Directory UMM :Data Elmu:jurnal:S:Structural Change and Economic Dynamics:Vol12.Issue1.Mar2001:

12 (2001) 1 – 28

Network technologies, communication

externalities and total factor productivity

Anna Creti

IDEI-LEMME,21Alle´e de Brienne,31000Toulouse,France Accepted 16 December 1999

Abstract

The main aim of this paper is to determine the factors which enhance or temper firms’ private incentives to use communications technologies that are characterised by network externalities and allow firms to influence their rate of technological change or total factor productivity (TFP). As regards the impact of the network effect on TFP, we find that when the externality parameter is low, a slightly negative effect appears, but this effect is reversed when the externality is higher. This relationship is valid regardless of the number of firms. Our result is particularly interesting because it offers a possible explanation for the Solow productivity paradox. We conclude that, in addition to the degree of network effects, market structure, consumer preferences and the number of users also have a very important influence on TFP and technological change. © 2001 Elsevier Science B.V. All rights reserved. JEL classification:D24; L96; O31

Keywords:Dynamic profit maximisation; Total factor productivity; Network externalities

www.elsevier.nl/locate/econbase

This paper is based on Chapter 6 of my dissertation at Toulouse University. I am very grateful to two anonymous referees, Erik Brynjolfsson, David Encaoua, Danny Quah and more particularly to my advisor Patrick Rey for their useful suggestions. I also benefited from comments by seminar participants at CREST-LEI, CNET, LSE-STICERD, Royal Economic Society Conference 1998, Warwick, Spring Meeting of Young Economists 1998, Berlin, EARIE Conference 1998, Copenhagen. All errors remain mine. Financial support by the CNET-Centre National d’Etudes des Te´le´communications and the European Commission through the TMR Marie Curie Programme is gratefully acknowledged.

E-mail address:creti@cict.fr (A. Creti).

0954-349X/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 5 4 - 3 4 9 X ( 9 9 ) 0 0 0 2 9 - 6

1. Introduction

It has been stressed that in a world of global economies it is likely that the challenge for firms may not always be among the first to produce new information, but may instead be how to recognise, obtain, employ and complement the relevant innovative information (De Bondt, 1996).

Information technologies (computers, mainframes, etc. henceforth, referred to as IT) and telecommunications (telephone, fax, modem, virtual private networks) play an important role in information processing: they increase the speed and accuracy of analysing, storing and transmitting information on input costs, production technologies and other strategic data (Toffelmire, 1992). Furthermore, they also enhance integration among firms, and foster the accessibility of large knowledge sources such as patent publications and databank networks.

Despite the recognised importance of IT-telecommunications technologies and the productivity gains expected by using them, empirical studies aimed at measuring their influence on economic performance have found disappointing results (see for example, Morrison and Berndt, 1991; Berndt et al., 1992, and, for an extensive review, Brinjolfsson and Yang, 1996). Robert Solow commented upon this situation by stating ‘we see computers everywhere except in the economic statistics’. This became known as the Solow paradox or information technology productivity paradox.

We believe that perhaps this disappointing empirical evidence should stimulate further analysis of the relationship between communication technologies and firms’ productivity. Without distinguishing between information technologies and telecommunications, we refer to communications technologies that contribute to information processing and are characterised by the existence of a community of users. The main aim of this work is in fact to determine factors that enhance or temper firms’pri6_ateincentives to invest in communication technologies. From this point of view, even if telecommunications could be considered as public services, our analysis also differs from recent work on returns to public infrastructure (see for example, Ashauer, 1989; Berndt and Hanson, 1992; Garcia-Mila and McGuire, 1992; Nadiri and Mamuneas, 1994; Morrison and Schartz, 1996).

In particular, we consider that the usage of communications technologies allows firms to obtain knowledge and therefore to influence their rate of technological change or total factor productivity. The role of IT investment goods in the production process differs, therefore, from the role of other capital goods: commu-nication technologies are used to smooth the process of production and to lower transaction costs (Van Mejil and Van Zon, 1933; Norton, 1992), while other capital goods are used together with labour to produce output.

Next to this direct effect on firms’ productivity, IT investment goods also have an indirect effect via their influence on the knowledge diffusion process among the users. We claim that IT advantages stem also from the fact that they create a network of firms, as for example, between component and material suppliers in assembly-type firms (as in the automobile industry) or in service firms (as in hotel and catering industries). Furthermore, the new flexibility and speed of

communica-tion systems and their links with data banks create the possibility for a wider variety of new value-added networks and new types of information services. Communication technologies are thus effective in the distribution of knowledge among firms, creating a spillover effect: in fact, there is some empirical evidence that excess returns in using computers and information technologies do exist (Brinjolfsson and Hitt, 1993; Oliner and Sichel, 1994). We think that these spillovers may be related to the fact that communication technologies involve a network of users, so they are likely to create a ‘network externality effect’, meaning that the individual usage of the network is a positive function of the existent community of users1_{. This effect is mostly neglected in discussion about IT. We}

propose an original way of modelling these externalities, stressing the importance of the network architecture, namely the existence of direct and indirect links among users.

The main contribution of this paper thus lies in the specification of the dynamic evolution of factor productivity that we assume to be affected by network technolo-gies usage and spillovers. We therefore focus on the impact of the extent of the network effect and the number of users on the steady state values of total factor productivity (TFP) and communication investment, in a partial equilibrium model. The main ideas of the paper are drawn from the literature on R&D spillovers (Dasgupta and Stiglitz, 1980; Spence, 1984; Judd, 1985; Levin and Reiss, 1988; De Bondt, 1996); in contrast with this kind of literature, we present a rather new specification of the network effects, which differs from the usual R&D spillover effects. The endogenous growth models based on innovation mainly drive our modelling of the TFP dynamics. However, when specifying the production function for the blueprints (Grossman and Helpmann, 1991), these models concentrate on a linear relationship between total investment in research and the knowledge stock it accumulates as a consequence. We focus on the possibility of a non-linear relation-ship between the innovative activity (in our case, the telecommunication invest-ment) and general knowledge, assuming that this relationship exhibits decreasing returns. This situation is similar to a crowding of information, because later contacts are less significant than earlier communications.

One of the most interesting results of the model, mainly obtained by numerical simulations, regards the impact of the network effect on the TFP. We find that when the externality parameter is low, a slightly negative effect appears, but this effect is reversed when the externality is higher. This result may offer a possible explanation for the Solow producti6_{ity paradox, as it can reconcile the very mixed} evidence found by the econometric models. Our simulations suggest that the impact of IT is non-neutral with respect to the characteristics of the firms analysed. Moreover, we show that an additional reason can explain the heterogeneous results obtained by the empirical analyses: all these studies estimate reduced forms models 1_{The idea of network externality effects has been widely explained by the literature on}

telecommuni-cations residential demand (Artle and Averous, 1973; Squire, 1973; Rholfs, 1974; Taylor, 1994) and on competition among firms producing goods likely to create network effects (Katz and Shapiro, 1985, 1986, 1992; Economides, 1991; Liebowitz and Margolis, 1994).

that do not distinguish between the general impact of communication technologies and the extent of the network effect.

One of the most popular explanations of the Solow productivity paradox is that output and inputs of information-using industries are not being properly measured by conventional approaches. In our view, this measurement error also depends on having neglected some important determinants of the IT/telecommunications usage: degree of network effects, market structure, consumer preferences and number of users. We thus suggest that some further empirical work is desirable, especially ranking industries according to the expected extent of spillovers. Following our simulation results, we could expect that firms selling low differentiated products, requiring a large and technologically efficient communication network and an intensive extent of communication externalities, would experience a positive impact of IT/telecommunications on total factor productivity and then have the highest private incentive in communication technologies investment.

