p,r) M , 2(p+r)+p2r2s < Q.

Finally, one piece of additional notation will be useful. Suppose M is a set of middle

managers. Define

V() = E[V(G ].(p,r)

Here, = M corresponds to the matrix organization, = MR corresponds to the R-hierarchy, = MP

corresponds to the P-hierarchy, and = corresponds to the flat structure.

Now consider the long run problem. This involves comparing the expected values of the various

structures, V(), to each other to find the best structure. We start by comparing the R-hierarchy and the

P-hierarchy using Assumption 3. The difference between their values is given by

This expression is clearly increasing in Q. If we define QRP as the value of Q for which the right hand

side of (8) is zero, it is easy to check that the R-hierarchy is better than the P-hierarchy if and only if Q >

QRP.

Under certain additional assumptions on the interaction probabilities, we can further characterize

the conditions leading either to an R-hierarchy or a P-hierarchy. In particular we show that the firm will

adopt an R-hierarchy when interaction probabilities are high (in the first-order stochastic dominance


E[Q(2r)2(1r2r)] > 0.

sense) and a P-hierarchy when these probabilities are low.

These two results are summarized in Proposition 1.

Proposition 1. (a) The R-hierarchy is better than the P-hierarchy if and only if Q > QRP, where QRP is

defined as the value of Q for which the right hand side of (8) is zero.

(b) Assume that the interaction probabilities and are such that > = for some > 0. Suppose Qp r p r

and the distribution of is such that the R-hierarchy is superior to the P-hierarchy. Then, for anyr

distribution of that stochastically dominates the original distribution, the R-hierarchy is superior to ther

P-hierarchy (for the initial value of Q).

Proof. Part (a) is obvious from (8). For part (b), using the assumed conditions in (8), the R-hierarchy is

preferred to the P-hierarchy if and only if

Since Assumption 3 guarantees that the function Q(2r+)-2(1+r2+r) is increasing in r, this expectation

increases with first-order stochastic dominant shifts in the distribution of .r

Q.E.D.

The intuition for this result follows by understanding the advantages and disadvantages of each

hierarchy. First, recall that the difference between the two hierarchies is which two middle managers are

available for discovering and coordinating interactions. In the R-hierarchy, the managers who can

analyze the low-probability interactions are available. In the P-hierarchy, it is the managers who can

analyze the high-probability interactions that are present. The advantage of the R-hierarchy is that the

firm need pay the opportunity cost, Q, of the CEO only if both the low-probability interactions are

discovered. The disadvantage is that, by starting with the low-probability interactions, one obtains a

lower expected benefit from the two middle managers and takes a larger chance of foregoing the benefits

of the other two interactions than if one had started with the high-probability interactions. This makes it

clear why the R-hierarchy is optimal for large Q, and the P-hierarchy is optimal for small Q.


V(matrix)V(R-hierarchy) QE[r2(1p2)] 2[pE(pr2)F]. (9)

V(matrix)V(P-hierarchy) QE[p2(1r2)] 2[rE(rp2)F]. (10)

V(matrix)V(flat) V(matrix) (sQ)E(p2r2) 2[(pr)2F]. (11)

For the second part, because of the assumption that and differ by a constant, whenp r

interaction probabilities are likely to be large, the net advantage of checking the low-probability

interactions first is higher.

We now compare the matrix organization to the two hierarchies and to the flat organization to

determine which is the best overall design. We start by comparing the matrix organization to the two-

middle-manager structures. The difference between the value of the matrix strategy and the R-hierarchy

(net of fixed costs in both cases) is given by

The difference between the value of the matrix strategy and the P-hierarchy (net of fixed costs in both

cases) is given by

The expressions in (9) and (10) are clearly increasing in Q and decreasing in F. Thus, we expect

organizations with CEOs whose opportunity cost within the firm (Q) is higher and organizations with

middle managers whose opportunity costs outside the firm (F) are lower, other things equal, to be more

likely to adopt the matrix form.

Using Assumption 3, the difference between the values of the matrix and flat organizations (net

of fixed costs) can be expressed as

Clearly, this difference is increasing in the CEOs value added, s, decreasing in the variable cost of the

CEO, Q, and decreasing in the fixed cost of middle managers, F. Thus, firms in which the CEO has high

value added and/or low variable cost, other things equal, should prefer the matrix organization to the flat

structure. Firms in which middle managers have low fixed costs, other things equal, should also prefer

the matrix organization to the flat structure.

