1

Communication flows and firms’ organisation:the impact of the network externality effect on the production function*

Anna Creti†

December 1998

AbstractThe aims of this paper are to analyse telecommunications demand and usage by firms and in

particular to define the effect of the network externalities on the production function, focusing oncommunication flows among production units. We consider the case of two business units, that can besymmetric or in a leader/follower situation regarding their communication decisions. We show that thesubstitution effect between labor and the input information is related to the network effect, or theadvantage that a production unit obtains by using an input whose costs are shared by another user. Themodel is then generalised, taking into account different connectivity functions and indirect links amongproduction units: here we analyse conditions under which the usage effect, or the factors that increaseunits’ individual telecommunications demand, overcomes the network effect. We also compare ourresults with some of the existing models on communication networks and firms’ organisation. Theconclusions of the paper suggest how this theoretical framework serves not only to explain several typesof business telecommunications usage, but also to shed light on the complex relationship between firms’organisation and communication flows.

JEL Classification: D21, L23, L96Key words: business telecommunications demand, network externality, firms’ organisation

*This paper is based on a chapter of my dissertation at Toulouse University. I am very grateful to Marc Aldebert,Alain Bousquet, Marc Ivaldi, Said Souam, Michel Wolkowicz and more particularly to my advisor Patrick Rey fortheir useful suggestions. I also benefited from comments by seminar participants at CREST-LEI, CNET, CMUR-Warwick University, York University, London Business School, STICERD, EEA and ESEM Conferences 1997,Toulouse, EARIE Conference 1997, Leuven. All errors remain mine.Financial support by the Centre National d’Etudes des Télécommunications is gratefully acknowledged. † LSE-STICERD, Houghton Street, London WC2A 2AE, e-mail: a.creti@lse.ac.uk

2

1. Introduction

Business telecommunications demand is one of the areas most in need of attention in the

evaluation of how the telecommunication services provision is changing. However, despite the

interest in understanding how firms use telecommunication added value services1 and a growing

number of business tariff options, both theoretical and econometric analyses of the basic firms’

telecommunications needs are hard to find.

In order to study telecommunications business demand, some authors have attempted to

focus on business telecommunications demand in aggregate models (Perez-Amaral et alii, 1995

and Curien-Gensollen,1989). The use of aggregate data is useful to understand the general trend

of business telecommunication demand, but « firms vary a great deal from one to another in

terms of their productive structure, and this requires, when possible, a disagregation in the study

of the telecommunication demand » (Perez-Amaral et alii, 1995). Nevertheless, micro-

econometric analysis of telecommunications usage by firms are even more difficult to find than

aggregate analyses: to our knowledge, only one study analyses firms’ data (Griffin, 1989),

focusing on substitution relationships among business intercity telecommunications service,

treated as multiple goods in the firms’ production function.

The difficulty of finding micro-econometric studies on business demand essentially relies

on two important factors: the first one is technical, since panel data or simply cross section

telecommunications usage by firms are very scarce; the second one is more conceptual, because,

once individual data are obtained, it is not clear how to model firms’ telecommunications

demand. « Attempts to model business telecommunications demand in terms of generic needs of

a generic firm, or analogously to the residential customers, are simply not very useful » (Taylor,

1994). It is however clear that the main focus of the business telecommunications demand is

how to model usage, since no business can be without basic telephone service. Accordingly, to

approach business demand, as residential demand, in an access/no access framework, is not

relevant, as also reminded by Taylor. However, the type of access to telecommunication services

(local or medium-long distance, single or multiple lines, leased lines) is an important

determinant of business demand.

In a previous paper (Creti-Wolkowitcz, 1998), we analysed business telecommunications

usage on cross-section data from the «Base Marketing Enterprises-France Telecom» relative to

the first two-monthly traffic2 of 1997. Minutes of traffic for local, national and international

calls, two-monthly bill, and firms’ characteristics of 3026 French firms were provided. We

treated local, national and international calls as multiple goods that are inputs in the firms’

1 As for example, virtual private networks, integrated services with data and voice transmission, leased lines, mobilecommunication, etc.2 This database includes firms from very small (with fewer than five employees) to big enterprises (with more thanfive thousand employees). The sample is representative of the France Telecom business customers having atelecommunications services budget of more than 300 000 FF by year and positive local, national and internationaltraffic.

3

production function, together with labor, considered as a fixed short-term input. Based on cost

minimisation and dual theory, a translogarithmic cost function was estimated. From that

function, the share equations gave the basic parameters to calculate price elasticities, allowing

for imperfect substitution among local, national and long distance traffic. The structural firms’

characteristics - i.e. firms’ sector, mono or multi-location, geographic plant dispersion, and

employment structure - were included in the analysis as dummies for the cost function and the

share equations.

Our estimation, which exhibited a good fit, suggested two main features. The first one is

that the structural characteristics are quite important in determining business telecommunication

demand, since compared to these determinants, telecommunications price appears less important

(estimated price elasticities are quite low in absolute value). The second result is that some

omitted variables not observed in our database (the information about internal communication

networks linking different plants and the number/type of access at the plant level) have lowered

the estimation goodness of fit, especially for national traffic. We also found that observed traffic

duration per employee did not increase when moving from medium to big firms. A possible

explanation could be that, even in presence of private network, in general only a fraction of

plants is « on-net », otherwise the virtual solution would be too costly.

If the impact of firms’ structural characteristics is quite intuitive and widely discussed

(Taylor, 1994), the second effect, namely the impact of firms’ internal communication networks

on business telecommunications demand, deserves particular attention. Internal communication

networks allow firms to create dedicated links between all, or also just a portion of their plants,

benefiting from particular tariffs. In general, virtual private networks are adopted by medium and

large firms that are in the categories where our estimation for national traffic seems to be the

most imprecise.

To our knowledge, the complex interaction between internal communication networks and

business telecommunications demand has never been analytically studied. The results of our

estimation stimulated us to suggest a new theoretical demand-side perspective, taking into

account call and network externalities generated by communication among firms’ plants. In the

standard literature on telecommunications residential demand, the idea of those externalities is

very well characterised: the network externality effect means that the value of a network

increases each time that an additional users joins the others, while the call externality is the

benefit of being called.

We thus study the effect of network externalities on the business telecommunications

demand, rather than on the consumer demand, as it has traditionally been done. Our approach is

a new one, in which we underline that network externalities may influence the production

function, acting as technical production externalities. The models on network externalities (see,

for example, Economides, 1991, Economides et alii, 1993, 1994; Katz and Shapiro, 1985, 1986,

1992; Liebowitz and Margolis, 1994; Capello, 1994), jointly with the traditional theory of

telephone demand (Artle and Averous, 1973; Rholfs, 1974; Squire, 1973) and its most recent

4

advances (Bousquet et Ivaldi, 1994, Taylor, 1994), allow us to set a formal framework where

telecommunications are an input of the production function.

We present a model, which considers the behavior of firms and their telecommunications

demand in the presence of network externalities. The analysis is developed first in the case of a

symmetric situation, where two firms’ units simultaneously decide their telecommunications

demand, and then assuming that one of these units acts as a leader in the communication

decisions, while the other one follows. The objective of Section I is to show how the

characteristics of the network externality effect - the interdependence among users and the non-

compensation of the benefits arising being linked trough a telecommunication network - have

important consequences on units’ telecommunications demand. Moreover, we analyse whether

sharing the input telecommunications with other users may allow a production unit to achieve

the same production level with lower quantity of labor involved in the production process.

Hence we study whether the substitution effect between labor and the input information is related

to the network effect, or the factors that stimulate a unit to substitute her own calls with contacts

done by the others units belonging the network.

In Section II, the model is generalised, taking into account different connectivity functions

and indirect links among units. Our goal is here to show that the network structure determines

the telecommunications usage. In particular, we analyse conditions under which the usage effect,

or the factors that increase units’ individual telecommunications demand, overcomes the

network effect.

We also discuss the main results of our model in the context of the literature on firms’

organisation and information processing (Radner, 1992 and 1993). A comparison with the

Bolton-Dewatripont (1994) model on hierarchical firms’ organisation and communication flows

is developed: we show that in our context the conveyor belt and the tree network are totally

equivalent communication structures in the set-up of an efficient communication network

(Section III).

The conclusions of this paper suggest how this theoretical framework serves not only to better

explain, and, possibly, to estimate, business telecommunications usage, but also to shed new

light on the complex relationship between firms’ organisation and communication flows.

Section I

I.1 Theory of production in presence of network externalities

In order to analyse the effect of network externalities on the production function, we apply a

mix of different approaches (the traditional theory of telephone demand, Artle and Averous,

1973; Squire,1973; Rohlfs, 1974 and some recent advances, Taylor, 1994) on residential and

business telecommunications demand to the theory of production and the suggestions of the

5

Capello’s (1994) descriptive analysis on the relationship between network externalities and the

production function.

We consider two symmetric production units belonging to the same firm. We assume that the

central manager of the firm has a specific production target for the two units: the production

process is thus totally decentralised. You can think of a bi-divisional corporation, where each of

the two units is in charge of the production of one car model.

The two units are also given a capacity constraint in the usage of telecommunication services:

for example, the central manager provides them with a leased line or a virtual private network.

This assumption is a realistic one: the decision regarding the set-up of a telecom network is

generally centralised by a decision-maker who chooses the type of access (local or medium-long

distance, single or multiple lines, leased lines) and of services (internet, electronic mail). The

equipment purchase as well as the usage intensity is then delegated to the single divisions.

Let us assume that the production for the unit i is characterised by a certain amount of labor

(L), and of a certain volume of information (N), obtained using a telecommunication network:

iiiiii NLY δβη= i=1,2 (1)

We assume constant returns to scale (β +δ =1)3.

Our idea is that the network externality effect is twofold. On one side, the individual value of

the network (the total volume of information available to one unit) is a positive function of the

number of users, as in the traditional theory on telephone demand. On the other side, the units

can substitute their direct calls with those of the other units belonging to the network in order to

get a specific information amount that is endogenously determined. Here the network externality

effect is dependent form the total usage of telecommunication services, like in Bousquet-Ivaldi,

1994, with the additional aspect that the total volume of information is also a unit’s choice

variable4. Therefore we assume that the volume of information obtained by unit i depends on

the number of contacts unit i has with unit j (obtained as the sum of contacts useful for unit i,

generated by unit i and by unit j). We specify the following networking functions:

N N N1 11 1 21 2= +ε ε (2)

1 011 21> > >ε ε

Symmetrically, for unit 2 :N N N2 22 2 12 1= +ε ε1 022 12> > >ε εIn the above networking functions, 11ε and 21ε represent the efficiency of the contacts in terms

of the volume of information, N 1 represents the number of calls generated by unit 1 toward unit

2, and N 2 is interpreted as the number of calls generated by unit 2 towards unit 1 (symmetrically

3 This hypothesis can also be neglected, but it simplifies the analytical results.4 Obviously, the volume of information could be also assumed to depend on a specific telecommunicationtechnology. However, since our aim is to find a way of separating the network effects from those effects that mayimprove the telecommunication technology, only one communication service is assumed to be available.

