CSIS Discussion Paper No. 54

THE BASICS OF A NETWORK ECONOMY
Yukihiro Kidokoro∗
Center for Spatial Information Science, University of Tokyo
Address: Faculty of Economics, Akamon Research Building 542, University of Tokyo,
7-3-1, Hongo, Bunkyo, Tokyo, 113-0033, Japan
Tel&Fax: +81-3-5841-5641
E-mail: kidokoro@csis.u-tokyo.ac.jp

Abstract
Using a simple general equilibrium model with an endogenously evolving
communications network, we demonstrate the basic characteristics of a network-based
economy. First, we characterize the first-best solutions, compare them with the solutions
under a laissez-faire economy, and show that in a laissez-faire economy, thecommunications network contracts and causes subsequent distortions to wages, firm size,
and the number of firms. Second, we examine the effects of representative policies, suchas a subsidy for the price of communications services, an income transfer, and a subsidy
to firms, and evaluate these policies in terms of the total social surplus.

∗

I would like to thank Tatsuo Hatta, Takuo Imagawa, Marvin Kraus, Yoshitsugu Kanemoto, Hideo Konishi,

Hajime Miyazaki, and David Wildasin for helpful suggestions and discussions. I also benefited fromdiscussions when presenting an earlier version of the paper at Boston College, the Ohio State University,and the University of British Columbia. Of course, I am solely responsible for any remaining errors andomissions. The financial aid from the Seimeikai foundation is gratefully acknowledged.

1

I

Introduction

The Internet grows inexorably. According to the Ministry of Public Management,
Home Affairs, Posts and Telecommunications [2002], the number of people worldwide toaccess the Internet surged from 26 million in 1995 to 544 million in 2002. A major reason
for this rapid increase is that the Internet enables us to communicate with one anothereasily and cheaply. Suppose that you want to read the mimeographed paper of a certain
professor. If you cannot access the Internet, you have to mail a letter asking the professor
to send the paper. This exchange of letters would take a long time. If the professor is toobusy to send the paper, either you write the letter again or you wait. Again, this process
would be time-consuming. If both the professor and you can access the Internet, thesituation changes dramatically. You simply have to e-mail the professor, who will send
the paper to you as an attached file within a very short time. You may not even need tosend an e-mail. If the professor has a homepage that includes the papers, you candownload them to read at your leisure.
This paper examines the basic characteristics of an economy with a communications
network and various policies that influence the development of communications
networks and economic performance. To do this, we build a simple general equilibrium
model with an endogenously evolving communications network. Our model has two
types of consumer, three types of firm, and the government. The type of consumer that
communicates with others using a communications service provided via a network is thetype-IT consumer, while the type that does not is the type-NIT consumer. An increase in
the number of type-IT consumers, that is, an increase in the network size, yields benefits
to all type-IT consumers, because such an enlargement expands communication
possibilities for each type-IT consumer. A type-NIT consumer can freely become a2

type-IT consumer by receiving education, but the cost of education to become a type-IT
consumer differs between individuals. The government partly subsidizes the education to
become a type-IT consumer. Firms are classified into those that produce the composite
consumer good, and a monopoly that provides the communications service. The former
consist of type-IT firms, which hire type-IT consumers, and type-NIT firms, which hire
type-NIT consumers, and produce the composite consumer good using consumers’
information as an input. Knowledge useful for type-IT firm production is assumed to be
increasing with network size, on the grounds that type-IT consumers accumulate
information by communicating with one another. Thus, communication between type-IT
consumers has positive externality on type-IT firm production, which increases theincome of type-IT consumers.
First, we characterize the first-best outcome. In our model, the size of the network is
determined by the price of the communications service and the government’s subsidy forthe cost of education required to become a type-IT consumer. In the first-best outcome,the total price of a communications service, which is the sum of the prices that a senderand receiver of information pay, equals the marginal cost. The subsidy for the cost of
education is positive, that is, the private burden of the cost of education must be lower
than the true cost of education, because an additional type-IT consumer generates positive
externalities on the utility of other type-IT consumers and on the production of type-IT
firms.
Second, as an extreme case, we characterize the solutions under the laissez-faireeconomy, in which there is no government intervention. In this case, the price of thecommunications service is higher than its marginal cost, due to monopoly pricing by theprovider of the communications service. The private burden of the cost of education3

equals its true cost, because the subsidy for the cost of education is zero. The high priceof the communications service and the low subsidy for the cost of education combine tocause the communications network to contract in the laissez-faire economy. The small
network size means a decrease in the number of type-IT consumers and an increase in thenumber of type-NIT consumers. Since the wage of type-NIT consumers and the numberof employees in each type-NIT firm do not depend on the network size, the number oftype-NIT firms, which is defined as the number of type-NIT consumers over the numberof employees in each type-NIT firm, increases. The small network size also implies
fewer externalities on type-IT firm production, which decreases the wage of type-IT
consumers and increases the number of employees in each type-IT firm. A decrease in
the number of type-IT consumers and an increase in the number of employees in each
type-IT firm result in a decrease in the number of type-IT firms, which is defined as thenumber of type-IT consumers over the number of employees in each type-IT firm.
Third, we examine the effects of various policies in a situation in which the existing
network is below optimal due to a higher price of the communications service or a lower
subsidy for the cost of education. The major policies we focus on are: i) a subsidy for the
price of the communications service; ii) an income transfer from type-IT consumers to
type-NIT consumers; iii) an increase in the government’s subsidy for employment in
type-IT firms; and iv) an increase in the government’s subsidy for employment in
type-NIT firms. We consider the effects of these policies on network size, income,
employment, and the number of firms, and we show whether these policies are beneficial
or not in terms of the social surplus.
Our model is unique in that it can deal with various changes in economic
performance brought about by an endogenous evolution of the communications network4within a consistent theoretical framework. However, our model does relate to otherpapers in the literature, and before proceeding, we briefly link our study to this existing
literature.
First, our study is related to the literature on network externalities. In our model, an
evolution of the communications network yields positive externalities in the form of
increased utility and income of type-IT consumers. These positive externalities justify asubsidy for the cost of education and make the optimal private burden of the cost of
education lower than its true cost. This result is an application of the studies of Squire[1973], Littlechild [1975], and Kanemoto [1990, 2001], which, in the context of
telecommunications industries, show that the optimal fixed tariff in a two-part tariffmechanism is lower than the true fixed cost.
Second, this paper is related to the literature on the relationship between information
technology (IT) and the labor market. For example, Krueger [1993], Doms et al. [1997],
Berman et al. [1998], and Machin and van Reenan [1998] find that an increase in the
demand for skilled workers coexists with an increase in the income of skilled workers in
the United States and other OECD countries. Autor et al. [1998] and Bresnahan [1999]
consider the possibility that IT generates wage inequality. Although our model does not
focus on IT investment as a whole but on the evolution of communications networks
through IT, our results are consistent with these arguments. In our model, evolution of thecommunications network, that is, an increase in the number of type-IT consumers
increases both the number of type-IT employees and their wages. Since an expansion of
the communications network does not change the wages of type-NIT consumers, anincrease in the wages of type-IT consumers through an evolution of the communications
network increases wage inequality. Acemogle [1998] and Galor and Moav [2000],5