Comparing the R-hierarchy and the flat organization, the difference between their values (net of


V(R-hierarchy)V(flat) V(R-hierarchy) sE(p2r2) QE(r2) 2[rE(pr2)F]. (12)

V(P-hierarchy)V(flat) V(P-hierarchy) sE(p2r2) QE(p2) 2[pE(rp2)F]. (13)

fixed costs) can be expressed as

By symmetry, the corresponding comparison between the P-hierarchy and the flat organization is

Again, these differences are increasing in the CEOs value added, s, decreasing in the variable cost of the

CEO, Q, and decreasing in the fixed cost of middle managers, F.

We are now in a position to characterize the optimal organization design as a function of the

variable cost of the CEO, Q, and the value added of the CEO, s, holding constant the fixed cost of middle

managers, F, and the distribution of interaction probabilities. It will be useful to have some notation for

the values of Q for which the matrix organization and each hierarchy have the same expected value.

Denote these by QMR and QMP, for the R and P hierarchies, respectively. Thus QMR is the value of Q for

which the right hand side of (9) is zero, and, similarly QMP is the value of Q for which the right hand side

of (10) is zero. Recall that QRP is the value of Q for which the R-hierarchy and P-hierarchy have the

same expected value.

Proposition 2. Case 1: QMP < QMR.

a. If F > and F > , then the values of (s,Q) for which each structure ispE(pr2) pr2

optimal are given in Figure 1a.

b. If F > and F < , then the values of (s,Q) for which each structure ispE(pr2) pr2

optimal are given in Figure 1b.

c. If F < and F > , then the matrix organization is optimal for (s,Q) such thatpE(pr2) pr2

the right hand side of (11) is positive, and otherwise the flat organization is optimal.

d. If F < and F < , then the matrix organization is optimal for all (s,Q).pE(pr2) pr2

Case 2: QMP > QMR.

a. If F > and F > , then the values of (s,Q) for which each structure isrE(rp2) pr2

9The constraints on s and Q implied by Assumptions 2 and 3 are not shown on Figures 1 and 2. These constraints may eliminate parts or even all of some regions.


optimal are given in Figure 2a.

b. If F > and F < , then the values of (s,Q) for which each structure isrE(rp2) pr2

optimal are given in Figure 2b.

c. If F < and F > , then the matrix organization is optimal for (s,Q) such thatrE(rp2) pr2

the right hand side of (11) is positive, and otherwise the flat organization is optimal.

d. If F < and F < , then the matrix organization is optimal for all (s,Q).9rE(rp2) pr2

Proof. Setting the right hand sides of equations (8), (9), (11) and (12) equal to zero and solving for Q as

a function of s yields the straight lines in Figures 1a and 1b. Similarly, setting the right hand sides of

equations (10), (11) and (13) equal to zero and solving for Q yields the straight lines in Figures 2a and

2b. It is easy to check that the indicated structures are optimal in their respective regions. It remains

only to show that if QMR > QMP, then QMP > QRP, and if QMR < QMP, then QMP < QRP. First suppose QMR >

QMP. If QRP A QMP, then for Q M [QMP,min{QRP,QMR}], V(MP) @ V(M) < V(MR) @ V(MP), which is

impossible. Similarly, if QMR < QMP and QMP A QRP, then for Q M [max{QRP,QMR},QMP], V(MR) < V(M) @

V(MP) @ V(MR), which is impossible.

Q.E.D.

Generally speaking, Proposition 2 shows that, holding the value added of the CEO fixed, as one

increases her variable cost, the optimal structure changes from the P-hierarchy to the R-hierarchy to the

matrix form to the flat structure. Some of these stages may be absent, however, depending on the level of

fixed costs, value added of the CEO, and the distribution of interaction probabilities. In particular, when

the value added of the CEO is relatively small, the matrix form is not optimal unless the fixed costs are

quite small. When the fixed cost of the middle managers is small relative to their expected benefit, the

flat structure is never optimal.


Intuitively, as Q increases, it becomes more important to economize on using the CEO. When

her time is not highly valuable, it pays to investigate the high probability interactions first. For somewhat

larger Q, it is better to refer projects to the CEO only when the low probability interactions are present.

