POSITIVE AND NEGATIVE
EXTERNALITY EFFECTS
ON PRODUCT PRICING
AND CAPACITY PLANNING
a dissertation
submitted to the department of engineering-economic systems
and the committee on graduate studies
of stanford university
in partial fulfillment of the requirements
for the degree of
doctor of philosophy
Susan Y. Chao
June 1996
c Copyright by Susan Y. Chao 1996
All Rights Reserved
ii
I certify that I have read this dissertation and that in
my opinion it is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Samuel Chiu
(Principal Advisor)
I certify that I have read this dissertation and that in
my opinion it is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Donald Dunn
I certify that I have read this dissertation and that in
my opinion it is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
John Weyant
Approved for the University Committee on Graduate
Studies:
iii
Abstract
Physically constrained subscription-based telephone network services can experience
opposing market forces which aect new product adoption. In such networks, a
positive externality due to increases in subscribership encourages more consumers to
sign up. As a result, the addition of users to the system then leads to an increase
in network load (measured in call minutes for the entire system). At some point,
call demand exceeds network capacity and subscribers are forced to wait for call
completion. This translates to a negative externality in the form of congestion and not
only reduces the consumption by current customers but also discourages subscriber set
expansion. These concurrent positive and negative externalities ultimately determine
demand dynamics, given subscriber attitudes and pricing changes.
A typical example of these subscription-based services can be found in the mobile
communications industry. In the past few years, metropolitan cellular telephone has
been plagued by massive congestion, the result of inadequate network capacity planning. As a result, industry experts are addressing the problem by devising schemes
to increase capacity for the allotted bandwidths. The emerging Personal Communications System industry can benet from this experience by predicting the evolution
of the customer base and recognizing the industry's inherent contrasting dynamics
and then optimizing pricing decisions and capacity expansions accordingly.
This research presents a general model of markets with positive and negative
externalities. Intended to be used as an analytical tool by company executives, consumer watchdog groups, or governmental regulatory bodies, the model maximizes
a weighted sum of company prot and Consumer Surplus. Through the use of the
Calculus of Variations and other optimization techniques, analytical conclusions are
iv
drawn about the trajectory of prices over time and the value of increasing capacity as
functions of consumer preferences, company costs, and available capacity. Examples
are solved numerically using MathematicaT M . These examples lead to observations
about optimal capacity and pricing decisions and about tradeos between revenue
function complexity and company prot.
v
Acknowledgments
First and foremost, I would like to thank the members of my reading committee,
Professors Samuel Chiu, Donald Dunn, and John Weyant. The time they invested
in making suggestions, giving advice, reading drafts, and attending the oral exam
was very much appreciated. Thank you also to the other two members of my orals
committee, Professors Michael Fehling and Bruce Lusignan.
In particular, I would like to commend Professor Chiu on the outstanding job he
did as my advisor. I would be hard-pressed to nd another faculty member at Stanford
who would be so generous with his time, support (both nancial and intellectual),
and wit. Certainly my experience at Stanford has greatly benetted from association
with him.
I would also like to thank various people in the Department of EngineeringEconomic Systems who have aided me over the years: Rosalind Morf, Danielle Herrmann, Susan Clement, Maggie Barstow-Taylor, and Nancy Florence. Their help in
avoiding administrative and bureaucratic mishaps minimized the bumps on my road
to scholastic success.
The original inspiration for this work came from a research opportunity provided
by Mr. Takahiro Ozawa and all of the other very hospitable folks at Info-Com Research Ltd. in Tokyo, Japan. I will always be extremely grateful for their generosity
and kindness which made my stay in Tokyo so enjoyable.
Thank you to my mother and father who have encouraged me to achieve what I
really desire and also to acknowledge that goals can change over time. How fortunate
I have been that they instilled in me from early on such a deep appreciation for
education. Thanks also to family members and friends who extended their unwavering
vi
support even when graduate school appeared to be a life-long vocation.
Finally, I would like to say a loving thank you to my husband, Earl Levine. I
am forever indebted to him not only for the many hours he spent aiding me in the
drafting of this thesis through his typing, LATEX ing, and gure generation, but also
for the joy and meaning he has brought to my life.
vii
Contents
Abstract
iv
Acknowledgments
vi
1 Introduction
1
2 The Market
8
1.1 Scope of Study : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
1.2 Overview : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
2.1 The Consumer : : : : : : : : : : : : : : : : : : :
2.1.1 Consumer Behavior : : : : : : : : : : : : :
2.1.2 Willingness to Pay : : : : : : : : : : : : :
2.1.3 Consumer Surplus Maximization : : : : :
2.1.4 System Dynamics : : : : : : : : : : : : : :
2.1.5 Congestion : : : : : : : : : : : : : : : : :
2.2 The Producer : : : : : : : : : : : : : : : : : : : :
2.2.1 Provider Characteristics : : : : : : : : : :
2.2.2 Costs and Capacity : : : : : : : : : : : : :
2.2.3 Taris : : : : : : : : : : : : : : : : : : : :
2.2.4 Producer Surplus Maximization : : : : : :
2.2.5 Optimal Consumption for Dierent Taris
2.3 The Regulator : : : : : : : : : : : : : : : : : : : :
viii
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3 The Static Problem
3.1
3.2
3.3
3.4
3.5
Introduction : : : : : : : : : : : : : :
The Flat-Rate and Two-Part Taris :
The Three-Part Tari : : : : : : : : :
The Flat-Rate/Two-Part Tari : : :
The Nonlinear Tari : : : : : : : : :
3.5.1 Problem Formulation : : : : :
3.5.2 Necessary Conditions : : : : :
3.5.3 Static Nonlinear Solution : : :
3.5.4 Results : : : : : : : : : : : : :
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4 The Dynamic Problem
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5 Examples
88
4.1 Subscriber Set Growth Function : : : : : : : : : : : : : : : : : : : : :
4.2 The Two-Part Tari : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.3 The Nonlinear Tari : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.1 Overview : : : : : : : : : :
5.2 Static Examples : : : : : :
5.2.1 Assumptions : : : :
5.2.2 Numerical Results
5.3 Dynamic Examples : : : :
5.3.1 Assumptions : : : :
5.3.2 Numerical Results
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6 Conclusions and Extensions
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6.1 Conclusions : : : : : : : : : : : : : : : :
6.1.1 General Conclusions : : : : : : :
6.1.2 Numerical Example Observations
6.2 Extensions : : : : : : : : : : : : : : : : :
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A Appendix
119
Bibliography
124
A.1 Regulated Rate of Return : : : : : : : : : : : : : : : : : : : : : : : : 119
A.2 Special Case: No Externalities : : : : : : : : : : : : : : : : : : : : : : 121
x
List of Tables
5.1 Flat-Rate, Two-Part, Three-Part, and Nonlinear Results if K = T ,
k0 = T w5 o , k1 = w4o , and = 0, without Congestion Eects : : : : : : : 91
5.2 Flat-Rate, Two-Part, Three-Part, and Nonlinear Results if K = K =
2:5T , k0 = T 5wo , k1 = w4o , and = 0, with Congestion Eects : : : : : 95
5.3 Flat-Rate, Two-Part, Three-Part, and Nonlinear Results if K = K s,
k0 = T w5 o , k1 = 0, and = 0 : : : : : : : : : : : : : : : : : : : : : : : 96
5.4 Flat-Rate, Two-Part, Three-Part, and Nonlinear Results if K = K s,
k0 = T w5 o , k1 = w4o , and = 0 : : : : : : : : : : : : : : : : : : : : : : 99
5.5 Flat-Rate Dynamic Results for Increasing K when k0 = wo5T , k1 = w4o ,
and = 0 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 101
6.1
6.2
6.3
6.4
Summary of Analytical Conclusions : : : : : : :
Summary of Analytical Conclusions, Cont. : : :
Summary of Observations from Examples : : : :
Summary of Observations from Examples, Cont.