utilizing a different approach to explain wage inequality, use growth theory todemonstrate that technological evolution causes wage inequality. Our model shows that
we can rationally explain wage inequality using simply a one-shot model, provided wetake into account the evolution of communications networks.
Third, our study is related to the literature on the relationship between IT and theorganization of the firm. Brynjolfsson et al. [1994] shows empirically that IT investments
reduce firm employment. Brynjolfsson and Hitt [1998], Brynjolfsson and Hitt [2000],
and Bresnahan et al. [2002] argue that IT, skilled labor, and work organization arecomplements, and suggest that the decentralization of work organization is necessary forthe utilization of IT. However, these arguments lack a theoretical foundation. The only
exception is Brynjolfsson [1994], which uses principal-agent theory to explain therelationship between IT and the number of employees in the firm. The problem with thisapproach is that it ignores effects on other aspects of firm performance, such as wages.
Our general equilibrium set-up shows that evolution of the communications network, that
is, an increase in the number of type-IT employees, decreases the number of type-IT
employees per firm (that is, makes work organization smaller), and raises the incomes oftype-IT employees. Thus, our model provides a theoretical foundation for the arguments
advanced in this branch of the existing literature.
The rest of the paper is structured as follows. In Section 2, we build the model. In
Section 3, we derive the first-best outcome and compare it to the solution under thelaissez-faire economy. In Section 4, we examine the effects of various policies. Section 5
concludes.

6

II Model
Our model consists of two types of consumer, two types of firm that produce thecomposite consumer good, a provider of a communications service, and the government.
Let us define these players in turn.

1. Consumers
Consumers are indexed by i ∈ [0,1] . We ignore the integer problem. Consumers areclassified into two types: type-IT consumers use the communications service, say e-mail,and the composite consumer good, while type-NIT consumers only consume thecomposite consumer good. We assume that type-NIT consumers can freely become
type-IT consumers by receiving education.
A type-IT consumer communicates with other type-IT consumers.

Type-ITconsumers are distributed in the range [0, n ] . The utility function of type-IT consumer i
is assumed to be separable in the quantity of information to be sent, xijs , the quantity of
information to be received, xijr , and the composite consumer good, z , the price of which
is normalized to be unity. The quantities of information to be sent and to be received, xijs
and xijr , can be thought of as the frequencies of sending and receiving information. The
utility function of type-IT consumer i is
n

n

j =0

j =0

U iIT = z + ∫ u s ( xijs )dj + ∫ u r ( xijr )dj ,(1)

where u s ( xijs ) is the utility obtained by type-IT consumer i when he or she sends
information, whose volume is xijs (or at frequency xijs ,) to consumer j , and u r ( xijr ) is i ’sutility when he or she receives information, whose volume is xijr (or at frequency xijr ,),7

from consumer j . For tractability, we assume that the sub-utility function regarding
sending information, u s ( xijs ) , has the same form as the sub-utility function regarding
receiving information, u r ( xijr ) , that is, u s (i) = u r (i) = u(i) where u (0) = 0 , u′(i) > 0 , andu′′(i) < 0 .

We further assume that the utility obtained by type-IT consumer i bycommunicating does not depend on who is communicated with; that is, for type-IT
consumer i , all other type-IT consumers are symmetric. These assumptions simplify theutility function of type-IT consumer i to
U iIT = z + n {u( xis ) + u( xir )} .

(2)

The budget constraint of type-IT consumer i is
n

n

j =0

j =0

wiIT = z + p s ∫ xis dj + p r ∫ xir dj + e(i ) − t(3)where wiIT , p s , p r , e(i ) , and t respectively denote the income of type-IT consumer i ,the unit price of sending information, the unit price of receiving information, i ’s cost of
education incurred in becoming a type-IT consumer, and the government’s education
subsidy. We simplify our analysis by assuming that all type-IT consumers receive thesame wage, that is, wiIT = w IT . Since we have assumed that all other type-IT consumers
are symmetric for type-IT consumer i , the unit prices of sending and receiving
information, p s and p r , are independent of the person communicated with, consumer j .
The unit prices of sending and receiving information, p s and p r , can be thought of as theunit access cost to connecting to a network, such as the Internet. Equation (3) implies that
if we want to send or receive a larger volume of information or if we want to send or
receive information more frequently, we have to pay more. For the sake of simplicity, theunit prices of sending and receiving information are assumed to be symmetric. Thus, the8unit price of sending information is the same as that of receiving information, that is,p s = p r = p . Consequently, (3) reduces tow IT = z + pn( xis + xir ) + e(i ) − t .

(4)Regarding i ’s cost of education, we make the following assumptions: i) i ’s cost of
education to become a type-IT consumer, e(i ) , equals its true social cost; and ii) i ’s costof education becomes higher as i approaches n , that is,ei > 0 .