6

for unit 2). Here we obviously refer only to contacts that contain the relevant information for the

production process.

The underlying idea for the assumption of the linear relation between the number of contacts

on a network and the volume of input information is that N 1 and N 2

are imperfect substitutes.

Moreover, we expect ε11 to be greater than ε21 (and consequently, ε11ε22 >ε21 ε12), since the

contacts generated by the interested unit have a higher probability of being more relevant for that

unit than the calls received. This formulation allows us to point out the concept of call

externalities or the benefit of being called.

As regards the cost function, units face the same unit costs for labour ( Lc ). The costs of

contacts are those related to the purchase of the necessary equipment to exploit the

communication network (depending on the volume of information, cnNi, where cn is the unit

equipment cost); and those related to the usage (as a function of the number of calls, c NN i ,

where cN

is the unit usage cost). The simplifying assumption of linear costs can be justified as a

consequence of behavior of a firm choosing among different kinds of telecom equipment, having

different quality and capacity. The unit will then allocate her resources proportionally to N and

N .

The unit costs of telecommunication equipment and contacts are equal for both firms: the

network available is unique. Moreover, it is realistic to assume that cn is greater than cN

.

According to the assumption on the equipment costs and the usage costs, the networking cost

function will be:

jii

jiNi

ii

NniNini N

cN

ccNcNcNC

εε

ε−+=+= )()( jii ≠= 2,1 (3)

where )/()(’ 1 iiNn ccNC ε+= is the marginal cost of the network and N 2 is viewed as a

subsidy on costs. This cost function thus reveals the strategic interdependence of the

telecommunications usage by the two productive units. On one side, a production unit has

incentives to increase her calls, in order to increase her production, but on the other side, she can

also free ride and let the other production unit to call, decreasing her telecommunication costs.

The production units act as perfect competitors in the output market. Our objective is then to

analyse the choice of an optimal resource allocation among a traditional production factor, labor,

and new inputs such as information and know-how gathered through the telecommunication

network. Each unit will find the optimal resource allocation among labor and information, taking

into account the interdependence between N 1 and N 2

.

I.2 The Nash Equilibrium

Under the previous hypothesis, unit 1 minimises its costs given the output level *1Y and the

networking function, considering N 2 as fixed. The unit 1’s minimisation program is:

7

0

0* ..

2211111

111

111,, 111

=−−=−

++=

NNN

NLYts

NcNcLcCMin

ii

NnLtotNNL

εεη δβ (4)

FOC of cost minimisation for the input information can be interpreted as follows:

1

1

111

1

1

1

N

Ncc

N

N

N

Y nN

∂∂

λλ∂∂

∂∂

+=⋅ (4a)

where λ1 is the lagrangian multiplier associated to the constraint of the fixed output level Y*.

Beside the usual equality of marginal productivity and marginal costs, the right hand side of (4a)

includes the evaluation of the marginal efficiency of contacts achieved by unit 1 ( 11 / NN ∂∂ ).

The demand function for the input information that guarantees the minimisation of costs is:1

11

1

1

11 )(’

*β

βδ

η

=

NC

cYN LD

At the equilibrium, since DN1 is independent of N 1 and N 2 , combining the network demand

with the networking function, we obtain the reaction function N 1 ( N 2 ) of the unit 1’s number of

contacts as follows:

211

21

1

11

1

111

11 )(’

*N

NC

cYN L

εε

βδ

ηε

β

−

= (5)

The reaction function of N 1 confirms that the contacts of unit 1 are imperfect substitutes for

contacts generated by unit 2 (∂ ∂N N1 2 0/ < )5.

Unit 2 will also minimise its costs and choose the number of contacts in the network (N 2 )

until this level guarantees a profit maximisation. Units play a simultaneous game for the optimal

choice of the number of calls.

The Nash equilibrium of this game is obtained by solving the system given by the unit 1’s-

unit 2’s reaction6 functions. The equilibrium is as follows7 :

)(1

*

)(1

*

11221112212211

2

22112212212211

1

DDN

DDN

NNN

NNN

εεεεεε

εεεεεε

−−

=

−−

=(6)

The Nash-equilibrium result points out an important characteristic of the externality effect:

the interdependence between the users of the telecommunication service. In fact, the difference

from the traditional model of production factors allocation, given a certain level of output, is that 5 The sufficient condition for the stability of the Nash equilibrium is that unit 1’s reaction curve is steeper than unit

2’s (∂ ∂N N1 2/ = ε21 / ε11 < ∂ ∂N N2 1/ = ε22/ ε21). We easily see that this condition means ε ε ε ε11 22 21 12> ,

as stated by hypothesis.6 The SOC are always satisfied, because the objective function is linear and the constraint functions are concave or

linear (respectively for the production and the networking functions).7 The necessary condition for the equilibrium to exist with positive N1

* and N 2*

is as follows:

/// 2221211211 εεεε >> DD NN . Note that if N ND D1 2= , this condition is always verified under our hypothesis

on the externality parameters.

8

the optimal number of calls depends not only on the choices of unit 1, but also on what unit 2

decides (and vice versa). The result in the presence of a leadership in the communication

decision-making is analysed in the next paragraph, where we will find out an additional feature

of the network externality effect: the non-compensation of the benefits obtained through the

telecommunications’ network.

I.3 The Stackelberg-sequential equilibrium

We now suppose that unit 1 is the leader and unit 2 the follower: this can correspond to the

behaviour of a marketing unit versus a production unit. The follower’s problem is still to

minimise the production costs for a given output, taking the decision of unit 1’s as given; the

leader’s problem is to achieve cost minimisation taking into account reaction function of unit 2.

The objective of this paragraph is to better illustrate the impact of the network externality effect

on the optimal number of calls, underlining the conditions under which a free-riding problem

may arise.

The unit 1’s FOC for cost minimisation imply:

)(21122211

221

111

εεεεε

β

δ

−+

=N

n

L

cc

LcN (7)

We thus obtain the same demand functions as in Paragraph 1.3, except for the fact that in this

case, the marginal costs of the network are nNS ccNC +−= )/()(’ 12212211221 εεεεε -where S stands

for Stackelberg. The demand equation for the number of contacts is obtained by again using the

networking function; the result is represented by the following equation, which gives the

Stackelberg solution for unit 1:

ANC

cYN

SLS

)()(’)(

**

12212211

2221

1

11

1

112212211

2211

εεεεεε

βδ

ηεεεεε

β

−−

−

= (8)

where A is:2

22

2

222

2

)(’

*β

βδ

ηε

=

NC

cYA L

Comparing the Nash and Stackelberg solutions, we obtain the following results:

Proposition 1

The leader’s demand for the volume of information is lower in the Stackelberg-sequential game

than in the Nash game.

Corollary 1

The number of calls generated by the leader (unit 1) in the sequential game is lower than the

number of contacts used by unit 1 in a symmetric market situation.

9

Corollary 2

The number of contacts used in the production function by the follower (unit 2) in the sequential

game is higher than the number of contacts used by unit 2 in a symmetric market situation.

Corollary 3

The sum of the optimal volume of information for unit 1 and 2 is higher in the simultaneous

game than in the sequential one.

Proof. See the Appendix.

The intuition behind these results is a very simple one. The temporal asymmetry allows unit 1

to lower her production costs. In fact, the presence of network externalities implies that the

higher the effort made by unit 2 to be connected in the sequential game, the higher the

advantages unit 1 achieves on its production function in terms of decrease in production costs in

the short term. Since the marginal costs of the network are higher than in the Nash game (see

proof of Proposition 1), unit 1 can easily change her amount of information by simply lowering

the number of contacts toward unit 2, as our model predicts. But in the long term, even if the

demand for labour increases, due to the reduction in the volume of information, the simultaneous

reduction of N1 and N1 compensates for the increase, so that total costs lower. The Stackelberg

outcome can be explained as a clockwise rotation in the leader’s reaction function, due to the

higher network externality effect. The greater the indirect or call externality effect (i.e. ε ε12 21)

compared to the direct network effect (ε ε11 22 ), the higher the shift, and hence the lower the

Stackelberg solution of unit 1, because the network marginal costs increase, and thus the demand

for input information decreases. Moreover, since in the Nash game the non-compensation

process plays a less important role than in the sequential game, when the units are symmetric

they would generate a higher aggregate demand for the network (Corollary 3).

Our results may also be interpreted as follows. The symmetric case is likely to represent an U-

form (product-oriented) organisation, where the two symmetric units are in charge of the

production of two car models. In the M-form (functional-oriented) organisation, where for

example one division is devoted to marketing and the other one to production, it is reasonable to

think that the marketing division could act as a leader in the communication decisions.

Proposition 1 and its corollaries would suggest that the U-form is likely to have higher total

communication levels than the M-form. Our simple results also point out that the marketing

division in the M-form would save the communication costs, inducing the production division to

increase her calls (for a more detailed analysis of communication flows, network externalities

and oligopolistic competition with U-form and M-form organisations, see Creti, 1998).

The results of Proposition 1 seem to contradict the traditional Cournot/Stackelberg analysis

(Spence, 1979; Dixit, 1980, Tirole, 1988). When two firms compete in a market and choose their

quantities, the capacity accumulated by firm 1 is a strategic substitute of unit 2’s capacity- as, in

our case, the contacts between unit 1 and unit 2. But the Cournot one stage simultaneous game

gives an output inferior to the Stackelberg game.

10

The difference between our model and the capacity game is mainly due to the impact of the

strategic substitutes on the firm’s objective functions. In the capacity model, an increase in the

capacity accumulated by the rival firm has a negative effect on profits. When firm 1 is the leader,

she accumulates more capital than it would have done in a simultaneous equilibrium. Capital

accumulation becomes a barrier to entry when the capital investment is difficult to reverse and

has a commitment value. The commitment effect is stronger the more slowly capital depreciates

and the more specific it is to the firm.In our model, N 2 is still a substitute of N1 , but its increase has a positive effect on the unit’s

objective function (∂ ∂C N1 2 0/ < ). This means that when unit 1 becomes the leader, she can

“free-ride”, forcing unit 2 to increase its number of calls. This equilibrium can be dynamically

unstable, because the telecom investment is neither a real barrier to entry (our firms are perfectly

competitive in the output market) nor irreversible, because we neglect network incompatibilities

and switching costs (only one network is available). System compatibility and quasi-

irreversibility of investment in specific touch-typing skills become very important in the

transition to another network. Analysing the technological choice among two incompatible

networks, we could explain the dynamic value of a specific investment in the telecommunication

input. This is left for further research.

