\documentstyle[11pt,harvard,doublesp]{article}   
\title{Network Competition with Reciprocal Proportional Access
Charge Rules   
\footnote{ We   
would like to thank  Tom Muench, Pradeep Dubey and Paul   
Teske for helpful comments. This is a very preliminary version of 
the   
paper and all errors are ours.}}   
   
\author{Toker DOGANOGLU\thanks{ Department of Economics,    
SUNY at Stony Brook, Stony Brook, NY 11794,USA.} and Yair   
TAUMAN\thanks{ Department of Economics,    
SUNY at Stony Brook, Stony Brook, NY 11794,USA and Faculty of
Management,    
Tel-Aviv University, ISRAEL.}}   
\begin{document}   
\maketitle   
    
 
 
 
 
\begin{center}   
{\bf SUNY at Stony Brook Discussion Paper\\ 
DP96-01} 
\end{center} 
 
 
\clearpage   
\begin{abstract}   
This paper presents a model of two competing local   
telecommunications    
networks, similar in spirit to the model of  \citeasnoun{tir3}. 
The networks have different attributes which we assume are fixed
and the consumers have idiosyncratic tastes for these attributes. 
The networks are mandated to interconnect and the access charges
are determined cooperatively in the first stage. In the second
stage, the two network companies are engaged in a price competition
to attract consumers.   
In the third stage, each consumer selects a   
network    
and determines the consumption of phone calls.    
   
\citeasnoun{tir3} have shown that except for restrictive scenarios,
the local price competition does not result in a pure strategy
equilibrium. In this paper, we assume that the two companies choose
access charge rules rather than simply access charges. These rules
determine the  access charges as a function of the future local
prices. We show that the family of reciprocal proportional access
charge rules generates a pure strategy equilibrium and we discuss
its properties.   
(JEL D4, K21, L41,43, L51, L96) 
\end{abstract}   
   
\clearpage   
   
   
   
   
   
\section*{Introduction}   
The telecommunications industry has been one of the fastest
developing   
industries of the last half century.   Traditionally, the telephone  
industry    
is considered to be   
a natural monopoly, since the cost structure consists of a large  
fixed cost component and a decreasing average operating cost   
(see \citeasnoun{noam}). For this reason, in most   
countries the   
telecommunications companies  are owned by the   
governments. The inefficiencies generated by  government ownership 
of a technologically very sophisticated industry has led to   
privatization in several countries and this trend is still in   
effect. In the US, while the telecommunications    
industry has always   
been privately owned,  it  has been subject to substantial
regulation.    

Over the last thirty years, with the emergence of entrants in
several   
segments   
of the market, the question of regulating the telecommunications  
industry has become even more complex. The  two   
well-known examples of such entrants    
are Microwave Communications Inc., MCI,   
in the US   
and Mercury Communications in the UK, which led to major policy
changes in their respective countries. Both of these companies have
provided  long distance service for the consumers using local
networks of the incumbent monopolists, AT\&T in the  US, and BT in
the UK. For the history of the evolution of competition in the
telecommunications industries in these two countries see
\citeasnoun{ingo}.   
In the US, the AT\&T monopoly was broken down into a long distance
company and seven Regional Bell Operating Companies (RBOCs) which
were awarded monopolies for local service in their operating areas.
The motivation behind this was that a vertically integrated 
monopoly would not have incentives to let a competitor enter in
some segment of their business.   
The RBOCs were not allowed to provide long distance service, while
MCI and AT\&T were banned from local access markets.   
The long distance companies had to pay the local network companies  
access fees to interconnect with their networks   
and the fees were determined by the   
FCC. In the UK, the monopoly stayed intact and an access charge
mechanism was designed for competitors to access BT's local
networks. OFTEL, the regulatory agency in the UK, has set the rates
for interconnection  
(see \citeasnoun{tir1}).   
 
The access charge mechanism design has been a subject of immense
discussion in the last decade, most of it in the context of one
vertically integrated firm competing in one segment of the market.
A competitor has to have access to the other segment and access
charges need to be determined.   
\citeasnoun{tir2}    
proposed a mechanism which yields the welfare   
maximizing access charges. The    
Efficient Component Pricing Rule (ECPR) of    
\citeasnoun{baumol}, despite its ease,    
generated a lot of discussion since it is    
efficient only under very strict assumptions, and seems to favor
the incumbent monopolies.   
\citeasnoun{nikki1} present a critique   
of ECPR, while \citeasnoun{arms} reinterpret   
the ECPR in the light of  \citeasnoun{tir2}.   
\citeasnoun{nikki2} examine the incentives of two long distance   
companies for   
interconnection   
when the bottleneck facility is owned only by one of them.   
They calculated the final   
prices for several different scenarios    
as well as the access charges.   
   
In recent years, breakthrough  technologies like the introduction
of the fiber optic lines, mobile communication networks, the
transformation of cable networks to carry phone calls and the
amazing growth of Internet, have questioned the necessity of
monopolies in the local access market (see \citeasnoun{noam}). The
current trend is to open the whole telecommunications market to
competition. Several countries are planning for a competitive
telecommunications industry and passing legislation to prepare the
legal grounds for  competition.   
The latest example of such policies is the Telecommunications Act
of   
1996 in the US, which essentially allows   
entry into the   
telecommunications market. These  developments bring new questions
to mind concerning the new policies that will be  
required to accord with  
the existence of several networks and their mutual interconnection.
  