(5)

The first assumption implies that there are no profits in the sector that provides theeducation required to become a type-IT consumer, and the second assumption means thatthe smaller the consumer’s index, i , the lower the cost of education and the greater thewillingness to enter the communications network.
A type-IT consumer maximizes his or her utility, (2), with respect to z and xis ,subject to the budget constraint, (4). In communication by e-mail, a person normallyreceives all incoming e-mails by paying his or her own access costs and spending his or
her own time, even if the messages include unsolicited e-mail (“spam”, or “junk e-mail”).
Thus, we assume that the receiver of information cannot sort out the incoming
information. Consequently, xir is chosen by other consumers, and hence cannot be
controlled by type-IT consumers, i . The first-order condition for type-IT consumer i ’s
maximization problem isnu′( xis ) − pn = 0 ,(6)

xis = x s ( p )(7)from which, we obtain

where
9

x sp ≡

∂x s 1= < 0.
∂p u′′(8)

From (7), we know that the volume of information to be sent (or the frequency of sending
information) is independent of i , and depends only on the unit price of thecommunications service. This implies
xis = xir ≡ x ( p ) .

(9)Type-NIT consumers, who are distributed in the range [n,1] , have the same utility
function, which includes only the quantity of the composite consumer good, that is,
U NIT = z .

(10)

A type-NIT consumer spends all of his or her income, w NIT , which is assumed to becommon to all type-NIT consumers, on the composite consumer good, and consequently,
U NIT = w NIT .

(11)

The total consumer surplus, denoted by CS , can be written as
n

1

i =0

i =n

CS = ∫ U iIT di + ∫ U iNIT di(12)

The marginal, that is, n th, subscriber to the communications network attains thesame utility level whether he or she becomes a type-IT consumer or a type-NIT consumer.
That is, U IT = U NIT , which yields
w IT − 2 pnx ( p ) − e( n ) + t + 2nu( x ( p )) = w NIT .
Denotingthemarginal

benefitofbecominga

type-IT(13)
consumerbyMB ≡ w IT + 2n {u ( x ( p )) − px ( p )} − {e( n ) − t} , we can rearrange (13) as

MB = w NIT .

(14)

To ensure a unique and stable equilibrium, the marginal benefit of becoming a type-IT
consumer, MB , must be decreasing with respect to network size, that is,10MBn < 0 .

(15)(See Figure 1.) If this condition is not met, there may be multiple equilibria.1 However,the purpose of this paper is not to show the existence of multiple equilibria, but to show
how the evolution of a communications network affects economic performance. Thus, inthe following analysis, we assume MBn < 0 , for which the increasing cost of education
relating to i , (5), is a necessary condition.

2. z-producing Firms
Two types of firm produce the composite consumer good, z . Type-IT firms are
assumed to employ only type-IT consumers and produce z using their knowledge.
Type-IT consumers’ knowledge level is k (n ) , where k ′(n ) > 0 . This assumption implies
that the exchange of information via communication extends type-IT consumers’
knowledge. We assume that knowledge that is useful for type-IT firm production per
type-IT employee, α , is increasing with the knowledge level, that is, α (n ) ≡ l (k (n )) ,where l ′(i) > 0 and consequently, α ′( n ) > 0 .
Type-NIT firms are assumed to hire only type-NIT consumers and produce z using
their knowledge. The knowledge level of type-NIT consumers is fixed, because these
consumers do not upgrade their knowledge by communicating with each other. Thus,
useful knowledge for type-NIT consumers is also constant.

For simplicity, usefulknowledge for type-NIT consumers is normalized to be unity.
Since the price of the composite consumer good is unity, the profits of the type-IT
firm can be written as

π IT = f (α (n ) LIT ) − w IT LIT ,(16)

where LIT is the number of employees that the type-IT firm hires, and consequently,11α (n ) LIT is the total amount of knowledge that the type-IT firm uses for its production.
The profits of the type-NIT firm are

π NIT = g ( LNIT ) − w NIT LNIT ,(17)

where LNIT is the number of employees that the type-NIT firm hires. Since usefulknowledge for type-NIT firms is assumed fixed at unity, LNIT is also the total amount ofknowledge that the type-NIT firm uses for its production.
The composite consumer good, z , can be costlessly imported from or exported to therest of the economy, which is not modeled here, and the z -producing industry is assumed
to be sufficiently competitive. Thus, in equilibrium, we obtain

π IT = π NIT = 0 .

(18)

Denoting the number of type-IT firms and the number of type-NIT firms by m IT andm NIT respectively, we obtainm IT ≡

n
,
LIT

(19)

1− n
.
LNIT

(20)

m NIT ≡

3. Provider of the Communications Service
For the sake of simplicity, the communications service is assumed to be supplied by amonopoly. Recall from (4) that each type-IT consumer pays 2 pnx ( p ) when connecting
to the network to send or receive information. We assume that the unit marginal cost ofsending information from i to j is independent of i and j and constant at c . Thus, theprofits of a monopolistic provider of the communications service, π COM , can be written
as

12

n

n

π COM = ∫ 2 pnx( p )di − ∫ cnx ( p )di − F
i =0

i =0

(21)

= (2 p − c )n x ( p ) − F ,2where F is the fixed cost of providing the service.2

4. Government
The government pays a subsidy for the education required to become a type-IT
consumer, t . The government’s surplus is
G = −tn .

(22)

As long as the government pays a positive subsidy, it has a budget deficit, which isassumed to be financed by non-distortional lump-sum transfers.

Thus, we ignore

distortions arising from collecting taxes.

5. The Total Social Surplus
From (12), (18), (21), and (22), the total social surplus, denoted by TSS , can be
written as
TSS = CS + π COM + G .

(23)

III First-Best vs. Laissez-faire
For the sake of convenience, let us begin by obtaining the relationship between the
network size, n , and the performance of z -producing firms. From profit maximization
by z -producing firms, we obtain the following Lemma 1.

13

Lemma 1

i)wnIT ( n ) > 0 and w NIT is constant.

ii)

LITn ( n ) < 0 and LNIT is constant.

iii) mnIT (n ) > 0 and mnNIT ( n ) < 0 .

The proof is shown in the Appendix.
Lemma 1-i) shows that the income of type-IT consumers is an increasing function of
network size, that is, the number of type-IT consumers, and that the income of type-NIT
consumers is fixed. Thus, the wage gap between type-IT and type-NIT consumers
increases as the communications network develops.

This results from α ′(n ) > 0 .