The remainder of the paper is organised as follows. After discussing the hypothe-ses of our model (Section 2), we analyse firms’ decision making on telecommunica-tion investment and technological level in a dynamic context (Sectelecommunica-tion 3). The second part of the paper focuses on the analysis of the steady state levels, together with the welfare analysis (Sections 4 and 5). The empirical implications of our model (Section 6) are proposed before the conclusions (Section 7).

2. Hypotheses of the model

The major feature of using a telecommunication network is captured by the externality effects. Here, we assume that the number of users and the externality parameter, i.e. the usability of the knowledge obtained by communicating with other users — are the determinants of the spillover effect. The total amount of knowledge obtained by communications can be thus interpreted as pure knowledge spillover.

The externality effects are specific for each network configuration. For the purpose of the present model, we choose a star shaped network (Fig. 1), where each firm has bi-directional links with the other n−1 users. The amount of spillovers depends positively on the number of firms belonging to the network and on the externality parameter.

We assume that each of then firms is like firm 1 in Fig. 1, i.e. each firm knows that she is called by n−1 users, and that she calls them. This hypothesis is more realistic than that of considering a full connected network, where each firm talks with all of the others and benefits from symmetric network effects. Firms are less interested than residential customers in network externality as simply the possibility of contacting everybody, but they are more reactive to the effective contacts they do or they receive.

Let us consider a representative firm i. In a first step, firm i decides her telecommunications usageNiand the direct contacts with otherjfirms, in order to

other firms, she adds a ‘piece’ of the total amount of their information to her own total amount of informationNSPi, gatheringjeNSPj. The total amount of

informa-tion obtained from contacting other firms is weighted by the parametereB1, which we interpret as the externality effect or the quality/benefit of exchange firm i-firm

j. The parametereis taken as exogenous by each firm.

For simplicity, we assume that NSPi is a linear function of Ni and of jeNSPj.

Moreover, because of indirect links, NSPj will depend on Nj and on the contacts

betweenj and the other firms linked to j, because:

NSPj=Nj+% k

eNSPk (1)

Taking into account the direct as well as indirect links, the networking function for firm ibecomes:

NSPi=Ni+% j

e

Nj+% k

eNSPk

(2)

And so on, if firmk has other indirect links, they will be integrated in the firmi’s networking function. This process will stop when every indirect link of the network has been considered2_.

In the case of the star-shaped network, the total amount of information spillovers gathered by the communication network is:

NSPi=

1

1−(n−1)e2

Ni+e % n

j=1 i"j

Nj

(3)

Fig. 1. Network interaction: the star-shaped network.

2_{In a topological approach, this process would corresponds to the evaluation of the oriented graph}

obtained in the star-shaped network, whereNiis the value associated with the nodeiandeNSPjis the

Externality effects are positive ife2₍

n−1)B1: we call this assumption the network stability condition.

Note that our formulation significantly differs from that of the R&D spillovers model (as for example Spence, 1984; Levin and Reiss, 1988; De Bondt et al., 1992), which simply model the intra-industry pool of knowledge as follows:

NSPiR&D=Ni+e %

n

j=1 i"j

Nj

The relevant intra-industry pool of knowledge for firmionly consists of the total amount of knowledge created with own R&D efforts and a part of the knowledge created by the other firms in the industry. Our claim is that, as IT/telecommunications are likely to create a web of links, the above formulation is not adequate to model the multidirectional interactions of a network of users. In fact, Eq. (3) indicates that the intra-industry pool of knowledge is a non-linear positive function of the extent of the spillover effect and the number of users; moreover, the sensitivity of the intra-industry pool of knowledge to both the parametereand the number of users

nis stronger than that considered by the standard R&D spillovers models. However, as in Levin and Reiss (1988), Romer (1990) and Grossman and Helpmann (1991), we include a double counting effect: each time that firmiuses the telecommunication network, she contributes to the stock of general knowledge capital.

We discuss the case in which a representative firm engages in telecommunications investment in order to increase the efficiency of her production process. We consider a simple Cobb-Douglas production function with constant returns to scale, as in Rebelo (1991):

Y=AK (4)

whereAtis the technical change or total factor productivity. Theminimum total cost

for producing Yt (Sato and Suzawa, 1984), once the optimum allocation of capital

has been performed, is as follows:

TC=(YcK)/A (5)

Producers engage in telecommunications investmentNi, to increase the productivity

of their input and to obtain competitive weapons in market rivalry, because a higher

Ayields lower prices and market expansion. Firms incur cnNias variable costs of

usage, which represents what managers frequently call the telecommunication budget. We thus consider that the evolution ofA, i.e. the dynamic change of the total factor productivity, is a non-linear function of telecommunications investment and spillovers3

.

3_{For a static model where the total factor productivity level is influenced by the communication}

technologies’ usage, see Creti (1998). In that model, it is shown that, under perfect competition, the substitution effectbetween traditional inputs, like capital or labour, and the input information is related to thenetwork effect, or the advantage that a firm obtains by using an input whose costs are shared by other users. Moreover, conditions under which theusage effects, or the factors that increase firms’ individual communications needs, overcome thenetwork effectare investigated.

This cost reducing aspect is often used in the literature on technological change and R&D. However, R&D investment and the R&D capital services and spillovers are embodied in the production function. The main difference with respect to these models is thus the hypothesis on technological change: we choose Hicks neutral technical change to better analyse the impact of communication technologies on TFP, without considering factor demand bias. For instance, Jones (1995) suggests that perhaps computers and other forms of capital play a complementary role in the discovery of knowledge. We think that, in some way, IT/telecommunication are more similar to R&D than to other forms of input: as R&D, they allow obtaining, processing and storing information useful to the production process. However, R&D is an internal source of information, while communication technologies are concerned with information channels external to the firms, and for this reason we believe that the impact of the IT usage on TFP is better modelled by a disembodied technological change. We thus assume that the telecommunication equipment is included in the variable capital, so it is not explicitly analysed. In our view what is new and interesting to model is IT/ tele-com usage and network externality effects on the technological decision of firms.

The TFP dynamic is given by: dAt

dt =At g

Nt

d NSPt

d0 _{d+d0BgB1 (6)}

The term on the right-hand side is analogous to a conventional production function, and, in that respect, it exhibits the standard neo-classical properties of positive marginal product and diminishing returns to communication technologies investment and spillovers. Over time, the marginal productivity advantages are decreasing with respect to IT/telecommunications expenditures and spillovers. This means that later contacts are less significant than earlier communications: there is a crowding of information.

Here d and d0 can be interpreted as elasticities of respectively the firm’s own communication investment and the spillovers to the TFP dynamics, and they have a different meaning than the parameter e or the extent of the externality effect. The parameters d and d0 have to be considered as the productivity or the effectiveness of the communication technologies and of their spillovers. We thus assume that a technical improvement of the communication is reflected in an increase of d and d0. For this reason, in the rest of the paper, we will refer to these parameters, as proxies of the technological opportunities linked to network usage.

Several special cases of the productivity generating function (6) are used in other studies. Ifg=1,d0 =0, we obtain Romer’s (1990) technological progress function; in our specificationgB1 avoids the so-called scale effects, as pointed out by Jones (1995), because it is no longer intuitive to assume that more resources are needed if one possesses a higher productivity level. With g, d0=0, we recognise the Dasgupta and Stiglitz (1980) specification. Whend, g=0 and d0=1, we derive the

Spence (1984) specification, which is also used by De Bondt et al. (1992). The Levin and Reiss specification can be obtained when g=0, and that of Sato and Suzawa (1984) results from settingg=1,d=d0 =0.