One of the most important articles in the 1996 act is the one which
addresses interconnection between networks.  It asserts that
interconnection should be provided on a nondiscriminatory manner to
everyone who wishes;   
the access to networks should be at a just and   
fair price; the access charges should   
be negotiated between interacting firms and binding agreements   
should be signed. These agreements are subject to the approval of 
FCC and   
Public Utility Commissions.\footnote{See Telecommunications Act of
1996 .}   
Like most laws, the 1996 bill uses vague language and it is   
subject to interpretation. Once the access fees are set by mutual  
consent, the   
networks act competitively, i.e. they compete   
in local pricing   
schemes, service quality, etc.   
   
\citeasnoun{tir3} (hereafter LRT) have analyzed a model of two
local network companies that possess different attributes for
consumers.\footnote{ \citeasnoun{carter} build on LRT and examine
the effects of brand loyalty.} In their model, the two companies,
given  access charges,  set the local prices competitively.    
The customers of a network are    
charged the same price independent of the network which    
completes their call. The networks compete  only in prices since
the   
other attributes   
are assumed to be fixed.   
Then consumers   
select their preferred networks \`{a} la Hotelling. It is shown in
LRT that   
equilibrium in local prices    
may not exist except for restrictive values of access   
charges.   
   
   
This paper  deals with a similar model. It differs from LRT in two
aspects: the determination of access fees and the determination of
consumer demands. In the first stage, the two companies negotiate
access charge rules rather than access charges.   
The access charges are functions of the local prices and they are 
determined  only after the second stage, when the local   
prices    
are determined. Thus, any change in demands and   
therefore in local prices will automatically result in new access
charges.    
The other difference between our model and that of LRT is in the
consumer preferences. The utilities of consumers in both models
consist of a deterministic part and a random part. While the
deterministic part in LRT generates  demands with constant price
elasticity, we use quadratic utilities which generate linear
demands.   
This avoids the unboundedness of the consumption for small prices 
and provides   
a satiation point, which is  natural for this type of   
services.\footnote{There is a limit to   
the time individuals spend on phone calls.} For the random
component    
we use the Weibull distribution, which resembles the Normal   
distribution and provides analytical convenience. Furthermore, it
is  easy   
to extend this model to deal with more than two network companies. 
 
   
   
It is shown here that if the network companies restrict their first
stage negotiation to reciprocal\footnote{Reciprocal here means that
both networks employ the same rule.} proportional access charge
rules (RPACR), which determine the access charges as  certain
proportions of the future local prices, then a pure strategy
equilibrium always exists and the local prices are explicitly
computed.    
If the companies are allowed to set the proportion factor to
maximize their joint profits then they will set access charges
smaller than local prices when the networks are not close
substitutes. However, in this case the resulting local price
coincides with the monopoly price. In other  words, the companies
charge the customers the monopoly price but charge each other a
lower price.   

If the two networks are close substitutes, they act competitively
and the result is  low local prices even under joint  profit
maximization. In this case, the access charges will coincide with
the local prices.    
The adoption of RPACR can be viewed as a mild and simple regulatory  
policy. It does not require information about the determinants of
the   
industry. If the services of the networks are close   
substitutes then the use of RPACR results   
in competitive prices. In addition, as noted   
above, RPACR generates access charges which are responsive to
future changes like demand shocks and technological innovations. 
However, if one believes that the competing companies will find a
way to increase differentiation then further government
intervention may be necessary.   
 
 
The paper is organized as follows.   
In Section 1, we   introduce the general model for the industry.  
In Section 2, we develop a model for consumer choice and derive
their demands. In Section 3, we analyze the competition between the
two network companies. Concluding remarks appear in Section 4.   
   
\section{The Industry Model}   
The telecommunications industry is a complicated industry to model.
We deal with a  simple case of two network companies which only
provide local telecommunications services. We  refer to them as
Network 1 and Network 2.  
The networks incur zero marginal cost for each call.    
 
    
There is a fixed cost associated with building a network. Operating
and maintenance costs are assumed to be independent of the amount
of service provided.   
Usually, operating and maintenance costs do depend on the number  
of  customers of a network. But to simplify   
the analysis, we assume that   
these costs are included in the fixed cost  
component.\footnote{ Computer simulations suggest that our results
will   
remain true   
without this assumption, at least when these costs are not too   
large   
relative to the fixed cost. However the methods we use to prove   
our result does not apply to this case.} The fixed cost of each
network  is   denoted by $F$.   
Each   
company    
faces two demand functions. The  demand function $X_{11}$, for
calls initiated and completed at Network 1, and the demand function
$X_{12}$, for the calls initiated at Network 1 and completed at
Network 2. The demand functions for Network 2, $X_{22}$ and
$X_{21}$, are defined similarly. The demand functions $X_{ij}$ will
be derived from utility maximization.  
This is done in the next section.    
   