This is accomplished by hiring the middle managers who can discover and exploit these events, i.e., those

in MR. For still larger Q, the firm economizes further on the CEOs time by first investigating all four

interactions, i.e., using the matrix form. Finally, for very large Q or small s, it is not worth involving the

CEO at all. This reduces the value of investigating individual interactions to the point where the middle

managers are not worth their fixed costs, so the flat structure is optimal. The optimality of this last case

requires that the fixed cost of the middle managers exceeds their expected contribution, assuming the

CEO will not be consulted.

A second interpretation of Proposition 2 involves holding Q constant while increasing s. For s

close to Q, the CEO will not be used for coordination of projects. This reduces the value of the middle

managers, so the flat structure is preferred. As s increases, the firm adopts either the matrix structure or

one of the hierarchies, depending on Q. The firm moves from the flat structure to the P-hierarchy, for

small Q, to the R-hierarchy for medium Q, and to the matrix form for large Q. Note that, holding Q

constant, changes in s result in at most one change of structure. This is because the comparisons among

the matrix form and the two hierarchies are independent of s, since the probability of obtaining s is the

same in all three.

Another set of comparative statics results involves the CEOs opportunity cost, Q, and the fixed

cost of the middle managers, F, holding s and the distribution of the interaction probabilities constant.

As one would expect, increases in F, holding Q and the other parameters constant, result in decreases in

the number of middle managers employed. For low fixed costs of middle managers, all four are

employed in the matrix form. As F increases, the firm moves to the flat structure possibly via one of the

hierarchies. Whether a hierarchy is used and if so which hierarchy for intermediate values of F depends


on Q and the distribution of the interaction probabilities. There is a tradeoff between economizing on the

opportunity cost of the CEO by having middle managers investigate potential interactions first and the

salaries of those managers. For large Q, the expected net value added of the CEO is small. It is optimal

to use the CEO only if it is certain that a company-wide interaction has occurred. This requires the

matrix structure with all four middle managers present and is optimal only if their fixed costs are small.

Otherwise, it is best not to use the CEO for coordinating projects, i.e., to use the flat structure. For

smaller Q and intermediate values of middle management salaries, it may be worthwhile to take a chance

on wasting Q by hiring only two middle managers, i.e., use a hierarchy structure. Which hierarchy

depends again on Q, and the intuition is the same as that presented following Proposition 1 above. The

formal result is stated as

Proposition 3. Let FR be the value of F for which V(MR) = V() when Q = QRP. Similarly, let FMR be

the value of F for which V(MR) = V(M) when Q = QRP. If FR > FMR, the values of (F,Q) for which each

structure is optimal are given in Figure 3. If FR < FMR, the values of (F,Q) for which each structure is

optimal are given in Figure 4.

Proof. Setting the right hand sides of equations (8), (9), (10), (11), (12) and (13) equal to zero and

solving for Q as a function of F yields the straight lines in Figures 3 and 4. The difference between the

two figures is accounted for by the inequality between FR and FMR. FR is the F at which the straight

lines determined by (12) and (13) intersect. Similarly FMR is the F at which the straight lines determined

by (9) and (10) intersect. It is easy to check that the indicated structures are optimal in their respective

regions.

Q.E.D.

We now turn to the empirical implications of the results.

10These results are consistent with the findings of the organization behavior literature cited inSection 1 which documents a positive relationship between size and extent of decentralization.


7. Empirical Implications

First, we consider the impact of changing the distribution of interaction probabilities. Recall that

in Proposition 1(b), we showed that large interaction probabilities result in hierarchies that are organized

to exploit the lower probability interactions. It seems reasonable to suppose that firms producing closely

related products have larger interaction probabilities. Thus, in such firms the divisional structure should

be oriented toward exploiting the least likely interactions. On the other hand, highly diversified

conglomerates are likely to have smaller interaction probabilities, so the divisional structure should be

oriented toward exploiting the most likely interactions. For such conglomerates, the most likely

interactions are those across functions within a given product line. Consequently, the model predicts that

highly diversified firms (at least those organized as hierarchies) will be organized as divisional

hierarchies, i.e., along product lines, as opposed to functional hierarchies.

Second, consider the result of changes in the opportunity cost of the CEOs time in coordinating

projects. Empirically, if we identify the CEOs variable cost with the size or complexity of the firm,

Proposition 2 makes a prediction regarding the life cycle of the firms organization structure. In

particular, it suggests that firms will start with a divisional structure oriented toward exploiting the most

likely interactions. As the firm becomes larger and/or more complex, the divisional structure will shift

toward exploiting less likely interactions, and the frequency with which projects are referred to the CEO

will decrease. As size or complexity increase further, the firm will adopt a matrix form, and the CEO

will be involved in coordinating projects even less often. Finally, if size or complexity increase still

further, the firm will become highly decentralized with little middle management, and the CEO will be

completely uninvolved in project coordination.10 Note that this description assumes that the fixed costs

of middle managers and the value added of the CEO in coordinating activities is held constant as the


size/complexity of the firm increases.