We assume monopolistic competition in the output market. This allows us to investigate the influence of product differentiation and the number of competitors on technological change and welfare. We consider two kinds of goods:n differenti-ated products and one homogeneous product. The utility is quasi-linear, i.e. it adds two kinds of sub-utility or two sectoral utility levels for the differentiated and the homogeneous goods. As in Grossman and Helpmann (1991), the utility of the differentiated goods is a` la Dixit-Stiglitz, which means that there is linear homo-geneity among the n number of varieties. The optimal allocation of expenditures across differentiated goods and homogeneous good yields the usual demand function for each differentiated good:

yi=pi −1_P

D a−b_X

0

b ₍₇₎

where4 _{a=1/(1−r)\1,b=1/(1−z)\1, with the following price index:}

PD=

% n

i=1

pi

1−a

n

1/(1−a)

(8) Note thatais the elasticity of substitution acrossn differentiated goods, whileb is the overall price elasticity of demand. We then logically assumea\b.

The last hypothesis we need is thedynamic of exogenous 6_{ariables, i.e. scale of} demand, and in particular, the costs ofNandK. As in Morrison and Berndt (1991), we assume that the exogenous variables increase at given constant rates (X0t=

X00e

−sXt_,c

nt=cn0e

−snt_,c

kt=ck0e

−sKt_).

3. Dynamic profit maximisation and balanced growth rate

The representative firm will maximise the following intertemporal profit: Max

P_it,N_t

&

t=0

e−rt_[

yit(pit,PD)pit−TCit(yit,Ait)−cnNit] dt (9)

under the constraints of: the demand for differentiated goods, the price index, the total costs, the dynamic of the TFP and the communication externalities, as well as the initial condition Ait=0=A0.

4_{In monopolistic competition, marginal revenue of a firm is equal torp (Grossman and Helpmann,}

1991). The conditiona\1 is needed so that the price elasticity of demand perceived by a firm will be larger than 1. This is required to avoid negative marginal revenue in a monopolistic situation.

The dynamic maximisation is solved by using optimal control techniques5_{. We}

are interested only in the symmetric (Nash) equilibrium, because all firms have the same cost and price conditions, and the same spillovers, as firm 1 in Fig. 1. This yields: pit=pt and Nit=Nt Öi=1 . . .n. In particular, apt=((yt/(pt)/(pt/yt), the

perceived price elasticity, obtained taking the price and telecommunications invest-ment decisions of other firms as given, becomesaP=a−(a−b)/n. The perceived

price elasticity is then always higher than 1.

Using the growth rates of the FOC at the symmetric equilibrium (see Appendix I), we are able to calculate the steady state growth rates of the system variables. Combining them we can look at the relationship between the growth rate of the control variableAit and the growth rate of the state variable we are interested in,

i.e.Nit,:

A. *=(d+d0 )(1−g)

g [cˆK(1−b)+dcˆn−dbX. 0]=(d+d0 )N. * (10)

where:

g= 1

[1−g−(d+d0)(b−1)]

We easily see that the autonomous scale demand parameter has a positive impact on the TFP growth rate, while marginal costs of capital and IT/telecommunications have a negative effect6_{. Reducing the growth rate of these marginal costs is then a}

measure that fosters the output growth rate. It could be argued that political measures in favour of deregulation of the telecommunications market and stronger competition among the IT suppliers, yielding lower marginal costs, have a direct impact on firms’ growth.

It is also interesting to look at the growth rate multiplierg. The elasticity of the own knowledge stock (1−g) has a negative effect on g: the more the own knowledge stock is efficient, the less investment is required to increase it. The impact of IT/telecommunications on TFP growth rate can be assessed looking at the term (d+d0)(b−1). The communication technologies positively contribute toA., both through a direct (d) and an indirect effect (d0), that stimulate further

5_{We have a unique state variableA}

it, two control variables,pitandNit, and the co-state variablemt.

Necessary conditions are also sufficient for global maximum if the Mangarasian sufficient theorem is satisfied. This theorem contains two conditions: (a) the profit functions and the TFP equations are differentiable and concave in the variables (Nt,At); and (b) in the optimal solution,mt]0,Öt[0,),

if dAt/dt is non-linear in N(t. The model satisfies the first condition; the second condition has to be

checked for each optimal solution, because dAt/dtis non-linear inNt(Chiang, 1992, pp. 215 – 221). 6_{The marginal cost of capital and IT/telecommunications has a negative influence on the balanced}

growth rate because it is inversely related to the steady state growth rate of IT/telecommunications investment. This effect could seem counter intuitive, as higher input prices imply that the potential benefits of telecommunications investment per unit of output increase. But in a situation with a price elastic demand curve (a,b\1), higher input prices also imply that the price level will increase, which decreases demand more than proportionally.

telecommunications investments by increasing the productivity of these efforts. Moreover, the increased productivity induced by the communication technologies decreases total production costs and then the price level (cost reducing effect). It follows that the firm’s revenue is increased by (b−1) times the productivity change. The externality parameteredoes not appear in the TFP growth rate. However, it plays an important role in the steady state values of the TFP and the telecommu-nications investment level, as we will see in the following Sections 4 – 6.

Manipulating the FOC, we also acquire the two differential equations dA/dt=

f(A,N), dN/dt=g(A,N), which are necessary to study the system dynamics. However, these differential equations depend on time: costs, price and the market scale factor grow in time at an exogenous rate. In order to apply the phase diagram technique, it is a prerequisite that the variable t does not enter into differential equations as a separate argument (the system has to be autonomous); otherwise, each point in a phase space can imply different directions of the system over time. When this is the case, it is not possible to make qualitative statements about the characteristics of a possible equilibrium (Chiang, 1984). In order to apply the phase diagram analysis to the inter-temporal maximisation with exponentially growing prices and costs, we have to remove the time component from the dynamic problem.

In the Appendix, we use a time elimination method to make the system autonomous: the endogenous variables will be deflated with their steady state growth rates; the appendix also describes the dynamic properties of the system. The two dynamic constraints in terms of the redefined or deflated variables are:

dAd

dt =Ad

g_N

d

d1

1+e(n−1)

1−e2_(n−1)

n

d0

−saAd (11)

dNd

dt =l1Nd

d1_(l

2Nd

1−d1_−l

3Ad

b+g−2

) (12)

where:

Ad=Ate

((d+d0)[cˆK(1−g)+cˆn−bX.0])/(g)t_=A

te −sat

Nd=Nte

((1−g)[cˆK(1−b)+bX.0−cn])/(g)t_=N

te −sNt

g= 1

[1−g−(d+d0)(b−1)]

d+d0 =d1

l1=

1 1−d1

, l2=r+sa(1−g)

l3=hX00

b

cn0

1+e(n−1) 1−e2_(n−1)

n

d0

d+ d0

1+e(n−1)

n

cK0

1−b

ap−1

ap

b

n(a−b)/(1−a)

(13)

Note that the subscriptd means that the steady state values are obtained after the time elimination method has been applied, and the depreciation factor sa is the

steady state growth rate of A. The initial values of the exogenous variables also appear (X00, cn0, ck0).

Equating the dynamic equations to zero, we obtain the steady state value ofAd

andNd

7_:

A*d=

X00bcK1−0b

cn0(r+sa)

aP−1

aP

b

na−b/(1−a)

SNS´N

n

1/(1−g+d₁(b−1))

sa

d1−1/(1−g+d1(b−1))

(14)

N*d=

X00b cK1−0 b

r+sa

aP−1

aP

na−b/(1−a)S´N

cn0

n

1/((1−g+d1(b−1))d1)

sa

2−g−b/(1−g+d1(b−1))

SN

d1(b−g)−2(1−g)/((1−g+d1(b−1))d1) ₍₁₅₎

where the externalities obtained by using a communication network are as follows:

SN=

1+e(n−1) 1−e2_(n−1)

n

d0

S´N=

d+

d0

1+e(n−1)

n

(16)

4. Analysis of the steady state TFP and IT/communication levels

In this section, we focus on the impact of the parameters defining the network externality effect (i.e. the number of firms,n, and the extent of the network effect,

e) on the steady state levels of TFP and telecommunications investment8

. Several, and sometime contrasting, forces are at work.