The two companies first negotiate access charge rules, $\hat{a}_1$
and  $\hat{a}_2$, where $\hat{a}_1$ is the per unit price that
Network 2  will pay Network 1 for each unit call that is completed
in Network 1. The term $\hat{a}_2$ is similarly defined.   
Then they are engaged in a price competition which   
determines   
the local prices $p_k$,    
($k=1,2$). The price $p_k$ is the  per unit charge   
of company $k$ to each of its customers whether their calls are
completed locally or  in the other network.   
In contrast to LRT, the  access fees may depend   
on the local prices $p_1$ and $p_2$. Thus, it is assumed that   
$\hat{a}_k=a_k(p_1,p_2)$ for $k=1,2$. We further restrict our   
attention to the simple case of the proportional rule   
\[\hat{a}_k=a_kp_k,\]      
where $0\le a_k \le 1$, for $k=1,2$.   
If $a_1=a_2=a$, then the access charge rule will be   
reciprocal, and it is called the Reciprocal Proportional Access
Charge Rule (RPACR).  Thus, we treat each network as a regular
customer who only buys partial service (just completion of a call)
and therefore pays only a proportion of the full service price.   
This is consistent with the nondiscrimination requirement of the
Telecommunications Act of 1996.   
The profit functions of the two networks are given by   
\begin{eqnarray}   
\Pi_1 & = & p_1X_{11}+(p_1-ap_2)X_{12}+ap_1X_{21}-F,   
\label{pi11}\\   
\Pi_2 &=&   
p_2X_{22}+(p_2-ap_1)X_{21}+ap_2X_{12}-F.\label{pi21}   
\end{eqnarray}   
Consider the following sequence  of events   
   
{\bf \it Stage 1.}    
The network companies  select  an RPACR by mutual consent.   
 
{\bf \it Stage 2.}    
The two companies choose their prices $p_1$ and $p_2$   
simultaneously and   
independently and announce them publicly.   
 
{\bf \it Stage 3.}    
After observing the prices $p_1$ and $p_2$, every consumer   
selects a network to subscribe to and chooses the   
amount of  calls.   
   
The Telecommunications Act of 1996 requires companies to negotiate  
and then to    
sign binding   
agreements concerning their access charges before they engage in
the competition for local prices, services, etc.   
In our model, the parties  negotiate  rules, not charges. The rules
then determine access prices as functions of their local prices. 
That is, the access charges are determined only after local prices
$p_1$ and $p_2$ are determined. Any change in local prices will
have an immediate impact on the access prices.   
A crucial point is to determine how to select access charge rules
in the bargaining stage.    
In our model it amounts to the selection of the proportion $a \in
[0,1]$. 
   
   
\section{Consumer Demand For Local   
Telecommunications Services}   
   
In this section, we will specify the consumers' utilities and model
the process by which they select their networks. Suppose that there
are N potential consumers.    
Each consumer  is assumed to have  idiosyncratic tastes which
depend on the various attributes of the networks (like specific
services offered by the companies, the intensity of advertising,
accounting methods, etc.).    
This allows us to explain   
the existence of several networks with similar   
products but different prices. Consumer $i$ who subscribes to   
network $k$ has  the following   
utility function,   
   
\begin{equation}   
u_i(k,x,y)=  ((r-sx)x +y)e^{{\sigma    
}\epsilon_{ik}}, \label{utility}   
\end{equation}   
where $x$ is the total consumption of telephone calls in minutes
and  
$y$ is the amount of numeraiare (money)  consumed. The term  
$\epsilon_{ik}$ measures the idiosyncratic taste variable of the
consumer.  
It is assumed that $\epsilon_{ik}$'s are Weibull distributed
(\cite{mcfadden}),  
statistically independent for all $i$ and $k$, and that they are
private information of the consumers.    
The  cdf of the    
Weibull distribution closely resembles that of the Normal
distribution.   
The term $\sigma$ is a measure of the dispersion of tastes , that
is, $\sigma$ measures the substitutability between the services of
the two networks. As $\sigma \rightarrow 0$ the networks become
perfect substitutes, while as $\sigma \rightarrow \infty$ the
networks become perfect complements.   
One important feature of this utility function is that the
deterministic part will result in a linear demand function. This
implies a bounded amount of calls demanded at prices close to zero. 
Another feature of the quadratic utility function is that it
provides a satiation point, which is natural  for such services as
there is a limit to the time an individual will spend on the phone.
  
Random utility models of this kind have been extensively employed   
in the   
literature, starting with  \citeasnoun{mcfadden}; see   
\citeasnoun{depalm}   
for  a   
wide variety of examples.     
   
        
   
Let $V_{ik}(x)$ be the deterministic part of the surplus   
of consumer $i$ who subscribes to Network $k$   
and consumes $x$ units.   
Then   
    
\begin{equation}   
V_{ik}(x)=(r-sx)x -p_kx. \label{value}   
\end{equation}   
This is maximized for     
\begin{equation}   
x_{k}={1 \over {2s}} (r-p_k), \label{inddemand}   
\end{equation}    
and the maximum is given by   
\begin{equation}   
V_{ik}={1 \over {4s}} (r-p_k)^2. \label{maxval}   
\end{equation}    
Observe that (as in LRT)  the demand of each consumer, $x_k$ does
not depend on $i$; thus we have dropped the index $i$.   
   
