Network Competition, Interconnection and PricingWith Partial Participation and

Call Externalities

Ramiro Camacho-Castillo 1

1Ph.D. Candidate, Department of Economics, University of Maryland-College Park

Abstract

This chapter extends the literature on competition among interconected telephonenetworks, in particular with respect to the regulation of the interconecction fee andthe choice of pricing regime. We present a model of two vertically differentiated tele-phone networks, which pay each other per-minute termination charges and competein final markets using nonlinear tarifs. We allow for the existance of call externali-ties and partial participation, and analize the optimal interconection fee under twopricing regimes: only callers pay, and both caller and receiver pay.

0.1 Introduction

This chapter analizes the determination of the efficient per-minute termination

fee that telephone networks should charge each other under a model of two vertically

differentiated networks, which compete by offering calling plans to final costumers,

using general nonlinear tariffs. We allow for the existance of call externalities under

partial participation of costumers, and analize the optimal interconection fee under

two pricing regimes: only callers pay (CPP), and both caller and receiver pay (RPP).

We build upon the work of Laffont (2004), Hahn (2003), Hermalin (2001) and

Rochet (2001), who studied competition in horizontally differentiated networks, to

instead study competition between two vertically differentiated networks. Con-

sumers are heterogeneous, have private information on their preferences for the ser-

vice and benefit from network and call externalities. We study the internalization of

call and network externalities and whether the regulator can intervene to bring the

coverage and quantity of calls closer to efficient levels. Three types of interventions

are considered: setting the interconnection rate, banning/limiting network-based

price discrimination and allowing networks to charge for incoming calls.

The dissertation will have and introduction and three chapters. In the intro-

duction, I will survey the literature and describe the issues surrounding interconnec-

tion, competition, pricing and regulation in the telephone industry. Chapter 1 will

study RPP by using a monopoly model with incoming call pricing. A welfare analy-

sis of the effects of moving from CPP to RPP will be carried through focusing in the

internalization of call externalities and the effects on participation rates (network

1

externalities) as well as the allocation of calls. Chapter 2 will study competition un-

der partial participation and the CPP. In this chapter, two interconnected vertically

differentiated networks compete to attract customers and may use the interconnec-

tion rate as an instrument of both collusion and internalization of externalities. The

welfare implications and possible regulatory intervention are analyzed. Finally, In

chapter 3, a model of competition with RPP will look into the determination of

prices and coverage under different assumption about the determination of the in-

terconnection fee: cost based, bill-and-keep, cooperative, non-cooperative, among

others.

Network effects in telephony services imply that it is effcient to have all cos-

tumers connected to a single network so that each of them can call every other

costumer. However, a single firm may charge monopoly prices, which implies that it

is also desirable to have competition by at least a few different networks to reduce

market power and achieve an acceptable level of efficiency. Both objectives may be

achieved to a greter extent by a policy of mandatory interconection at regulated,

efficient rates.

Interconection is generally regulated worldwide because, under voluntary inter-

connection, a large dominante network may deny interconnection to smaller rivals or

charge them high termination rates. Network effects, possibly helped by economies

of scale, will cause the market to ber monopolized. Free contracting of interconnec-

tion terms, would give an advantage to the player with the most costumers, which

by refusing interconection, will become a monopolist and foreclose the market from

potential entry. Mandatory and regulated interconection is seen as the solution for

2

internalizing network effects and, at the same time, mantaining competition.

In practice, operators sign an interonection agreement specifying the terms of

exchange among them such as price, place of delivery of calls, capacity, reliability

parameters, among others. The ”‘interconection rate”’ or ”‘termination rate”’ is the

per-minute price an operator charges for completing calls within its network. The

Termination rate is perceived, both as a source of revenue by the operator receiving

the call, and as a cost by the operator sending the call. In this way, networks

see their revenue as coming from selling calls to costumers and selling termination

services to rivals.

The determination of the interconnection rate has been a contentious issue

in regulatory agencies ever since deregulation in the 1980s. The literature on the

subject has raised concerns that an inapropriate rate may be responsible for low

coverage, calling distortions, and collusion among operators. More specifically, the

literature has identified three efficiency effects:

(1)The rise-each-other-cost effect, which may be used as a commitment device

to facilitate price fixing in retail markets. Operators may agree to charge each other

a high interconection tariff to raise each others perceived costs and credibly commit

to keep high prices and low quantities in final markets. (2) the foreclosure effect,

according to which a dominant player sets a high rate to foreclosure small com-

petitors, restoring the natural monopoly tendencies of unregulated markets. Small

network will have an incentive to ration trafic to the large network, which makes

their service less atractive to costumers; and (3) a double marginalization effect from

a failure to cooperate. Each network unilaterally increases the interconection rate

3

in an attempt to maximize its margin over the competitors customers. Rates may

end up above the monopoly level.

Although all three innefciency efects may be present to some extent, it is

the second effect (foreclosure effect) that has received more attention by regulators

and competition agencies. A small network that is forced to pay high rates for

terminating calls in a large network, may respond by either charging high prices for

calls to the larger network, or charge the same prices and absorbe the losses. In the

first case, costumers may have and incentive to switch to the large network to save

on calls because more costumers are located there. In the second case, the loses may

force the small network to exit the market. The small firms may exit, or Potential

networks may avoid entry, after observing the low rate of proffitability of operating

under those conditions.

In practice, regulators may impose the tarif directly or indirectly via a mech-

anism that let operators reach an agreement by themselves. In the first case, the

regulator must estimate the right tarif and mandate its adoption. In the second

case, operators generally negotiate a rate and the regulator intervenes only to settle

disagreements after negotiations fail. The rate may be subject to ex-ante restricc-

tions such as reciprocity or uniformity (non-discriminatory among networks). It is

important to notice that, even when the regulator ”only” settles disagrements about

the rate, it is really providing a treath point to negotiators. Networks in a weak

position may chose not to accept the other operators proposal and take intead the

regulators rate. For this, the regulator needs also to provide a value for the efficient

level of the interconection rate.

4

In any case, the regulator needs to estimate an efficient rate, and a correct

theory of how that interconection rate ultimate produces the right trade offs in

several variables: quantity of calls, costumer choice of provider, entry of new players,

quality, patterns of calls, among others. These rate are different under CPP and

RPP regimes.

In prctice, some regulatory regimes assume that effcient interconecction rates

are equal to cost (cost-based regulation), which then is estimated by a cost model

such as the Total Element Long Run Incremental Cost (TELRIC). Other regulatory

regimes, take a rate of zero (bill-and-keep regulation), and let operators recover cost

from their costumers by charging both senders and receiver per minute rates for

calls.

Network effects are an important component of the overall value of the net-

works. When the percentage of potential costumers covered by all networks is less

than 100%, the optimal regulatory intervention must take that into account and,

posibly, choose a particular value of the interconection rate. The network effect

component of the interconection rate has been controversial among regulators,and

there is sustantial disagreement as to the convenience of using interconection rates

to cover ”universal service” goals, intead of other policy tools.

The effects of a certain interconnection rate depend on the set of pricing re-

striction adopted by the regulator for the final telephone market. The regime may

allow or prohibit: (1) Network-based price discrimination where On-net calls may

be priced differently than off-net calls; (2)Callers Pay Principle (CPP), where cus-

tomers pay only for calls made, not for calls received; or the Receivers Pay Principle

5

(RPP), where both callers and receivers pay.