The termSN={[1+e(n−1)]/[1−e 2

(n−1)]}d0 _{represents the productivity level of}

the firm’s telecommunication expenditures or the pure network effect, while the termS%

N={d+d0/[1+e(n−1)]} is the elasticity of the telecommunications

invest-ment on the TFP dynamics at the symmetric equilibrium. A higher e and/or an increased number of users n increase the pure network effect: this enhances the productivity of the firm’s own network usage which results in higher marginal benefits of communication technologies. However, a larger extent of spillovers as well as a larger number of users mean that the appropriability of the firm’s own

7_{Note that in equilibriumA}

dandNdare constant and positive, but the original variables grow at a

constant steady state growth rate, as defined by the FOC of the original non-deflated system.

8_{As regards the other parameters, note that the interest rater,c}

nhave a negative effect on both TFP

and telecommunications steady state levels. SincebB1,cKhas a negative influence on both the steady

state levels. The autonomous parameter of scale of demand (X) has a positive influence onAd* andNd*.

The depreciation ratesahas a negative impact onAd* (becaused1B1): the faster the depreciation rate,

the lower the steady state total factor productivity, while its effect onNd* depends on the ‘adjusted

elasticity of demand’: if b+gB2, sa has a positive impact on the steady state telecommunications

usage, but if the elasticity of demand is high, the effect could be reversed. Remember that 1−g+d1(b−

communications usage declines (or S%

N decreases). We therefore find two typical

effects studied in the models on R&D and spillovers. On the one side, when e

and/or n increase, the intra-industry spillover pool is fostered, which results in a benefit for the firm, but on the other side, less appropriability of knowledge spillovers has a disincentive effect on telecommunications investment, as free riding and exploiting incoming contacts becomes more appealing to the firms. Although these two effects are similar to those analysed by the R&D models, they differ in the intensity: as we also noted in Section 2, in our model the impact of the parameterse and n on the intra-industry pool of knowledge is stronger than that considered by the standard R&D spillovers models. Indeed, this will have substan-tial implications in the comparative statics analysis.

The parameter n has also a positive impact on the steady state TFP and telecommunication investment level through the perceived price elasticity: a greater number of firms implies a higher perceived price elasticity, which increases the perceived change in demand by a productivity improvement. However, an increase of the number of users also exhibits a negative effect onAd andNd, as it increases

what we call the competition effect, i.e. the termn(a−b)/(1−a)_{. This effect is driven}

by the Dixit-Stiglitz specification of the demand for differentiated goods, which at the symmetric equilibrium is a negative function of the number of available varieties. The Dixit-Stiglitz specification has been mainly used by the endogenous growth models and indeed differs from the linear demand specification, quite common to those R&D models focusing on quantity competition. Both of these opposite effects are stronger when product differentiation is low (the terma−b is high, and/or there is a great difference between the elasticity of substitution among goodsa and the inter-industry price elasticity b).

As it is analytically quite difficult to disentangle the above-mentioned effects, we mainly use numerical simulations9

. Whenever appropriate, we also compare our results with those of the literature on R&D and spillovers.

4.1.The extent of the network externality effect (e): comparati6_{e statics}

4.1.1. The impact of the extent of network externality on N*d

The impact of the network extent parameter on Nd* is ambiguous, as the

following derivative shows: sign

(N*d

(_e

9_{The figures in this section are obtained from simulations performed using ‘Mathematica 3’. The}

values of the parameters and exogenous variables are:a=2.5,b=1.5,cno=0.8,cKo=1.35,X00=100,

r=0.007,sa=0.005,g=0.05,d0=0.1,d=0.3,e=0.2. Only if different values for some parameters are used, we report them.

=sign

!

−d0 (n−1)SN

(d₁(b−g)−2(1−g))

(1+e(n−1))2

B0

+d0SN(d1(b−g)−2(1−g))S%NSN

d0(d1(b−g)−2(1−g))₋₁ (

(_e

1+e(n−1) 1−e2

(n−1)

n

"

\0

Our simulations yield:

Simulation result1: when the number of firms is low, an increase of the extent of network effects leads to a decrease of the steady state communication level. However, when the number of firms increases, a higher network effect causes telecommunications investment to decline, and then to rise.

With a specification similar to that of Spence (1984) — which, in terms of our model, would imply at the symmetric equilibrium dA/dt=N[1+e(n−1)], S%_N= [1+e(n−1)]−1 _and

SN=1 — the positive effect of the network spillovers

disap-pears, while the negative one prevails. In our model, the possibility of a positive impact of network extent on telecommunications investment depends on the double counting method10_{, and more specifically on our modelling of the network}

external-ity effect. Another important determinant of the sign of the derivative is the general price elasticity (b). The value of this elasticity is an indicator for the degree of product differentiation with regard to the homogeneous good. In our model, a higher level ofbor less differentiated products increases the possibility of a positive influence ofeon Nd*. This effect also differs from the findings of De Bondt et al.

(1992), who conclude that a positive effect is more likely when product differentia-tion is moderate to high. In their model, moderate to high product differentiadifferentia-tion implies that appropriability is larger. In our model, higher product differentiation has an opposite effect: it lowers the perceived price elasticity of demand and therefore it results in less competition, which in turns lowers the incentives to invest in cost-reducing innovation. Therefore, the positive effect of spillovers on the cost-reduction innovation is more likely to appear under low product differentiation.

Our analysis confirms the results of the literature on R&D that high technologi-cal opportunities (i.e. highd and d0 ) will cause a positive influence of eon Nd*.

Summarising the results of this sub-section, we can say that, in our model, a higher number of firms in the product market raises the total volume of spillovers and, together with high technological opportunities linked to communication networks and low differentiation, creates the conditions for greater appropriability of telecommunications investment. These three conditions are therefore the prereq-uisites for a positive effect of network externality on the steady state telecommuni-cations level.

¿¹¹¹¹¹¹¹¹¹¹¹Ë¹¹¹¹¹¹¹¹¹¹¹À

10_{It must be stressed that, in our model, the number of firms determines the stability of network}

effect: to have positive network effect we assumee2_{B(n−1). With many firms, the positive effect is then}

limited to a low range of the externality parameter.

Í

Ã

Ã

Ã

Ã

Ã

Ã

Fig. 2. The impact of extent of network effect on productivity level.

4.1.2. The impact of the extent of network externality on A*d

The technological performance of a firm can be measured by its steady state productivity level. In this sub-section, we investigate the relationship between technological performance and the extent of network effects. While the impact ofd

and d0 on steady state TFP is always positive, the parameter e has again an ambiguous effect:

sign

(A*d

(_e

=sign

!

−

S%_Nd0_(n−1) (1+e(n−1))2

B0

+d0SN −1_S%

N

(

(_e

1+e(n−1) 1−e2

(n−1)

n

\0

"

This yields:

Simulation result2: when the externality parameter is low, an increase of e lowers

A*, but this effect is reversed when the externality is higher. Fig. 2 shows that thed

steady state productivity level first decreases then increases, until the ‘stability condition’ is reached. This effect appears regardless of the number of users. To our knowledge the impact of the spillover parameter on the technological performance of a firm has been neglected not only by the literature on R&D, but also by empirical studies on productivity and usage of information technologies. The rationale for simulation result 2 is that, as long as the extent of the network effect is too low, there is no incentive to invest in IT/telecommunication usage, which exhibits a negative or zero impact on TFP. When the quality of the exchanges increases, namely when the information gathered by the community of users is more valuable to the firm, then a positive impact of network effect can be predicted. Another interesting explanation of our result should probably take into account the consideration of a learning function, because it is recognised that extensive experience is needed by firms and organisations to exploit IT gains, as it is for most radically new technologies. Probably, positive learning effects may cause an increase in the extent of the interaction among communication technologies

Í

à Ã

à Ã

users, thus endogenising the shift from the negative to the positive part of the curve we depicted in Fig. 2. The investigation on this latter point is left for further research.