Finally, one may think that, given the posited externalities in the communication process

between the two units, a centralised decision would be better. In fact, straightforward

calculations would show that, jointly minimising the production costs of the two units, the

presence of externalities in a market implies a lower amount of demand compared to the

optimum. A network whose size was determined by equating private benefits with marginal

costs would be too small from a social point of view. However, we can rule out this problem,

emphasising that the central manager is assumed to decide only the type of access for

communication channel between the two units. Then our hypothesis is that the exchange of

strategic information through the network, as well as the production process, has to be totally

decentralised. We will better analyse the relationship between the centralised/decentralised

organization and the communication decisions in Section III.

I.4 Information and labor: the “substitution effect” in the Nash/Stackelberg games

The equilibrium solutions in both the Nash/Stackelberg games show that there is a

substitution effect between information and labour, since if unit labour costs for a unit increase,

the contacts will increase. This property is, of course, related to the specific form of the

production function assumed, namely the Cobb-Douglas technology. For this reason it is

interesting to compare the TRS (technical rate of substitution) in the case where network

externalities are not present with the model modified by network externalities. The TRS is equal

to the ratio of the unit cost factors, as obtained by the FOC. We then measure the slope of the

11

isoquant of production at the equilibrium point (in absolute value). For this comparison, we

consider the output as fixed.

We obtain the following result:

Proposition 2

The network externality effect increases the technical rate of substitution between labour and

volume of information with respect to the case where this effect is not present.

Proof. See the Appendix.

This confirms the result obtained comparing N1 *N to N1 *S. In the Nash game, even tacking

into account the networking function, unit 1 is no longer able to exploit the total network

externality in her decision (as we show in the Appendix) only the direct effect ε11 appears in the

TRS). When the indirect effects (ε21 and ε12) are also exploited by unit 1, network externalities

allow a further gain, ceteris paribus.

Nevertheless, this effect must be better elucidated. We know from Proposition 1 that the input

demand function *1NN is always higher than *

1SN . As a consequence, in the Stackelberg game,

unit 1 lowers her number of contacts. We argue that higher substitution effect in the sequential

game is not due to an increased number of calls by unit 1, but to the mechanism of network

externality. That effect is similar to the hicksian technological progress, which allows units 1 to

achieve the same production level with lower input information, but its nature is different: here

the productivity effect is related to the network effect. Proposition 2 thus completes the results of

Proposition 1: not only the advantage unit 1 gains in the production function is due to the

interdependence between the agents, but also the non compensation of the network effect here

takes the specific form of a productivity effect not paid by the firm.

Section II

II.1 A generalisation with different network configurations

So far, the analysis of Section 1 has been focused only on bidirectional contacts among two

decision-makers. We now extend the model to the case of more than 2 units: once again, this

question arises when firms decide whether to connect their production units or their branches on

to a virtual private network, or to provide them with a specific type of access to the

telecommunication services. Since the solution of totally linking all the units may be very costly,

firms have to consider the trade-off between a larger externality effect (then allowing a larger

number of units linked to the network) and the cost of creating a “full connected”

telecommunications network. The production units take now into consideration the role of

indirect, as well as, direct links.

The substitution effect among contacts actually requires a deeper analysis. For example, if

three units are all linked to the network, we expect that an increase in the contacts between firm i

12

and j decreases the number of calls between unit i and k. This effect can take place for two

reasons. The first one is related to the technical capacity of a network. If it is fixed, an increase in

the contacts between firm i and j may create congestion in the network and influence the number

of telephone calls which unit i may have with other units. A second, and more important, reason

may be related to the interest of unit i in getting into contact with other units.

When deciding on their use of the network, units will then consider indirect links as well as

direct links. In this case, the volume of information unit i gets is dependent not only on the direct

link between unit i and j, but also, when unit j is linked to unit k, on the indirect link between i

and k, through j. Nevertheless, the link between j and k affects the volume of information

obtained by unit i in a more limited way than a direct link between unit i and k.

The number/ importance of those indirect links will change telecommunications demand by

units belonging to the network: our main idea is that the entire structure of the communication

network affects the demand for each given link. Said in another way, the structure of the

network determines its usage. The objective of this section is thus to analyse the intensity of the

telecommunications usage by each node (in terms of total volume of information gathered and

number of direct calls), when he belongs to different network structures.

As in Section I, we derive the networking functions taking into account direct as well as

indirect links, and then we will solve the unit’s cost minimisation problem. To start as simply as

possible, we focus on the case of 3 units, although most insights can be generalised for n units.

In our network structure, the three units can be linked together, directly or indirectly, in

different ways. We consider below three different kinds of linkage8 (Figure 1):

• serial connectivity, where unit 1 is directly connected with unit 2, and unit 2 with unit 3, but

no direct linkage between firms 1 and 3 exists;

• partial connectivity, where unit 1 is directly linked to both firms 2 and 3 but there is no direct

link between units 2 and 3;

• full connectivity, where all units are directly connected to each other.

8We assume, as in Section I, that when a link between two firms exists, it always allows bidirectional contacts.

13

Figure 1

Network configurationsSerial connectivity

Unit 1

Partial connectivity

Unit 1

Full connectivity

Unit 1

How can these structures affect units’ networking functions?

We now assume that each unit knows the structure of the network to which she belongs; she

obtains an amount of information, specific to the network configuration.

For example, let us consider one of the three units, unit i. In a first step, unit i decides hertelecommunications usage Ni

and the direct calls with other units, in order to obtain the amount

of information Ni . Each time that unit i contacts other firms, she adds a “piece” of the totalamount of their information to Ni and gathers E Nij j

j∑ , where the total amount of information

obtained from contacting other units j is weighted by parameters Eij <1, meaning the externality

effect or the quality/benefit of exchange unit i-unit j. These parameters are taken as exogenous

by each unit and are associated with the direction of exchange among networked units.

Unit 2

Unit 2

Unit 2

Unit 3

Unit 3

Unit 3

14

For simplicity, we assume that Ni is a linear function of Ni and of E Nij jj

∑ . For example,

consider unit 1 in the simple serial connectivity case: she is only linked to unit 2, while a direct

link between units 2 and 3 exists. The hypothesis discussed above means that:

N N E N1 1 12 2= +where N1 is the decision variable of unit 1, and E N12 2 is the total amount of knowledge in

efficiency terms obtained by directly contacting the unit 2 through the network.

So far, Ni is only a function of Ni and of direct contacts. Because of indirect links, Nj will

depend on N j , and on the contacts between j and the other units linked to j, since:

N N E Nj j jkk

k= + ∑Taking into account the direct as well as indirect links, the networking function for unit i

becomes:

N N E N E Ni i ij jj

jkk

k= + +∑ ∑( )

And so on, if unit k has other indirect links, they will be integrated in the unit i’s networking

function. This process will stop when every indirect link of the network has been considered9. In

this mechanism of "diffusion of knowledge", in addition to Eij , the externality effect of the

exchange unit i- unit j, other externality parameters appear (all less than 1):E E Eij ji i

j= as the benefit from the two-way exchange unit i - unit j

E E Eij jk ijk= as the benefit from the indirect exchange between unit i and j, through unit k

E E E Eij jk ki iikj= as the benefit from the circuit among unit i, j, k

Moreover, we assume that the sum of all these parameters is lower than one10.

Finally, unit i’s networking function, taking into account the entire structure of the network, is

a linear function of her own telecommunication usage (Ni), the telecommunication usage of

units contacted directly (N j), and indirectly trough j (Nk ), and also of the various externality

parameters (E E E Eij ij

ijk

iikj, , , ).

Hence, N j and Nk are substitutes of Ni , but their degree of substitutability depends, in a

complex way, not only on the externality effects generated by direct contacts between unit i and

unit j, but also on those benefits arising from the indirect links unit i- unit k, through unit j.

9In a topological approach, this process would be the evaluation of the oriented graph obtained in the serial, partialor full connectivity structure, where Ni

is the value associated with the node i and E Nij j is the value of the arcoriginating from the node i toward the node j. Furthermore, our networking function has a meaning very similar tothe “value function” of a network as used by some papers in the cooperative game theory (Jackson-Wolinsky, 1996,Dutta-Mutuswami, 1997), to indicate the output of the agents when they are organised according to a particulargraph.10 As we will see, this hypothesis allows having positive Ni and marginal network costs.

15

Let us to go back to the serial connectivity example. Unit 1’s networking function is obtained

by solving the system which takes into account the entire structure of the network, i.e., direct

link unit 1- unit 2, and indirect link unit 1- unit 3:

N N E N

N N E N E N

N N E N

1 1 12 2

2 2 21 1 13 3

3 3 21 2

= += + += +

(9)

Solving this system of equations determines unit 1’s network N1 as a function of N1 and

N j ’s:

NN E E N E N

E E11 2

312 2 13

23

12

23

1

1=

− + +− −

( )(10)

32 also and NN - although unit 1 is not directly linked to unit 3- are substitutes of N1; their

degree of substitutability depends on the efficiency of direct contacts between unit 1 and unit 2

(E12) but also on the two-way exchange between units 2 and 3 (E23), and on the externality

generated by the contact unit 1- unit 3, through unit 2 (E132 ).

We similarly compute the networking functions for the partial and full connectivity

configurations.

Once we have obtained the networking functions in the serial/partial/full connectivity cases,

we must solve the cost minimisation problem. To fix ideas, we will analyse the impact of

different network configurations on unit 1’s costs minimisation. As in Section I, unit 1

minimises her production costs given the output level Y* and the networking function, as a

function of N1, N 2 , N 3 and the externality parameters.

Under the different network structures, unit 1 decides her demand for input, and then, in

particular, on N1 and N1, taking as given N 2 andN 3 , and the externality parameters. All units

move simultaneously and achieve Nash equilibrium in their decision variables Ni .

In paragraph II.2, we derive the network connectivity and cost functions associated with each

network structure and we solve unit 1’s cost minimisation problem under different network

configurations.

In paragraph II.3, we compare the different reaction functions and the Nash equilibrium

associated with the full/serial/partial connectivity cases, for better understanding the interaction

between network structure and telecommunications usage.

II.2 Cost minimisation under different network configurations

We start from the most complex situation: full connectivity. Once we have obtained the

network connectivity and the cost functions, we minimise production costs. Serial and partial

connectivity are obtained by eliminating some two-way links from the full connectivity network

configuration.

16

II.2.1 Full connectivity

In order to simplify the networking function, we use the following notations:

D

EEc

D

EEb

D

Ea

EEEEED

’=

=

1=

121313

31212

32

3211

2311

32

31

21

++−

−−−−−=

The networking function then becomes:N aN bN cN1 1 2 3= + + (11)

where a,b,c are exogenous parameters in the cost minimisation.

As in Section I, we assume that networking costs consist of the general costs of achieving theamount of information N1 gathered by the network (such as equipment and/or subscription

costs, cnNi) and the costs related to the use of the network (usage costs, as a function of the

telecommunications usage, c NN

). We also keep the simplifying assumption of linear costs.

The networking cost function is then:

C N c N c N cc

aN

c

abN cNn N n

N N( ) ( ) ( )1 1 1 1 2 3= + = + − + (12)

where C N cc

anN’( ) ( )1 = + is the marginal cost of the network.