The adoption of these pricing restrictions may have consecuences for efficency

and the determination of the interconecction rate. For instance, pricing based on

CPP gives raise to ”‘call externalities”’, defined as the benefit a receiver gets from

receving a call for which he/she does not pay a price. Similarly, network based

price discrimination may give rise to foreclosure of small operatos, trough predation

or price squeeze tactics, because large operators may charge too low prices for on

net calls and too high prices for off net calls. A concern with RPP may be that by

charging receivers, network coverage may be too low, which implies loses in potential

network effects.

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Chapter 1

Monopoly Model

1.1 Basic set up

A monopoly offers a two-way telephony service to heterogeneous customers

represented by a taste parameter θ ∈ [θ, θ], which are distributed according to c.d.f.

F (θ) and density f(θ). The monopolist knows the distribution but cannot directly

observe individual types. The cost structure of the monopoly is composed of two

parts: a fixed cost k of adding and servicing a new customer independently of

intensity of use; and a marginal per-call cost 2c, which includes the cost of sending

a call from the customer to the switching facility and then to the receiver (each step

with a cost of c).

Customer θ makes x(θ, θ′) calls to custumer θ′ and receives y(θ′, θ) calls from

costumer θ′. The total amount of calls made and received by costumer θ are respec-

tivelly:

X(θ) =∫ θθLx(θ, θ′)f(θ′)dθ′

Y (θ) =∫ θθLy(θ′, θ)f(θ′)dθ′

where θL is the lowest type of costumer that subscribes to the monopoly. Customers

types in the interval [θ, θL] do not subscribe, while those in the interval [θL, θ] do

subscribe. We define the rate of coverage of the market as n = 1− F (θL).

For a given level of coverage θL, the monopolist maximizes profits by offering

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each costumer θ a nonlinear price schedule, T (X(θ), Y (θ)), as a functions of total

calls made by that costumer. The choice of T() as a function of totals imply that

operators cannot set specific prices for individual sender-receiver pairs. In practice,

these means that users pay three rates: a fixed rate for being subscribed, a per-

minute rate for calling independent of the identity of the receiver, and a per-minute

rate for receiving calls independent of the identity of the sender. The monopolist

solves:

π(θL) = maxT (∗,∗)∫ θθL

[T (X(θ), Y (θ)) − cX(θ)]f(θ)dθ − nk where Then, the

monopolist optimazis for coverage θL.

Costumer θ gets gross utility from calling θ′ equal to θu(x(θ, θ′)), and gross

utility from being called by θ′ equal to α(θ)u(y(θ′, θ)). Here, α(θ) 6= 0 means that

receivers obtain benefits from being called; α(θ) = 0 means no benefit, while α(θ) =

θ, means sender and receiver enjoy the same benefits from a call. We assume that

the valuation of incoming calls is non-negative (no-nuisance calls), weakly higher for

higher types, and weakly convex, i.e. α(·) ≥ 0, α′(·)geq0, α′′(·) ≥ 0. With respect

to the utility function, we assume that u′(·) > 0, and u′′(·) < 0.

On the other side, each costumer maximizes his utility from sending and re-

ceiving calls minus payments to the monopolist:

v(θ) = maxx(θ,θ′),y(θ′,θ)

∫ θ

θL[θu(x(θ, θ′))f(θ′)dθ′ + α(θ)u(y(θ′, θ))]f(θ′)dθ − T (X(θ), Y (θ))

(1.1)

8

subject to:

x(θ, θ′) ≤ y(θ, θ′)

y(θ′, θ) ≤ x(θ′, θ)

The first restriction means that, in general, callers are willing to make a certain

number of calls, but receivers may not be willing to receive that many calls. Simi-

larly, the second restriccion means that receivers may be willing to receive a certain

amount of calls, but callers may not be willing to make that many calls. Senders

may be restricted in the number of call they can make by unwilling receivers, while

receivers may accept more calls than the sender is willing to make.

1.2 solution

For a given tarif structure and coverage level (thetaL), the costumer optimiza-

tion problem has the FOC’s:

θU ′(x(θ, θ′))− dTds

+ λθ′ = 0

and

α(θ)U ′(y(θ′, θ))− dTdr

+ γθ′ = 0

where λθ′ = 0 if x(θ, θ′) < y(θ, θ′) and γθ′ = 0 if x(θ, θ′) > y(θ, θ′).

while for a given θL, the monopolist problem gives FOCs:

9

1.3 The choice of coverage

Once the amounts of calls as a function of type is determined, the monopolist

has to choose the level of coverage, θL, or the miminum type of costumer who wants

subscribed to the network. Defining H(θ) as the surplus extracted by the monopolist

from customer θ (to be determined below), the monopoly solves:

maxθL

π(θL) =∫ θ

θL[H(θ)− ncx(θ)− k]f(θ)dθ (1.2)

If a local maximum exists, the FOC is given by:

H(θL)−∫ θ

θL

dH(θ)

dθL

f(θ)

f(θL)dθ = cs(θ) + cnx(θL) + k (1.3)

which says that the surplus extracted by the monopolist from adding the

marginal customer θL, plus the gain of surplus extracted from inframarginal cus-

tomers as a result of the new traffic,calls made from and to θL must equal the

additional cost.

A global optimum θL∗ generally exists for standard utility functions, taking

one of three forms: full coverage when θL = θ; partial coverage when θL ∈ (θ, θ); or

no coverage θL = θ. Figure 1.1 depicts the shapes for π(θL) in each case: C1, C2

and C3, respectively.

10

Figure 1.1:

1.4 CPP

We assume that the gross surplus obtained from sending a call is the same

regardless of the identity of the receiver. This means that senders demand an equal

number of minutes for each potential receiver.1. Similarly, customers receive the

same gross surplus from incoming calls regardless of the identity of the sender. ut,

in general, different amounts of calls are received from different senders. Following

Hahn (2003), this is expressed by the inclusion of a linear (in incoming calls) term

in the additive utility specification.

Under CPP, no user has an incentive to refuse calls and x(θ, θ′) is independent

of θ′, which means that every recipient θ′ ∈ [θL, θ] receives the same amount x(θ)

of calls from θ, so the restriction in 1.2 disapears and both variables are reduced to

y(θ, θ′) = x(θ, θ′) = x(θ).

Simplifying in the original problem we have the FOC’s:

u′(x(θ)) = dTds

substituting the optimal value x(θ) in 1.2 we have

v(θ) = nθu(x∗(θ)) + α(θ)∫ θθLu(x(θ′))f(θ′)dθ′ − T (s)

which from the envelope theorem

v′(θ) = nu(x∗(θ)) + α′(θ)∫ θθLu(x(θ′))f(θ′)dθ′

1This is equivalent to assuming that each costumers has an ex ante equal probability of calling

any other costumer

11

By the revelation principle, we can see the monopoly problem as one in which

the monopoly chooses directly the quantities x(θ) for all θ while guaranteeing a

surplus v(θ) to every individual θ ∈ [θL, θ]. As a borderline condition, it must be

the case that v(θL) = 0.

π(θL) = −∫ θθLv(θ)f(θ)dθ +

∫ θθL

[nθu(x(θ)) + α(θ)∫ θθLu(x(θ′))f(θ′)dθ′]f(θ)dθ −

∫ θθLcx(θ)f(θ)dθ

Eliminating v(θ) using integration by parts, we can write π(θL as:

π(θL) = maxθL π(θL) =∫ θθL

[ψ(θ)us + φ(θ)ur − cs− k]f(θ)dθ

where

ψ(θ) = [θ − 1−F (θ)f(θ)

]

φ(θ) = [[α(θ)− 1−F (θ)f(θ)

α′(θ)]

are the coeficients representing the proportion of the surplus extracted from user θ.

We assume that the hazard rate 1−F (θ)f(θ

, which implies ψ(θ), ψ(θ)/θ, φ(θ) and φ(θ)/θ

are Increasing for α().