xi
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113
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116
List of Figures
1.1 Model Overview : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
6
Subscribers vs. Non-Subscribers : : : : : : : : : : : : : : : : : : : : :
Willingness to Pay for Consumer : : : : : : : : : : : : : : : : : : :
Willingness to Pay for Dierent Consumers : : : : : : : : : : : : : : :
Willingness to Pay with Dierent Congestion Levels : : : : : : : : : :
Willingness to Pay for Dierent Subscription Set Sizes : : : : : : : :
Consumer Equilibrium : : : : : : : : : : : : : : : : : : : : : : : : : :
Consumer Surplus for the Marginal Subscriber : : : : : : : : : : : : :
Flat-Rate Tari : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
Two-Part Tari : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
Three-Part Tari : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
Optimal Tari Selection : : : : : : : : : : : : : : : : : : : : : : : : :
Flat-Rate/Two-Part Tari : : : : : : : : : : : : : : : : : : : : : : : :
Finding Consumption with the Three-Part Tari Using Marginal Willingness to Pay : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
Consumption With the Three-Part Tari When Only R1 is Used : : :
Consumption With the Three-Part Tari When Only R2 is Used : : :
Consumption With the Three-Part Tari When Neither R1 Nor R2 is
Used : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
Consumption With the Three-Part Tari When R1 and R2 Are Used
Optimal Consumer Usage for the Three-Part Tari : : : : : : : : : :
9
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21
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
xii
23
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2.19 Consumption With the Flat-Rate / Two-Part Tari When Only the
Flat Cost Portion is Used : : : : : : : : : : : : : : : : : : : : : : : :
2.20 Consumption With the Flat-Rate / Two-Part Tari When Only the
Variable Cost Portion is Used : : : : : : : : : : : : : : : : : : : : : :
2.21 Consumption With the Flat-Rate / Two-Part Tari When Both the
Flat and Variable Cost Portions are Used : : : : : : : : : : : : : : : :
2.22 Optimal Consumer Usage for the Flat-Rate / Two-Part Tari : : : :
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
5.17
5.18
5.19
5.20
5.21
5.22
5.23
28
29
29
30
x and p for Dierent Taris, without Congestion Eects : : : : : : : 91
CS and PS for Dierent Taris, without Congestion Eects : : : : : : 92
Optimal Taris, without Congestion Eects : : : : : : : : : : : : : : 92
x and p for Dierent Taris, with Congestion Eects : : : : : : : : : 93
CS and PS for Dierent Taris, with Congestion Eects : : : : : : : 93
Optimal Taris, with Congestion Eects : : : : : : : : : : : : : : : : 94
Comparison of Optimal Capacities, = 0, K1 = 0 : : : : : : : : : : : 95
Flat-Rate Tari Optimal Capacity Comparison, = 0, K1 = w4o : : : 97
Two-Part Tari Optimal Capacity Comparison, = 0, K1 = w4o : : : 98
Three-Part Tari Optimal Capacity Comparison, = 0, K1 = w4o : : 98
Nonlinear Tari Optimal Capacity Comparison, = 0, K1 = w4o : : : 99
Fixed Tari for Varying Maximum Capacity : : : : : : : : : : : : : : 102
CS and PS for Varying Maximum Capacity : : : : : : : : : : : : : : : 102
x, d , p, and K for the No Congestion Case : : : : : : : : : : : : : : 103
CS, PS, and TS for the No Congestion Case : : : : : : : : : : : : : : 104
Consumption Trajectory for the No Congestion Case : : : : : : : : : 105
x, d , p, C , Total Usage, and K for the Congestion Case, K = K = T 106
CS, PS, and TS for the Congestion Case, K = K = T : : : : : : : : : 106
Consumption Trajectory for the Congestion Case, K = K = T : : : 107
x, d , p, C , Total Usage, and K for the Congestion Case, K = K = 4T 108
CS, PS, and TS for the Congestion Case, K = K = 4T : : : : : : : : 108
Consumption Trajectory for the Congestion Case, K = K = 4T : : : 109
x, d , p, C , Total Usage, and K for the Congestion Case, K = K = 10T 110
xiii
5.24 CS, PS, and TS for the Congestion Case, K = K = 10T : : : : : : : 110
5.25 Consumption Trajectory for the Congestion Case, K = K = 10T : : : 111
xiv
Chapter 1
Introduction
Suppose for the moment that you are new to an area and would like to go out
for dinner. You decide to drive around and scope out the choices before making a
decision. You pass by Restaurant A which looks full, with patrons milling about on
the sidewalk in front waiting for their turn to go in. Restaurant B, on the other
hand, has only a couple of people inside. Quickly deducing that the quality of food is
probably higher at Restaurant A than Restaurant B, you park and go into Restaurant
A. You then learn that the wait will be 2 hours. Unfortunately, you're too hungry to
wait 2 hours and since you surmise that Restaurant B's food isn't good enough for
the sit-down dinner price, you stop at Taco Box on the way home for a cheap meal.
Imagine now that you have nally subscribed to an Internet service. You nd
it so useful in saving on long-distance phone bills, speeding up letter delivery, and
looking up information, that you convince your friends and family to join you on the
Information Superhighway. Interestingly enough, though, as time goes on, you notice
that your email is taking hours or days to be delivered and service is slowing down
overall. Who's the culprit? Well, you are, along with others.
Finally, let's say you have a cellular phone. It has greatly aided communication
between you and your clients and colleagues. In fact, the more competitors, clients,
and colleagues that rely on the service, the more you value your phone. The problem
is that, more and more, either your calls are getting dropped or else you can't place
a call altogether.
1
Chapter 1: Introduction
2
As you may have already guessed, the scenarios just described to you are examples
of markets which have the opposing forces of positive and negative externalities.
The positive externalities encourage the potential consumer to purchase a particular
product. The negative externalities, on the other hand, tend to dissuade someone from
consuming the same product. If you happen to be the restaurant owner or Internet
or cellular service provider described above, you would probably like to know how
to best price your product and increase your business's capacity to make the most
money.
This research will therefore address the optimization model which was developed
to assess product pricing and capacity planning in markets with positive and negative externalities. The providers for these markets are assumed to be monopolistic
and, furthermore, the services are assumed to have no substitutes. Various model
details (e.g. revenue functions and producer costs) have been tailored to the Personal
Communications Service (PCS) or cellular service examples; however, the model also
maintains enough generality to accommodate other implementations.