Type-IT consumers’ increased knowledge level, facilitated by evolution of thecommunications network, benefits type-IT firm production. Consequently, the wage oftype-IT employees, that is, the income of type-IT consumers, increases as thecommunications network expands. Since knowledge that is useful for type-NIT firm
production is fixed at unity per type-NIT employee, the income of type-NIT firms isirrelevant to network size.
Lemma 1-ii) shows that the number of employees per type-IT firm is a decreasing
function of network size and the number of employees per type-NIT firm is fixed. This
results from Lemma 1-i). As the network expands, the wage paid by type-IT firms rises,
and consequently, the number of employees per type-IT firm decreases. Since the wage
paid by type-NIT firms is irrelevant to network size, the number of employees per
type-NIT firm is also independent of network size.
Lemma 1-iii) focuses on the number of firms. As the network expands, the numberof type-IT consumers increases and the number of employees per type-IT firm decreases.
14

This implies that the number of type-IT firms increases. By contrast, since an expansionof the network reduces the number of type-NIT consumers without affecting the numberof employees per type-NIT firm, the number of type-NIT firms decreases.
Next, we obtain the effects of the unit price of the communications service and thegovernment subsidy for education on the network size as given by Lemma 2.

Lemma 2i)np ≡∂n( p, t )
<0
∂p

ii)

nt ≡

∂n( p, t )>0∂tLemma 2 shows that the communications network shrinks as the unit price of thecommunications service rises, and expands as the government subsidy for education
increases. The high unit price of the communications service lowers the marginal benefit
of entering the network, and consequently causes the network to contract, as shown in
Figure 2. The low burden of the cost of education from a high government subsidy raises
the marginal benefit of entering the network, and therefore expands the network as shownin Figure 3.
Now let us derive the first-best value of the unit price of the communications service
and the government subsidy for education by maximizing the total social surplus, (23).

Proposition 1

In the first best, we obtain:
i)

p* =

1
c,
2
15

x

ii)

1 

t * = 2n ∫  u x − c dx + nwnIT > 0 ,2 
x =0 

where the superscript ∗ denotes the first best outcome.

In the first-best outcome, the unit price of the communications service is half of its
marginal cost, that is, the sum of the prices that a sender and a receiver pay equals themarginal cost of the communications service. This result is an application of the studies
of telecommunications by Littlechild [1975] and Kanemoto [1990, 2001], which showthat the optimal unit price per communications service is equal to its marginal cost,
although these models do not explicitly consider the benefits of communication to thereceiver.
Second, we focus on the government subsidy for education. In the first-best outcome,the government subsidy for education is positive. This is because an additional entrant tothe communications network generates two positive externalities, one of which affects
the utility, the other the income, of existing type-IT consumers. If consumer i becomes a
type-IT consumer, existing type-IT consumers can communicate with consumer i , andconsequently, existing customers obtain greater utility both as senders and receivers of
information. That is, new type-IT consumers generate a positive externality in the formx

1 
of higher utility for all type-IT consumers, which is given by 2n ∫  u x − c dx . The
2 
x =0 

other externality affects the income of type-IT consumers. As a result of accumulating
information by communicating with consumer i , type-IT consumers are paid a higher
wage by type-IT firms, and thus have a higher income. This means that new type-IT
consumers generate a positive externality in the form of greater income for all type-IT

16

consumers, which is given by nwnIT . Due to these two positive externalities, the socially
optimal subsidy for education is positive.
Next, we consider the laissez-faire economy, in which there is no government
intervention. In this case, the total payment for the communications service is higher thanits marginal cost due to monopoly pricing, and the government subsidy for education iszero.

Denoting the equilibrium under laissez-faire economy by, we derive thefollowing proposition.

Proposition 2

When p̂ > p* and tˆ = 0 ,i)n < n* ,ii)

w IT < w IT * , w NIT = w NIT * ,iii) LIT > LIT * , LNIT = LNIT * ,iv) m IT < m IT * , m NIT > m NIT * .

(See Appendix for proof.)
Since the price of the communications service is higher and the government subsidy
lower in the laissez-faire economy than in the first-best outcome, the network is below its
optimal size in the laissez-faire economy. The smaller network lowers the income oftype-IT consumers and increases the number of employees per type-IT firm. Since in thelaissez-faire economy, the number of type-IT consumers decreases and the number of
employees per type-IT firm increases, relative to the first-best outcome, the number oftype-IT firms decreases. By contrast, the income of type-NIT consumers and the number17

of type-NIT consumers per type-NIT firm is irrelevant to the size of the network. A
constant number of employees per type-NIT firm and a larger number of type-NIT
consumers in the laissez-faire economy mean that the number of type-NIT firms is larger
than in the first-best outcome.

IV Effects of Various Policies
In Proposition 2, we have shown that in the laissez-faire economy, which ischaracterized by monopoly pricing and no education subsidy, network size decreases andconsequently, distortions arise. Although Proposition 2 focuses on an extreme case, thisresult holds as long as either the total payment for the communications service is higher
than its marginal cost or the government subsidy for education is inadequate. In reality,
we have no reason to believe that the provision of communications services iscompetitive enough to result in marginal-cost pricing or that government subsidies foreducation are determined by taking into account positive externalities on type-IT
consumers and type-IT firms. We then look for desirable policies when the size of thecommunications network is below optimal due to price exceeding marginal cost and alower subsidy for education.
First, we consider a subsidy for the communications service. Here, we focus on a
subsidy for the unit price of the communications service, τ 1 , to decrease type-IT
consumers’ communication costs. In this case, the unit price of the communications
service for type-IT consumers becomes p − τ 1 . The results are stated as Proposition 3.