We also define:

us = nu(x(θ))

ur =∫ θθLu(x(θ))f(θ)dθ

are θs gross utility of sent and received calls respectively.

The FOC’s are:

u′(x ∗ (θ))[ψ(θ) + 1n

∫ θθHφ(θ)f(θ′)dθ′] = c

The right side of the equation represents an ”‘adjustement”’ to marginalcost

made by the monopolist in order to partially internalize call externalities. It says

that surplus extraction must take into account the fact that receivers are benefited

12

from calls although only senders pay a marginal price. In particular, for the highest

type θ the marginal price is reduced by exactly the amount of gross surplus from

reception given to the lowest subscriber θL. Surplus extraction from reception of

calls is limited to a fixed amount for each costumer regardless of type, while surplus

extraction from sending calls is ψ(θ) varies across costumer.

Finally, the amount of surplus extracted by the monopolist from costumer θ

is: H(θ) = ψ(θ)us + φ(θ)ur

1.5 Benchmarks

The first best allocation of calls and subscription under observable θ is the

same for a social planner or a monopolist, considering the perfect discrimination

capacity for the latter as a result of knowing θ. In both cases, the quantity of calls

has to be determined based on:

[θ + 1n

∫ θθHα(θ)f(θ)dθ]u(x(θ)) = c

From this, the price of calls is going to be decreasing on θ

pe(θ) = θu′(x(θ)) = c

1+ 1θn

∫ θθLα(θ′)f(θ′)dθ′

From this, it is not surprising that, pm(θ) > pe(θ) for every θ. Under monopoly,

even the highest type makes an inefficient quantity of calls. Thus the efficiency at

the top principle of the standard screening model is not veryfied.

The possibility of subscribers who do not make any call and only receive calls is

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present depending on the pricing scheme. The opposite is also possible. Subscribers

who only make calls but do not receive.

1.5.1 example

Here we assume that costumer are in the interval [0, 1] and have unit demand,

so that U(0) = 0 and u(x) = 1 for x ≥ 1. Assume also that α(θ) = aθ, with a ≤ 1 .

Under CPP, whenever a positive coverage is profitable, the monopolist chooses

θL to equate marginal revenue to marginal cost. Costumer θ ≥ θL receives gross

surplus equal to nθ(1 + a) from sending and receiving calls to each of the n other

subscribers. In this example, scheening of costumers by the monopolists is reduced

to separating subscribers from no subscribers. Therefore the mononopolist charges

all subscribers the same subscription fee T = nθL(1 + a) equal to the surplus of

the marginal type, θL. The monopolist maximizes the total amount collected minus

costs:

π(θL) = nT − n2c− nk

solving this problem produces

θL = 1−(1 + a− c) +

√((1 + a− c)2 − 3(1 + a)k)

3(1 + a)

Unsurprisingly, the monopolist restsricts overal participation and quantity of

calls made for each θ. The maximum coverage achievable when k = c = 0 is equal

to n = 2/3 or θL = 1/3. In contrast, in the socially optimal allocation when an

interior solution exists, the marginal costumer is given by:

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θL = 1−(1 + a− c) +

√((1 + a− c)2 − (3/2)(1 + a)k)

(3/2)(1 + a)

which is always lower than in the monopolist case(higher coverage). For c

and k small enough full coverage (θL = 0) is the welfare maximizing choice. This

however requires pricing equal to zero, which could be implemented if financing

through non-distortionary public funds were available. If not, the Ramsey pricing

level θL defined as π(θR) = 0 may be the social second best:

θL = 1−(1 + a− c) +

√((1 + a− c)2 − 4(1 + a)k)

(2)(1 + a)

Figure shows the shape of the profit functin and welfare as a functon of θL.

Point A is the choice of the monopolist, while point B is the choice under Ramsey

pricing. Notice that the level of coverage that maximizes the social walfare is closer

to full coverage.

With the posibility of charging for receiving, the monopolist has two instru-

ment to screen costumers: price for sending and price for receiving calls. This means

that pricing may separate subscriners in three groups:2 those in the interval [0, θL]

are no subscribers, those in [θL, θL] are call receivers only, while those in [θL, 1] sand

and receive calls.

To implement this separation, the monopolist chooses two tariffs: TL equal to

2for some combinations of k, c and a RPP is the same as CPP, because the monopolist may

not find proffitable to have no call-making subscribers. The specific condition is XX PONER

CONDICION. Under continuos utility function the no call making costumers becamo all costumers

below the θ

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(θl)(1 − θh), the gross utility of costumer θL; and TL the gross utility of θL. PL,

which is obtained by equating the gross surplus of L from belonging to one or the

other interval:

aθL(1− θL)− TL = θL(1− θL)− θL(1− θL)

This reduces to: TL = θL(1− θL) + aθL(1− θL).

Coverage is determined by the solution of:

maxθL,θL π(θL, θL) = (1− θL)TL + (θL + θL)TL − (1− θL)(1− θL)c− (1− θL)k

With two instruments to screen costumers, the monopolist unambiguosly ex-

tracts more rents. However, coverage and monopoly proffits are unambiguosly larger,

but costumers, welfare may be larger or smaller.

In the RPP case the optimal choice of coverage implies the existence of call-

receiving only costumers. pricing according to RPP increases screening capacity

and profit extraction for the monopolist, but also coverage and social welfare.

1.6 RPP

1.6.1 The Model

We extend the previous model by giving the monopoly the possibility of charg-

ing for call reception. As I mentioned above, when receivers cannot refuse or limit

the duration of incoming calls, charging per incoming calls can be accomplished

trough the the fixed part of the tariff which was already available to the monopolist

when only senders pay.

The interesting case arises when receivers can refuse or limit the duration

16

of calls. In such case the monopoly will have an additional instrument to screen

customers and a different possibly welfare improving pattern of calls and subscription

rate is possible. Under certain circunstances, allowing reception pricing may be

irrelevant because the monopoly would not choose it. It is also possible that, even

if chosen, it may not be welfare improving because by increasing the monopoly’s

ability to screen customers, the deadweight loss of monopoly may increase.

As before, customers are offered the nonlinear tariff T (s, r) where s and r are

the total number of sent and received calls by individuals θ. The firm cannot charge

different prices for calls to different receivers and senders (receivers) are indifferent

as to the identity of the receiver (sender) of the calls. For receivers who do not refuse

or limit incoming calls because reception charges are small, senders will choose the

same number of calls x(θ) to each of those recipients. If receiver θ′ refuses calls or

limits its number then sender θ sends x(θ, θ′) < y(θ, θ′) number of calls to receiver

θ′. The limit on reception y(θ, θ′) = y(θ′) set by θ′ is uniform across senders due to

the indifference over the identity of the sender.

In the first case x(θ, θ′) = x(θ) and in the second case x(θ, θ′) varies with θ′.

This is equivalent to x(θ, θ′) having the form:

x(θ, θ′) =

x(θ) if θ is not restricted by θ′

y(θ′) if θ is restricted by θ′

In this way if individual θ sends x(θ, θ′) to θ′ and receives x(θ′, θ) from θ′ then

his/her utility is given by:

U(x(θ, θ′), x(θ′, θ)) =∫ θθL

[θu(x(θ, θ′)) + α(θ)u(x(θ′, θ)]f(θ′)dθ′ − T (s, r)

where s =∫ θθLx(θ, θ′)f(θ′)dθ′ and r =

∫ θθLx(θ′, θ)f(θ′)dθ′

17

The amount of sent calls x(θ, θ′) will be determined by either the sender or

the receiver in the following way: The monopoly may choose marginal prices for

sent and received calls as to give either the sender or the receiver the choice over

determining x(θ, θ′). If the sending price p(θ) is high and the receiving price r(θ) is

low, sender θ will be the one who decides x(θ, θ′). If r(θ′) is high and p(θ) low, then

receiver θ′ will choose to limit the number of received calls thus choosing x(θ, θ′).