This marginal cost is an indirect function of E E E E12

13

1123

1132, , , , and E2

3 , through 1−a : an

increase of these externality parameters will then lower the marginal costs of the network. The

second and the third term of total costs show that 32 and NN act as subsidies on the unit 1’s cost

function. But, since b/a varies directly with respect to E23 and E E12

312, (and similarly, c/a

depends positively on E23 and E E13

213, ) an increase of one of these externality parameters

increases the networking costs, since it decrease the value of N N2 3 and as subsidies. We will

later discuss the consequences of externality parameters on the units’ behaviour.

Anyway, the idea of the network effect is well captured by the impact of the externality

parameters on costs. Unit 1 could gain not only from the efficiency of the direct links, as in

Section I, but also from the link between unit 2 and 3, which creates a real network externality

effect.

The unit’s minimisation problem now takes into account the production constraint and the

networking function, as defined by (14). Solving the Lagrangian associated with the firm

minimisation program, we obtain the same structure of input demand as in paragraph I.2. Here,

N D1 does not depend on the decision variables of the other firms belonging to the network, i.e.,

32 and NN . Nevertheless, some externality parameters due to the exchange with unit 2 and unit

3 affect N D1 through the marginal costs of the network C N’( )1 , which are an indirect function of

E E E E12

13

1123

1132, , , , and E2

3 .

Combining the network demand with the networking function, we obtain the unit 1 reaction

function under the full connectivity network structure, N F1 , as a function of 32 and NN .

17

NN

a

bN

a

cN

aF

D

1

1 2 3= − −(13)

or

NE

E E E E E N E E N E E NF D1

23 1

213

23

1123

1132

1 12 123

2 13 132

3

1

11=

−− − − − − − + − +[( ) ( ) ( ) ]

N 2 and N 3 are still strategic substitutes for N1, but their degree of substitutability depends, in

a complex way, not only on the efficiency of direct contacts between unit 1 and unit 2 or 3, but

also from the two-way exchange between units 2 and 3. For example, the elasticity of

substitution between N1 and N2 (measured as ∂ ∂N N1 2/ , in absolute value) increases with

greater benefit of the direct contact between unit 1 and unit 2 (E12), but also on the

quality/efficiency of the two-ways exchanges unit 2- unit 3 (E23 ) and of the communications unit

1- unit 2 through firm 3 (E123 ).

II.2.2 Serial and partial connectivity

We follow the same method as in paragraph 2.1 in order to calculate the unit ’s reaction

functions in the serial and partial connectivity network configurations.

In the case of serial connectivity, unit 1’s connectivity function is as follows:

N a N b N c NSS S S1 1 2 3= + +

where:

SS

Ss

SS

S

D

Ec

D

Eb

D

Ea

EED2

131232

32

21

1

1

==−=

−−=

Even if firm 1 is not directly linked to unit 3, the existence of a link between units 2 and 3

allows unit 1 to benefit from N3 as a strategic substitute for N1 . Nevertheless, we note that the

externality parameters (the terms a b cS S S, , ) are lower than those of the full connectivity case.

Since the number of units belonging to the network does not change, this effect is due to the

missing link between unit 1 and unit 3.

The networking cost function is:

).()()( 3211 NcNba

cN

a

ccNC ss

S

N

s

Nn +−+=

The marginal cost of the network [ )/()(’ 1 SNn accNC += ] is positive by hypothesis; it varies

indirectly with respect to E12 , and E2

3 , through as−1 ; the second term (in absolute value) is an

indirect function of E23 and E12 through SS ab / (similarly, this term depends positively on E2

3

and E132 through SS ac / ). The externality parameter associated with the exchange unit 2-unit 3

again has an ambiguous effect: if an increase in E23 decreases marginal costs, it also decreases

the value of N 2 andN 3 as subsidies on costs11.

11 We also note that, ceteris paribus, an increase in the externality parameter associated with the two-way exchangeunit 1- unit 2 E1

2 decreases marginal costs, while an increase in the one-way externality unit 1- unit 2 E12 will have

18

The serial connectivity marginal costs are higher than those of full connectivity. Moreover,

the second term of the networking cost function (in absolute value), decreases. These two effects

increase the cost function with respect to the full connectivity case, because of the lower

network externality effect. When there is only an indirect link unit 1- unit 3, it is clear that thesubsidy on unit 1’s costs arising from N 2 andN 3 as substitutes for N1 is lower.

The network demand (N DS1 ) is similar to that of the full connectivity case, except for the

marginal costs and is still independent from N 2 andN 3 . Combining the network demand with

the networking function, we obtain the unit 1’s reaction function of the serial connectivity case( N S

1 ) :

[ ]NE

E E N E N E NS DS1

23 1

223

1 12 2 132

3

1

11=

−− − − +( ) ( ) (14)

Even if a direct link unit 1- unit 3 does not exist,N 2 andN 3 are both substitutes for N1. Their

substitutability rate depends not only on the efficiency of direct contacts between unit 1 and unit

2, but also on the two-way exchange between unit 2 and 3.

In the partial connectivity configuration, the networking function is as follows:N a N b N c NP

P P P1 1 2 3= + +

where:D E E

aD

bE

Dc

E

D

P

PP

PP

PP

= − −

= = =

1

1

12

13

12 13

Now unit 1 is directly linked to f unit 3, but since the link between unit 2 and unit 3 ismissing, the externality parameters ( a b cP P P, , ) are lower than those of the full connectivity

case. Since the number of units belonging to the network does not change, the lower network

effect is due to the missing link not involving unit 1, namely the connection unit 2- unit 3.

The partial connectivity networking cost function is:C N c c E E N c E N E Nn N N

( ) [ ( )] ( ).1 12

13

1 12 2 13 31= + − − − +

where the marginal costs of the network are higher than the case of full connectivity.

Moreover, the second term of the networking cost function (in absolute value), decreases. These

two effects again increase of the cost function with respect to the full connectivity case: unit 1

has only direct links, and she cannot gain from indirect links involving the other units connected

to the network. Combining the network demand with the networking functions, we obtain the

N P1 reaction function, as a function of N 2 andN 3 , as follows:

N E E N E N E NP DP1 1

213

1 12 2 13 31= − − − +( ) ( ) (15)

N 2 andN 3 are both strategic substitutes for N1, and, in this case, their degree of

substitutability only depends on the efficiency of direct contacts from firm 1 toward units 2 and

unit 3.

an opposite effect on total costs. An increase of E13

2 , the externality parameter associated with the link unit 1- unit 3

through unit 2, also increases total costs, lowering the value of N3 as a substitute for N1 .

19

II.3 Comparison of reaction functions and Nash equilibrium under the different

network structures

To understand the interdependence between network structure and telecommunications usage,

it is useful to compare unit 1’s reaction functions for different network configurations and the

corresponding Nash equilibria.

As explained in Section I, the network externality has an impact on:

• the demand of input information, through the marginal costs of the network;

• the decision variable of a unit, the telecommunications usage, through the strategy of

substitution with the decisions variables of the other units belonging the network .

The first effect can easily be analysed, obtaining the following result:

Proposition 3

In the full connectivity configuration, each node has the highest demand for input information.

Proof. See the Appendix.

The effect of network externality on the telecommunications usage N is less clear-cut. On

one hand, the increase in network demand associated with the full network connectivity case will

increase N , but, on the other hand, a unit can be tempted to free ride and to benefit from the

telecommunications usage of the others networked.

This problem becomes evident looking at unit 1’s reaction functions. When comparing the

full to the serial and partial connectivity cases, the first term on the right-hand side (involving

only the externality parameters) increases, the second term (the demand for input information)decreases, and the substitution rate of N N2 3 and in absolute value decreases (or their value as

subsidies on costs decreases).

We have then to compare the relative magnitude of the pure network effect, linked to the

interaction between firms, compared to the usage effect, as the sum of factors, which could

increase N1, i.e., the demand and the substitution effects.

For better understanding the relative importance of the usage and network effects, we will use

some further simplifying hypothesis. We assume that the three units are identical, so we can

write the demand of input information as follows:

β

η)’(

’

l

l

CN D =

(16)

where :

PSFicY L

,, 3,2,1 1 )( ’*

==<= lβηβ

δη

β

The only differentiating factor among units is the network marginal cost (C’l), associated

with the different network configurations they belong to and is still positive by hypothesis.

20

We now assume that the externalities are isotropic, meaning that a unique externality

parameter is associated with each interaction among firms12: E E Eij ji= = < 1

Under these hypotheses, the problem can easily be analysed as a two-variable problem, where

we compare Ni and the average usage of the other units belonging to the network

(( ) /N N N i2 3 2+ = − ).

Two situations are representative of all the possible network configurations. In the first case,

all units are symmetric (full connectivity), and in the second one, units are asymmetric, since

unit 1 is in the partial connectivity configuration, while the others are in serial connectivity. The

system of reaction functions corresponding to the full connectivity case can be written as

follows:

i

F

Fi

i

F

Fi

NEEC

EEN

NE

E

EC

EEN

−+−−=

−−

−−−=

−

−

)1()(

)231(’

)1(

2

)1()(

)231(’

’

32

2’

32

β

β

η

η

(17a)

where

2

32’

1

)231(

E

EEccC NnF −

−−+=

The reaction functions system corresponding to partial connectivity for unit i and to serial

connectivity for the other units is:

i

S

Si

i

P

Pi

NE

E

EC

EN

NEC

EN

)1(

2

)1()(

)21(’

2)(

)21(’

22’

2

’

2

+−

+−

=

−−

=

−

−

β

β

η

η

(17b)

where

)21(1

)21( 3’2

3’ EccC

E

EccC NnPNns −+=>

−−+=

Analysing the two systems of reaction functions, we obtain the following result on the

equilibrium levels:

Proposition 4

When a stable, unique and positive Nash equilibrium exists for the both the full and the partialconnectivity configurations ( E ∈( , )0 γ with γ < 05. ), we have

N N N NiS P F

iF

− −< =* * * *1 1 a n d . Moreover, the following inequalities hold:

N N N NP FiS

iF

1 1* * * *,< <− −

Proof. See the Appendix.

We observe that the solutions associated with the full connected network are equal, because

firms are totally symmetric. Moreover, unit 1’s partial connectivity reaction function is steeper

12 This hypothesis also means that E E E E Eij ji i

jijk= = = 2 and

E E E Eij jk ki = 3 .

21

than that associated with full connectivity ( ∂ ∂ ∂ ∂N N N NFiF P

iS

1 1* * * */ /− −> ), and the reaction

function of the other units is flatter in the case of serial connectivity

( ∂ ∂ ∂ ∂N N N NiF F

iS P

− −>* * * */ /1 1 ).