By (\ref{utility}) and (\ref{maxval}) consumer $i$ prefers Network  
$k$ to   
Network $\bar{k}$ if and only if   
\[    
V_{ik}e^{\sigma \epsilon_{ik}} \ge    
V_{i\bar{k}}e^{\sigma\epsilon_{i\bar{k}}}.\]   
Therefore, the network companies assign  probability $P_{ik}$    
that $i$ will    
choose Network $k$ over $\bar{k}$, where   
\begin{equation}   
P_{ik}=Probability(V_{ik}e^{{\sigma}\epsilon_{ik}}>V_{i\bar{k}}e  
^{{\sigma}\epsilon_{i\bar{k}}}). \label{prob}   
\end{equation}     
Since the   
$\epsilon_{ik}$'s are independent and Weibull distributed, this   
probability    
is given by \cite{depalm}   
\begin{equation}   
P_{ik}={1 \over {1+ ({V_{i\bar{k}} \over V_{ik}})^{1 \over   
\sigma}}}.   \label{mshare}   
\end{equation}   
For the derivation of (\ref{mshare}) see \citeasnoun{depalm}.   
   
Applying (\ref{maxval}) and (\ref{mshare}) to the case k=1, we have
  
\begin{equation}   
P_{i1}= {(r-p_1)^\tau \over {(r-p_1)^\tau+(r-p_2)^\tau}}, 
\label{prob1}   
\end{equation}   
where $\tau={2 \over \sigma}$. Thus, this probability does not
depend on $i$, and we can drop the index $i$ from $P_{i1}$ and
write $P_1$.   
From the point of view of the two companies, every consumer will  
select    
Network 1 with probability $P_1$. The expected   
number of consumers who will subscribe to Network 1 is therefore
$NP_1$. Notice that $P_k$   
can be viewed   
as the market share of Network $k$. Consequently by (\ref{prob1}),  
the expected 
market share, $m(p_1,p_2)$, of Network 1 is given by   
\begin{equation}   
m(p_1,p_2)= {(r-p_1)^\tau \over {(r-p_1)^\tau+(r-p_2)^\tau}}   
.\label{share}   
\end{equation}   
The expected market share of Network 2 is obviously
$P_2=1-m(p_1,p_2)$.   
Observe that the aggregate subscriber demand faced by Network $k$  
is given by,   
\begin{equation}   
X_k=X_k(p_1,p_2)= {N \over {2s}}(r-p_k)P_k,   
\ \ \ \  \ \ k=1,2. \label{aggd}   
\end{equation}   
   
Next let us find the expected number of calls initiated in Network
$k$, and completed  in Network $j$ where $k,j \in \{1,2\}$.    
From the companies' point of view, the probability that a   
customer of Network $k$, $k\in   
\{1,2\}$,  calling a customer of Network $j$  is $m(p_1,p_2)$ if  
$j=1$ and    
$1-m(p_1,p_2)$ if $j=2$. Therefore by (\ref{share}) and   
(\ref{aggd}),   
we conclude that,   
\begin{eqnarray}   
X_{11} & = & {N \over {2s}}(r-p_1) (m(p_1,p_2))^2, \label{x11}\\  
X_{12} & = & {N \over {2s}}(r-p_1) m(p_1,p_2)(1-m(p_1,p_2)) ,\\  
X_{22} & = & {N \over {2s}}(r-p_2) (m(p_1,p_2))^2, \\  
X_{21} & = & {N \over {2s}}(r-p_2) m(p_1,p_2)(1-m(p_1,p_2))
\label{x21}. 
\end{eqnarray}   
We use these demand functions in the next section to analyze the
competition between the two companies.   
   
   
\section{Network Competition with RPACR}   
In this section, we analyze the price competition when the access  
charge rules   
are reciprocal proportional rules. This   
means that both networks use the same rule,    
$\hat{a}_k=ap_k, \ \ 0\le a \le 1, \ \ k\in{1,2}$,   
to calculate access charges.   
The number $a$ is   
determined in the first stage of the game.    
To simplify notation, we use $p_1=p$ and   
$p_2=q$. Using this rule we obtain, by (\ref{pi11}), (\ref{pi21}), 
and (\ref{x11})-(\ref{x21}) 
\begin{eqnarray}   
\Pi_1 & = & {N \over {2s}} \{p(r-p)m(p,q)+ (p-ap)(r-p)m(p,q)     
(1-m(p,q)) + \nonumber \\   
\  & \  & +ap(r-q)m(p,q)(1-m(p,q)) \}-F, \label{pi1} \\   
\Pi_2 & = & {N \over {2s}} \{q(r-q)(1-m(p,q))+ (q-aq)(r-q)   
m(p,q)(1-m(p,q)+ \nonumber \\   
\  & \ & +aq(r-p)m(p,q)(1-m(p,q)) \}-F.  \label{pi2}   
\end{eqnarray} 
Given $a, \ 0\le a \le 1$, the two network companies are engaged  
in a price competition. 
We find that when $N$ is sufficiently large to cover the fixed
costs, 
a symmetric equilibrium exists\footnote{Although we did not succeed 
in  proving the uniqueness of this equilibrium, computer  
simulations suggest that it is indeed unique.} and can be computed 
explicitly. \\ 
{\bf Theorem 1:}   
For every  $0\le a \le 1$, $\tau >0 $ and $r > 0$,     
there exists one and only one symmetric equilibrium.   
It is given by:   
\[p*=q*={{(2+a)r} \over {\tau+4}}\]   
{\it \bf Proof.} See Appendix.   
   