Further implications of Proposition 2 are available if we identify more specifically which

interactions are most likely and which are least likely. For example, if interactions between functions

relating to a given product are more likely than economies of scale from combining a function across

products, then the firms organization structure will progress from a divisional hierarchy to a functional

hierarchy (followed by other stages, as described above) as the size/complexity of the firm increase.

Third, we examine the effects of changes in the incremental benefit of coordinating company-

wide interactions, s. Possible empirical proxies for s include tightness of resource constraints, the extent

to which incentive schemes focus on unit performance, the extent of regulation, and the stability of the

environment. When units must compete for scarce corporate resources, the gains to company-wide

coordination of the allocation are likely to be large. Similarly, when compensation schemes do not give

unit managers an incentive to take account of the effects of their choices on the company as a whole,

there should be greater benefits to coordination by the CEO. On the other hand, severe regulation may

allow little scope for the CEO to improve performance through coordination of activities. Likewise,

stable environments do not require frequent intervention by the CEO to reap coordination benefits. Thus,

firms with weak resource constraints, compensation schemes that reward company-wide performance,

strong regulatory constraints, and/or stable environments should have a flat, highly decentralized

organization structure. Firms whose characteristics are the opposite of these should have either matrix

(those whose CEOs have high opportunity cost for coordinating projects) or hierarchy structures (those

whose CEOs have low opportunity cost).

Finally, consider the impact of changes in the fixed cost (salaries) of the middle managers, i.e.,

their productivity in the next best alternative employment. From Proposition 3, as salaries increase,

perhaps because of increased demand for middle managers, one expects firms to move toward flatter

structures. This might involve changing from a matrix form to a hierarchy or to a flat structure. Note,


however, in testing such implications, it is important to control for changes in the other parameters. In

particular, it is likely that when salaries increase so do the benefits provided by middle managers,

presenting a difficult identification problem.

8. Conclusions

This paper attempts to explain organization structure based on optimal coordination of

interactions among activities. The main idea is that each middle manager is capable of detecting and

coordinating interactions only within his limited area of expertise. Only the CEO can coordinate

company-wide interactions. The optimal design of the organization trades off the costs and benefits of

various configurations of managers.

The model provides a number of empirical predictions regarding firms organization design. In

obtaining these results, we made a number of simplifying assumptions. Perhaps the most important of

these is that middle managers have no opportunity cost of coordinating interactions. This assumption

allows us to ignore a large number of solutions that would be optimal for various levels of this

opportunity cost. Since these solutions are rarely observed in practice, we believe that ignoring the

opportunity cost of middle managers is justified.

A more important abstraction embedded in the model is the absence of incentive problems.

These introduce a large set of considerations revolving around providing incentives to transfer

information truthfully across managers within the organization structure. In particular, centralization of

decisions will, no doubt, be more costly in such situations. This will bias the organization design toward

flatter structures.

D:\Userdata\Research\Hierarchies\Organization Design2.wpd 28 July 29, 1999 (4:00PM)

Figure 1a: Optimal Organization Design (Proposition 2, Case 1a)

This figure is drawn using the following example: F = 1.3, , and is uniformly distributed onp ' r%0.05 r[0.5,0.95]. For this example, QPR = 2.105, QMP = 8.585, and QMR = 11.454.


Figure 1b: Optimal Organization Design (Proposition 2, Case 1b)



Figure 2a: Optimal Organization Design (Proposition 2, Case 2a)



Figure 2b: Optimal Organization Design (Proposition 2, Case 2b)



F

Q

0 0.5 1 1.5 2 2.5 3 3.5 4

0

2

4

6

8

10

R-hierarchy

P-hierarchy

FlatMatrix

Figure 3: Optimal Organization Design (Proposition 3, FRi > FMR)

This figure is drawn using the following example: s = 10, , and is uniformly distributed onp ' r%0.05 r[0.5,0.95].


Figure 4: Optimal Organization Design (Proposition 3, FRi < FMR)

This figure is drawn using the following example: s = 8, , and is uniformly distributed onp ' r%0.05 r[0.2,0.5].


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