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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: [email protected]
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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
, ,> .