As regards the sensitivity of the optimal solutions to the externality parameter, the following

result is obtained:

Proposition 5

In the case of full connectivity, an increase of the externality parameter has a negative effect on

the optimum solution. An increase (decrease) of the externality parameter increases (decreases)

the partial connectivity Nash equilibrium of unit 1 and decreases (increases) the optimal

solution of the other units.

Proof. See the Appendix.

For the optimal set-up of an internal communication network, our result implies that, when

units are identical, totally symmetric and externalities are isotropic and quite low, the full

connected network ensures the higher number of contacts: the network effect logically

overcomes the usage effect. A higher substitutability effect leads a unit to decrease her usage

when all the others are connected and then all have the same Nash equilibrium. It is also logical

that, in the serial case, the average telecommunications usage of unit 2 and unit 3 is lower than

that of the full connectivity case, since their number of direct link increases. More interesting is

the result for unit 1: her number of direct links is the same in the full and in the partial

connectivity cases. Nevertheless, the existence of an indirect link between firms 2 and 3

stimulates further telecommunications usage. But when this indirect link exists, an increase of

the externality parameter has the above-mentioned free riding effect: unit 1 lowers her

telecommunications usage, gaining from the higher network externality generated by the link

unit 2 - unit 3.

Conversely, in the partial connectivity case, an increase in the externality parameter has a

positive effect on unit 1’s telecommunications usage, since there is higher benefit in

communicating directly with the other units connected.

Section III

III.1 Communication and hierarchical firms’ organisation

We already suggested (page 9) that the results of Proposition 1 could shed some light on the

relationship between communication and firms’ organization. This motivates the interest of

further discussing our results in the context of the literature à la Radner (1992,1993) focusing on

22

communication and the hierarchical organisation of administrations, clerical work, and

production inside firms13.

This is not the place to review the wide literature on this subject. We briefly summarize its

main issues only. Radner’s analysis focuses on the structure of efficient hierarchies (defined as

‘ranked threes’) and proves that the lower bound to the number of processor used is

approximately linear in the size of the problem (or the total amount of information to be

processed). In Radner’s framework, the size of the problem determines both the numbers of

processors and delay.

Marshack and Mc Guire (1971) were the first to propose the model of a finite automaton as a

formalism of the notion of a boundedly rational decision-maker, who has a limited capacity of

information processing. Marshack and Radner (1972) then explored the model of a decision-

making organisation as a network of information processors, but their analysis was concerned

more with the decentralisation of information than with information processing. In a similar

spirit, Marshak and Reichlestein (1987) studied conditions under which a hierarchical structure

of decision-making would be efficient in a broader set of structures. In their model, every

processor is also responsible for the final decision about some action variable, and the only cost

of processing is that of communication.

Two papers by Keren and Levhari (1983, 1989) offer a complementary view to the above

mentioned works. They analyse the problem of minimising the costs of information processing

in an organisation in order to explain the increase of long-term average production cost function,

that is traditionally depicted in a U-form. They assume that a manager’s work time depends on

the number of managers from whom he receives information. The output of the firm is an

increasing function of its size (the number of the lowest level managers) and a decreasing

function of its delay (the times it takes to complete the information processing). The average cost

is defined as the cost of information processing divided by the output. Keren and Levhari

provide sufficient conditions under which the average costs is eventually increasing.

More recently, Pratt (1997) generalises this model, allowing managers to have different

ability or capacity of information processing. Processors are remunerated with a wage that is a

function of their ability. Hence, average cost is the ratio of total rental cost of processors (the

sum of the wages paid to all processors hired by the firms) on the total number of information to

be processed, as a proxy for firm’s size. Pratt offers an additional explanation of the increasing

part of the long-term firm’s cost function: he shows that if the wage function is strictly convex in

the capacity, the average cost of information processing is increasing in the hierarchy size.

All those models focus on information processing and on the process of producing a decision

from an amount of data, so large that the information-processing task has to be decentralised

among a number of separate processors. The decision is the output of a single processor, for

example, the top of the hierarchy.

13The models developed in that literature aconcern a subset of firms (managerial units, production units), as well asbureaucracies, or, in other context, central and local jurisdictions (Caillaud-Julien-Picard, 1995).

23

However, in a firm, there are many different decisions to be made, and it is totally impractical

for them to be put out by the same processor, or as a function of the same information. This

situation, in which different decisions are based on different pieces of information, implies the

« decentralisation of information ».

For this reason, we think that the issue of centralisation or decentralisation should be further

considered. A first limit to the centralisation solution is the fact that information is costly both to

transmit and to process. With communication costs, centralisation may be inefficient and the

form of the organisation must optimise the communication network so as to induce efficient

usage of information. Moreover, we showed that there are network externalities involved in

information processing.

In order to focus on the importance of communication costs, we choose to briefly review the

Bolton-Dewatripont (henceforth B-D) model, which extends the Radner’s basic model including

costly communication. The focus on communication costs more easily allows us to compare the

B-D results with those of the model developed in Section II (paragraph III.2). We also apply our

network externality model to pyramidal forms slightly different from those suggested by B-D,

analysing some configurations of communication networks linked with firms’ organisation

(paragraph III.3).

III.2 A comparison with the "firm as a communication network"

In the B-D model, the internal organisation of firms is seen as a communication network that

is designed to minimise both the costs of processing new information and the costs of

communicating this information among its agents. The central idea of the paper is that “the

benefits of greater specialisation achieved by having more agents team up within the same

organisations (each one handling more specialised information) are partly (and sometimes

entirely) offset by the increased costs of communication within the enlarged group of agents” (B-

D, 1994). Repeatingly processing information, agents become specialised, and the more they are

specialised, the more communication is needed in order to coordinate the agents' activities and to

aggregate the available information. The trade-off between specialisation in processing

information and communication as aggregating information is then one of the determinants of

this model.

The B-D model analyses pyramidal networks, i.e., the communication structure where a

single agent receives all the processed information, and each agent sends his information to at

most one another agent. The authors concentrate on two kinds of pyramidal networks: one form

is the hierarchy in which each agent has an equal number of subordinates, and another one that

they call the conveyor belt, in which each agent (except the bottom agent) has only one

subordinate.

24

Figure 2

Two forms of hierarchical networks

2

2

1

3

1

3

C onveyor belt R egular pyram yda l ne twork

Source: Bolton-Dewatripont, 1994

B-D show that if the objective is just minimising delay, pyramidal networks are efficient.

Moreover, if the objective is exploiting returns to specialisation, i.e., minimising total labour

time spent per processed cohort, the result is that there exists a trade-off between returns to

specialisation due to high frequency and communication costs. It may pay for an agent to

delegate part of the job to another agent in order to increase frequency, but delegation induces

communication costs. Delegation is minimised in that the receiver works at least as much as the

sender in order to economise on communication costs. Overall, the shape of returns to

specialisation can lead to any outcome from no delegation to maximum delegation. In presence

of returns to specialisation and in the absence of any concerns about delay, it may be efficient to

have several agents processing any given cohort despite the increased time cost in

communication.

In general, to fully exploit gains from specialisation, it may be efficient to have more than two

agents per cohort. Then the question arises as to which communication structure between these

agents is best suited to exploit the gains from specialisation.

The main result of the B-D work is that "an efficient network resembles a regular pyramid

when it is efficient to have agents fully specialised in either processing or aggregating

information. But an efficient network may also resemble an assembly line ("conveyor belt")

when it is efficient for most agents to be involved in both processing or aggregating information

[...] In most cases, the efficient network is similar to either or a combination of these two

structures" (B-D, 1994).

A strong analogy exists between the conveyor belt and regular pyramidal network with the

network configurations analysed in Section II. To introduce the idea of hierarchical network in

our model, we follow Radner’s definition, i.e., that each node can be ranked to represent a

hierarchical level (or a production unit), which only communicates with the superior level. The

serial connectivity structure will then correspond to the regular pyramidal network, with unit 1 at

the top level. The partial connectivity is the conveyor belt in terms of B-D model.

25

We now compare some results obtained in Section II with the main conclusions of the B-D

model, while underlining the differences between the two approaches.

The first difference is that, in our model, each node produces output combining labor, and

processing information through interactions with other units belonging to a communication

network. The unit’s objective is minimising production costs, taking into account the existence

of communication possibilities among nodes.

A second difference is that the B-D model deals with acyclic networks: that is, networks

where no agent receives, directly or indirectly, information for any of his direct or indirect

superiors. On the contrary, we consider that each time a link between two nodes exists, a two-

way communication is involved. This also means that, in the network structure, indirect links

matter, as becomes evident when considering communication costs.

The third and most important difference is the modeling of communication costs. In the B-D

model, communication costs are due to reading time, as in Radner (1993). These costs involve a

fixed cost of communication (λ>0) and a variable cost proportional to the item communicated

(ami , a>0)14: C m ami i( ) = +λ . The positive returns to scale are due to the possibility of

aggregating a given number of items processed (C m mi i( ) / is a decreasing function of mi). In our

case, the communication costs involve costs of network capacity (c Nn i) and variable costs

depending on the contacts made by unit i with directly linked units (c Nn i ).

The returns of information enter the cost function through the network externality effect. In

fact, unit i’s total volume of information (Ni ) depends on her telecommunications usage, but

also on the entire structure of the network, namely the telecommunication usage of firms directly

and indirectly linked to firm i, and the various externality parameters. Returns of information are

then specific to each network configuration.

Despite these differences, our model can easily deal with the same issues as those analysed by

B-D. In particular, we can find the optimal communication network, as the structure where

production and networking costs are minimised.

Imagine that a decision-maker minimises the sum of the production and networking costs of

units 1, 2, 3, and he has to choose between two structures, i.e. the conveyor belt (serial

connectivity case) or the regular pyramidal network (partial connectivity).

The following result is easily obtained:

Proposition 6

Minimising the sum of the production and networking costs of each node, the regular pyramidal

network and the conveyor belt are equally efficient.

Proof (sketch). The proof is immediate: the total networking costs are the same in the two

structures: in each case, two firms are in the serial connectivity, and only one in the partial

14The authors note that communication makes sense only if λ+a<1; that is, reading a processed item takes less timethan processing it oneself.

26

connectivity case. Since, obviously, production costs are equal for each node in the two

structures, the objective functions are the same in both the regular pyramidal network and

conveyor belt.

Note that, to show Proposition 6, we do not need to assume that neither units are identical nor

do we restrict the value of the externality parameters. We are then tempted to find other

properties of pyramidal networks that are suited to specific forms of corporate organisation.

III.3 Pyramidal communication networks and corporate organisation

As our empirical work showed (Creti-Wolkowicz, 1998), there is a strong link between the

firm’s organisation and the structure of the telecommunication network. The objective of the

telecommunication decision maker is certainly to organise an efficient telecommunication

network that guarantees the minimisation of production and networking costs, but this must

often be reconciled with the informational needs of the top level hierarchy.

The underlying idea is that a telecommunication network is efficient if the top level

minimises its production and the networking costs, in terms of our model, while taking into

account the interdependence among units belonging to the network. Nevertheless, it has to be

stressed that, due to the network externalities, the private choice of top level should be

inefficient compared to the objective of minimisation of the sum production and networking

costs of all units belonging to the network.