Proposition 3

A subsidy for the unit price of the communications service, τ 1 , expands n , increases18

w IT and m IT , and decreases LIT and m NIT . This subsidy raises the total social surplus,
1
as long as τ 1 < p − c .
2(See Appendix for proof.)
A subsidy for the unit price of the communications service enlarges thecommunications network, that is, increases the number of type-IT consumers, because it
increases the benefits of entering the network. From Lemma 1, an evolution of thecommunications network increases the incomes of type-IT consumers and reduces thenumber of employees per type-IT firm, and consequently increases the number of type-IT
firms. The incomes of type-NIT consumers and the number of employees per type-NIT
firm remain unchanged, and consequently the number of type-NIT firms decreases.
The net social benefits of a subsidy for the unit price of the communications service
are classified into two types. These are, first, the direct benefits caused by a decrease in
the price of the communications service for type-IT consumers, and second, the indirect
benefits that derive from an evolution of the communications network. The direct
benefits are those caused by an increase in the volume of information being sent or
received (or the frequencies of sending or receiving information) brought about by adecrease in the price of the communications service from p to p − τ 1 . These direct
benefits are positive as long as the total payment for the communication service,12( p − τ 1 ) , is higher than the marginal cost, c , that is, τ 1 < p − c . The indirect benefits
2
are those due to an expansion of the network, which are positive when the size of thecommunications network is below optimal. Thus, when the size of the communications
network is not sufficiently large, a subsidy for the price of the communications service is19

welfare-improving as long as the total payment for the communications service is higher
than the marginal cost.
The essence of Proposition 3 is an expansion of the network through a decrease in theprice of the communications service. Obviously, the same result holds when the price ofthe communications service is regulated by a price-cap, p . The results are stated as
Corollary 1.

Corollary 1

Price-cap regulation expands n , increases w IT and m IT , and decreases LIT andm NIT . This regulation raises the total social surplus, as long as p >1c.
2

The difference between Proposition 3 and Corollary 1 is in the distribution of surplus.
In Proposition 3, in which a subsidy is given to the unit price of the communications
service, type-IT consumers gain benefits from a decrease in the price in exchange for thegovernment deficit. On the contrary, in Corollary 1, in which a price-cap regulation is
imposed, type-IT consumers also gain benefits, but these benefits are due to a decrease in
the profits of the provider of the communications service. In our general equilibrium
model, these two policies yield the same total social surplus. In reality, however,price-cap regulation would be a more desirable policy, because it would generate
incentives for providers of communications services to reduce costs, and thereby
additional benefits. (These cost-reducing incentives are not modeled here.)
Second, we deal with an income transfer from type-IT consumers to type-NIT
consumers. For the sake of simplicity, we focus on the following mechanism: each20

type-IT consumer pays a tax, τ 2 , on his or her income, w IT , and the total amount of thecollected tax, nτ 2 w IT , is shared equally between all type-NIT consumers, who each
receivenτ 2 w IT. The after-tax incomes of a type-IT consumer and a type-NIT consumer
1− nare denoted by w ITA ≡ (1 − τ 2 ) w IT and w NITA ≡ w NIT +nτ 2 w IT
respectively. The results
1− nare stated as Proposition 4.

Proposition 4

An income transfer from type-IT consumers to type-NIT consumers causes n tocontract, increases LIT and m NIT , and decreases w ITA and m IT . Whether w NITA increases
or not is ambiguous. This income transfer lowers the total social surplus.

(See Appendix for proof.)
An income transfer from type-IT consumers to type-NIT consumers decreases incomes oftype-IT consumers and increases those of type-NIT consumers at first. This income
transfer implies that the benefits of becoming a type-IT consumer fall and those ofremaining a type-NIT consumer rise. As a result, the communications network shrinks.
Subsequently, a contraction of the communications network further reduces the incomesof type-IT consumers, which results in a lower income transfer per type-NIT consumer,nτ 2 w IT. In sum, an income transfer from type-IT consumers to type-NIT consumers
1− nalways reduces incomes of type-IT consumers, but its effect on the incomes of type-NIT
21

consumers is ambiguous. A decrease in the incomes of type-IT consumers results in an
increase in the number of employees per type-IT firm, which together with a decrease in
the number of type-IT consumers, implies a decrease in the number of type-IT firms.
Since the number of employees per type-NIT firm remains unchanged, a contraction of
the communications network means an increase in the number of type-NIT firms. An
income transfer from type-IT consumers to type-NIT consumers decreases the total social
surplus through a contraction of the network when network size is below optimal.
Proposition 4 suggests that such an income equalization policy is harmful for the
total social surplus, because it doubly distorts the incentives to become a type-IT
consumer: a tax for type-IT consumers decreases the benefits of entering the network anda subsidy for type-NIT consumers increases the opportunity costs of entering the network.
This policy may even harm type-NIT consumers, who are supposed to receive benefits if
the income transfer is large. A contraction of the network reduces the incomes of type-IT
consumers, and consequently reduces the income transfer to a type-NIT consumer,nτ 2 w ITnτ w IT, given τ 2 . When τ 2 is high enough, a decrease in 2
by a contraction of
1− n
1− n
the network is likely to upset an increase in it by an increase in τ 2 , and in this case, anadditional income transfer to type-NIT consumers lowers their incomes and utility. The
upshot is that this policy further magnifies the distortions stemming from a small network
size, without guaranteed benefits even to type-NIT consumers.
Third, we consider policies for the labor market, which are aimed at improving thetotal social surplus. We here focus on a subsidy, τ 3 , per employee in type-IT firms. The

22

results are stated as Proposition 5.

Proposition 5

An increase in the government’s subsidy per employee in type-IT firms, τ 3 , expands
n , increases w IT and m IT , and decreases LIT and m NIT . This subsidy increases the totalsocial surplus.

(See Appendix for proof.)
The government’s subsidy per employee in type-IT firms, τ 3 , increases the wage
paid by type-IT firms, that is, the incomes of type-IT consumers, by the full amount, τ 3 .
Initially, the net wage paid by type-IT firms is unchanged, and hence so too is the numberof employees per type-IT firm. Subsequently, however, the increase in the wage oftype-IT consumers increases the incentive to enter the communications network, which
thus expands. The development in the communications network further raises the wageof type-IT consumers and decreases the number of employees per type-IT firm. Since thetotal number of type-IT consumers increases while the number of employees per type-IT
firm decreases, the number of type-IT firms increases. A decrease in the total number oftype-NIT consumers reduces the number of type-NIT firms, because the incomes oftype-NIT consumers and the number of employees per type-NIT firm remain unchanged.
The essential point in Proposition 5 is that an increase in the incomes of type-IT
consumers through this subsidy in effect lowers the private burden of the cost of
education from e(i ) − t to e(i ) − t − τ 3 . Thus, this subsidy is equivalent to a subsidy foreducation. This subsidy increases the total social surplus when the size of the network is
below optimal. Proposition 5 suggests that subsidies for IT-related workers are justified23

as a substitute when the subsidies for the cost of education are inadequate.
On the contrary, for comparison, let us examine the case where the government gives
the subsidy, τ 4 , per employee in type-NIT firms. The results are stated as Proposition 6.