1.1 Scope of Study
Markets with positive demand externalities are rst studied in one form or another
by the following. Artle and Averous [1973] investigate the telephone service paradigm
in which the basic issue is access or no access. Squire [1973] studies the positive externalities arising from the telephone system for which consumers do not directly pay.
Rabenau [1974] applies Artle and Averous [1973] to the urban growth issue. Communication demand externalities, new services, and the equilibrium subscriber set
are explored in Rohlfs [1974]. Littlechild [1975] studies the eects of the interactions
of the two-part tari and consumption externalities. Finally, while both Oren and
Smith [1981] and Oren, Smith, and Wilson [1982] present a case study which forms
the basis for the examples presented later in this research, the former source studies
multi-part tari results whereas the latter concentrates on results using a nonlinear
tari.
Chapter 1: Introduction
3
Literature on dynamic pricing begins with an analysis of product diusion and
moves into the importance of dynamic pricing along with optimality conditions under
various conditions. Product growth or diusion is discussed by Bass [1967]. New
product adoption is investigated by Midgley [1976] and Mahajan and Muller [1979].
Robinson and Lakhani [1975] demonstrate the importance of dynamic price planning
in optimizing long-term prots. Optimal dynamic pricing and new product planning
are studied in Dolan and Jeuland [1981] and [1982].
Various sources were used for the basis of pricing structures in this research. Leland and Meyer [1976] investigate consumer discrimination with nonuniform pricing.
In Willig [1978], nonlinear pricing and Pareto superiority in terms of prot and consumer preference are studied. Roberts [1979] discusses nonlinear pricing and welfare
analysis. Monopoly pricing is covered in Brito and Oakland [1980] and Mirman and
Sibley [1980]. Nonlinear or nonuniform pricing is the subject of Goldman, Leland
and Sibley [1984] and Oren, Smith, and Wilson [1983]. Oren, Smith, and Wilson
[1984] investigate pricing similar or overlapping products and applications to market
segmentation. Finally, a thorough study of various tari schedules used in utility
pricing is presented in Brown and Sibley [1986].
Information from Harte [1994] on PCS/cellular service costs including those due to
licensing, equipment, operations, sales and marketing, and nancing, form the basis
for the cost functions assumed in this work.
The static nonlinear tari problem and the dynamic problems all depend on the
Calculus of Variations or, equivalently, optimal control for their solutions. General
discussions on these topics which proved to be immensely instructional include Gregory and Lin [1992] and Kamien and Schwartz [1991]. This research built upon
and extended the results presented in these and other works to accommodate the
characteristics of the static and dynamic market with positive and negative demand
externalities. In addition, MathematicaT M was used extensively to numerically solve
examples; Wolfram [1988] provided discussions on the algorithms and techniques used.
Dhebar [1983], Dhebar and Oren [1986], and Dhebar and Oren [1985] treat problems which integrate demand dynamics, nonlinear pricing, and eects from a positive
Chapter 1: Introduction
4
externality. Careful attention in particular is paid to market equilibria and critical
mass. Xie and Sirbu [1995] take this work one step further by considering markets with multiple providers, by drawing on game theory to compare monopoly and
duopoly results.
Finally, this research draws largely upon work by Dhebar and Oren but makes
one signicant distinction: it investigates markets which exhibit both positive and
negative externalities. As discussed in Marshall [1995] and Piller [1994], congestion
has become a powerful force in many physically constrained markets. Current examples include the Internet and cellular service and perhaps PCS in the future. Note
that congestion in these systems becomes a factor long before the positive externality
eects of new Internet users or cellular service customers wears o, hence posing the
need for concurrent analysis of the two externalities. Emphasis is placed on deriving
results for the static problem under varying conditions; results are also derived for
the dynamic problem.
This research models markets with positive and negative externalities and their
resulting system dynamics to enable the decision-maker to determine an optimal
pricing and capacity strategy. In nding the optimal xed and variable price schemes,
the decision-maker might need to consider a variety of environmental factors as well
as overall objectives. Several such issues which are addressed include:
Regulatory policies: In markets characterized by the allocation of nite nat-
ural resources (e.g. the electromagnetic spectrum), the redistribution of such
resources can profoundly aect system capacity and its resultant congestion.
Technology changes: Markets with binding physical constraints benet from
technological innovations which increase capacity. Examples of these innovations in the cellular market which increase virtual capacity for a given spectrum
allocation include Time Division Multiple Access (TDMA), Frequency Division
Multiple Access (FDMA), and Code Division Multiple Access (CDMA).
Objectives: Depending on the optimizer, the objective can be formulated as
Chapter 1: Introduction
5
Consumer Surplus maximization, prot (Producer Surplus) maximization, regulated prot maximization, or weighted Producer-Consumer Surplus maximization. All of the above can be analyzed for short-term to long-term optimization
periods.
The approach to solving the optimization problem is summarized in Figure 1.1.
The black box represents the optimization model. The inputs to the black box include
some representation of consumer preferences, the regulatory environment including
the prevailing spectrum allocations and pricing restrictions, and technology in terms
of actual system implementation. Subject to these inputs, consumers will then make
some decisions, namely \should I consume" and, if the answer is yes, \how much
should I consume". At the same time that their decisions depend on the level of
congestion and the number of other subscribers, their decisions determine these two
variables. So within the optimization problem's black box is a separate equilibrium
problem. The black box's output will include optimal values for the number of subscribers, their individual consumption levels, the level of congestion, and the optimal
taris.
1.2 Overview
Chapter 2 introduces the players in the market aected by positive and negative
externalities, namely the consumer, the producer, and the regulator. Section 2.1 discusses consumer behavior including purchasing heterogeneity, willingness to pay, and
Consumer Surplus optimization. The concept of congestion is also dened. Producer
costs and revenue structures are covered in Section 2.2. Discussion of revenue structures extends to depict consumer buying patterns for the at-rate tari, the two-part
tari, the three-part tari, and the at-rate/two-part tari. Finally, in Section 2.3,
the notion of Weighted Surplus is introduced, given that some regulatory body, corporate executive, or consumer watchdog group would like to optimize system values
in order to maximize producer prot, Consumer Surplus, or a weighted average of
the two.
Chapter 1: Introduction
6
Consumer
Preferences
Regulatory Policy: Technology:
Spectrum Allocation
Pricing Regulation
Congestion
AMPS,
CDMA,
FDMA, etc.
Number of
Subscribers
Consumer
Decision-Making:
Subscribe?
Consume how much?
Optimal values:
Tari, Number of Subscribers,
Consumer Consumption Levels, Congestion
Figure 1.1: Model Overview
Chapter 1: Introduction
7
Chapter 3 presents complete analytical solutions to the static Weighted Surplus
maximization problem. First order necessary conditions are given for the two-part,
three-part, and at-rate/two-part taris. In Section 3.5, the static nonlinear tari
problem is solved using the Calculus of Variations.
The dynamic problem is solved in Chapter 4. Complete solutions are derived for
the general nonlinear tari problem as well as the two-part tari problem.