4.2.The number of users (n): comparati6_{e statics}

4.2.1. The impact of the number of users on N*d

In this subsection, we discuss the impact of the club of users on the steady state IT/telecommunication level. When the number of firms increases, the cost-reduction investment is negatively affected by two effects: the lower appropriability and the competition effect. However, an increase ofnhas a positive impact on the network effect and on the perceived price elasticity:

sign

(N*d

(_n

=sign

!

−

S%

Neu1

(1+e(n−1))2

B0

+u2dSN

d1(b−g)−3−2g (

(_n

1+e(n−1) 1−e2₍

n−1)

n

\0

%+bu3

(

(_n

1−

1

aP

\0

+u4

a−b

1−an

a−b/(1−a)−1

B0

"

u1=SN

d1(b−g)−2(1−g)_(1−1/a

P)n

(a−b)/(1−a)

u2=SN(1−1/aP)n

(a−b)/(1−a)

u3=SN

d1(b−g)−2(1−g)_S

Nn(a−b)/(1−a)

u4=SN

d1(b−g)−2(1−g)_S

N(1−1/aP)

We then find two patterns:

Simulation result3: when the extent of the externality effect is low, an increase of the number of users has a negative impact on Nd*; when e is high, this effect is

reversed.

The increasing pattern is another element that does not appear in the standard literature on R&D and R&D spillovers. An increasing pattern is more likely to appear when products are less differentiated (higher values ofb), the technological opportunities linked to the communication network increase (higher d0) and the difference between the elasticity of substitution (a) and the inter-industry price elasticity (b) is high.

4.2.2. The impact of the number of users on A*d

The impact of an increased number of users on the steady state productivity level has the same determinants as discussed in the Section 4.2.1:

Í Ã Ã Ã Ã Á Ä Í Ã Ã Ã Ã Á Ä

Í

Ã

Ã

Ã

Ã

Á

Ä

Í

Ã

Ã

Ã

Ã

Ã

Ã

Á

Ä

sign

(A*

(_n

=sign

!

−v1

S%

N

(1+e(n−1))2

B0

+v2d0SN

(

(_n

1+e(n−1) 1−e2_(n−1)

n

+v4

(

(_n

aP

aP−1

n

\0 +v3

a−b

1−an

a−b/(1−a)−1

B0

"

v1=SN(1−1/aP)n

(a−b)/(1−a)

v2=S%N(1−1/aP)n

(a−b)/(1−a)

v3=S%NSNn(a−b)/(1−a)

v4=S%NSN(1−1/aP)

A few simulations help to illustrate the complex interplay between market structure and technological change in our model. First, we analyse the impact of elasticity of substitution and inter-industry price elasticity.

Simulation result4: when spillovers are low, the productivity level decreases with an increased number of users. This decreasing pattern is also found when the difference between a and b is very low. When spillovers are high, a duopoly always obtains the highest productivity level. The total factor productivity decreases, then increases with an increased number of users, until the stability condition has been reached. The higher the difference betweenaandb, the faster the decrease.

The increasing pattern is an interesting one and, to our knowledge, a new pattern to the literature. With higher spillovers and less differentiation, entry determines first a decrease then an increase of the steady state productivity level. The intuition behind these results is that the lower market shares as a result of a higher number of users on the communication network drive the productivity level down. This negative effect decreases when the number of firms increases and at a certain intermediate level of rivalry, the positive influence of a larger investment in IT/telecommunication and then a larger intra-industry knowledge stock becomes more important. The positive pattern is then driven by the same forces that determine the positive impact of the number of users on the telecommunications steady state level: high network effect and low differentiation.

If we combine all the results of these sub-sections, we can expect that the most fa6_{ourable condition for a firm to increase her in}6_{estment in network technologies and}

to obtain a more efficient total factor producti6_{ity is: to be in a market with low} product differentiation (and/or with a great difference between the elasticity of substitution among goods and the inter-industry price elasticity), an important number of users, an intensive network externality extent and considerable techno-logical opportunities linked to the telecommunication network.

Í

Ã

Ã

Ã

Ã

Á

Ä

Í

Ã

Ã

5. Consumer surplus, total profits and welfare

The last section of this paper concentrates on the impact of network externalities and number of users on consumer surplus, total profits and welfare. Using the FOC of the deflated system and the definition of the consumer utility, the consumer surplus can be written as:

CS=nTC

b−1

aP

aP−1

(17)

where:

TC=X00bcK0

1−b_na−b/(a−1)

Adb−

1

aP

aP−1

−b

Remembering that a, b\1, a\b, aP\1, is it easy to see that the consumer

surplus depends positively on the number of firms, reflecting love of variety, and on the perceived price elasticity, because if aP increases, the mark-up over costs

decreases. The technological level or total factor productivity also has a positive impact on consumers’ surplus.

Using FOC, total profits can be written as follows:

np=TC

<

1 aP−1

−

sa

d+

d0

1+e(n−1)

r+(1−g)sa

=

(19) Note that, as for consumer surplus, throughTC, the technological level of firms has a positive effect on total profits. In order to have a viable situation in monopolistic competition, the long run total profits have to be equal to zero. This condition defines the no-entry condition and the optimal number of firms present in the market (n*).

The first term of Eq. (19) can be interpreted as the gross profit’s ratio, while the second one represents the IT fixed investment ratio. The impact of an increase of the number of users on total profit is ambiguous, because two opposite effects are present. On one side, the entry of new users increases the perceived price elasticity

aP, which has a negative effect on profits; on the other side, it lowers the fixed cost

ratio of the investment in the communication technology with network effects, which increases profits. We find that the first effect dominates the second one:

Simulation result5: profits decrease with entry of new users and the appearance of a negative profit is more likely when technological opportunities (d, d0) are larger.

This situation is analysed by the traditional literature on R&D. The long-run equilibrium is attained at different optimal numbers of firms, depending on different values of elasticity of substitutionaand the productivity of the networkd. Once the no-entry condition is determined, the stability condition gives us a condition on network effect associated with that scenario: e=n*−1. This link between the optimal number of firms and the extent of the network effect ensures the coherence of the model. We find a result similar to that of Dasgupta and Stiglitz (1980): industries with larger technological opportunities tend to have few firms.

A higher extent of network effect decreases the second term in the brackets (see Eq. (19)) and thus reduces the probability of a negative profit rate. If network externality is high, then the number of viable users in a market will increase. A higher degree of product differentiation (lower level ofa), decreases the perceived elasticity and hence makes a negative profit less likely. Whenadecreases, the number of viable users increases. The model therefore predicts that industries with high differentiation have more IT/telecommunication users than the industries with more homogeneous goods.

Combining these findings we can conclude that traditional industries with high technological opportunities, a low extent of communication spillover and a low degree of product differentiation allow a low equilibrium number of network technologies users. Highly differentiated industries with low technological opportunities and high network effects are characterised by a high equilibrium number of users.

5.1.Welfare

We now analyse the total impact of the number of users, and the extent and prod-uctivity of the network effect on total welfare, as the sum of consumer surplus and total profits. We easily see that technological opportunities increases welfare, because both consumer surplus and total profits depends on the productivity levelAd*.

The impact of the network effect extent on welfare is not clear-cut. First, parameter

ehas a positive impact on total welfare, because it increases total profits. Second, in the previous section (simulation result 2) we found that a positive impact ofeon the steady state productivity levelAd* is likely to appear when this parameter is high,

regardless the number of firms.

Wheneis low, its positive effect on the total profit to total costs ratio does not easily counterbalance the negative impact on total costs, that enters both consumer surplus and total profits. The impact of the extent of network effects thus depends on its impact on the steady state technological level. This yields the following result:

Simulation result6: when the extent of network effect is high, welfare is a positive function of the network externality extent.