Proposition 6

An increase in the government’s subsidy per employee in type-NIT firms, τ 4 ,contracts n , increases w NIT , LIT , and m NIT , and decreases w IT and m IT . This subsidy
decreases the total social surplus.

(See Appendix for proof.)
The government subsidy of employment in type-NIT firms, τ 4 , increases the wage
paid by type-NIT firms, that is, the incomes of type-NIT consumers, by the full amount,τ 4 . Since the net wage paid by type-NIT firms is unchanged, the number of employees
per type-NIT firm remains the same. Since an increase in the wage of type-NIT
consumers means an increase in the opportunity costs of becoming a type-IT consumer,
the communications network shrinks. Applying Lemma 1, we know that with regard to
the wage of type-IT consumers, the number of employees per type-IT firm, the number oftype-IT firms, and the number of type-NIT firms, the government subsidy per employee
in type-NIT firms yields opposite effects to those of Proposition 5.
An increase in the incomes of type-NIT consumers through this subsidy in effectraises the private burden of the cost of the education required to enter the network frome(i ) − t to e(i ) − t + τ 4 . Accordingly, this subsidy adversely affects the size of thecommunications network and decreases the total social surplus when the network size isbelow optimal.
24

V Concluding Remarks
In this paper, we built a general equilibrium model in which evolution of thecommunications network is explicitly considered. We also compared the solutions underlaissez-faire with the first-best outcomes. In addition, we examined the effects of various
policies on network size, income, the number of employees, and the number of firms.
These policies included a subsidy for the price of the communications service, an income
transfer from type-IT consumers to type-NIT consumers, and subsidies to type-IT and
type-NIT firms. Here, we relate our results to actual policies and conclude our analysis.
First, policies in telecommunications markets are very important. As we showed in
Proposition 3 and Corollary 1, a decrease in the price of communications services
increases the total social surplus, as long as price exceeds marginal cost. Price-cap
regulation for a monopolistic provider would be more desirable due to additional
potential benefits stemming from cost-reducing efforts, but even in the case of a subsidy
for the price of the communications service, the total social surplus increases. By
providing this subsidy, the government runs a budget deficit. However, in terms of thetotal social surplus, this deficit is completely offset by the reduced payments made by
consumers to the provider of the communications service. The remaining effect is an
expansion of the communications network, which increases the total social surplus. Thus,
lowering the price of communications services is beneficial. Although our model does
not explicitly correspond to the fixed (or lump-sum) payment system that has become
more common in broadband communications, the mechanisms described in this paper
would be applicable even in that case, at least approximately. Thus, we could suggest
desirable policy directions for broadband communications.

25

If broadband networks

prevail, the total social surplus remains low, as long as the price of communications
services via broadband networks is high. To increase the total social surplus, the accessprice to broadband networks should be kept low, at the marginal cost if possible, throughcompetition policies and subsidies.
Second, our results provide a perspective on the so-called digital divide problem.3
As we showed in Proposition 2, in the laissez-faire economy, in which the price of thecommunications service is higher than its marginal cost and in which consumers have to
pay the entire cost of education to become type-IT consumers, the size of thecommunications network is below its optimal. Since evolution of the network has
positive externalities on the utility and income of all type-IT consumers, policies to
evolve communications networks are needed for a higher social surplus. This result
suggests that preventing the digital divide is justified on the grounds of efficiency, rather
than on the grounds of equity.
Third, our results suggest desirable policy directions for labor markets and education.
To relieve the distortions stemming from an underdeveloped network, not only is a low
price of communications services needed, but so too is a subsidy for the cost of educationin order to reduce the private burden of the cost of education. Proposition 5 demonstrates
that a subsidy for employees in type-IT firms increases the total social surplus and can beused as a substitute for a subsidy for the cost of education, because it raises the benefits ofbecoming a type-IT consumer. On the contrary, Proposition 6 shows that this mechanism
is completely reversed in the case of a subsidy for employees in type-NIT firms. An
income equalization policy in Proposition 4 produces a worse outcome, because it doubly
lowers the benefits of becoming a type-IT consumer by transferring income from type-IT
consumers to type-NIT consumers. To develop communications networks and increase

26

the total social surplus, government policy must be designed to increase the benefits ofbeing a type-IT consumer and/or decrease those of being a type-NIT consumer.

27

Appendix
Proof of Lemma 1
From profit maximization by the type-IT firm, we obtainα ( n ) f ′(α ( n ) LIT ) − w IT = 0 .

(A1)

The zero-profit condition, (18), gives us
f (α ( n ) LIT ) − w IT LIT = 0 .

(A2)Totally differentiating (A1) and (A2) yields
wnIT ( n ) = f ′(α (n ) LIT )α ′(n ) > 0 ,(A3)α ′( n ) LIT
L (n) = −
< 0.
α

(A4)

IT
n

Profit maximization by the type-NIT firm yieldsg ′( LNIT ) − w NIT = 0 .

(A5)From the zero-profit condition, (18), we obtaing ( LNIT ) − w NIT LNIT = 0 .

(A6)Solving (A5) and (A6), we obtain the fixed value of w NIT and LNIT .
The number of type-IT firms, m IT , and the number of type-NIT firms, m NIT ,respectively satisfies
mnIT (n ) =LIT − nLITn

mnNIT ( n ) = −

LIT

2

> 0,

1
< 0.
L
NIT

(A7)

(A8)
Q.E.D.

Proof of Lemma 2
28

Totally differentiating (13), we obtainn p ( p, t ) =

2nx
< 0,
MBn

(A9)

nt ( p, t ) = −

1
> 0.
MBn

(A10)
Q.E.D.