Given that receivers get the same utility for calls regardless of the identity of

the sender but high type senders prefer to send a higher number of calls, we can

divide the space (θ, θ′) in two regions separated by curve l(θ) (see figure 1.2). Region

A is formed by sender-receiver pairs in which sender s is restricted in the number

of calls he/she makes to receiver r. Region B is formed by sender-receiver pairs,

inwhichenders choose the amount of calls and receivers accept all of them. The

possibility of no call making subscribers, no call receiving subscribers and receivers

who make all the calls they wish or receive all the calls sent are possible for different

forms l(θ) may take.

For instance, θ1 such that l(θ1) = θL is the lowest type of caller that is restricted

by receivers. It is possible that for some type of customer below θ0, x(θ) be reduced

to zero (no call making customer) who, noneteless may be subscribed because of

call reception.

On the receiver side, θ1 is the marginal type of the restricting receivers. θ > θ1

are customers who receive any call they are sent. The curve separating A from B is

characterized by the equality y(θ′) = x(θ). Here sender θ and receiver θ′ agree on

the amount of calls they want to make or receive respectively. So while customers in

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l(2)

C A

B

2L

2L

Figure 1.2: improve graph

[θ, θ0] are not restricted by receivers, customers in [θ1, θ] do not restrict any receiver.

1.6.2 The customers problem

Customers θ solves the following maximization problem taking y(θ′) as given:

maxx(θ),y(θ)

θUS(θ) + α(θ)UR(θ)− T (s, r) (1.4)

where

US(x(θ), y(θ)) = ∫ l(θ)θL

u(y(θ′))f(θ′)dθ′ + u(x(θ))∫ θl(θ) f(θ′)dθ′ for θ ∈ [θL, θ]

19

UR(x(θ), y(θ)) =

∫ l−1(θ)θL

u(x(θ′))f(θ′)dθ′ + u(y(θ))∫ θl−1(θ) f(θ′)dθ′ for θ ∈ [θ0, θ1]

u(y(θ))∫ θθLf(θ′)dθ′ for θ ∈ [θL, θ0]

The FOC’s for the customer maximization problem are:

x(θ) : θ dUsdx(θ)

= dT (s,r)dx(θ

y(θ) : α(θ) dUrdy(θ)

= dT (s,r)dy(θ

or simply:

x(θ) : θu′(x∗(θ)) = dTds

y(θ) : α(θ)u(y∗(θ)) = dTdr

Substituting x∗(θ) and y∗(θ) in 1.7 and derivating using the envelope theorem.

We obtain the consumer surplus function.

v′(θ) = uθ + α′(θ)uθ

where

uθ =∫ θθLθu(x∗(θ, θ′)f(θ′)dθ′

and

uθ =∫ θθLx(θ′, θ)f(θ′)dθ′

For given θ0 θL and θ1 the monopoly choose x(θ), y(θ) to solve

πRPP (θL) = maxx(θ),y(θ)

∫ θθL

[ψ(θ)us + φ(θ)ur − cs]f(θ)dθ

where c and s are as defined above and

ur = Ur(x∗(θ), y∗(θ))

and

us = Us(x∗(θ), y∗(θ))

from which we obtain the marginal conditions for x(θ) and y(θ).

20

ψ(θ)u′(x(θ)) + u′(x(θ))α(l(θ)) = c for θ ∈ [θL, θ]

u′(y(θ))l−1(θ) + φ(θ)u′(y(θ)) = c for θ ∈ [θ0, θ1]

u′(y(θ))θL + φ(θ)u′(y(θ)) = c for θ ∈ [θL, θ0]

For the remaining of this section we assume u(θ) = u(θ), that is, the asymetry in

preferences for a customer acting as sender and as receiver is captured by α(θ).

by taking the simpler case u(x) = u(x), the FOC are:

u′(x(θ))[ψ(θ) + α(l(θ))] = c for θ ∈ [θL, θ]

u′(y(θ))[l−1(θ) + φ(θ)] = c for θ ∈ [θ0, θ1]

u′(y(θ))[θL + φ(θ)] = c for θ ∈ [θL, θ0]

We have a system of two equa-

tions in x(θ) and y(θ). To solve it, let’s ”‘invert”’ equation the second equation by

substituting θ for l(θ) we get

u′(x(θ))[θ + φ(l(θ))] = c for θ ∈ [θL, θ]

By comparing with the first equation, we conclude that l(θ) must be the

solution to

θ + φ(l(θ)) = ψ(θ) + α(l(θ))

and also that θ1 must be equal to θ as implied by the definition l(θ) = θ1 in the

preceding equation. Furthermore, the function l(θ) = y−1(x(θ)), being the compo-

sition of two increasing functions , is also increasing. Figure 1.3 shows the shape of

l(θ) in the (θ, θ′) space.

For pairs of customers in A and C, senders determine the amount of calls. for

customers in B, receivers decide. In particular, for pairs in C receivers never limit

incoming calls.

Lets take u(x) = ln(1 +x), θ uniform in [1, 2] and α(θ) = a+ bθ. We calculate

21

l(2)

C A

B

2L

2L

Figure 1.3: improve graph

l(θ):

l(θ) = (2b+ 1− θ)/b

which satisfy l(2) = 2

Substituting l(θ) back in our three FOC’s we obtain:

x∗(θ) = 3θ−4+a+2bc

− 1

y∗(θ) =

3θb−4b+a+2

c− 1 for θ ∈ [θ0, θ]

θL+2θb−2b+ac

− 1 CHECK THIS for θ ∈ [θL, θ0]

x∗(θ) and y∗(θ) are linear and increasing everywheere in [1, 2] which is necessary

for the global ICC. Moreover, marginal prices are decreasing: p(θ) = θu′(x∗(θ)) and

r(θ) = (a+ bθ)u′(y∗(θ)).

22

1.6.3 The Choice of θL

We have seen how quantities of sent and received calls are determined and

hence the pattern of calls. This pattern is conditional on the coverage level delimited

by the marginal participating customer θL. The profit function for both the CPP

and RPP cases after substituting the x∗(θ) and y∗(θ), respectively are given by:

πRPP (θL) =∫ θθL

[ψ(θ)u∗s + φ(θ)u∗r − cs∗]f(θ)dθ

where u∗s, u∗r and s∗ are us, ur and s respectively evaluated at the optimal choice of

x(θ) and y(θ).

Figure 1.4: improve graph

The following proposition shows the divergence in preferred pricing policies for

the monopoly and the regulator. Basically, the regulator increases social welfare by

imposing CPP over the monopoly’s preferred RPP policy. The regulator’s preferred

23

policy results in a lower coverage level.

Proposition 1 Under symmetric preferences for sending and receiving calls, α(θ) =

θ and u(x) = u(x):

-The monopolist prefers RPP over CPP.

-The social planner prefers CPP over RPP.

-The level of coverage under RPP is below the lever under CPP.

Figure 1.4 shows the profit function for our example. Although profits under

both pricing policies appear to be very similar, the calling patterns are radically

different. Under RPP, customer θ sends and receives equal amounts of calls x(θ) =

y(θ). Under CPP, both high and low type customers send a higher amount of

calls than under CPP but low types receive more under CPP than under RPP.

The marginal customer θL sends the same amount of calls under both regimes but

receives more under CPP.