This logic offers us a criterion to link network structures with different forms of organisation

within firms and with some examples of business telecommunication networks. We determine

the telecommunication network where the top-level hierarchy (node 1, in our network design)

has the highest demand of input information. We then simply find the network, which

minimises the production and the networking marginal costs of node 1, under the hypothesis that

all units simultaneously minimise their production and networking costs. In order to focus on

network configuration, we still assume, as in Section II, that each information exchange among

units involves the same externality parameter. However, we do not need to assume that units are

identical: that hypothesis is useful when analysing network and usage effects, which jointly are

the determinants of node 1’s optimal telecommunications demand.

We can start analysing the conveyor belt and regular pyramidal networks, simply using the

results obtained in Section II. The efficient network configuration that minimises unit 1’s

production costs taking into account the externality generated by the interactions with other

units, is the serial connectivity case. So we have the following result: the top-level hierarchy

prefers the conveyor belt structure to the regular pyramidal network, with some restriction on

the value of the externality parameter (see Proposition 5).

This result is also obtained by B-D. They show that if the variable cost of communicating an

additional item does not exist, a conveyor belt is more efficient than the regular pyramidal

27

network for some special values of the frequency of processing items. Once the number of

agents at the bottom layer is fixed, and each layer has an equal workload, the conveyor belt is the

network that minimises the number of communication links.

This "decentralisation effect" can also be obtained in the B-D model, allowing a decrease in

communication costs. It is interesting to note that there is also some empirical evidence that the

computerisation of firms leads to smaller and flatter organisations, and thus to greater

decentralisation (Brinjolfsson et alii, 1993).

III.3.1 The decentralisation effect

In the B-D model, the efficiency of the conveyor belt structure is based on quite strong

hypotheses, namely a fixed number of layers at the bottom level (and no communication variable

cost), or a decrease of the communication costs. In our model, these aspects are linked: the

number of nodes at the bottom layer directly affects the communication cost via the network

externality effect. We now generalise the comparison of the regular pyramidal network and the

conveyor belt, increasing the number of the nodes at the bottom level. We consider a finite

number of nodes, in total n, which represent, for example, the number of division inside a firm,

or the number of firms belonging to the same holding company. The n nodes are a community of

interest, very important to state the communication needs of firms (Taylor 1994).

The regular pyramidal network is then the structure where the top level directly

communicates with the remaining n-1 nodes. The decentralised network is a configuration where

the top level communicates with an intermediate node and this latter with the other n-2 nodes.

Generalising the network configuration already analysed in Section II, at the bottom level we

have at least 3 nodes in the regular pyramidal network, and at least 2 in the decentralised

structure. This means that ∞ > ≥n 4 .

The idea of generalising the network structure to n nodes is suggested by Radner (1992) and,

in an empirical context, by Bousquet-Joram-Le Paih (1994)15. We focus on the star-shaped

telecommunications networks (corresponding to the regular pyramidal network) and the

decentralised network (corresponding to the conveyor belt).

15 This work analyses traffic and tariffs of "Colisée Numéris", a specific service offered by France telecom fortelecommunications networks inside firms. Based on a sample of 37 firms (in total 1200 plants), this study alsoattempts to establish some typologies of network structures coupled with firm’s organisation.

28

Figure 3

The regular pyramidal network and the decentralised network

Which network configuration minimises node 1’s production and networking costs?

Following the methodology described in paragraph II.2.1, node 1’s networking function in the

regular pyramidal configuration is:

NN

E

E

EN

i

n

i

n ii

n

11

2

1

12

1

121 1

=−

+−

=

−

=

−=∑ ∑∑( )

(18)

or:

NN

n E

E

n ENi

i

n

11

2 221 1 1 1

=− −

+− − =

∑( ) ( )( )

In order to have a positive and finite network externality effect, which means positive and

finite N1, we have to limit the values of the externality parameter to E n< −1 1/ .16

Unit 1’s networking function in the decentralised network is:

[NE

EN

E

EN E Ni

n

i

n

i

n ii

n

1

2

1

2

2

1

1 12

1

1 23

1

1 1=

−

−+

−+

=

−

=

−

=

−=

∑

∑ ∑∑

( )( )

(19)

or:

[Nn E

n EN

E

n EN E Ni

i

n

1

2

2 1 2 23

1 2

1 1 1 1=

− −− −

+− −

+

=∑( ( ) )

( ) ( )( )

In that case, positive externality effect results when 0 1 1< < −E n/ and 1 1 2> > −E n/ .17

It follows that (for the proof, see the appendix):

Proposition 7

When the externality effect is positive (E n< −1 1/ ), the top level demand of input information

is higher in the regular pyramidal configuration than in the decentralised network.

16 Note that 1 1 1/ n − < for n>2.17 Note that 1 2 1/ n − < for n>3.

2 3 4 n5 ...

1 1 1

2

3 4 5 ... n

29

Corollary 7.a

Minimising the sum of the production and networking costs of each node, the pyramidal and

decentralised networks are equally efficient.

Proof (sketch). As for Proposition 4, the two network configurations have the same total

networking costs, since, in both structures, there is a unit in the partial connectivity structure,

while the remaining ones are in serial connectivity. Production costs being equal, the objective

function is the same in both pyramidal and decentralised network.

Corollary 7.b

In the regular pyramidal network, each unit belonging to the bottom level has a lower demand

of input information than unit 1.

Proof (sketch). The position of node 1 in the decentralised network is the same of the nodes

at the bottom level in the regular pyramidal network. This means that in the regular pyramidal

network, each unit belonging to the bottom level has higher networking costs than unit 1. It

immediately follows that their demand of input information will be lower.

Our results are confirmed by empirical evidence: in the star-shaped network there is a strong

concentration of the traffic from the bottom toward the top level, which acts as a collector of

information (Bousquet-Joram-Le Paih, 1994). It is also shown that firms in the financial and

banking sectors often choose this kind of network structure.

IV. 1 Summary and conclusions

This paper focused on telecommunication demand by firms, when introducing the network

externality effect in the theory of production (Section I). We thus presented a model that

considers the optimal resource allocation in the presence of network externalities arising from

the "input information". The analysis is developed first in the case of a symmetric situation, and

of a leader/follower situation. We analysed how the interdependence among users through bi-

directional contacts affects telecommunications demand. Moreover, we showed that the

productivity effect of telecommunications is related to the network effect, or the non-

compensation effect that a unit obtains by using an input whose costs are shared by many users.

In Section II, the model is generalised, taking into account different connectivity functions

and indirect links among units. Three firms and three network configurations are considered

(partial, serial and full connected network). We showed that a crucial interdependence exists

between network structure and telecommunications usage. In particular, we analyse conditions

under which the usage effect, or the factors that increase units’ individual telecommunications

demand, overcome the network effect. Moreover, under the hypothesis that the externality

parameter does not depend on the direction of calls (isotropic externality), we characterised the

Nash equilibria associated with the different network configurations analysed.

30

In Section III, we compared our results with some recent papers dealing with communication

and intra-firm organisation. We showed that the existence of links among business partners

reflects particular firms’ organisation, and that in a hierarchical firm there is a strong

concentration of the traffic from the bottom toward the top level, which acts as a collector of

information. Communication networks do foster the decentralisation effect, as Bolton-

Dewatripont (1994) also show. In addition to that, our model shows that the decentralisation

effect depends not only form the structure of the network, but also from the extent of the

externality associated with the information exchange among firms.

We think that our model yields different avenues for future research, both empirical and

theoretical. As regards the empirical implications, our paper suggests a view on the network

externality effect that is quite new and serves to explain several types of business

telecommunication demand, oriented toward the value added telecommunication services. Our

model would help in understanding the relationship between the architecture of internal

communication networks and the overall firms’ telecommunication demand. If data on virtual

private network were available at the plant level, in addition to data on the usage of

telecommunication network outside firms, an interesting model for business telecommunications

demand could be estimated, taking into account internal communication networks. The first step

to implement this method would be to aggregate the plants by each firm, and to weight them by

the plant size, in order to obtain the percentage of inter-plant calls on total firms’ calls. The

number of inter-plant calls can be normalised by dividing them for the total number of potential

connections among firms’ units. Total communication costs become the sum of the costs of

using POTS plus the internal communication costs. This cost function could then be estimated,

perhaps by a translogarithmic model, allowing the production function to be more flexible than

the standard Cobb-Douglas we used in our theoretical model. This method is being investigated.

Our model could also offer some insights about inter-firm telecommunications networks. For

example, a firm with multiple suppliers has a telecommunication structure like the partial

connectivity network, and then a medium telecommunications demand level. Her

telecommunications demand can increase if her suppliers are networked, i.e., if a full

connectivity structure is used. This situation is likely to appear when, for example, the

production process is just-in-time: in that case, high telecommunications demand and usage can

be attained, in order to continuously co-ordinate the production process with the stock control.

Concerning the theoretical perspective, a very interesting extension of Section III is also the

analysis of hierarchical organisation of firms in a context of asymmetric information between the

top level authority and the bottom levels (as in Caillaud-Julien-Picard (1995), or Aghion-Tirole,

(1997) ), introducing not only top-down communication flows, but also information exchange at

the same hierarchical level. An extermely fascinating question is to understand how authority

relationships interact with the incentives to communicate within a firm. This is also being

investigated.

31

Our model shows that, in addition to the existence of access to other subscribers, the network

externality effect encompasses in a stronger way the quality of the information exchange and of

the externality, which spreads from the links among firms’ unit. In that perspective, the existence

and the strength of the network externality effect is not only due to the number of firms

belonging the network, but also to their capability to process and exploit the information

communicating with others. Our model therefore emphasises that "it is not just what you know

but whom you know" (Brynjolfsson-Van Alstyne, 1996).