Proof of Proposition 1
The total social surplus, which is defined in (23), can be reduced ton

TSS ( n ( p, t ), p ) = nw + (1 − n ) w
ITNIT+ n {2u( x ( p )) − cx ( p )} − ∫ e(i )di − F .
2

(A11)

i =0

from (2), (4), (9), (11), (12), (21), and (22). The first-order conditions are
TSS p = TSS1n p + TSS2 = 0 ,(A12)TSSt = TSS1nt = 0 ,(A13)where TSSi is the derivative of TSS with respect to the i th argument. From (A10) and(A13), we obtain
TSS1 = 0 .

(A14)

TSS2 = 0 ,

(A15)

1
c.
2

(A16)

Eqs. (A12) and (A14) yieldfrom which we havep* =Rearranging (A14), we obtainx

1 t * = 2n ∫  ux − c dx + nwnIT > 0 ,2 
x =0 

29

(A17)

1
1
where ux − c ≥ 0 from ux ≥ p* = c and nwnIT > 0 from (A3).
2
2
Q.E.D.
Proof of Proposition 2
When p̂ > p* and tˆ = 0 , from (A9) and (A10), we haven < n* .

(A18)

w IT < w IT * ,(A19)

w NIT = w NIT * ,(A20)

LIT > LIT * ,(A21)

LNIT = LNIT *(A22)m IT < m IT * ,(A23)m NIT > m NIT * .

(A24)

From Lemma 1 and (A18), we obtain

Q.E.D.

Proof of Proposition 3
Defining the unit price of the communications service after a subsidy, p , byp ≡ p − τ 1 , (A9) now holds in terms of p , that is,n p ( p, t ) < 0 .

(A25)

From (A3), (A4), (A7), (A8), and (A25), we havenτ1 = n p ( p, t ) pτ1 = −n p ( p, t ) > 0 ,(A26)wτIT1 ( n( p ( p,τ 1 ), t )) = wnIT n p pτ1 = − wnIT n p > 0 ,

(A27)

30

LτIT1 ( n ( p ( p,τ 1 ), t )) = LITn n p pτ1 = − LIT
n np < 0 ,(A28)mτIT1 (n ( p( p,τ 1 ), t )) = mnIT n p pτ1 = − mnIT n p > 0 ,

(A29)

mτNIT
( n ( p( p,τ 1 ), t )) = mnNIT n p pτ1 = − mnNIT n p < 0 .
1

(A30)

The total social surplus in this case, TSS 1 , can be written asn

TSS 1 = nw IT + (1 − n ) w NIT + 2n 2u( x ( p )) − 2 pn 2 x ( p ) + (2 p − c )n 2 x ( p ) − ∫ e(i )di − F − 2τ 1n 2 x ( p )i =0n= nw IT + (1 − n ) w NIT + n 2 {2u( x ( p )) − cx ( p )} − ∫ e(i )di − F .
i =0

(A31)
From (A31), we obtaindTSS 1 ( n( p, t ), p )= TSS11n p pτ1 + TSS21 pτ1
dτ 1
x


1 
1 
1 



= −  2nx  p − c  + 2n ∫  ux − c dx + nwnIT − t  n p − 2n 2  p − c  x p .
2 
2 
2 


x =0 

(A32)

Since TSS11 > 0 when n < n∗ , the term inside the square brackets is positive. Thus, from(A25), we obtaindTSS 1
>0
dτ 1

(A33)

1
1
as long as p − c > 0 , that is, τ 1 < p − c .
2
2
Q.E.D.

Proof of Corollary 1
We focus on the case in which price-cap regulation is binding, that is, p = p . In thiscase, from (A3), (A4), (A7), (A8), and (A9), we have31

n p = n p ( p, t ) < 0 ,(A34)

w pIT ( n( p, t )) = wnIT n p < 0 ,(A35)

LITp ( n ( p, t )) = LITn np > 0 ,(A36)

m pIT ( n ( p, t )) = mnIT n p < 0 ,(A37)

m pNIT ( n ( p, t )) = mnNIT n p > 0 .

(A38)

From (A11), we obtaindTSS ( n ( p, t ), p )= TSS1n p + TSS2
dp
x


1 
1 
1 



=  2nx  p − c  + 2n ∫  ux − c dx + nwnIT − t  n p + 2n 2  p − c  x p .
2 
2 
2 


x =0 

(A39)

Since TSS1 > 0 when n < n∗ , the term inside the square brackets is positive. Thus, from(A34), we obtain
dTSS
<0
dp
as long as p >

(A40)

1
c.
2
Q.E.D.

Proof of Proposition 4
An income transfer, τ 2 , changes (13) tow ITA − 2 pnx ( p ) − e( n ) + t + 2nu( x ( p )) = w NITA .
Totally differentiating (A41) yields

32

(A41)

w IT
nτ 2 ( p, t ,τ 2 ) = 1 − n < 0 ,MBn + ∆
where ∆ = −τ 2 {wnIT (1 − n ) + w IT }(1 − n ) 2

(A42)

< 0.

From (A3), (A4), (A7), (A8), and (A42), we havewτITA
( n ( p, t ,τ 2 ),τ 2 ) = − w IT + (1 − τ 2 ) wnIT nτ 2 < 0 ,
2
NITA

wτ 2

(A43)

IT
IT
nw IT nτ 2τ 2 {w + (1 − n )nwn }( n( p, t ,τ 2 ),τ 2 ) =+,1− n(1 − n ) 2(A44)LτIT2 ( n ( p, t ,τ 2 )) = LITn nτ 2 > 0 ,(A45)mτIT2 (n ( p, t ,τ 2 )) = mnIT nτ 2 < 0 ,

(A46)

mτNIT
( n ( p, t ,τ 2 )) = mnNIT nτ 2 > 0 ,
2

(A47)

is ambiguous, because the first term is positive but the second
where the sign of wτNITA
2
term is negative.
The total social surplus in this case, TSS 2 , can be written asn

TSS 2 = nw ITA + (1 − n ) w NITA + n 2 {2u( x ( p )) − cx ( p )} − ∫ e(i )di − F
i =0n= nw + (1 − n ) w
IT

NIT

(A48)

+ n {2u( x ( p )) − cx ( p )} − ∫ e(i )di − F2

i =0From (A48), we obtaindTSS 2 ( n( p, t ,τ 2 ))= TSSn2nτ 2
dτ 2
x


1 
1 


=  2nx  p − c  + 2n ∫  ux − c dx + nwnIT − t  nτ 2 .
2 
2 

x =0 


(A49)

Since TSSn2 > 0 when n < n∗ , the term inside the square brackets is positive. Thus, from33

(A42), we obtaindTSS 2
<0.
dτ 2

(A50)
Q.E.D.