Chapter 5 presents numerical results from examples of markets exhibiting positive
and negative externalities. Static solutions to examples using the at-rate, two-part,
three-part, and nonlinear taris are given. Next, a dynamic example using the atrate tari is solved for dierent values of a particular system parameter (maximum
capacity). Trajectories for all system values are given.
Finally, Chapter 6 summarizes conclusions from both analytical and numerical
results and suggests ideas for further work.
Chapter 2
The Market
2.1 The Consumer
2.1.1 Consumer Behavior
The consumers in this research are assumed to be distinguishable only by income
or propensity to purchase. Depending on income level and various external factors,
consumers have an observable maximum amount each is willing to pay for service.
Assume that we can reverse-order them in terms of income or maybe even likelihood
of service consumption. Therefore, without loss of generality, rank the consumers
uniformly and continuously on an index, , from 0 to 1, as shown in Figure 2.1. The
consumer with the highest salary, or the greatest likelihood of purchasing the product,
has index equal to 0, implying that customers will sign up for service in order of
their index. So, for example, the customer with equal to 1 will sign up last, if at
all.
2.1.2 Willingness to Pay
The willingness to pay function which reects both positive and negative externalities
diers from that of the positive externality only case with the following observations
in mind:
8
Chapter 2: The Market
9
Non-Subscribers
Subscribers
0
x
Higher income /
More likely to consume
1
Lower income /
Less likely to consume
Figure 2.1: Subscribers vs. Non-Subscribers
W (q)j
qmaxj
q
Figure 2.2: Willingness to Pay for Consumer Assumption 2.1. The willingness to pay function, W (q; ; x; C ), is dierentiable
over x, , C , and q. x represents the total number of subscribers, q() is the amount
customer consumes, and C is the level of congestion.
Assumption 2.2. Given some subscriber set size and congestion level, consumer will exhibit a willingness to pay which is non-decreasing over q up until q = qmaxj , that
consumer's maximum consumption level or saturation consumption. For q > qmaxj ,
willingness to pay is non-increasing. See Figure 2.2.
Chapter 2: The Market
10
W (q)jx;C
1 < 2 < 3
Wq j1 > Wq j2 > Wq j3
1
2
3
qmaxj3 qmaxj2 qmaxj1
q
Figure 2.3: Willingness to Pay for Dierent Consumers
Assumption 2.3. Consumer willingness to pay is non-increasing over when the
number of subscribers and congestion level are held constant, as shown in Figure 2.3.
@W (q; ; x; C ) 0
@
Assumption 2.4. Willingness to pay for any particular consumer, , is non-increas-
ing over C when x is held constant, as shown in Figure 2.4.
@W (q; ; x; C ) 0
@C
Assumption 2.5. Willingness to pay for consumer is non-decreasing over the
number of subscribers when C is held constant, as illustrated in Figure 2.5.
@W (q; ; x; C ) 0
@x
@W
Particular attention must be paid to @W
@x and @C . These partial derivatives are
precisely where the positive and negative externalities are realized. Because diners
will likely believe a popular restaurant's food is better and an Internet user values
Chapter 2: The Market
11
W (q)j;x
C1 < C2 < C3
Wq jx1 > Wq jx2 > Wq jx3
C1
C2
C3
q
Figure 2.4: Willingness to Pay with Dierent Congestion Levels
W (q)j;C
x1 < x2 < x3
Wq jx1 < Wq jx2 < Wq jx3
x3
x2
x1
q
Figure 2.5: Willingness to Pay for Dierent Subscription Set Sizes
Chapter 2: The Market
12
her connection more when she can reach more associates and friends, @W
@x is greater
than or equal to zero. Similarly, since a long restaurant wait encourages diners to go
elsewhere and congestion problems lower the value a cellular customer assigns to his
service, @W
@C is less than or equal to zero.
2.1.3 Consumer Surplus Maximization
In order to decide whether to subscribe and how much to consume, each customer
maximizes his/her Consumer Surplus.
Assumption 2.6. Consumers maximize their individual Consumer Surplus which is
dened as the dierence between their willingness to pay and the price they pay for
service.
CS (q; ; x; C ; R) = W (q; ; x; C ) ? R(q)
(2.1)
Proposition 2.1. Consumers will set their marginal willingness to pay equal to the
marginal tari rate in choosing an optimal consumption level.
Proof By the rst order necessary condition, dierentiate Consumer Surplus with
respect to q and set equal to 0.
@CS (q; ; x; C ; R) = @ W (q; ; x; C ) ? R(q)
(2.2)
@q
@q
= Wq ? Rq
Wq = Rq
(2.3)
(2.4)
The subscriber set size, x, can be found by locating the consumer whose willingness
to pay exactly equals his cost.
W (q(x); x; x; C ) = R(q(x)
(2.5)
Essentially, people will subscribe if their willingness to pay is greater than or equal
to their cost. Thus, in Figure 2.1, subscribers lie to the left of x and non-subscribers
lie to the right.
Chapter 2: The Market
Tari
13
How much
Should I consume?
Number of
Subscribers
Congestion
Figure 2.6: Consumer Equilibrium
2.1.4 System Dynamics
The dynamics of service/product adoption by consumers are as follows. Consider a
typical prospective consumer A. A rst evaluates the current pricing plan, number
of subscribers, her income, the quantity she would consume if she signed up, and
the total congestion of all consumers. If her willingness to pay exceeds the price she
would pay for her anticipated consumption, then she will subscribe. This evaluation
is carried on by all prospective consumers, as summarized in Figure 2.6.
In particular, the marginal subscriber B , who has a lower income than that of
A, will also consider the prevailing prices, number of customers, etc., leading to his
specic willingness to pay. In contrast to A, however, B will have a willingness to pay
that does not exceed his anticipated consumption cost; rather it equals it. Since it has
been assumed that consumers subscribe in reverse order of income, all prospective
consumers with income levels lower than that of B will not subscribe, given the
current pricing scheme and number of subscribers. Once the marginal consumer is
determined, the subscriber set has been established and consists of all consumers with
incomes no less than that of B .
The remaining prospective consumers reevaluate their willingnesses to pay using
the new total number of consumers and the subscription process reiterates. For a
particular pricing scheme, xed-point analysis can be used determine system conditions which will induce equilibrium values of consumption, subscriber set size, and
Chapter 2: The Market
?
14
CS q(); ; x; C; R j=x
0 A
Original Curve
Increase Congestion
Reduce Congestion and Rates
B0
A0
B
B00
1
= Consumer Index
x = Subscriber Set Size
x = Equilibrium x
C = Congestion
q() = Consumtion by CS = Consumer Surplus
R = Consumer Tari, Rate
Figure 2.7: Consumer Surplus for the Marginal Subscriber
congestion.
For any subscriber set size, let us now consider the last subscriber's Consumer
Surplus function, that is, the Consumer Surplus of marginal subscriber x given the
subscriber set size is x. The middle curve in Figure 2.7 corresponds to a typical
graph of the last subscriber's behavior. In it, you can observe the opposing forces of
positive and negative externalities, as well as the tendency for consumers with larger
to have a lower willingness to pay. For example, people between points A and B
have non-negative Consumer Surplus and will choose to subscribe. After point B,
however, the eects of congestion and increasing consumer index make it undesirable
for anyone to subscribe.