Proposition 3 assumes identical tastes for send and receive calls. We can argue

that a customer θ gets more (less) surplus as sender than as receiver. In fact, the

previous result is valid as long as the derivative of α(θ) be equal to 1. That is,

α(θ) = θ + c if θ + c is positive.

Under a general function α(θ), the choice of CPP vs RPP will depend only

on α′(θ) because the function l(θ) depends only on the derivative of α(θ) reflecting

the fact that nonlinear pricing tends to extract any constant surplus received by all

customers under both regimes.

In our example, figure 1.5 depicts the profit function for both CPP and RPP

24

Figure 1.5: improve graph

policies assuming α′(θ) < 1. In this case, the monopoly chooses RPP and a higher

level of coverage compared to CPP and it is socially preferable to do so. On the

other hand, figure 1.6 depicts the case for α′(θ) > 1. Here the opposite happens.

CPP is superior both for the monopoly and society. Coverage is greater under CPP

than under RPP.

Although the choice of pricing policy depends only on the derivative of α(θ),

coverage and the calling pattern are influenced by the the ”‘fixed part”’ of α(θ).

25

Figure 1.6: improve graph

1.6.4 IC Constraints

1.6.5 Second degree IC constraint

1.6.6 No Call Making/Receiving Customers

1.6.7 Benchmarks: θ,θ′ observable

The efficient price allocation under first degree price discrimination satisfies:

θu′(x(θ)) + 1n)

∫ θθLα(θ′)u′(x(θ′))f(θ′)dθ′ = c

or in terms of prices

pe(θ) + 1n

∫ θθLre(θ′)f(θ′)dθ′ = c

26

which is the marginal price of sending a call plus the average of the marginal

prices for reception of all the subscribers.

The solution of this integral equation is technically complex. but, for instance,

in the case α(θ) = θ a solution is p(θ) = r(θ) = c/2. Senders and receivers share

half of the costs of a call. by comparing with eq(**), this coincides with the RPP

allocation. We have then efficiency at the top for the symetric case. In general:

Proposition 2 The monopoly allocation under RPP does not satisfy the efficiency

at the top property of the standard screening model.

By comparing eq ** and eq** at θ = θ we have

θu′(xe(θ)) + 1n)

∫ θθLα(θ′)u′(xe(θ′))f(θ′)dθ′ = c

θu′(xrpp(θ)) + α(θ)u′(xrpp(θ)) = c

from which we conclude that xrpp(θ) < xe(θ).

27

Chapter 2

Competition with CPP

2.1 The Model

There are two competing firms selling vertically differentiated two-way telecom-

munications services. The cost structure of firm i, i = 1, 2 can be decomposed in

the following parts: a fixed cost ki representing the cost of adding and servicing

a new customer regardless of intensity of use and a marginal one-way cost ci per

standarized call which represents the transit cost of a call from a customer location

to the switching facility, the marginal cost of a call to a customer in network j is

thus ci + cj. in particular, on-net calls cost 2ci.

Customers derive utility from both making and receiving calls. We assume that

the gross surplus obtained from sending a call is the same regardless of the identity

of the receiver. This means that, from the point of view of a sender, receivers who

are identically priced are perfect substitutes and therefore each sender distributes

his/her calls uniformly across receivers unless calls are priced differently for different

receivers. Similarly, customers receive the same gross surplus from an incoming call

regardless of the identity of the sender but, in general, different amounts of calls

are received from different senders. Following Hahn (2003), this is expressed by the

inclusion of a linear (in incoming calls) term in the additive utility specification.

Let ni be the size of network i and xij the number of calls a subscriber of

28

network i makes to each subscriber of network j. The total number of calls a

subscriber of network i makes is then n1xi1 + n2xi2. Calls are priced according to

the nonlinear tarif Ti(n1xi1, n2xi2) where n1xi1 and n2xi2 are the total number of calls

made by a subscriber of network i to subscribers of networks 1 and 2 respectively.

As in standard models of adverse selection, potential subscribers are repre-

sented as a continuum of types θ in an interval [θ, θ] and are distributed according

to F (θ) with density f(θ). The distribution of types is public information but firms

cannot directly observe individual types. The intensity of a customers’ preferences

for telephone calls is represented by his/her θ.

A customer of type θ who is subscribed to network i of quality qi gets an

utility from sending xij calls to a customer in j equal to θqiu(xij) while he/she gets

α(θ)qiu(xji) from receiving xij from customer of network j.

The gross utility of customer θ in network i is thus

U(xi1, xi2; θ) = θqi[n1u(xi1)+n2u(xi2)]+qiα(θ)[∫ θ

θHu(x1i(θ)f(θ)dθ+

∫ θH

θLu(x2i(θ))f(θ)dθ]

(2.1)

Here α(θ) 6= θ expresses the possibility of a customer valuing calls differently

when acting as a sender and as a receiver. Quality qi is the same across both types of

calls reflecting the assumption that quality is interpreted as mobility. For instance,

qi can be high (mobility) or low (fixed line). A custumer with a mobile phone gets

the same high quality for sent and receive calls. In contrast, a fixed line phone gets

both types of call of low quality.

29

Notice that∫ θθHu(x1i)(θ)f(θ)dθ+

∫ θHθL

u(x2i(θ))f(θ)dθ is independent of θ given

that all subscribers receive the same number of calls regardless of type and that the

utility of received calls is an externality because the sender has no control over the

variable xji(θ′) except in the case θ′ = θ and j = i which is a set of measure zero. We

assume that the valuation of incoming calls is positive (no-nuisance calls), higher

for customers with higher values of θ and weakly convex, i.e. α(·) > 0, α′(·) > 0,

α′′(·) ≥ 0. With respect to the utility function, we assume u′(·) > 0, u′′(·) < 0.

In this specification, n1 and n2 reflect the effect of network externalities. Larger

total coverage n1 + n2 benefits customers already suscribed. Additionaly, in the

presence of tarif-mediated network externalities, the size of the own network matters.

When on-net calls are cheaper than off-net calls, a subscriber benefits if his/her

network is bigger. If off net calls are cheaper subscribers benefit if the other network

is bigger.

2.1.1 The timing of the game

Firms, customers and (possibly) the regulator play a three stage game with

the following timing

Stage 1: Firms or regulator choose interconnection rates a1,a2.

Stage 2: Firms offer nonlinear tarif T (s1, s2) as a function of number of on-net and

off-net calls.

Stage 3: Customers make a subscription decision by choosing Firm 1, Firm 2, or

the outside option (no participation). They also decide the amounts xij(θ) of

30

calls to purchase.

At stage 2, tarifs cannot be a function of particular pairs of customers other

than the on-net/off-net distinction. Calling a customer of higher type is priced the

same as calling a lower type as long as both receivers are on the same network. This

may be justified because, in thios setting, customers value calls the same regardless

of receiver. So, if firms offer customer θ a menu of tariffs p(θ, θ′) ∀ θ′ receiver with

price possibly different for different receivers. Because customers do not know ex-

ante when they are going to call once subscribers to a network they decide on the

based on etc etc 9THIS WAS MISLEADING AS ALREADY SEEN) and therefore,

firms have no ex-ante information about pairs of customers when they choose the

tarif.

Firms and consumers base their pricing, subscription and call consumption de-

cisions on their expectations of network sizes n1, n2. We assume that, at equilibrium,

their expectations are fulfilled.

The fact that the networks have different qualities is interpreted, not as quality

of audio or reliability, but as the degree of mobility. In this sense, a wireline phone

can make and receive the same quality calls but provide a low quality service because

of restrictions on mobility.