Proof of Proposition 5
A subsidy per employee in type-IT firms changes the profits of a type-IT firm to

π IT = f (α ( n ) LIT ) − ( w IT − τ 3 ) LIT .

(A51)

From profit maximization by the type-IT firm, we obtainα ( n ) f ′(α ( n ) LIT ) − ( w IT − τ 3 ) = 0 .

(A52)

The zero-profit condition, (18), yields
f (α ( n ) LIT ) − ( w IT − τ 3 ) LIT = 0 .

(A53)Totally differentiating (A52) and (A53) gives us
w1IT ( n,τ 3 ) = f ′(α ( n ) LIT )α ′(n ) > 0 ,(A54)

w2IT ( n,τ 3 ) = 1 > 0 ,(A55)

α ′( n ) LIT< 0,
L (n) = −α(A56)where wiIT is the derivative of w IT with respect to the i th argument.

TotallyITndifferentiating (13), we havenτ 3 ( p, t ,τ 3 ) = −

1
>0
MBn

(A57)

from (A54) and (A55). Thus, from (A54) to (A57), we obtainwτIT3 ( n( p, t ,τ 3 ),τ 3 ) = w1IT nτ 3 + w2IT > 0 ,(A58)

LτIT3 ( n ( p, t ,τ 3 )) = LITn nτ 3 < 0 .

(A59)

34

From profit maximization by the type-NIT firm and the zero-profit condition, (18),
we obtain (A5) and (A6), which gives us the fixed value of w NIT and LNIT .
From (A7), (A8), and (A57), we obtainmτIT3 ( n ( p, t ,τ 3 )) = mnIT nτ 3 > 0

(A60)

mτNIT
( n ( p, t ,τ 3 )) = mnNIT nτ 3 < 0
3

(A61)

The total social surplus in this case, TSS 3 , can be written asn

TSS = nw + (1 − n ) w
3

IT

NIT

+ n {2u ( x ( p )) − cx ( p )} − ∫ e(i )di − F − τ 3n (A62)2i =0From (A55) and (A62), we obtaindTSS 3 ( n( p, t ,τ 3 ), p,τ 3 )= TSS13nτ 3 + TSS33
dτ 3
x


1 
1 


=  2nx  p − c  + 2n ∫  u x − c dx + nwnIT − t  nτ 3 + nw2IT − n
2 
2 

x =0 


x


1 
1 


=  2nx  p − c  + 2n ∫  u x − c dx + nwnIT − t  nτ 3 .
2 
2 

x =0 


(A63)
Since TSS13 > 0 when n < n∗ , the term inside the square brackets is positive. Thus, from(A57), we obtaindTSS 3
> 0.
dτ 3

(A64)
Q.E.D.

Proof of Proposition 6
A subsidy per employee in type-NIT firms changes the profits of a type-NIT firm to

π NIT = g ( LNIT ) − ( w NIT − τ 4 ) LNIT .

35

(A65)

From profit maximization by the type-NIT firm, we obtaing ′( LNIT ) − ( w NIT − τ 4 ) = 0 .

(A66)

The zero-profit condition, (18), yieldsg ( LNIT ) − ( w NIT − τ 4 ) LNIT = 0 .

(A67)

Totally differentiating (A66) and (A67) gives us the fixed value of LNIT andwτNIT(τ 4 ) = 1 > 0
4

(A68)

Totally differentiating (13), from (A3) and (A68), we havenτ 4 ( p, t ,τ 4 ) =

1
<0
MBn

(A69)

From (A3), (A4), and (A69), we obtainwτIT4 (n ( p, t ,τ 4 )) = wnIT nτ 4 < 0 ,(A70)

LτIT4 ( n ( p, t ,τ 4 )) = LITn nτ 4 > 0 .

(A71)

From (A7), (A8), and (A69), we havemτIT4 (n ( p, t ,τ 4 )) = mnIT nτ 4 < 0 ,

(A72)

mτNIT
( n ( p, t ,τ 4 )) = mnNIT nτ 4 > 0 .
4

(A73)

The total social surplus in this case, TSS 4 , can be written asn

TSS = nw + (1 − n ) w
4

IT

NIT

+ n {2u ( x ( p )) − cx ( p )} − ∫ e(i )di − F − τ 4 (1 − n ) (A74)2i =0From (A68) and (A74), we obtain

36

dTSS 4 ( n ( p, t ,τ 4 ), p,τ 4 )= TSS14nτ 4 + TSS34
dτ 4
x


1 
1 


=  2nx  p − c  + 2n ∫  ux − c dx + nwnIT − t  nτ 4 + (1 − n ) wτNIT− (1 − n )
4
2 
2 

x =0 


x


1 
1 


=  2nx  p − c  + 2n ∫  ux − c dx + nwnIT − t  nτ 4 .
2 
2 

x =0 


(A75)
Since TSS14 > 0 when n < n∗ , the term inside the square brackets is positive. Thus, from(A69), we obtaindTSS 4
<0.
dτ 4

(A76)
Q.E.D.

37

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Squire, L., [1973], “Some Aspects of Optimal Pricing for Telecommunications,” Bell
Journal of Economics 4, 515-525.

40

Notes
1. See Rohlfs [1974], Oren and Smith [1981], and Economides and Himmerlberg[1995] for analyses that focus on the existence of multiple equilibria.
2. For the sake of simplicity, we implicitly assume that a provider of communications
services does not hire either type-IT consumers or type-NIT consumers. That is, wedisregard employment by the provider of communications services.
3. According to the OECD [2001, p. 5], the term “digital divide” is defined as “thegap between individuals, households, businesses, and geographic areas at different
socio-economic levels with regard both to their opportunities to access information andcommunication technologies and to their use of the Internet for a variety of activities.”

41

Figure 1

MB

MB

w NIT

Determination of n

42nFigure 2

MB

MB′

MB
w NIT

The effect of an increase in p

43n

Figure 3

MB

MB

MB′
w NIT

The effect of an increase in t

44n