Imagine the subscriber set size is now A. If the person with just greater than
A also subscribes, then she will have a positive Consumer Surplus. For that matter,
everyone else up until B will also join. But if the subscriber set size is now B, the
person with just greater than B will not sign up because his Consumer Surplus
would be negative. Point B is the system's stable equilibrium and point A is the
unstable equilibrium. Point A can also be called a critical mass (see Dhebar [10] for
Chapter 2: The Market
15
more details) because it is the point which must be reached in order to attain the
next stable equilibrium.
Contrast this scenario with one in which our network capacity is suddenly reduced.
What would happen? Essentially the curve would shift downwards, creating new
equilibrium points. Notice that the stable equilibrium has decreased to point B0 .
Also, the critical mass has increased to A0 . This means that not only do we now have
to leap to a larger critical mass, but our potential equilibrium market size has shrunk.
Now, what would result if we decided to lower the rates and the congestion was
magically alleviated? The curve would shift upwards. If we reduced rates and congestion enough, in fact, we could push the eventual subscriber set size further out
towards the limit equals one. This could be the situation for a PCS in which we
would like to have as large a consumer base as possible, so would have to allow for
enough capacity and keep rates down.
2.1.5 Congestion
Dene congestion to be a relationship between the total usage by consumers and
system capacity, K .
Assumption 2.7. Congestion changes in response to the total number of consumers,
each customer's consumption, and system capacity. A congestion function adhering
to these restrictions could be
C=
Z x
0
q()d ? K .
(2.6)
Apply this congestion model to a highway example. Drivers generally perceive
freeway congestion as some function of the total road space taken up by vehicles.
Hence, x number of cars driving along a stretch of a four-lane highway creates a
higher level of congestion than the same number of cars on an eight-lane highway.
Furthermore, the worst congestion occurs when bumper-to-bumper conditions persist
all along the length of a road, regardless of the size of the freeway. Congestion is
alleviated when K , or the number of lanes, is increased.
Chapter 2: The Market
16
Note in particular that C has been designed to be nonpositive. Less severe congestion translates to more negative values for C . When capacity exactly meets the
total usage, C is zero. This condition will be called \critical congestion". One system experiencing critical congestion is the highway with constant bumper-to-bumper
trac. At rst glance, this may not appear to be an optimal solution, but results in
subsequent sections will show that under certain conditions (e.g. very low available
capacity) \critical congestion" is the best choice.
2.2 The Producer
2.2.1 Provider Characteristics
The rm is assumed to be a prot-maximizing entity which is subject to federal and
state regulation. The service provider charges each customer a quantity-sensitive
price, which is typical of the current revenue schemes for several telecommunications
services. Subscribers periodically pay a xed subscription fee to be able to consume
the product; their consumption quantity then determines an additional variable cost.
The provider maximizes the rm's Producer Surplus, or prot, which is dened as
the dierence between total revenues and total costs.
2.2.2 Costs and Capacity
This research lumps the rm's costs into three groups: those solely dependent upon
total consumed quantity, the total number of users, or system capacity. More specifically, the costs for a cellular service or PCS (see Harte [1994] for a more complete
list) could include:
1. Usage-sensitive costs: Local Exchange Carrier (LEC) taris, engineering, and
customer service.
2. Subscriber set size sensitive costs: billing, sales, and administration. etc.
Chapter 2: The Market
17
3. Capacity sensitive costs: switching equipment, leased communication lines, radio towers, radio equipment, installation and construction costs, FCC licensing,
and relocation of existing spectrum users.
Call the rst group of costs C1(q), the second group C2 (x), and the third group
C3 (K ). The total capacity available to the system, K , might depend on such variables
as technology, FCC regulations, and special environmental constraints. By design,
system capacity is constrained by the following equation.
K ? K 0
2.2.3 Taris
The service provider charges a consumer R(q) for product amount q. Tari R(q) can
take on many forms such as the at-rate, two-part , three-part, at/two-part, and
nonlinear taris, as described below.
The at-rate tari is quantity insensitive and only involves a xed charge, the
subscription rate. Figure 2.8 illustrates the at-rate tari.
R(q) = p ; q 0
(2.7)
The two-part tari allows for a xed charge plus a linear usage charge, as shown
in Figure 2.9.
R(q) = p + rq ; q 0
(2.8)
The three-part tari charges two dierent marginal tari rates, depending on the
level of consumption. The three-part tari is depicted in Figure 2.10 and then in
Figure 2.11. One representation of the three-part tari combines two two-part taris.
Tari R1 has a lower xed rate p1 but a higher marginal rate r1 whereas R2 has a
higher xed rate but a lower marginal rate r2 .
Chapter 2: The Market
18
R(q)
p
q
Figure 2.8: Flat-Rate Tari
R(q)
1
r
p
q
Figure 2.9: Two-Part Tari
Chapter 2: The Market
19
R(q)
1
1
p2
r1
r2
p1
q
q^
Figure 2.10: Three-Part Tari
R
Slope 1= r1
R
Slope 2= r2
p2
p1
q^
Use R1 Use R2
q
Figure 2.11: Optimal Tari Selection
Chapter 2: The Market
20
Proposition 2.2. Revenue function R1 is preferable for 0 q < q^ whereas R2 is
optimal for q q^.
Proof Consumers have the choice to pay R1 = p1 + r1q or R2 = p2 + r2q for some
q 0. For 0 q < q^, p1 + r1 q < p2 + r2 q, so R1 is the less expensive option. For
q q^, however, p1 + r1 q > p2 + r2 q, so R2 is the better alternative.
Hence, consumers will divide themselves into two groups: those who consume
more, q, but at marginal rate r2 , and those who consume less, q , at a marginal rate
r1 . Furthermore, we could also add a third tari to attract those with an even smaller
optimal consumption level. Of course, each addition of subscribers due to a new tari
adds to the congestion level, so really we have to reformulate the problem with an
n-part tari and solve for the resulting equilibrium values. Notice that the eective
tari here is composed of the lower curve segments due to subscriber self-selection, as
shown in Figure 2.11. If we added other tari choices ad innitum the result would
be the drawing out of the general nonlinear tari contour in segments.
The three-part tari is summarized as follows.
R(q) = p1 + r1 q ; 0 q < q^
= p2 + r2 q ; q q^
Or, equivalently, eliminate the subscript for the xed rate p1 , resulting in
R(q) = p + r1 q ; 0 q < q^
?
= p + r1 ? r2 q^ + r2 q ; q q^ .
(2.9)
(2.10)
The at-rate/two-part tari charges the same rate for consumption up to q = q^ and
adds a marginal rate for q > q^. Figure 2.12 illustrates the at-rate/two-part tari.
R(q) = p ; 0 q < q^
?
= p + r q ? q^ ; q q^
(2.11)
(2.12)
The last revenue function investigated in this research is the most general. The
nonlinear tari may include a xed subscription rate, p, in addition to its quantitysensitive portion, r(q).