The existence of network externalities creates a coordination problem because

consumers may have different beliefs as to the behavior of the other consumers and

the networks. Consumers form expectations about the size of each network ne1 and

ne2 and make subscription/quantity decisions based on those expected sizes. As

31

pointed out by Katz and Shapiro (1985), multiple equilibria are posible by allowing

different expectations of network sizes. Here, we simplify the analysis by assuming

a rational expectation equilibrium. That is, consumers’ expectations about the size

of both networks are the same and expectations are fulfilled in equilibrium based

on the corresponding consumption/subscription choices: ne1 = n1 and ne2 = n2.

Given subscription rates n1, n2, 0 < n1, n2 < 1 and n1 + n2 ≤ 1 and tarif

T (·, ·) a type-θ customer subscribed to network i chooses outgoing-call quantities by

solving:

V (θ) = maxx11,x12

[U(xi1, xi2; θ)− Ti(n1xi1, n2xi2)] (2.2)

Asume that q1 > q2 so that network 1 is the high quality network. Under

the appropriate assumptions on f(θ), we can focus on the situation in which higher

types adopt network 1 and lower types adopt network 2 which means that there

are two cut-off values θL and θH such that individuals in Ai = [θH , θ] subscribe to

network 1 and those with values in A2 = [θL , θH ] subscribe to network 2 while

individuals in [θ, θL] do not subscribe to any network.

Type θH individuals are indiferent between subcribing to networks 1 or 2

while type θL individuals are indifferent between subscribing to network L and not

subscribing at all.

V1(θH) = V2(θH) (2.3)

V2(θL) = 0 (2.4)

32

Throughout the paper and for ease of notation, I will refer intercheangeably to

cut-off types θH and θL or market shares n1 and n2. Market shares are defined as:

n1 = 1− F (θH) (2.5)

n2 = F (θH)− F (θL) (2.6)

Intead of choosing Ti(n1xi1,n2xi2) firm i choose xi1(θ) and xi2(θ) for each θ

and then calculate Ti(·, ·) by using the FOC’s

θqin1u′(xi1(θ)) =

dTidxi1

i = 1, 2. (2.7)

θqin2u′(xi2(θ)) =

dTidxi2

i = 1, 2. (2.8)

We can then reduce the problem to one in which firm i calculates xi1(θ) and xi2(θ)

for each θ and then recovers Ti(·, ·) from the above FOC’s.

For given n1, n2, firm 1 solve

Π1(n1, n2) = maxx11x12

∫ θ

θHT1(n1x11, n2x12)−2c1n1x11−n2(c1+a2)x12+[a1−c1]x21−k1}f(θ)dθ

(2.9)

and similarly firm 2 solve:

Π2(n1, n2) = maxx21x22

∫ θH

θLT2(n1x21, n2x22)−2c2n2x22−n1(c2+a1)x21+[a2−c2]x12−k2}f(θ)dθ

(2.10)

xij =∫ θHθL

x2i(θ)f(θ)dθ

T (·, ·) is obtained from the indirect utility function 1.4

33

Ti(n1x11, n2x22) = U(xi1, xi2, θ, n1, n2, )− Vi(θ) (2.11)

Substituting Ti(·, ·) into 1.11 and integrating by parts we obtain:

Π1(n1, n2) = maxx11x12

−n1V2(θH) +∫ θ

θH[n1φ11(x11) + n2φ12(x12)]f(θ)d(θ) +

∫ θH

θLn1φ21f(θ)dθ − n1k1

(2.12)

where:

φ11(x11) = [ψ(θ) + α(θH)]q1u(x11)− 2c1x11 (2.13)

φ12(x12) = ψ(θ)q1u(x12)− (a2 + c1)x12 (2.14)

φ21(x21) = [α(θH)q1u(x21(θ)) + (a1 − c1)x21(θ) (2.15)

Here, V1(θH) is the across-the-board surplus firm 1 has to give each of its

customers. That is, every customer of network 1 has to have at least the surplus

of its lowest type. ψ(θ) = θ − 1−F (θ)f(θ)

is the coefficient representing the fraction

of consumer θ’s baseline surplus from sending calls captured by the firm. In other

words, the firm captures ψ(θ)/θu(xij(θ)) from customer θ in iwho sends xij(θ) callsto

any recipient in j. Consequently, φij(θ) is the per-capita surplus extracted from

customer θ calling from i to j net of marginal costs.

The next-to-last term in expression 1.14 is composed of the net gain made

by network 2 for each incoming call (a1 − c1) and the surplus extracted from the

receiving customers α(θH) multiplied by the number of incoming calls.

In the case of on-net calls, the specification of utility used implies that the firm

captures a fraction of consumer’s surplus from outgoing and incoming calls equal to

34

ψ(θ)+α(θH). Surplus from outgoing calls is captured at a different rate for different

customers while surplus from incoming calls is captured at a constant rate equal to

the incoming call surplus of the lowest type.

For off-net calls, the monopoly only captures surplus from calls sent while sur-

plus from reception by the other network custumers can only be captured indirectly

via the interconnexion rate or by modifying θH through market stealing.

Firm 2’s proffits are given by:

Π2(n1, n2) = maxx21x22{n1V2(θH)−(n1+n2)V2(θL)+∫ θHθL

[n2φ22(x22)+n1φ21(x21]f(θ)

+∫ θ

θHn2φ12f(θ)d(θ)− n2k2} (2.16)

φ21(x21) = ψ(θ)q2u(x21)− (a1 + c2)x21 (2.17)

φ22(x22) = [ψ(θ) + η2]q2u(x22)− 2c2x22 (2.18)

φ12(x12) = q2η2u(x12) + (a2 − c2)x12 (2.19)

2.2 solution

The firs order conditions of the preceding maximization problems are:

x11 : [ψ(θ) + α(θH)]q1u′(x11) = 2c1 (2.20)

x12 : ψ(θ)q1u′(x12) = c1 + a2 (2.21)

x21 : ψ(θ)q2u′(x21) = c2 + a1 (2.22)

x22 : [ψ(θ) + η2]q2u′(x22) = 2c2 (2.23)

35

where: η2 = α(θL)− n1/n2[α(θH)− α(θL)]

As in the standard model of adverse selection, 1−F (θ)f(θ)

is assumed to be non-

negative and decreasing, which makes φ(θ) increasing.

Depending on the values of θ, θH and θL, the FOC’s may have cut-off points

θij in which either 2c1ψ(θHH)+α(θH)

= q1u′(0) or 2c2

ψ(θLL)+η2= q2u

′(0) or ci+ajψ(θij)

= qiu′(0).

In each case, the corresponding xij(θ) will be zero for types θ ≤ θij. This represents

the possibility that low types may choose not to make a certain type of call. For

instance if θ in network L is such that θ < θLH , then He/She will make xLH(θ) = 0

calls to network H. We can even have no vall making subscribers in which xLH(θ) =

xLL(θ) = 0 which are subscribed merely for call reception.

Additionaly, we assume the necessary conditions to guarantee the satisfaction

of the IC constraint.

Efficiency in the amount of on-net calls requires that we take into account the

effect of call and network externalities. For θH and θL given:

θq1u′(xij) + qiu

′(xij(θ))∫s∈Aj

sf(s)ds = ci + cj

or

θq1u′(xij =

ci + cj

1 + E(s|s∈Aj)θ

which comparing to the FOC above for xij(θ) above, implies that on-net calling

is inneficiently low for every customer. There is no efficiency at the top as in the

standard adverse selection model without externalities.

Proposition 3 Inneficiency of call allocation. Given θH and θL subscribers con-

36

sume suboptimal quantities of on-net calls.

Proposition 1 implies that network externalities working through θH , θL will

play a role in reducing this inefficiency. Additionaly, efficiency in the amount of off-

net calls will generaly depend on the interconnection charge, which is determined

at an earlier stage.