The influence of an increase in the number of users is again difficult to understand. First, new users increase the perceived price elasticity, which increases consumer surplus and decreases total profits. Second, the influence of new users on the productivity level is ambiguous (see Section 4.1.2). The difference betweenaandb, and the extent of the network effect determines whether the influence is positive or negative. Third, when consumers show love of variety, entry directly increases the total surplus. Our simulations show that:

Simulation result7: with a high differentiation, welfare increases with entry of new users.

crucial importance, because with low differentiation (high inter-industry price elasticitya) and low e, welfare decreases with the entry of new users. This result is coherent with the analysis of the impact of entry on the steady state TFP: in Section 4.2, we said that A* decreases with entry if the differentiationd and the extent of

spillover are low. The decreasing welfare is then driven by this negative effect on

Ad*, which decreases consumer surplus and total profits. We obtain the following

result:

Simulation result8: when differentiation and extent of network effect are low, total welfare decreases with entry.

6. How is this model linked to empirical evidence?

The question of the impact of the network effect on the TFP is closely related to the Solow productivity paradox. Indeed, the major part of the empirical literature on the Solow productivity paradox measures the effect of productivity investment on TFP at the industry or firm level, while our model analyses TFP at the steady state (i.e. when the TFP is constant). Nevertheless, we think that the relationship we found between the extent of network effect and the productivity level offers some interesting insights about the impact of information technologies on TFP.

As stated in the introduction of this paper, in recent years, the relationship between information technology and productivity has become a source of debate. As consequence, studies attempting to measure the impact of IT on total factor productivity are very heterogeneous.

Aggregate level studies are quite disappointing: the overall negative correlation between productivity and the advent of computers underlies many of the arguments that information technology has not helped the United States productivity, or it has been counterproductive (Jorgenson and Stiroh, 1995; Baily, 1996). Indeed, the relationship between white-collar productivity and information technology is very hard to measure, especially with aggregate data (Brinjolfsson and Yang, 1996).

As going down to the industry level and emphasizing cross-sectional effects should help to control many problems that arise from aggregation (Siegel and Griliches, 1991), we have investigated the micro-economic conditions that foster investment in communication technologies. We claim that telecommunications and information technologies are characterised by the network externality effect, distin-guishing them from a simple substitute for labour or other kinds of capital inputs. This opens the possibility of a more accurate specification of the impact of IT on total factor productivity via the network effect, as a proxy for the spillovers obtained by using the communication technologies. Our results show that, ignoring the complex interplay between market conditions, number of users, extent and productivity of network effect may bias the measurement of the IT contribution to the firms’ performance.

Our simulation results suggest that the impact of IT is non-neutral with respect to the characteristics of the firms analysed. This is confirmed by Brinjolfsson and Hitt (1993), who find that up to half of the excess returns imputed to IT could be

attributed to firm-specific effects. It is thus worthwhile looking at the empirical results and ranking firms or industries according to the expected extent of spillovers. This expected extent of network effect could be considered as important as the Yale technology flows matrix, that represents the technological closeness among sectors, and is used in R&D studies as a tool to measure the pure knowledge spillover intensity (Van Mejil and Van Zon, 1933).

Following our predictions, we could expect that activities requiring a large communication network, an intensive extent of spillovers, and good technological opportunities, like for example, retailing and banking services, would experience a positive impact of IT on total factor productivity11_{. In general, at the firm level, the}

results of IT on productivity seem to be more clearly associated with the kind of industry to which the firms’ sample belongs, as our results confirm.

The results of our model regarding the positive impact of the network effect on the consumers’ surplus are consistent with Bresnahan (1986), who estimated the benefits to consumers of declining computer prices using the hedonic price index method.

Finally, our results on total profit’s analysis offer a possible answer to the debate frequently raised by management science in attempting to understand whether the usage of IT leads to more concentrated or less concentrated industries. The argument that favours the first hypothesis is that network technologies have the advantage of organising the production process with a lower number of subordi-nates, higher control, and just-in-time scheduling (Mintzberg, 1982). The second view refers to the decentralisation effect that is possible when efficient communica-tion networks are used (Hubey, 1990; Brinjolfsson, 1993). The results of our model show that the causality between centralisation/decentralisation and communication technologies is not driven simply by the usage, as the above mentioned studies claim. We find that the equilibrium number of users is determined by combining market conditions, productivity of the communication network, and extent of the network externality effect.

7. Conclusions

The main aim of this paper was to determine factors that enhance or temper firms’ incentives to invest in communication technologies that process information and are characterised by the existence of a community of users. In particular, we focused on usage of communications technologies allowing firms to obtain knowl-edge and therefore to influence their rate of technological change or total factor productivity in a dynamic context.

11_{For instance, Diewert and Smith (1994) provide an interesting case study of a large Canadian retail}

distribution firm. They found that the astounding 9.4% quarterly multifactor productivity growth experienced by this firm is made possible by the computer revolution which allows a firm to track accurately its purchase and sales items and to use the latest computer software to minimise inventory holding costs.

We solved an inter-temporal profit maximisation, under the constraints of the demand derived from the Dixit-Stiglitz consumer’s utility function, the price index (as a measure of the intensity of competition) and the dynamic factor productivity. The modelling of the TFP is the most original contribution of this paper to the literature on endogenous technological change. We showed that the communication technologies’ usage is characterised by a community of users, which creates a network externality effect. We thus focused on the impact of the parameters defining the network externality effect-mainly the extent of network spillover and the number of users-on the steady state values of TFP and telecommunications investment. Finally, we calculated consumer surplus, total profits and welfare, and we analysed again their sensitivity to a change of the network effect extent and the competition intensity.

Our model predicts that the most favourable condition for a firm to increase her investment in network technologies and to obtain a more efficient total factor productivity is as follows: to be in a market with low product differentiation (and/or with a great difference between the elasticity of substitution among goods and the inter-industry price elasticity), to share the usage of network technologies with an important number of users. Moreover, an intensive network externality extent and considerable technological opportunities linked to the telecommunica-tion network also are suitable conditelecommunica-tions to better exploit the advantages of communications technologies.

However, as in each case where firms’ decisions concern strategic economic variables exhibiting externalities, there are some conflicts between the private incentives to telecommunications usage and their effects on total welfare. In particular, if a high number of users and low differentiation may stimulate private decision of telecommunications investment and usage, the same conditions, to-gether with a low extent of network effects, give welfare decreasing with the intensity of competition.

Our model could be extended to take into account more complex network configurations. In that case, we may find other interesting patterns for telecommu-nications and total factor productivity steady state levels. For example, if we consider the full connected network, where each firm has symmetric spillovers, the conditions for a positive network effect will limit the value of the externality parametereto discontinuous intervals. We then expect more irregular results than those analysed in the present work.

Appendix I. Optimal control for the dynamic profit maximisation

To solve the dynamic profit maxmisation, we use the current value Hamiltonian, which allows us to disregard the discount factor (Chiang, 1984):

Hc₍

pit,Nit,mt)=[pityit(pit)−TCit(Ait,yti)−cnNit]=mt(hAit

g Nit

d NSPit

d0 _{) (I.1)}

Under the symmetric Nash equilibrium conditions, the price index, demand and spillovers can be written as follows:

PDt=[npt

1−a_]1/(1−a)

yt=X0bpt−bna−b/(1−a)

NSPt=Nt

1+(n−1)e

1−(n−1)e2

The first order conditions for profit maximisation are then as follows: (_Hc

(_pt=0U

aPt

aPt−1

TCt

yt

=pt (I.2)

where

apt=

(_yt (_pt

pt

yt

=a−a−b

n

(_Hc

(_N

t

=0Ucn=

( (_N

t

(dAt/dt)

=mt

hA

g

Nd+d0−1

1+e

2_(n−1)

1−e2_(n−1)

d0

n

d+ d0

1+(n−1)e

n

(_H1c

(_At= −

dm

dt=mrU−

dm

dt+mr= TCt

At

(_Hc

(_m

t

=0UdAt dt =hAt

g_N

t

d+d0

1+e(n−1)

1−e2_(n−1)

n

d0

The transversality condition is: lim

t“

mte−

rt =0

In order to calculate the steady state growth rate of the system, we first consider the growth rates of the demand function and total costs at the symmetric equi-librium (for simplicity we drop the time subscript):

yˆ=b(X. 0−pˆ)

TC. =yˆ−A. +cˆK

(I.3) We then differentiate the first order conditions with respect to time to obtain the equilibrium rates of growth, assuming that the growth rate of the system will be constant in the steady state (the changes in growth rates over time are therefore equal to 0 and dmˆ/dt=dA. /dt=0):

pˆ=TC. −yˆ cˆn=mˆ+A. −N.

mˆ=TC. −A.