Chapter 2: The Market
21
R(q)
1
r
p
q^
q
Figure 2.12: Flat-Rate/Two-Part Tari
2.2.4 Producer Surplus Maximization
The rm's prot, or Producer Surplus, can now be dened as the dierence between
revenues and costs. Revenues are integrated over all consumers from 0 to to get
total revenue. Usage-sensitive costs are also integrated over all consumers and added
to subscriber set sensitive costs and capacity sensitive costs to obtain the total cost
to the producer.
Producer Surplus =
Z x
0
[R(q) ? C1 (q)]d ? C2 (x) ? C3(K )
(2.13)
R(q) is any one of the revenue functions discussed in the last section: the at-rate
tari, the two-part tari, the three-part tari, the at-rate/two-part tari, or the
nonlinear tari.
2.2.5 Optimal Consumption for Dierent Taris
Given any one of the revenue functions described previously, consumers will choose
their optimal consumption levels for some x and C by maximizing Consumer Surplus.
This Consumer Surplus maximization eectively equates marginal willingness to pay,
Chapter 2: The Market
22
Wq (q; ; x; C ), with the marginal tari. Since the marginal tari is 0 for the at-rate
tari, all consumers up to x will purchase their saturation or maximum consumption.
The two-part tari consumer will set marginal willingness to pay equal to r.
The Consumer Surplus maximization process is a bit more complicated for the
three-part tari and at-rate/two-part tari, because consumers must choose between
tari parts.
Proposition 2.3. A unique consumer ^ exists who is indierent between revenue
functions R1 and R2 in the three-part tari.
Proof From Proposition 2.2, consumers divide themselves into those consuming q
using R2 and those choosing q at marginal rate r1 . From Proposition 2.1, This would
mean that Wq (q; ; x; C ) = r2 and Wq (q; ; x; C ) = r1 .
Figure 2.13 shows this with dots at the intersections of Wq and r1 for q and Wq and
r2 for q. Recall from Assumption 2.2 that marginal willingness to pay for consumer
is non-increasing over q. Thus for subscribers who use R2, the intersection dots
move to the left for increasing . For consumers who prefer R1 , the dots shift to the
right with decreasing . As the dots move towards the region bounded by q = qu and
q = ql, the preference between R1 and R2 decreases.
Since Assumption 2.1 asserts that willingness to pay is continuous over , there
exists a consumer ^ who is indierent between R1 and R2. This person's Consumer
Surplus is a follows:
W (qu; ^; x; C ) ? (p2 + r2 qu) = W (ql ; ^; x; C ) ? (p1 + r1 ql )
(2.14)
where qu and ql are given by Wq (qu; ^; x; C ) = r2 and Wq (ql; ^; x; C ) = r1 , respectively.
Consumers optimally choose their consumption level by setting marginal willingness to pay equal to marginal tari when Consumer Surplus is non-negative. The
following graphs illustrate under which conditions only portions of the three-part and
at-rate/two-part taris will be used. If, for example, p2 in the three-part tari is
set too high, the switching consumption q^ will be so large that consumers will never
want to consume in the R2 portion of the revenue function as seen in Figure 2.14.
Chapter 2: The Market
Wq
23
^
r1
r2
ql
use R1
q^
q
qu
use R2
Figure 2.13: Finding Consumption with the Three-Part Tari Using Marginal Willingness to Pay
Chapter 2: The Market
24
W (q)j
R1
R2
Use R1 Only
q^
q
Figure 2.14: Consumption With the Three-Part Tari When Only R1 is Used
Figure 2.15 shows the result of setting r1 too high. Consumption level q^ becomes relatively small and consumers immediately nd that R2 is the more attractive option.
If R1 and R2 are both simply priced too high for the market, no consumer will have
a willingness to pay which exceeds the cost of service, so no one will subscribe. See
Figure 2.16. Finally, both R1 and R2 may be set so that some consumers prefer the
former and others with smaller index and higher willingness to pay prefer the latter,
as shown in Figure 2.17.
Figure 2.18 shows the optimal consumer usage pattern when the revenue function
is the three-part tari, and, in particular, both portions of the revenue function
are used. If R1 is not selected by any consumer then the lower usage curve will
disappear. Similarly, too high an R2 would eliminate the upper usage curve. Note
that the indierence between R1 and R2 by consumer ^ causes a jump in the optimal
usage pattern at that consumer's index. This occurs because the marginal taris, r1
and r2 , are not equal and because consumers choose their optimal consumption level
by setting marginal willingness to pay equal to marginal tari.
Similar treatment can be given to the at-rate/two-part tari. If the consumer
Chapter 2: The Market
25
W (q)j
R1
R2
Use R2 Only
q
q^
Figure 2.15: Consumption With the Three-Part Tari When Only R2 is Used
W (q)j
R1
R2
=0
Use neither R1 nor R2
q
Figure 2.16: Consumption With the Three-Part Tari When Neither R1 Nor R2 is
Used
Chapter 2: The Market
26
W (q)j
R1
R2
Use R1 and R2
q^
q
Figure 2.17: Consumption With the Three-Part Tari When R1 and R2 Are Used
Chapter 2: The Market
27
q ()
qu
ql < q^ < qu
ql
q (x)
^
x
Figure 2.18: Optimal Consumer Usage for the Three-Part Tari
Chapter 2: The Market
28
W (q)j
R(q)
p
q^
q
Figure 2.19: Consumption With the Flat-Rate / Two-Part Tari When Only the Flat
Cost Portion is Used
marginal willingness to pay shown in Figure 2.19 never quite reaches the marginal
tari r when Consumer Surplus is non-negative, consumers will consume only in the
free usage portion p of the curve. Furthermore, if even the last consumer wishes to
purchase at least q^ , all subscribers will consume exactly q^.
If instead consumer marginal willingness to pay for the last consumer does reach
or exceed the marginal tari r, all customers will choose to consume in only the
variable portion of the revenue function. See Figure 2.20. Here, the xed tari p
is set too high. The revenue function in Figure 2.21 appeals to consumers in both
revenue regions. Finally, an optimal consumption pattern for the at-rate/two-part
tari is graphed in Figure 2.22. The at region 2 [^; ~] corresponds to consumers
who would like to consume at least q^ but whose marginal willingness to pay does not
exceed r. The lower curve represents those who consume in the at-rate portion of
the revenue function. The upper curve represents customers who choose to consume
in the variable rate portion of the revenue function.
Chapter 2: The Market
29
W (q)j
R(q)
p
q
q^
Figure 2.20: Consumption With the Flat-Rate / Two-Part Tari When Only the
Variable Cost Portion is Used
W (q)j
R(q)
p
q^
q
Figure 2.21: Consumption With the Flat-Rate / Two-Part Tari When Both the Flat
and Variable Cost Portions are Used
Chapter 2: The Market
30
q
q^
q~
q (x)
^
~
x
Figure 2.22: Optimal Consumer Usage for the Flat-Rate / Two-Part Tari
Chapter 2: The Market
31
2.3 The Regulator
This research has attempted, as was done in Dhebar [1983], to allow for a wide
spectrum of optimizers. In other words, the models presented in the next couple of
chapters are versatile enough to maximize quantities from Total Consumer Surplus to
prot to some weighted sum of the two. Hence, this model can be used by consumer
watchdog groups, company executives, and governmental regulatory bodies like the
Federal Communications Commission (FCC) or California Public Utility Commission
(CPUC).