2.3 No call making subscribers

Before continuing to interconexion and coverage issues, we have to point out

that some customers may be subscribers but make zero calls at least to some sub-

scribers. There are three possibilities, zero on-net calls, zero off-net calls or zero of

both (no-call-making subscribers).

We assume that no-call-making subscribers are present only in the low quality

network. That is, the high quality network has all its customers making a positive

number of calls in equilibrium. To delimit the extent of no call making subscribers

we assume two cut off points θLL and θLH ∈ [θL, θH ] such that customers in [θL, θLL]

make zero off-net calls (xLL(θ) = 0) and customers in [θL, θLH ] make zero off-net

calls (xLH(θ) = 0) and as a consequence customers in [θL,min(θLL, θLH)] do not

make any call whatsoever.

Alhtough it is in principle posible that some customers above θH subscribe to

network 1 and choose to make zero number of calls on or off-net, we are going to

assume away this case based on monotonicity of call making with respect to type. 1

1the conditions for this will be discused in an apendix

37

The values θLL and θLH are defined by:

[ψ(θLL) + η2]]q2u′(0) = 2c2 (2.24)

and

[ψ(θLH)]q2u′(0) = a1 − c2 (2.25)

No call making subscribers exist only if θLL or θLH are above θL. Whenever

no call making subscribers are discussed we will assume θL < θLL < θLH . That is,

the lowest type customers make zero calls to either network then higher types make

only on-net calls while those above θLH make both types of calls. This condition

is equivalent to on-net calls being cheaper so that lower types drop off-net calling

before droping on-net calling.

Customers in [θL, θLL] make zero calls of both types and therefore must be

required to pay a fixed subscription fee equal to the utility of the lowest type θL.

That is

α(θL)x where x is the sum of calls received from every sender:

x =∫ θθHx∗HL(θ)f(θ)dθ +

∫ θHθLL

x∗LL(θ)f(θ)dθ

customers in [θLL, θLH ] additionaly pay according to the amount of off-net calls sent.

In this way, firm 2 obtains profits from both types of calls:

π2(n1, n2) =∫ θHθLL

n2φ22(x∗LL(θ))f(θ)dθ +∫ θHθLH

n1φ21(x∗LH(θ))f(θ)dθ − n2k

38

Similarly, Firm 1 obtains profits:

π1(n1, n2) =∫ θθHn1φ11(x∗HH(θ))f(θ)dθ +

∫ θθHn2φ21(x∗LH(θ))f(θ)dθ − n1k

2.4 The Choice of xij(θ)

Before tackling the choice of coverage, let’s see how the monopoly would choose

its pricing schedule under 3 different cases:

Case 1: Knowledge of both both sender and receiver types (θ, θ′ known)

Case 2: Knowledge of only senders type.

Case 3: Knowledge of neither sender or receiver’s type.

In the first case the monopolist can observe pairs of sender-receiver (θ, θ′) and

price calls individually. In the second case, the monopolist can observe senders types

but cannot price discriminate according to receivers of a call. The tarif T (s, θ) is

a function of types and total number of calls sent only. This implies that although

high θ′s will call more, they will make no distintion as to revceiver of the same

network and will end up calling each one of them the same amount. Receivers will

end up receiving the same amount of total calls regardless of type.

In the second case, the monopolist charges tarifs T (s, θ) under CPP and

T (s, r, θ) under RPP.

In the third case, knowledge of sender and receivers is not assumed. Tariffs

are: T (s) and T (s, r) for CPP and RPP respectively

39

In the second and third cases the total amount of calls received is the same

across types under CPP, while different types generally receive different amounts of

calls under RPP.

2.5 Case 1: Fully Efficient Allocation under CPP

The monopolist chooses xij(θ, θ′) such that MU = MC as follows

θqiu′(xij(θ, θ

′)) + α(θ)qju′(xij(θ, θ

′)) = ci + cj (2.26)

This can be replicated by charging only outgoing calls at:

pij(θ, θ′) =

ci + cj

1 + cjα(θ′)ciθ

(2.27)

which for θ = θ′, i = j and α(θ) = θ we have p = c, price is equal to half the

marginal cost.

[qiθi + qjθj]u′(xij(θiθj) = ci + cj (2.28)

In an efficient assignment of customers to firms, the high type firm will serve cus-

tomers in [θH , θ] while the low type firm serve customers in [θL, θH ]

2.6 Case 1: Fully Efficient Allocation under RPP

Because of the full information assumption, charging for receiving calls does

not change the allocation xij(θ, θ′) of calls. In general we can choose prices:

p(θ, θ′) = θqiu′(xij(θ, θ

′)) (2.29)

40

r(θ, θ′) ≤ θqju′(xij(θ, θ

′)) (2.30)

or

p(θ, θ′) ≤ θqiu′(xij(θ, θ

′)) (2.31)

r(θ, θ′) = θqju′(xij(θ, θ

′)) (2.32)

For all p(θ, θ′) that satisfy the condition. So, we can support the efficient

allocation with multiple pricing strategy

2.7 Efficient allocation when charges depend only on total number

of calls sent (CPP second case)

The previous section assumed that the planer was able to identify and set a

tarif for each sender-receiver pair. In this section we assume that the palner knows

the type of each customer but cannot price discriminate for callssent to different

receivers other than the on/off-net disctintion. This can be explained as the result

of regulation or information costs or asymetries. For instance, the fact that new

customers do not know who are they going to call after subscribing makes them

expect to be given simple tarifs sucha as the marginal tarifs for on-net and off-net

calls. Otherwise a tarif contingent on the receivers type may be too complicated

or create the incentive for firms to cheat customers. This is a realistic assumption

given that cellphone firms price discriminate more usurd on sending than receiviong

of particular calls.

This is equivalent to assume that T (s1, s2) is a function of total amount of

sent to customers on-net and off-net.

41

Now under this restriction, the planer determines xij(θ) such that

θqiu′(xij(θ)) + qj

1

nju′(xij(θ))E(α(s)|s ∈ Aj) = ci + cj (2.33)

Where Aj is either the interval [θL, θH ] for j = 1 or [θH , θ] for j = H. Equations

(**) reflects the fact that marginal utility of sending a call plus the expected marginal

utility of receiving it across all receivers equals cost.

Notice that pij(θ) = qiu′(xij(θ)) is increasing in θ but u′(xij(θ)) is decresing in

θ so xij is increasing

pij(θ) =ci + cj

1 + qjqi

E(α(s)|s∈Aj)θ

(2.34)

From equation ** we can see that, in the case α(θ) = θ (symetry between

senders and receiver utilities) on-net calls are marginally priced at

pii =2ci

1 + E(s|s∈Ai)θ

(2.35)

while off-net calls are priced according to

pij(θ) =ci + cj

1 + qjqi

E(s|s∈Aj)θ

(2.36)

Prices for on-net calls are high for high types and low for low types. There is

efficiency at θ = E(s|s ∈ Aj) at which point pii(θ) = ci but inneficiently everywhere

else. For off-net calls

The efficient amount of calls sender θ makes under unobservable recipients is

equal to the amount of calls θ himself will send to an ”‘average”’ recipient in each

42

network under full observability. Such recipients θj, for j = H,L. are defined by by

E(α(s)|s ∈ Aj) = α(θj) (2.37)

This implies a pattern of too few calls to high types and too many for low

types compared to the full observability case.

Comparing case 1 and case 2, we can see that, while marginal price in general

depends on θ and θ′ under case 1 it is constant w.r.t. θ′ under case 2. Graph **

shows that, under case 2, customer θ in A1 pay the price he/she would have to pay

under case 1. Customers θ makes too many calls to low type receivers (θ′ low) and

too few calls to h igh type receivers (θ′ high).