The main contribution of Theorem 1 is that pure strategy
equilibrium in local prices always exists with RPACR.    
The equilibrium prices have intuitive properties.   
They are  decreasing in substitutability rate,   
$\tau$.    
The higher the substitutability rate   
$\tau$, the stronger is the   
competition between the networks, and, therefore,   
the lower are the prices.    
Second, the prices are increasing in the demand intensity, $r$, in
a linear way. Finally they are increasing in the access charge
proportion factor $a$. The proportion $a$ can be viewed as the
marginal cost of a call that is completed in the other network.   
The result asserts that the higher this cost is, the higher is the
price that the company charges to its customers.  
 
The value of $a$ is determined in the first stage by negotiation.
The bargaining between the two parties may result in essentially
any number in $[0,1]$ with the restriction that the revenue of a
company covers the fixed cost.     
Let us first examine the case where the companies set    
$a$ to maximize   
total profit, $\Pi_T$. Since $m(p*,p*)=1/2$, by Theorem 1,   
\begin{equation}   
\Pi_T= \Pi_1 + \Pi_2 ={N \over {2s}}{{(\tau+2-a)(2+a)r^2}    
\over {(\tau+4)^2}}-2F.\label{ssp}   
\end{equation}   
This function is concave in $a$ and it obtains its unconstrained
maximum at 
$a={{\tau} \over 2}$. To guarantee that  
$a\le 1$, the companies should select  
$a=min(\frac{\tau}{2},1)$ provided that they cover fixed costs.  We
summarize 
this in the next theorem.   
 
{\bf Theorem 2.}   
If $a$ maximizes joint profits then   
\begin{equation}   
a=\left\{    
\begin{array}{cc}   
1 & \tau\ge 2, \\   
\frac{\tau}{2} &  \tau\le 2,   
\end{array}   
\right.   
\end{equation}   
and   
\begin{equation}   
p^*=q^*=\left\{    
\begin{array}{cc}   
\frac{3r}{\tau+4} & \tau\ge 2, \\   
\frac{r}{2} &  \tau\le 2.   
\end{array}   
\right.   
\end{equation}   
Consequently, when $\tau\ge 2$ then $a=1$ and the access fee
coincides with the local price.    
If the rate of substitutability is  large, consumers will be
charged a small price (reflecting strong competition)  
and each company will be treated as any other customer by the other
company. 
If $\tau\le 2$ (reflecting low   
substitutability),   
the access fee is $\frac{\tau p}{2}$ and this   
is smaller than the local price.   
Therefore, in equilibrium the local price is $\frac{r}{2}$, which
is the monopoly price.   
 
In this paper the attributes of the networks (other than the
prices) are exogenously given. These attributes determine the value
of $\sigma$ and therefore $\tau$. One could add another stage to
the game, for instance before $a$ is determined. In this stage the
companies compete in attributes and their selections determine the
value of $\tau$. Then Theorem 1 determines the equilibrium prices
in terms of $a$ and these attributes. The equilibrium choice of
attributes will be a function of their costs. If the costs of
different attributes are quite similar and $a$ is selected to
maximize joint profits, then they will choose their attributes so
that  $\tau$ will be sufficiently small to guarantee monopoly local
prices (See Theorem 2).  This will eliminate the effectiveness of
the price competition and will justify government intervention. An
extreme case is when government sets $a$ to maximize total social
surplus, $SS$, subject to the constraint that the companies cover
fixed costs.   
The social surplus is the sum of the total industry  profits and  
the total consumer surplus, $CS$, where   
\[CS=\frac{N}{4s}(r-p*)^2.\]   
It is easy to verify that in equilibrium   
\[SS=\frac{Nr^2}{4s(\tau+4)^2}(\tau+2-a)(\tau+6+a)-2F.\] This
function is a decreasing function of $a$ for $0\le a\le 1$.   
Hence   
the social surplus is  maximized for the   
lowest level of $a$  which still covers fixed costs.   
Observe that if $a=0$ then $p^*=q^*=\frac{2r}{\tau+4}$ and  
\[N\ge \frac{(\tau+4)^2s}{(\tau+2)r^2}F\]  
should hold to cover fixed costs. Hence fixed costs may not be
covered  
for large $\tau$. This suggests that for a competitive industry the
"Bill and Keep" policy may jeopardize the viability of the
industry.   

\section{Conclusion}   
We have analyzed the competition between two network companies
which choose  
access charges using RPACR. The most important result is that an
equilibrium 
 in local prices always exists. The equilibrium prices exibit  
desirable properties when the services of the networks are close
substitutes  
and  we believe that this will be the case in a competitive  
telecommunications industry.   
The imposition of RPACR can be viewed as a mild regulatory policy. 
One important advantage is that implementation of RPACR does not
require  
information about the industry parameters.  Since RPACR is
responsive to  
future changes of the determinants of the market, it provides
flexibility  
for the companies to react to changes in the environement and in
strategies.  
 For low differentiation between the networks, the equilibirium
local prices  
are low, provided that both companies cover their costs. If one
believes  
that this industry will not be as competitive and the companies
will be  
able to differentiate themselves considerably, then RPACR may not
be useful.  
It may serve as a collusion device for the companies to induce  
monopoly prices. Further regulations should then  be imposed  
to prevent such a case.   
    