The quantity which will be maximized is the Total Weighted Surplus. Let be
some value between 0 and 1, inclusive. Then Total Weighted Surplus is
Total Weighted Surplus = Total Consumer Surplus + (1 ? )Producer Surplus
or
WS = CS + (1 ? )PS.
A company executive interested in maximizing company prot would set equal
to 0. An FCC regulator wanting to equally weigh the benets of consumers with
those of producers would choose equal to 1/2.
The basic problem investigated in the next two chapters is the maximization of a
weighted sum of Consumer Surplus and company prot subject to some constraints:
the denition of congestion;
the description of the subscriber set size;
each subscriber chooses her optimal consumption quantity;
non-negative system variables; and
non-negative Producer Surplus.
The problem will vary as dierent revenue functions are implemented in both the
static and dynamic frameworks.
Chapter 3
The Static Problem
3.1 Introduction
3.2 The Flat-Rate and Two-Part Taris
The at-rate and two-part taris, as dened in the last chapter, will be discussed
here together. Because the at-rate tari is a special case of the two-part tari with
the marginal rate r equal to zero, results in this section are applicable to the atrate tari. The only exception is the omission of the rst order necessary conditions
(FONC) directly related to r.
Begin by dening the three surpluses for the two-part tari. Recalling that the
two-part tari is expressed as R(q) = p + rq, Total Consumer Surplus is the dierence
between total willingness to pay and the total revenues paid by all consumers.
CS =
=
Z x
Z0
0
x
[W ? R]d
[W (q; ; x; C ) ? rq]d ? px
The components of Producer Surplus include the total revenues along with the costs
32
Chapter 3: The Static Problem
33
due to consumer usage, subscriber set size, and capacity.
PS =
=
Z x
Z0
0
x
[R ? C1]d ? C2 (x) ? C3 (K )
[rq ? C1(q)]d + px ? C2 (x) ? C3(K )
Weighted Surplus is a combination of the Producer and Consumer Surpluses, as is
stated below for the two-part tari.
WS = CS + (1 ? )PS
=
Z x
0
[W (q; ; x; C ) ? (1 ? )C1 (q) + (1 ? 2 )rq]d
+ (1 ? 2 )px ? (1 ? )C2 (x) ? (1 ? )C3 (K )
(3.1)
The static two-part tari problem will maximize Equation 3.1 subject to the constraints in Equation 3.2. These include the denition of congestion as the dierence
between total usage and capacity, the upper limit to capacity, non-negativity constraints, description of the last consumer's zero Consumer Surplus, individual Consumer Surplus maximization FONC, and a non-negativity constraint for Producer
Surplus.
Z x
Z x
0
K + C ? qd = 0
0
K?K 0
C0
K0
x0
p0
r0
1?x 0
W (q(x); x; x; C ) ? p ? rq(x) = 0
Wq (q; ; x; C ) ? r = 0; 0 x
(3.2b)
(3.2c)
[rq ? C1(q)]d + px ? C2 (x) ? C3(K ) 0
(3.2d)
(3.2a)
Chapter 3: The Static Problem
34
Optimization of WS constrained by the equalities and inequalities in Equation 3.2
is equivalent to maximizing I (p; r; x; C; K ). Variables 1 through 8 are Lagrange
multipliers for the non-negativity constraints and 1 and 2 are the multipliers for
the equality constraints.
I (p; r; x; C; K ) = WS(p; r; x; C; K ) + 1 K + C ?
Z x
0
qd
+ 2[W (q; x; x; C ) ? p ? rq(x)] + 1 (K ? K ) ? 2 (C )
+ 3(K ) + 4 (x) + 5 (p) + 6(r) +
7 (1 ? x) + 8
Z x
0
(3.3)
[rq ? C1 (q)]d + px ? C2 (x) ? C3 (K )
FONC for the maximization of I (p; r; x; C; K ) require its rst-order partial derivatives
with respect to p, r, x, C , K , 1 , and 2 to be zero. Furthermore, Kuhn-Tucker
conditions must be satised for the non-negativity constraints and their multipliers.
0 = Ip = Ir = Ix = IC = IK = I1 = I2
0 = 1(K ? K ) = 2 (C ) = 3 (K ) = 4 (x) = 5(p) = 6 (r)
= 7(1 ? x) = 8
Z x
0
(3.4a)
[rq ? C1 (q)]d + px ? C2 (x) ? C3(K )
K ? K; ?C; K; x; p; r; 1 ? x; PS 0
1; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 0
(3.4b)
(3.4c)
(3.4d)
First solve for the partial derivative with respect to the xed price.
0 = Ip
=
Z x
0
[Wq qp ? (1 ? )C1q qp + (1 ? 2 )rqp ]d + (1 ? 2 )x
? 1
+ 8
Z x
Z0
qpd + 2 [Wq (x)qp (x) ? 1 ? rqp (x)] + 5
x
0
[rqp + C1q qp ]d + x
So the optimality condition becomes
Ip = 0 = (1 ? 2 )x ? 2 + 5 + 8 x .
(3.5)
Chapter 3: The Static Problem
35
Next nd the partial derivative with respect to the variable price.
0 = Ir
=
Z x
0
[Wq qr ? (1 ? )C1q qr + (1 ? 2 )q + (1 ? 2 )rqr ]d
? 1
Z x
Z0 x
qr d + 2 [Wq (x)qr (x) ? q(x) ? rqr (x)] + 6
[q + rqr ? C1q qr ]d]
1
=
(1 ? )(r ? C1q )
Wqq + (1 ? 2Z)q d
0 Z
x 1
x
? 1 W d ? 2 q(x) + 6 + 8 [q + (r ? C1q ) W1 ]d
qq
qq
0
0
+ 8 [
Z x 0
The optimality condition from partially dierentiating I (p; r; x; C; K ) with respect to
the system capacity is
0 = IK = ?(1 ? )C3K + 1 ? 1 + 3 ? 8(C3K ) .
(3.6)
Maximizing I over congestion leads to
0 =IC
=
Z x
0
[Wq qC + WC ? (1 ? )C1q qC + (1 ? 2 )rqC ]d
? 1
Z x
Z 0x
qC d ? 1 + 2 [Wq (x)qC (x) + WC (x) ? rqC (x)] ? 2
[rqC ? C1q qC ]d
Z x
W
W
qC
qC
=
?(1 ? )(r ? C1q ) W + WC d + 1 1 + W d +
qq
qq
0
0
Z x
qC
2 WC (x) ? 2 ? 8 [(r ? C1q ) W
(3.7)
Wqq ]d .
0
The next optimality condition requires the partial derivative of I with respect to
+ 8
Z x 0
Chapter 3: The Static Problem
36
the number of subscribers to be zero.