The receivers θH and θL are such that any sender calling them are calling an

efficient amount as in the full information case regardless of the on-net/off-net origin

of the call. Although θH and θL are constant w.r.t. θ, they are increasing in θH and

θL respectively.

In the special case α(θ) = aθ + b both lines intersect at θ = E(s|s ∈ A1) for

every a. A smaller a (weaker call externalities) moves both curves upwards but leaves

θ∗ unafected so that for a tends to 0 (uniform call externalities) lack of knowledge of

the recepient is irrelevant for efficiency purposes (the number of calls is efficient for

every θ) and the marginal price in both cases pij = ci+cj

1+qjqi

bθ

. Furthermore, if b = 0 (no

call externalities then the marginal price is constant p(θ) = ci+ cj so that total tarif

is T (x11, x12, x21, x22) = Fθ +∑j∈{1,2} pijxij this is true for every choice of coverage

θH , θL which enters in the tarif trough nj.

43

2.8 Pricing inder CPP

The planer/monopoly 2 observes the type θ and chooses p(θ) and r(θ) as

to maximize the total surplus from calling. The choice of prices will involve call

rationing by receivers such that high types calling low types will normaly be rationed

more compared to low types calling high types. This is expressed in the the solution

by the existence of a curve l(θ) which gives the marginal recipient who rations calls

sent by θ. That is, if θ′ < l(θ), sender θ is rationed by receiver θ′ such the amount

of calls exchanged y(θ′) will be determined by θ′ actiuj upon reception price r(θ′).

On the other hand, if θ′ > l(θ) then recipiend θ′ receives all x(θ) calls sent by θ.

The monopoly chooses x(θ), y(θ) by maximizing the sum of the surplus gen-

erated by each θ. For a fixed θ such surplus is:

θ∫ l(θ)

θHu(y(s))f(s)ds+ [1− F (l(θ))]θu(x(θ)) + α(θ)u(y(θ))[1− F (l−1(θ))]+ (2.38)

α(θ)∫ l(θ)

θHy(s)f(s)ds−(c1+c2)x(θ)[1−F (l(θ))]−(c1+c2)

∫ l(θ)

θHu(y(s))f(s)ds (2.39)

The F.O.C. are:

p(θ) = θu′(x(θ)) =c1 + c2

1 + E(α(s)|s>l(θ))θ

(2.40)

r(θ) = α(θ)u′(y(θ)) =c1 + c2

1 + E(s|s>l−1(θ))α(θ)

(2.41)

where l(θ) is such that y(l(θ)) = x(θ). This is a nonlinear sistem of equations in

3 variables u′(x(θ)), u′(y(θ)) and l(θ). They express the fact that l(θ) crosses the

2Given that θ is observable and no price discrimination according to receivers is posible. The

monopoly and planer are going to choose the same allocation

44

45o line at θ. This is verified by substituting θ, l(θ) above. Type θ customers are

rationed by every receiver θ′ < θ but they do not but they do not ration any lower

type custumer.

The slope of l(θ) will, in general, depend on the slope of α(θ) and wheather

such slope is >< 1 as shown in the linear example: α(θ) = aθ+b. The F.O.C. reduce

to l(θ) = θ−θ(1−a)a

, whioch is linear and does not depend on b. l(θ) = θ and lies below

the 45o line for a < 1 and above if for a > 1. For a = 1, l(θ) = 0. Furthermore, for

a < 1 there are some customers at the bottom who are never rationed as sender no

mather how θH is choses. Also for a > 1 some customers at the bottom never reject

a call from anyone.

Now in this example, p(θ) is always increasing while r(θ) is increasing only for

a < 2− bθ

In the special symetric case a = 1 and b = 0 prices are:

p(θ) = r(θ) =c1 + c2

1 + θ+θ2θ

(2.42)

Notice that only θ custumers calling their own type, share the costs equally. Calls

between arbitrary θ and θ′ are such that:

p(θ) + r(θ′) =c1 + c2

1 + θ+θ2θ

+c1 + c2

1 + θ+θ′

2θ′

< c1 + c2 (2.43)

unless both θ = θ′ = θ otherwise, marginal price will sum up to less than marginal

cost.

Alternatively, if a = 0 and b > 0 so that recipients are equal in the surplus

45

they get from being conected then

p(θ) =c1 + c2

1 + bθ

(2.44)

r(θ) =c1 + c2

1 + θb

(2.45)

which implies a constant reception charge for every recipient. If b → to 0 then

r(θ)→ to0. Also receivers pay less than half the marginal cost because b < θ while

sender pay more/less the marginal cost if they are higher/lower than the b type.

Proposition 4 When markets are young CPP is comparatively more atractive than

RPP with respect to the mature phase. That is, dθHdc

<? > 0[VER ESTO] and

d[wRPP−wCPP ]dc

> 0

In terms of real world pricing this implies that when costs of billing customers

are present (it is costlier to bill customers under RPP than under CPP) then CPP

may be preferrable when the market is in the early stages of development (low

coverage and high c). As long as c falls and the spectrum of customers for which the

it is efficient to consume the services widens (even if billing cost remains constant),

then RPP will be more desirable so that at some point the switching fron CPP to

RPP is necessary.

2.9 Fully efficient allocation:The choice of θH and θL

Let φi,j(θ, θ′) be the surplus generated by a single call from customer θ in i to

customer θ′ in j.

φi,j(θ, θ′) = qiθu(xij(θ)) + qjα(θ′)u(xij(θ))− (ci + cj)xij(θ) (2.46)

46

Let sij(θ) be the surplus net of marginal cost customer θ in network i obtains from

calling customers of network j.

sij(θ) =∫Ajφi,j(θ, θ

′)f(θ′)dθ′ (2.47)

rij(θ′) =

∫Aiφi,j(θ, θ

′)f(θ)dθ (2.48)

where Aj is the interval [θL, θH ] if j = 2 and [θH , θ] if j = 1. Now the total surplus

is given by:

S =∑

i,j∈H,L

∫Ajsij(θ)f(θ)dθ (2.49)

Now to find θH we get the F.O.C.

[s11(θH)+s12(θH)]−[s21(θH)+s22(θH)]+[r11(θH)r21(θH)]−[r12(θH)+r22(θH)] = k1+k2

(2.50)

or

SH − SL = k1 − k2 (2.51)

where

SH(θ) = s11(θ) + s12(θ)] + [r11(θ) + r21(θ)] (2.52)

SL(θ) = s21(θ) + s22(θ)] + [r12(θ) + r22(θ)] (2.53)

Are the total surplus added by θ (his or others) by sending and receiving calls. The

equality above expresses that the increase in surplus θH creates by moving from

network L to network H must equal the increase in fixed cost of belonging to H

rather to L.

Similarly we can deduct the condition for θL as:

SL(θL) = k1ifθL > θ (2.54)

47

which is simply the condition that total surplus added by θL must be equal to fixed

cost.

In the case θL = θ, then only the fist condition applies

From the conditions above, we can see that for the particular case of monopoly,

θ satisfies the simpler condition s11(θL) + r11(θL) = k1

2.10 Competition: The Choice of θL∗ and θH∗

Firm 1 and firm 2 solve the respective problems:

maxθH

π(θH : θL, a1, a2) (2.55)

maxθL

π(θL : θH , a1, a2) (2.56)

from which we obtain θH(θL) and θL(θH) respectively. The equilibrium choice of

coverage θH∗ and θL∗ is the mutual best response:

θH(θL∗) = θH ∗ (2.57)

θL(θH∗) = θL∗ (2.58)

48

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