   
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\end{thebibliography}                                             
 
\newpage   
   
\section*{Appendix}   
{\bf Proof of Theorem 1.}   
Let $\alpha=r-p$ and $\beta =r-q$. Then the profit functions of the  
two companies are   
     
\[   
\Pi_1(\alpha,\beta) =\frac{N}{2s} {\alpha^x \over {(\alpha^x+   
\beta^x)}} \left[   
\alpha(r-\alpha)+a(\beta-\alpha)   
{\beta^x \over {(\alpha^x+ \beta^x)}} \right]-F,   
\]   
and   
\[   
\Pi_2(\alpha,\beta) =\frac{N}{2s} {\beta^x \over {(\alpha^x+   
\beta^x)}} \left[   
\beta ( r-\beta) +a(\alpha-\beta)    
{\alpha^x \over {(\alpha^x+ \beta^x)}} \right]-F.   
\]   
Let   
\[p^*=q^*=\frac{(2+a)r}{\tau+4}.\]   
Then   
\[\alpha^*=\frac{(\tau+2-a)r}{\tau+4}=\beta^*.\]   
We have to prove that $\Pi_1(\alpha,\beta^*)$ is maximized for
$\alpha=\alpha^*$ and that $\Pi_2(\alpha^*,\beta)$ is maximized for
$\beta=\beta^*$. By symmetry, it is enough to show this for
$\Pi_1(\alpha,\beta^*)$.   
Since $\Pi_1(\alpha,\beta)$ is continuous in the parameter $\tau$,
it is sufficient to prove our claim for positive rational values of
$\tau$.   
Let $\tau=\frac{m}{n}$, where $m>0$, $n>0$ are integers. Let \[t
\equiv \alpha^{\frac{1}{n}} \ \ \ and \ \ \ s\equiv
\beta^{\frac{1}{n}}.   
\]      
Then $t^*=(\alpha^*)^{\frac{1}{n}}=s^*$, where   
\begin{equation}   
s^*=(\alpha^*)^{\frac{1}{n}}=\left( \frac{m+2n-   
an}{m+4} r\right)^{\frac{1}{n}},\label{sstar}   
\end{equation}   
and    
\[   
\Pi_1(t,s) \equiv\frac{N}{2s} {t^m \over {(t^m+ s^m)}} \left[
t^n(r-t^n)+ar(s^n-t^n)   
{s^m \over {(t^m+ s^m)}} \right]-F.    
\]   
Since $t^n$ is an increasing function of $t$, it is sufficient to
prove that $\Pi_1(t,s^*)$ is maximized for
$t^*=(\alpha^*)^{\frac{1}{n}}$ when $s^*=(\beta^*)^{\frac{1}{n}}$. 
 