References

AGHION P. and J. TIROLE (1997) « Formal and Real Authority in Organisations », RandJournal of Economics, Vol. 24, pp. 1-29ARTLE R. and AVEROUS C (1973), « The Telephone System as a Public Good : Static andDynamic Aspects », Bell Journal of Economics and Management Science, vol. 4, pp. 89-100BIPE Conseil (1992), «La Prévision de Demande de Transfix», mimeo, CNETBOLTON P. and M. DEWATRIPONT (1994) « The Firm as a Communication Network »,Quarterly Journal of economics, Vol. CIX, pp 809-839BOUSQUET A. and M. IVALDI, BOUSQUET A. and M. IVALDI (1994) « Optimal Pricing ofTelephone Usage : an Econometric Approach », Working Paper, Tenth Annual ITS Conference,SidneyBOUSQUET A., JORAM D. and P. Le PAIH (1994), « Analyse du Trafic et de la Tarificationdu Service Colisée Numeris », mimeo, CNETBRINJOLFSSON E. and M. van ALSTINE (1993) « The Impact of Information TechnologyMarkets and Hierarchies », Working Paper MIT Sloan School of Management, n. 2113-93CAILLAUD, B. JULLIEN and P. PICARD (1995) « Hierarchical Organization and Incentives »,mimeo, LEI-CERAS, ParisCAPELLO, R. (1994) Regional Economic Analysis of Telecommunications NetworkExternalities, Elsevier Science Publisher, AmsterdamCRETI A. (1998) “Firms’ organisation and efficient communication networks”, cha.5, PhDDissertation, Toulouse UniversityCRETI A. and M. WOLKOWICZ (1998) « Analyse de la demande de telecommunications : uneapplication sur données en coupe », Working Paper, France Télécon-Division Marketing n. 29CURIEN N. and M. GENSOLLEN, (1989) Prévision de la Demande deTélécommunications, Eyrolles, ParisDIEBOLD France (1993), « Etude sur les Besoins Télécoms à Moyen/Long Terme pour lesPME- PMI », mimeo, CNETDIXIT, A.K. (1980) The Theory of International Trade, Cambridge Press, CambridgeDUTTA B. and S. MUTUSWAMI (1997) « Stable networks », Journal of Economic Theory,Vol. 76, pp. 322-344ECONOMIDES, N. (1991) « Compatibility and Market Structure », Working Paper New YorkUniversity, n321 Leonard Stern School of BusinessECONOMIDES, N. and C. HIMMELBERG (1994) « Critical Mass and Network size »,Working Paper New York University, n. 243, Leonard Stern School of BusinessECONOMIDES, N. and L. WHITE (1993) « One-way Networks, Two-Way Networks,Compatibility and Antitrust », Working Paper New York University, n. 44, Leonard SternSchool of Business

32

GRIFFIN J. and B. EGAN (1989) « Business Intercity Telecommunication Services », ReviewofEconomics and Statistics, Vol. 30, pp. 520-524JACKSON M. and A. WOLINSKI (1996) « A Strategic Model of Social and EconomicNetworks », Journal of Economic Theory, Vol. 71, pp. 44-74KATZ, M and C. SHAPIRO (1992) « Product Introduction with Network Externality », TheJournal of Industrial Economics, Vol. XL, pp. 55-84KATZ, M. and C. SHAPIRO (1985) « Network Externalities, Competition and Compatibility »,The American Economic Review, Vol. 75, pp. 424-440KATZ, M. and C. SHAPIRO (1986) « Technology Adoption in Presence of NetworkExternalities », Journal of Political Economy, vol. 94 pp. 822-841LIEBOWITZ, S. and S. MARGOLIS (1994) « Network Externality: an Uncommon Tragedy »,Journal of Economic Perspectives, vol. 8, pp. 12-23MARSHACK J. and B. Mc GUIRE (1971) « Lecture Notes on Economic Models forOrganisation Design », mimeo, University of MinnesotaMARSHAK, J. and R. RADNER Economic Theory of Teams, New Haven, CT, Yale U. Press,1972Mc GUIRE B. and R. RADNER (1986), Decision and Organisation, Minnesota Press,MinneapolisPEDREZ-AMARAL T., ALVAREZ-GONZALEZ F., and B. M. JIMENEZ (1995) « BusinessTelephone Traffic Demand in Spain : 1980-1991, an Econometric Approach », InformationEconomic and Policy, Vol. 36, pp. 60-75PRATT A (1997) « The average cost of information processing », Working Paper, presented atESEM, ToulouseRADNER R (1992) « Hierarchy : the economics of Managing », Journal of EconomicLiterature, Vol. XXX, pp. 1382-1415RADNER R (1993) « Information processing in firms and returns to scale », Annalesd’Economie et Statistique, Vol. 25/26, pp. 265-298RHOLFS, J . (1974) « A Theory of Interdependent Demand for Communications Service », BellJournal of Economics and Management Science, Vol. 5, pp. 16-37SAH J. and STIGLITZ J. (1986) « The Architecture of Economic Systems : Hierarchies andPolyarchies », American Economic Review, Vol. 76, pp. 716-727SPENCE, A.M. (1979), « Product Selection, Fixed Costs, and Monopolistic Competition »,Review of Economic Studies, Vol. 43, pp. 217-235SQUIRE, L . (1973) « Some Aspects of Optimal Pricing for Telecommunications », Bell Journalof Economics and Management Science, Vol. 4, pp. 515-525TAYLOR, L.D. (1994) Telecomunications Demand in Theory and Policy, Kluwer AcademicPublishers, DordrechtTIROLE J., (1988) The Theory of Industrial Organisation, MIT Press, Boston, Massachusetts

33

APPENDIX

Proof of Proposition 1

It is straightforward to note that the only differences between the firm 1’s Nash (henceforth N)

and Stackelberg (henceforth S) demand functions are the following terms of marginal costs of

the network:

)()(’ )(’12212211

221

111 and

εεεεε

ε −+=+=

NnSN

nN ccNC

ccNC (20)

Under the assumptions on the externality parameters, we have 112212212211 εεεεεε <− , which

means that )(’ )(’ 11SN NCNC < or SN NN 11 > , since the demand of input information is

negatively related to the network marginal costs.

Proof of Corollary 1

At the Stackelberg equilibrium, unit 1’s volume of information is as follows:

( ) +)- (*=]*[*)*(**22

21

22

1221111112

22

2111112211111 bNNbNNNNN SNSSSS

εε

εεεεε

εεεεε −+=+=

where:D

Nn

L Nc

c

cYb 2

2

222

2

2

2 ]

)(

[*

=+

= β

εβ

δη

(21)

Since the term - (ε ε ε ε11 21 12 22/ ) is always positive by hypothesis, a decrease of N S1* implies

a decrease of N S1*

Proof of Corollary 2

In this case, we have to compare the following pay-offs:

( ) ]*N[1

* 112222

2SDN NN ε

ε−= (22a)

( ) ]*N[1

* 112222

2NDS NN ε

ε−= (22b)

Since the first term of RHS is the same in both equations, and we know from the previousproof that N N

1* > N S

1* , the second term of RHS of equation (22a) will be lower than in (22b).

We then conclude that N N2* < N S

2*

Proof of Corollary 3

At the Nash-game equilibrium, total demand for information is:N N N N NN N

TDN N N

1 2 11 12 1 22 21 2* * * * *( ) ( )+ = = + + +ε ε ε ε (23)

Similarly, at the Stackelberg-game equilibrium,:N N N N NS S

TDS S S

1 2 11 12 1 22 21 2* * * * *( ) ( )+ = = + + +ε ε ε ε

Their difference is then:

34

N N N N N NTDN

TDS N S N S* * * * * *( )( ) ( )( )− = + − + + −ε ε ε ε11 12 1 1 22 21 2 2

We know from corollaries 1 and 2 that:N N

N N N N

N S

N S N S

1 1

2 212

221 1

0

0

* *

* * * *( )

− >

− = − − <εε

We can write:

N N N NTDN

TDS N S* * * *[( ) ( )]( )− = + − + −ε ε

εε

ε ε11 1212

2222 21 1 1

This difference will be positive if (and only if):

( ) ( )ε εεε

ε εεε

εε11 12

12

2222 21

12

22

11

21

0+ − + > ⇔ <

The condition means that ε ε ε ε11 22 21 12> , as stated by hypothesis.

Proof of Proposition 2

In the case where network externalities are not present, the firm 1’s TRS is as follows:

L

nNashLN c

cTRS =, (24)

Taking into account network externalities and Nash behaviour, the TRS becomes:

11,

11

11,

)(

εεε

L

NLN

L

NnNashLN c

cTRS

c

ccTRS +=

+=

When firm 1 is the leader, the TRS will be the following:

1)/(B’ where

’’

)’(

1122122111

,,

<<−=

+=+

=

εεεεεBc

cTRS

cB

cBcTRS

L

NLN

L

NngStackelberLN

It immediately follows that: TRSN,LStackelberg > TRSN,L

Nash >TRSN,L

Proof of Proposition 3

The demand for information is given by the equation

The marginal networking costs are positive (by assumption), and lower in the full connectivity

configuration than in the partial or serial cases:

))1(()(’

)1

1()(’

)1

1()(’

)(’),(’)(’0

31

21

32

32

21

32

3211

2311

32

31

21

since

NnPi

NnSi

NnFi

Si

Pi

Fi

cEEcNC

E

EEccNC

E

EEEEEccNC

NCNCNC

i

−−+=

−−−

+=

−−−−−−

+=

<<

(25)

35

We easily see that the marginal cost advantage depends on the network externality effect:

0<1

1

1

111

213

23

1123

1132

23

12

23

23 1

213− − − − −

−< − −

−− −E E E E E

E

E E

EE E,

We then conclude that:N N Ni

DFiDS

iDP> ,

Proof of Proposition 4

This proof requires some preliminary steps: in both the full connectivity and the partial/serial

connectivity configurations, we find the conditions for stable and positive solutions, and we

compare the Nash equilibrium of unit 1 with that of the other units (step 1 and 2). Each of

these steps gives conditions on the value of the externality parameter, which have to be taken

into account, when proving Proposition 4 (step 3).

Step 1: Conditions on the full connectivity Nash equilibrium

CS: Stability

A sufficient condition for the Nash equilibrium to be stable and unique1 in the full networkconnectivity configuration is that the externality parameter is sufficiently low ( E ∈( , . )0 0 5 ).

A sufficient condition for the Nash equilibrium to be stable is that the unit’s 1 reaction

function is steeper than the other firm’s reaction function. This conditions means:2

1

12 1 02E

E EE E

−< ⇔ + − < (26)

Since E is always positive, this condition is satisfied when E ∈( , . )0 0 5 .

CN: Positive equilibrium

A necessary condition for the Nash-full connectivity equilibrium solutions to be positive isthat the externality parameter is low ( E ∈( , . )0 0 5 ).

The necessary condition for the equilibrium to exist with positive N F1* is as follows:

1)1()(

)231(’

2)1()(

)1)(231(’2’

32

2’

32

<⇔+−−>

−−−−

EEC

EE

EEC

EEE

FFββ

ηη

This condition is always satisfied by hypothesis.Positive N i

F−* will be obtained if:

2

11

)1()(

)231(’

)1()(

)231(’’

32

2’

32

<⇔+−−<

−−−

EEEC

EE

EC

EE

FFββ

ηη

We then conclude that, when the externality parameter is lower than 0.5, both N NFiF

1* *and − are

positive.

1 Since the reaction functions are linear, if a stable equilibrium exists, it is unique.

36

Nash equilibrium solutions

When a stable and positive Nash equilibrium solution exist in the full connectivity case, theoptimal solution for unit1 is equal to that of the other firms ( N NF

iF

1* *= − ).

The system of the reaction functions gives the following Nash equilibrium solutions:

β

η)(

)21(’’

**1

F

Fi

F

C

ENN

−== −

Step 2: Conditions on the partial/serial connectivity Nash equilibrium

CS: Stability

A sufficient condition for the Nash equilibrium to be stable and unique in the partial network

connectivity configuration is that the externality parameter be sufficiently low (E ∈( , . )0 0 6 ).