(1−g)A. =(d+d0 )N.

(I.4)

rates. We then solve the system, in order to obtain the following steady state growth rates:

A. *=(d+d0 )

g [cˆK(1−b)+dcˆn−dbX.0] N.*=(1−g)

g [cˆK(1−b)+bX. 0−cˆn] yˆ*=b

g[X.0(1−g+d)−Z. (1−g)−dcˆn] pˆ*=1

g[Z. (1−g)+dcˆn−dbX.0]

where:

g= 1

[1−g−(d+d0)(b−1)]

Appendix I. Time elimination method

In order to apply the phase diagram analysis to the inter-temporal maximisation with exponentially growing prices and costs, we have to remove the time compo-nent from the dynamic problem by a redefinition of the variables.

The variablesAt,Nt,yt,ptgrow at constant rates that we note respectivelysa,sN,

sy, sp. These growth rates are directly linked to the constant growth rates of the

exogenous variables and to the various elasticities of the system and are easily obtained by differentiating the FOC of the dynamic maximisation problem with respect to time. When we deflate the endogenous variables with their steady state growth rates, we will obtain an autonomous system of differential equations (Lucas, 1988; Barro-Sala i Martin, 1995). We then define the following new variables:

Ad=Ate

−((d+d0)[cˆK(1−g)+cˆn−bX.0])/(g)t_=A

te

−sat _(II.1)

Nd=Nte

−((1−g)[cˆ_K(1−b)+bX.0−cn])/(g)t_=N

te

−sNt _(II.2)

yd=yte

−(b[X.0(1−g+d)−cˆK(1−g)−dcn])/(g)t_=y

te

−syt _(II.3)

pd=pte

−([cˆ_K(1−g)+dcˆ_n−dbX.0])/(g)t_=p

te −s_pt

(II.4) We will redefine the profit maximisation problem in terms of these new variables. In order to obtain a new dynamic constraint in terms of the TFP, we can differentiate Eq. (I.1) with respect to time:

dAd

dt =

dAt

dte

−oat_−s

aAte

−oat₌dAt dt e

−oat_−s

Differentiating the FCO of the original non-deflated system with respect to time gives sd=gsa+(d+d0 )sN; we thus have (at the symmetric equilibrium):

dAd

dt =Ad

g_N

d

d₁

1+e(n−1)

1−e2

(n−1)

n

d0

−saAd (II.6)

where, for simplicity, we replaced (d+d0) with d1.

Using the assumption that all the exogenous variables grow at constant rate and the Eqs. (II.1 – II.6), the profit maximisation problem can be rewritten in terms of the new discounted variables. The starting values of the exogenous variables X00,

cK0, cn0 all enter the current Hamiltonian of this redefined problem.

In order to have a meaningful situation we require that r+sa(1−g)\0. This

condition is automatically satisfied when the transversality condition of the re-defined problem is met.

Appendix I. Dynamic properties of the system

Looking at the Eqs. (14) and (15) in the text, we see that the dAd/dt=0 locus is

positively sloped and convex. The slope of the dNd/dt=0 locus is dependent on the

value of the elasticity of demand together with the elasticity of the firm’s own TFP with respect to the productivity generation process. We call the termb+gadjusted elasticity of demand because of the presence of the elasticity of dAd/dtto TFP level.

We thus analyse three different situations: low, moderate and high adjusted price elasticity of demand12_{. When b+gB2, the dN}

d/dt=0 locus is negatively sloped,

and it is horizontal when b+g=2. With moderate price elasticity of demand (2Bb+gB3−d1 or 3−d1Bb+gB(1+d1)/d1) the locus is positively sloped.

When the price elasticity is high (b+g](1+d1)/d1), the qualitative analysis cannot

be easily applied. We leave the answer to the analysis of the local stability equilibrium using a linearisation of the non linear differential system around the steady state.

As an example of the transition to the equilibrium path, we analyse the case of the low adjusted price elasticity of demand (b+g=2) (Fig. 3).

When dNd/dt=0, we havel2Nd

1−d₁

=l3Ad

b+g−2_{. Looking at the definition ofl} 2

andl3, this equality can be interpreted as the equality between the marginal costs

of increasing the state variable (left hand side) and the marginal benefits of TFP (right hand side). Whenb+g=2, an increase of the TFP the dNd/dt=0 locus is

the saddlepath.

On the dAd/dtlocus, the depreciation of the TFP level is equal to the

telecommu-nications level that generates TFP. This curve has a positive slope because a higher TFP implies more depreciation, which has to be met with more TFP. This objective can be achieved with more investment in telecommunications level.

12_{The detailed analysis of phase diagram and transition to the equilibrium is available upon request}

We assume an initial productivity level of A0BAd*. At the steady state

equi-librium, the firm will choose a constant level of discounted communications investment level, which is always equal to the constant discounted steady state level

Nd*. This implies that the change in the productivity level is always the same.

Because the initial discounted communications investment level of the firm is lower than the constant discounted steady state levelNd* and the change in productivity

level is the same, the growth rate of the discounted productivity level of the representative firm will be larger than the constant steady state growth rate. In the next period, our firm will have A1\A0. Given A1, the firm will choose the same

discounted communications investment level, which results in a higher growth rate than the constant steady state growth rate. The process will continue but the speed of the adjustment slows down, because the difference between the constant steady state growth rate (A*) and the initially discounted productivity level is smaller.d

Moreover, the speed of adjustment is lower that in the case of b+gB2. Higher adjusted price elasticity decreases the speed of convergence toward the discounted steady state level of communications investment.

The qualitative analysis of system dynamic suggests that, when the adjusted price elasticity of demand gets higher, the dNd/dt=0 locus rotates counter-clockwise.

The system stays saddle point stable as long as the slope of the dAd/dt=0 locus is

steeper than the slope of the dNd/dt=0 locus.

This conclusion is also verified by checking the stability of the system by characteristic roots, as follows. In fact, the existence of a saddle point can be examined by a linearisation (first order Taylor expansion) of the non-linear differential equation system near the steady state (Chiang, 1984).

The Jacobean matrix evaluated at the steady state (Ad*,Nd*) is:

gAdg−1Nd

d1_S

N−sa d1AdgNd

d1−1

SN

−l₁l₃(b+g−2)Nd

d1_A

d

b+g−3 _d

1l1Nd

d1−1_(l

2Nd

1−d1_−l

3Adb+g−2)+l1l2(1−d1)

n

N*,A*

(III.1)

rates. We then solve the system, in order to obtain the following steady state growth rates:

A. *=(d+d0 )

g [cˆK(1−b)+dcˆn−dbX.0] N.*=(1−g)

g [cˆK(1−b)+bX. 0−cˆn] yˆ*=b

g[X.0(1−g+d)−Z. (1−g)−dcˆn] pˆ*=1

g[Z. (1−g)+dcˆn−dbX.0]

where:

g= 1

[1−g−(d+d0)(b−1)]

Appendix I. Time elimination method

In order to apply the phase diagram analysis to the inter-temporal maximisation with exponentially growing prices and costs, we have to remove the time compo-nent from the dynamic problem by a redefinition of the variables.