It can be easily verified that   
\[\frac{\partial \Pi_1}{\partial t}=
\frac{t^m}{t(t^m+s^m)^3}P(t,s),\]   
where the polynomial $P(t,s)$ is given by   
\begin{eqnarray}   
P(t,s)& = &  ms^mt^n(r-t^n)(t^m+s^m)+arms^m(s^n-t^n)(s^m-t^m)
\nonumber \\ \ & \ & +
(rnt^n-2nt^{2n})(t^m+s^m)^2-arnt^ns^m(t^m+s^m).   
\end{eqnarray}   
After rearranging the terms we have    
\begin{eqnarray}   
P(t,s)&=&
s^{2m}\left[-(m+2n)t^{2n}+(rm-arm+rn-arn)t^n+arms^n\right]
\nonumber \\   
\ & \ & s^m\left[-(m+4n)t^{m+2n}+(rm+arm+2rn-arn)t^{m+n}-   
armt^ns^n\right] \nonumber\\   
\ & \ & -2nt^{2m+2n}+rnt^{2m+n}.    
\end{eqnarray}   
It is easy to verify that $t=t^*=s^*$ is the  root of the
polynomial   
$P(t,s^*)$. Therefore   
\begin{equation}   
P(t,s^*)=(t-s^*)g(t,s^*), \label{P}   
\end{equation}   
for a certain polynomial $g(t,s^*)$. It can be verified that
\begin{eqnarray}
g(t,s)&=&-2n\sum_{j=0}^{n-1}s^jt^{2m+2n-j-1}+n(r-2s^n)
\sum_{j=0}^{2m-1}s^jt^{2m+n-j-1}\nonumber\\   
\ & \ & -ws^m\sum_{j=0}^{n-1}s^jt^{2n-j-1}+[y-ws^n+n(r-2s^n)]s^{2m}  
\sum_{j=0}^{n-1}s^jt^{n-j-1}\nonumber\\   
\ & \ & -vs^m\sum_{j=0}^{n-1}s^jt^{m+2n-j-1}+(z-vs^n)\sum_{j=0}^{n-  
1}s^jt^{m+n-j-1}\nonumber\\   
\ & \ & +(z-vs^n-arm)s^{m+n}   
\sum_{j=0}^{n-1}s^jt^{m-j-1}, \label{division}   
\end{eqnarray}   
where $s=s^*=\left(\frac{m+2n-   
an}{m+4n}r\right)^{\frac{1}{n}}$ and where   
\begin{eqnarray}   
\label{dfs}   
w&=&m+2n, \nonumber\\   
v&=&m+4n, \nonumber\\   
y&=&rm-arm+rn-arn, \\   
z&=&rm+arm+2rn-arn. \nonumber   
\end{eqnarray}     
It can be easily checked that   
\begin{equation}   
z-vs^n-arm=0,\label{ident0}   
\end{equation}   
and   
\begin{equation}   
y-ws^n-n(r-2s^n)=-arm.\label{ident}   
\end{equation} \newline   
{\bf Lemma 1.} let $0\le t \le r^{\frac{1}{n}}$. Then, 
for $0 \le t \le r^{1/n}$, $g(t,s^*)<0$.\newline   
{\bf Proof.}   
Let us sum up the geometric series appearing in   
(\ref{division}) and apply (\ref{dfs}), (\ref{ident0}),
(\ref{ident}).   
We obtain   
\begin{eqnarray}   
\label{a3}   
g(t,s)&=&-2nt^{2m+2n-1} \frac{1-(\frac{s}{t})^n}{1-\frac{s}{t}}   
+n(r-2s^n)t^{2m+n-1} \frac{1-(\frac{s}{t})^{2m}}{1-\frac{s}{t}}   
\nonumber\\   
\ & \ & -(m+2n)s^mt^{2n-1} \frac{1-(\frac{s}{t})^n}{1-\frac{s}{t}}  
arms^{2m}t^{n-1} \frac{1-(\frac{s}{t})^n}{1-\frac{s}{t}} \\ \ & \
& -(m+4n)s^mt^{m+2n-1}
\frac{1-(\frac{s}{t})^n}{1-\frac{s}{t}}+arms^mt^{m+n-1}
\frac{1-(\frac{s}{t})^n}{1-\frac{s}{t}},\nonumber  
\end{eqnarray}  
where $s=s^*$. Let us first show that the sum of the last three
terms of the right-hand side of (\ref{a3}) is negative.   
Consider  the case where $t<s^*$.  Observe that   
\[   
arms^{2m}t^{n-1}>arms^mt^{m+n-1},\label{cond0}   
\]   
for all $s>t$. Therefore the sum of the last three terms of
(\ref{a3})  is negative.   
Consider next the case where   
$t>s^*$. It is sufficient  to show that    
\[   
(m+4n)s^mt^{m+2n-1}\frac{1-(\frac{s}{t})^n}{1-\frac{s}{t}}
>arms^mt^{m+n-1} \frac{1-(\frac{s}{t})^n}{1-\frac{s}{t}}.\]   
Therefore, it is  sufficient to show that    
\begin{equation}   
(m+4n)s^n-arm<0. \label{condf}   
\end{equation}   
It is easy to check that   
(\ref{condf}) holds for $s=s^*$. Consequently,    
the sum of the last three terms of (\ref{a3}) is  negative for
every $0\le t \le r^{1/n}$, and $s=s^*$.   
Returning to (\ref{a3}), we are left to show that when $s=s^*$   
\begin{eqnarray*} 
g(t,s^*)&<& -2nt^{2m+2n-1}
\frac{1-(\frac{s}{t})^{n}}{1-\frac{s}{t}}  
+n(r-2s^n)t^{2m+n-1}  
 \frac{(1-\frac{s}{t})^{2m}}{1-\frac{s}{t}}\\  
\ & \ & -(m+2n)s^mt^{2n-1}
\frac{1-(\frac{s}{t})^n}{1-\frac{s}{t}}<0.  
\label{cond2} 
\end{eqnarray*} 
Let    
\begin{equation}   
h(t,s)=-2t^{2m}-2s^{2m}+(r-2s^n)\frac{t^{2m}-s^{2m}}{t^n-s^n}.   
\label{h}   
\end{equation}   
It is easy to check that    
\[g(t,s)\le \frac{n(t^n-s^n)}{(t-s)}h(t,s).\]   
Therefore, it is sufficient to show that $h(t,s)<0$ for all   
$0 \le t \le r^{1/n}$ and for $s=s^*$. To this end, we need the   
following lemma.\\   
{\bf Lemma 2.}   
Let    
\[k(t,s)=\frac{t^{2m}-s^{2m}}{t^n-s^n}.   
\]   
Then, $\frac{\partial k }{\partial t}(t,s)>0 $ if and only if
$(2m-n)(t-s)>0$.\\   
{\bf Proof.}   
It can be easily verified that    
\[\frac{\partial k }{\partial t}(t,s)=   
-\frac{2mt^{2m-1}}{(t^n-s^n)}\left[\frac{n}{2mt^{2m-n}}   
\frac{t^{2m}-s^{2m}}{t^n-s^n}-1\right].\]   
By the mean value theorem   
\begin{equation}   
\frac{t^{2m}-s^{2m}}{t^n-s^n}=\frac{2m}{n}c^{2m-n},\label{mv}   
\end{equation}   
where $min(t,s)\le c \le max(t,s)$.   
Therefore   
\[\frac{\partial k }{\partial t}(t,s)=   
-\frac{2mt^{2m-1}}{(t^n-s^n)}\left[(\frac{c}{t})^{2m-n}-1\right],\]
and it is 
 now easy to verify that the condition of Lemma 2 holds.\\  
We will use Lemma 2 to prove that $h(t,s)<0$ for every $0\le t \le
r^{1/n}$ and $s=s^*$. First observe that by (\ref{h})    
\begin{equation}   
h(t,s) \le \left[(r-2s^n)   
\frac{t^{2m}-s^{2m}}{t^n-s^n}-2s^{2m}\right]. 
\label{case1}   
\end{equation}\\   
Consider the following four cases.   
   