The sufficient condition for the Nash equilibrium to be stable is:

21

23 1 0

22E

E

EE< + ⇔ − < (27)

Since E is always positive, this condition is satisfied when E ∈( , / )0 1 3 or E ∈( , . )0 0 6 .

CN: Positive equilibrium

∃ ∈ (0,0.5)γ such that a positive partial connectivity Nash equilibrium exists. The lower is

the unit variable cost of network compared to the unit cost of capacity, the higher γ will be.

The necessary condition for the equilibrium to exist with positive N P1* is as follows:

2’

’

2’

2

’

2

1

2

)1()(

)21(’

)(2

)21(’

E

E

C

C

EC

E

CE

E

P

S

SP +<

⇔

+−>−

β

ββηη

Positive N iS

−* will be obtained iif:

EC

C

E

E

EC

E

C

E

P

S

SP 2

1

2

)1(

)1()(

)21(’

)(

)21(’’

’2

2’

2

’

2

<

⇔+

+−>−

β

ββ

ηη

We then have to find the conditions under which the following inequalities are satisfied:

EC

C

E

E

P

S

2

1

1

2’

’

2<

<

+

β

A qualitative study of the function ( )β’’1 / PS CCf = is required in the interval E ∈( , . )0 0 6 ,

where the Nash solutions are stable.We know that C CS P

’ ’> , and then f1 >1. That function depends positively on the unit variable

cost cN

and negatively on the unit cost of capacity cn , as the following derivatives show:

021

1

21

1

)()(

3

2

32’

11

1 >

−−−

−= −

E

E

EC

cf

c

f

p

N

n

ββ∂∂

The sign of the derivative depends on the last term in the brackets, which is positive, sinceE<1 by hypothesis. The same consideration applies to the derivative of f1 with respect to c

N:

021

1

21

1

)()(

33

2

2’

11

1 <

−

−−−= −

EE

E

C

cf

c

f

p

n

N

ββ∂∂

37

where the terms in brackets is negative.

We can also show that in the interval E ∈( , . )0 0 5 , f1 depends positively on the externality

parameter:

[ ] [ ])2(2)41)(()21)(1()1(

2)( 42

2222

11

1NnNn

Nn

NccEEcc

EEcEc

Ecf

E

f++−+

−−+−= −ββ

∂∂

The sign depends on the last term in the brackets. Since ( )1 4 2− E is positive in

E ∈( , . )0 0 5 ,and all the other terms are positive, we conclude that the derivative is positive as

well in that interval.

Unfortunately, the sign of the derivative of f1 with respect to E in the interval E ∈( . , . )0 5 0 6

and the convexity/concavity with respect to E are quite difficult to prove, without giving anumerical value to c

N and cn . Numerical simulations show that the less cn increases with

respect to cN

, the more the function is convex.

The function fE

E2 2

2

1=

+ is an increasing function of E, and it attains 1 when E=1. This means

that the inequality β

<

+ ’

’

21

2

P

S

C

C

E

E is always verified in the interval E ∈( , . )0 0 6 .

The function fE3

1

2= is a decreasing function of E, and it attains 1 when E=1/2.

Comparing the plots of f3 and f1, we see that there exists a value of E, γ ∈(0,0.5), such as the

inequality is EC

C

P

S

2

1’

’

<

β

is satisfied.

Graph 1

Comparison of ( )β’’1 / PS CCf = with f

E3

1

2= and f

E

E2 2

2

1=

+

( )β’’1 / PS CCf =

0 0.1 0.2 0.3 0.4 0.5

0.2

0.4

0.6

0.8

1

1.2

1.4

γ.

fE3

1

2=

fE

E2 2

2

1=

+

E

Nash equilibrium solutions

When a stable and positive Nash equilibrium solution exists in the partial connectivity case,the optimal solution for unit 1 is higher than that of the other units (N NP

iS

1* *> − ).

The system of the reaction functions gives the following Nash equilibrium solutions:

+−

−−= βββ

η)()(

2

)(

1

)13(2

)12(’’

2

’’2

2*

1PSP

P

C

E

C

E

CE

EN

38

−

−−=− ββ

η)(

2

)(

1

)13(2

)12(’’’2

2*

PS

Si

C

E

CE

EN

Their difference is then:

−+

−+

−−=− − βββββ

η)(

1

)(

1

)(

1

)(

12

)()13(2

)12(’’’’’’

2

2

2**

1SPSPP

Si

P

CCCCE

C

E

E

ENN

We know that for the stability condition, ( )3 12E − <0, and then also ( )2 12E − <0. We have

only to study the sign of the last term on the right hand side.

Simple manipulations show that this term is positive iif :β

>

++

’

’2

)12(

)1(

S

P

C

C

E

E

This inequality is always verified, since )12/()1( 2 ++ EE is an increasing function of E and

always higher than one, while ( )β’’ / SP CC is a decreasing function of E, and always less than 1.

We conclude that the sign of the difference N NPiS

1* *− − is positive, and therefore N NP

iS

1* *> − .

Step 3: Comparison of the full versus partial/serial solutions

The full and partial connectivity Nash equilibrium for unit 1 can be factorised as follows:

β

η)(

)21(’’

*1

F

F

C

EN

−= (28)

[ ]ββββ

η)()(2)(

)13()(2

)12(’ ’2’’2’’

2*1 SPS

SP

P CECECECC

EN +−

−−=

Their difference is positive iif:

)21)(13(

)12(

)()(2)(

12

2

’2’’’

’’

EE

E

CECECC

CC

SPSF

SP

−−−>

+−

βββ

β

It is easy to show that when E ∈( , . )0 0 5 , the right hand side of the inequality is negative. The

first term of the left-hand side is positive by hypothesis; as regards the second term, we have:β

βββ

<

+⇔>+−

’

’

2’2’’

1

20)()(2)(

P

SSPS C

C

E

ECECEC

This is exactly the condition that has to be satisfied in order to have positive N iS

−* solutions.

The difficulty now arising is that we do not have the exact value of the externality parameter γ<0.5 Unfortunately, this value depends on the shape of ( )β’’

1 / PS CCf = , and cannot be

calculated without giving numerical values to the unit cost of capacity and to the unit variable

cost. Our numerical simulations showed that, under a reasonable hypothesis on the value of

unit variable costs and unit cost of capacity of the network, f1 is slightly higher than 1 at least

in the interval E<0.5, and that it increase very slowly. This means that the point at which f1

39

and Ef 2/13 = meet is very close to 0.5 (see Graph 1), as required to prove that the sign of the

term analysed is positive.

We then conclude that the last inequality always verified in the interval (0, γ) where γis very

close to 0.5, and then N NP F1 1* *< .

Moreover, we know from the previous steps that N N N NiS P F

iF

− −< =* * * *1 1 and . Once it is

shown that N NP F1 1* *< , we easily have N N N Ni

S P FiF

− −< =* * * *1 1 < , and then N Ni

SiF

− −<* * .

Proof of Proposition 5The derivative of N F

1* with respect to E is:

−+

−=−

E

CE

CE

N F

F

F

∂∂η

∂∂ β

β

)()21(

)(

2’

’

’

*1 (29)

It is easy to show that this derivative is negative iif:

1)23()1(

)(’ 222

12’ <−−−

− EEE

EC F

ββη

This inequality is always verified, since in E ∈( , )0 γ with γ < 05. , the term ( )E E2 3 2− − is

negative, and all the other terms are positive.For the partial/serial case, for computational convenience, we study ∂ ∂N Ei

S−* / . Remembering

that:

−

−−=− ββ

η)(

2

)(

1

)13(

)12(’’’2

2*

PS

Si C

E

CE

EN

and

E 0 13

12

0.5<<E 0)13(

)12(

2

2

2

2

when

∀>

−−

>−−

E

E

E

E

E

∂∂

γ

We study the sign of the derivative of the terms in square brackets.

We first prove two preliminary results:

Result 1:

E

C

E

C

E

C PPS

∂∂

∂∂

∂∂ βββ −−−

<<)(

2)()( ’’’

(30)

ProofCalculating the derivatives /)( and /)( ’’ ECEC SP ∂∂∂∂ ββ −− we have:

22’

’

221’1’)1(2

)1()(

2

)(

41

EC

C

EC

cE

C

cE

S

P

S

N

P

N −<

⇔

−>

+

++

β

ββ

ββ

since:1

’

’22 1)1(2

+

>>−

β

S

P

C

CE

40

The inequality always holds in E<0.5, and then also in the interval of interest (E<0.33).Note that the right hand side of Result 1 is always true, since the derivative ECP ∂∂ β /)( ’ − is

positive.

Result 2:[ ]

E

CE

E

C PP

∂∂

∂∂ ββ −−

< )()(2

’’

(31)

Proof

The above inequality can be written as:

β

ββ

∂∂

∂∂

)(

1)()(2

’

’’

P

PP

CE

CE

E

C+<

−−

We then have:

[ ] nNP

P cEEcCE

CE >−+−⇔<−

−

18)21(2)(

1)()2( 2

’

’

ββ∂

∂β

β

This is true because:[ ] 118)21(2 2 >−+− EE ββand c c

N n> by hypothesis

We can now prove Proposition 3. Combining (30) and (31), we have:

[ ] [ ]E

C

E

C

E

CE

E

CE SPPP

∂∂

∂∂

∂∂

∂∂ ββββ −−−−

>>>)()(

2)()(

2’’’’

then[ ]

E

C

E

CE SP

∂∂

∂∂ ββ −−

> )()(2

’’

It means that:

0)(

2

)(

1’’

<

− ββ∂

∂

PS C

E

CE,

or ∂ ∂N EiS

−* / <0.

Similar arguments apply to show that ∂ ∂N EP1 0* / > .

Proof of Proposition 7

We must exclude values of E that would yield negative or infinite externality effect in both

configurations. We then limit our attention to values such that: 0 1 1< < −E n/

The networking costs of node 1 in the regular pyramidal network are (where the apex rp refers

to regular pyramidal network):

[ ] )())1(1()(2

12

1 ∑=

−−−+=n

iiNn

rp NENEnccNC (32)

and the marginal networking costs:[ ]))1(1()(’ 2

1 EnccNC Nnrp −−+=

41

Total and marginal networking costs for the decentralised network are respectively (here d

means decentralised network):

)()2(1

))1(1()(

3212

2

1 ∑=

+−

−−−−+=

n

iiNn

d NENENEn

EnccNC

and:

C N c cn E

n Ed

n N’( )

( ( ) )

( )1

2

2

1 1

1 2= +

− −− −

It immediately follows that: C N C Nrp d’( ) ’( )1 1< . The demand of input of information is (ceteris

paribus) a decreasing function of the networking marginal cost (see Proposition 1). This

implies that: N ND rp D d1 1

, ,> .