The variablesAt,Nt,yt,ptgrow at constant rates that we note respectivelysa,sN, sy, sp. These growth rates are directly linked to the constant growth rates of the

exogenous variables and to the various elasticities of the system and are easily obtained by differentiating the FOC of the dynamic maximisation problem with respect to time. When we deflate the endogenous variables with their steady state growth rates, we will obtain an autonomous system of differential equations (Lucas, 1988; Barro-Sala i Martin, 1995). We then define the following new variables:

Ad=Ate−((d+d0)[cˆK(1−g)+cˆn−bX.0])/(g)t_=Ate−sat _(II.1) Nd=Nte

−((1−g)[cˆ_K(1−b)+bX.0−cn])/(g)t_=N te

−sNt _(II.2)

yd=yte−(b[X.0(1−g+d)−cˆK(1−g)−dcn])/(g)t_=yte−syt _(II.3) pd=pte

−([cˆ_K(1−g)+dcˆ_n−dbX.0])/(g)t_=p

te −s_pt

(II.4) We will redefine the profit maximisation problem in terms of these new variables. In order to obtain a new dynamic constraint in terms of the TFP, we can differentiate Eq. (I.1) with respect to time:

dAd

dt =

dAt

dte

−oat_−saAte−oat₌dAt

dt e

Differentiating the FCO of the original non-deflated system with respect to time gives sd=gsa+(d+d0 )sN; we thus have (at the symmetric equilibrium):

dAd

dt =Ad

g_N

d

d₁

1+e(n−1) 1−e2

(n−1)

n

d0

−saAd (II.6)

where, for simplicity, we replaced (d+d0) with d1.

Using the assumption that all the exogenous variables grow at constant rate and the Eqs. (II.1 – II.6), the profit maximisation problem can be rewritten in terms of the new discounted variables. The starting values of the exogenous variables X00,

cK0, cn0 all enter the current Hamiltonian of this redefined problem.

In order to have a meaningful situation we require that r+sa(1−g)\0. This

condition is automatically satisfied when the transversality condition of the re-defined problem is met.

Appendix I. Dynamic properties of the system

Looking at the Eqs. (14) and (15) in the text, we see that the dAd/dt=0 locus is positively sloped and convex. The slope of the dNd/dt=0 locus is dependent on the value of the elasticity of demand together with the elasticity of the firm’s own TFP with respect to the productivity generation process. We call the termb+gadjusted elasticity of demand because of the presence of the elasticity of dAd/dtto TFP level.

We thus analyse three different situations: low, moderate and high adjusted price elasticity of demand12_{. When b+gB2, the dNd/dt=0 locus is negatively sloped,} and it is horizontal when b+g=2. With moderate price elasticity of demand (2Bb+gB3−d1 or 3−d1Bb+gB(1+d1)/d1) the locus is positively sloped. When the price elasticity is high (b+g](1+d1)/d1), the qualitative analysis cannot be easily applied. We leave the answer to the analysis of the local stability equilibrium using a linearisation of the non linear differential system around the steady state.

As an example of the transition to the equilibrium path, we analyse the case of the low adjusted price elasticity of demand (b+g=2) (Fig. 3).

When dNd/dt=0, we havel2Nd

1−d₁

=l3Ad

b+g−2_{. Looking at the definition ofl} 2 andl3, this equality can be interpreted as the equality between the marginal costs of increasing the state variable (left hand side) and the marginal benefits of TFP (right hand side). Whenb+g=2, an increase of the TFP the dNd/dt=0 locus is the saddlepath.

On the dAd/dtlocus, the depreciation of the TFP level is equal to the

telecommu-nications level that generates TFP. This curve has a positive slope because a higher TFP implies more depreciation, which has to be met with more TFP. This objective can be achieved with more investment in telecommunications level.

12_{The detailed analysis of phase diagram and transition to the equilibrium is available upon request}

We assume an initial productivity level of A0BAd*. At the steady state equi-librium, the firm will choose a constant level of discounted communications investment level, which is always equal to the constant discounted steady state level

Nd*. This implies that the change in the productivity level is always the same.

Because the initial discounted communications investment level of the firm is lower than the constant discounted steady state levelNd* and the change in productivity

level is the same, the growth rate of the discounted productivity level of the representative firm will be larger than the constant steady state growth rate. In the next period, our firm will have A1\A0. Given A1, the firm will choose the same discounted communications investment level, which results in a higher growth rate than the constant steady state growth rate. The process will continue but the speed of the adjustment slows down, because the difference between the constant steady state growth rate (A*) and the initially discounted productivity level is smaller.d

Moreover, the speed of adjustment is lower that in the case of b+gB2. Higher adjusted price elasticity decreases the speed of convergence toward the discounted steady state level of communications investment.

The qualitative analysis of system dynamic suggests that, when the adjusted price elasticity of demand gets higher, the dNd/dt=0 locus rotates counter-clockwise.

The system stays saddle point stable as long as the slope of the dAd/dt=0 locus is

steeper than the slope of the dNd/dt=0 locus.

This conclusion is also verified by checking the stability of the system by characteristic roots, as follows. In fact, the existence of a saddle point can be examined by a linearisation (first order Taylor expansion) of the non-linear differential equation system near the steady state (Chiang, 1984).

The Jacobean matrix evaluated at the steady state (Ad*,Nd*) is:

gAdg−1Nd

d1_S

N−sa d1AdgNd

d1−1 SN −l₁l₃(b+g−2)Nd

d1_A

d

b+g−3 _d 1l1Nd

d1−1_(l

2Nd

1−d1_−l

3Adb+g−2)+l1l2(1−d1)

n

N*,A*

(III.1)

The last term reduces tol1l2(1−d1), because (l2Nd 1−d1_−l

3Ad

b+g−2_{) is 0 at the} steady state equilibrium. Moreover, at steady state, we also have:Adg

Ndd1_SN=saAd. The first term on the left hand side can be rewritten as: −sa(1−g).

To check the dynamic stability of the equilibrium, we have to know the signs of the characteristic roots (r1,r2). In order to have a stable saddle point equilibrium, the two characteristic roots must have opposite signs. The Jacobean matrix contains all the relevant information.

detJ=r1r2= −sa(1−g)(1−d1)l1l2+l1l3d1sa(b+g−2)Nd

d₁−1

Adb+g−3 (III.2) In order to have a locally stable saddle point, detJ has to be negative. The first term of detJ is negative, because g, d1B1. The second term is positive if (b+g−2)\0, and negative if (b+g−2)B0. In this latter case, we are sure that the dynamic system has a locally stable saddle point.

Using the steady state equation: l2Nd 1−d1_=l

3Ad

b+g−2

, the determinant can be rewritten as follows:

detJ=r1r2= −sal1l2[1−g−d1(b−1)] (III.3)

The determinant is positive if 1Bg−d1(b−1), negative if 1\g−d1(b−1). In the latter case, the system has a stable saddle point. When the determinant is positive, we are not able to determine the local stability of the system. In order to calculate this inference, we must calculate the trace of the Jacobean matrix(trJ), which is equal to the sum of the characteristic roots:

trJ=r1+r2= −sa(1−g)+(1−d1)l1l2 (III.4)

Replacing the parametersl1=1/(1−d1),l2=r+sa(1−g), the trace is rewritten as:

trJ=r1+r2=r (III.5)

The trace is always positive becauser\0 in order to meet the transversality condition of the original non-deflated dynamic system.

A positive determinant associated with a positive trace describes an unstable equilibrium. We are in presence of an unstablenode if:

trJ2_]4JU(r)2_]4s

a

(r+sa(1−g))

(1−d1)

[1−g−d1(b−1)] (III.6)

It then depends on the value of various parameters of the model. If this condition does not hold, we are in presence of an unstablefocus (Chiang, 1984, p. 64).

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