   
\noindent {\bf Case 1.}   
$2m-n<0$ and $t<s^*$. \\   
By Lemma 2, $\frac{\partial k}{\partial t}(t,s)>0$. Hence, it is
sufficient to show   
that the right-hand side of (\ref{case1}) is negative at $t=s^*$.
Indeed for $s=s^*$   
\[h(s,s)\le \frac{2m}{n}(r-2s^n)s^{2m-n}-2s^{2m}=s^{2m}\left[   
\frac{2m}{n}\frac{(r-2s^n)}{s^n}-2\right].\]   
Using (\ref{sstar}) , we just need to show that   
\begin{equation}   
\frac{2m}{n}\frac{(2an-m)}{(m+2n-an)}-2<0,\label{c1} \end{equation}
holds for all $m$ and $n$ such that $2m-n<0$.   
It is sufficient to show that this is true for $a=1$ since   
the left-hand side of (\ref{c1}) is increasing in $a$.  The last
inequality   
holds if and only if   
\[m^2+n^2>mn.\]   
Clearly this is true for all $m,n>0$; hence $h(t,s^*)$  is negative
for $t<s^*$.   
   
\noindent {\bf Case 2.}   
$2m-n>0$ and $t<s^*$. \\   
Again by Lemma 2, $\frac{\partial k}{\partial t}(t,s)<0$. Thus, we
need to show that    
the right-hand side of (\ref{case1}) is negative at $t=0$. Indeed
for $s=s^*$   
\begin{equation}   
h(0,s)\le (r-2s^n)s^{2m-n}-2s^{2m}<s^{2m}\left[   
\frac{(r-2s^n)}{s^n}-2\right].\label{c2}   
\end{equation}   
The right hand side of  (\ref{c2}) is negative at $s=s^*$ if and
only if \[\frac{2an-m}{m+2n-an}-2<0,\] for all $m$ and $n$ such
that $2m-n>0$. This is certainly true for $a=1$ and for all
$m,n>0$.   
 
 
\noindent {\bf Case 3.}   
$2m-n<0$ and $t>s^*$.\\   
Since $\frac{\partial k}{\partial t}(t,s)<0$, we need to show that
the right-hand side (\ref{case1}) is negative at $t=s^*$. For
$s=s^*$ \[h(s,s)\le \frac{2m}{n}
(r-2s^n)s^{2m-n}-2s^{2m}=s^{2m}\left[
\frac{2m}{n}\frac{(r-2s^n)}{s^n}-2\right],\] and this was shown to
be negative in Case 1.   
 
 
\noindent {\bf Case 4.}   
$2m-n>0$ and $t>s^*$. In this case, by (\ref{h})   
\begin{equation}   
h(t,s) \le \left[(r-2s^n)   
\frac{ t^{2m}-s^{2m}}{t^n-s^n}   
2t^{2m}\right].            
\label{case4}   
\end{equation}   
By (\ref{mv}), for all $s<t$ there exists $c$, $s \le c \le t$,
such that   
\begin{equation}   
\frac{t^{2m}-s^{2m}}{t^n-s^n}=\frac{2m}{n} c^{2m-n}.\label{c5}   
\end{equation}   
If  $2m-n>0$, $c^{2m-n}$ increases in $c$.  Therefore,    
whenever $s<t$,    
\[c^{2m-n}<t^{2m-n}<\frac{t^{2m}}{s^n}.\]   
Then by (\ref{case4}) and (\ref{c5}) we have   
\begin{equation} 
\label{38} 
h(t,s)\le (r-2s^n)\frac{2m}{n}t^{2m}s^{-n}-2s^{2m}=t^{2m}\left[   
\frac{2m}{n}\frac{(r-2s^n)}{s^n}-2\right]. 
\end{equation} 
The right-hand side of (\ref{38}) is negative if and only if      
\[\frac{2m}{n}\frac{(r-2s^n)}{s^n}-2<0,\]   
holds for $s=s^*$. But in Case 1, it was shown that this condition
is satisfied.   
This completes the proof of Lemma 1.   
   
\noindent    
By Lemma 1 and (\ref{P}), $P(t,s^*)>0$ whenever $0\le t \le s^*$  
and $P(t,s^*)<0$ whenever $s^* \le t \le r^{1/n}$. Therefore,   
$t=s^*$ is the unique maximizer of   
$\Pi_1(t,s^*)$ and hence $\alpha^*$ is the unique   
maximizer of $\Pi_1(\alpha,\beta^*)$. This completes the proof of
Theorem 1.    
   
   
\end{document}   
   
\\   
   