0 = Ix
=
Z x
0
[Wq qx + Wx ? (1 ? )C1q qx + (1 ? 2 )rqx ]d
+ [W (x) ? (1 ? )C1 (q(x)) + (1 ? 2 )rq(x)]
+ (1 ? 2 )p ? (1 ? )C2 x ? 1
Z x
qxd + q(x)
@q
(
x
)
@q
(
x
)
+ W (x) + Wx(x) ? r
+ 4
+ 2 Wq (x)
@x
@x
Z x
? 7 + 8 [rqx ? C1q qx]d + [rq(x) ? C1 (q(x))] + p ? C2x
0
Z x
W
qx
=
?(1 ? )(r ? C1q ) W + Wx d
qq
0
+ [W (x) ? (1 ? )C1 (q(x)) + (1 ? 2 )rq(x)]
Z x
W
qx
+ (1 ? 2 )p ? (1 ? )C2 x + 1
d ? q(x)
0 Wqq
+ 2 [W (x) + Wx(x)] + 4 ? 7
Z x
W
qx
+ 8 ? (r ? C1q ) d + rq(x) ? C1(q(x)) + p ? C2x
Wqq
0
The nal two optimality conditions (Kuhn-Tucker conditions listed in Equations
3.4b through 3.4d will not be repeated) merely recover the equality constraints.
0
I1 = 0 = K + C ?
Z x
0
qd
(3.8)
I2 = 0 = W (q; x; x; C ) ? p ? rq(x)
(3.9)
Proposition 3.1. The at-rate tari p will decrease over x if negative externality
eects outweigh those of the positive externality for the last consumer.
Proof: A larger contribution by the negative externalities (including index eects)
than by the positive externality for the last consumer implies that
Wx(q(x); x) W (q(x); x) + WC dC
dx
=x
.
Chapter 3: The Static Problem
37
Equation 3.9 gives
I2 = 0 = W (q; x; x; C ) ? p ? rq(x) .
(3.10)
Rearranging terms and dierentiating with respect to x yields
dp = d W (q; x; x; C ) ? rq(x)
dx dx
dC
dq
(x) dq
(x)
= Wq
dx + W + Wx + WC dx ? r dx =
=x
dC
W + Wx + WC dx =x
.
(3.11)
Hence,
dp 0 .
dx
Proposition 3.2. The marginal value of relaxing the constraint in Equation 3.2a
is related to the marginal cost of capacity and the marginal value of relaxing the
constraint restricting capacity to be smaller than the maximum available capacity.
Proof: From Equation 3.6,
0 = ?(1 ? )C3K + 1 ? 1 + 3 ? 8 (C3K ) .
The Lagrange multiplier for Equation 3.2a, 1 , has the interpretation of being the
marginal increase in the objective function due to a relaxation in the corresponding
equality constraint. Since the nontrivial case of K > 0 is assumed, 3 must be zero by
the Kuhn-Tucker condition for non-negative K . When Producer Surplus is optimized
( = 0), 8 is zero so the above expression becomes 1 = C3K + 1 . If some other
Weighted Surplus is desired ( 2 (0; 1]), the marginal value of relaxing Equation 3.2a
is equal to the marginal value of relaxing the constraint restricting capacity to be
smaller than the maximum available capacity plus the marginal cost of capacity times
Chapter 3: The Static Problem
38
the sum of (1 ? ) plus the marginal value of relaxing the non-negativity constraint
for Producer Surplus. In other words,
1 = (1 ? ) + 8 C3K + 1 .
Proposition 3.3. The marginal value of relaxing the constraint in Equation 3.2b is
related to the total number of subscribers, if the at rate p is positive.
Proof: From Equation 3.5,
0 = (1 ? 2 )x ? 2 + 5 + 8 x .
The Lagrange multiplier for Equation 3.2b, 2 , has the interpretation of being the
marginal increase in the objective function due to a relaxation in the corresponding
equality constraint. Since p is positive, 5 must be zero by the Kuhn-Tucker condition
for non-negative p. When Producer Surplus is optimized ( = 0), 8 is zero so
the above expression becomes 2 = x. If some other Weighted Surplus is desired
( 2 (0; 1]), the marginal value of relaxing Equation 3.2b is equal to the total number
of subscribers times the sum of (1 ? 2 ) plus the marginal value of relaxing the nonnegativity constraint for Producer Surplus. In eect,
2 = (1 ? 2 ) + 8 x .
Proposition 3.4. Assume the maximum available capacity K is not fully utilized
and Producer Surplus is the quantity to be maximized ( = 0). If an increase in
congestion leads to a positive change in the sum of Weighted Surplus and the two
equality constraints multiplied by their respective Lagrange multipliers, the system
will operate optimally at critical congestion, or C = 0.
Proof: Capacity does not equal the maximum available capacity.
K 6= K
Chapter 3: The Static Problem
39
Since the Kuhn-Tucker condition requires that
1 K ? K = 0 ,
1 must be zero. Setting = 0 in Equation 3.7 leads to
Z x
(r ? C1q )qC d + 1 1 ?
Z x
qC
]d .
2 + 8 [(r ? C1q ) W
W
qq
0
0=
Z x
0
0
qC d + 2 WC (x) ?
Because the objective is to maximize Producer Surplus, 8 equals zero by the KuhnTucker condition for non-negative Producer Surplus. Furthermore, for = 0, 1 =
C3K by Proposition 3.2 and 2 = x by Proposition 3.3. Note that 3 has been set to
zero because the nontrivial case of K 6= 0 has been assumed. The above expression
then becomes
2 =
Z x
0
(r ? C1q )qC d + C3K 1 ?
Z x
0
qC d + xWC (x) .
Finally, because an increase in congestion leads to a positive change in the sum of
Weighted Surplus and the two equality constraints multiplied by their respective Lagrange multipliers,
Z x
0
(r ? C1q )qC d + C3K 1 ?
Z x
0
qC d + xWC (x) 0
and so 2 > 0. Hence, critical congestion (C = 0) is optimal by the Kuhn-Tucker
condition for non-positive congestion.
Proposition 3.5. Assume that there is a positive value for relaxing the constraint
restricting capacity to be less than the maximum available capacity, that is, 1 > 0.
Furthermore, assume that Producer Surplus is the quantity to be maximized ( = 0).
If congestion is optimally set below critical congestion, 1 is equal to the change
in the sum of Weighted Surplus and the two equality constraints multiplied by their
respective Lagrange multipliers due to an increase in congestion, all divided by 1 minus
the change in total consumption by all users due to a change in congestion.
Chapter 3: The Static Problem
40
Proof: Congestion is set below critical congestion.
C0
Since the Kuhn-Tucker condition requires that
2 C = 0 ,
2 must be zero. Setting = 0 in Equation 3.7 leads to
0=
Z x
(r ? C1q )qC d + 1 1 ?
0
+ 8
Z x
0
[(r ? C1q )qC ]d .
Z x
0
qC d + 2 WC (x) ?
Because the objective is to maximize Producer Surplus, 8 equals zero by the KuhnTucker condition for non-negative Producer Surplus. Furthermore, for = 0, 1 =
C3K + 1 by Proposition 3.2 and 2 = x by Proposition 3.3. Note that 3 has been
set to zero because the nontrivial case of K 6= 0 has been assumed. Rearranging the
above expression then yields
1 =
1?
"Z x
1
R
x
0 qC d
C3K 1 ?
Z x
0
0
?(r ? C1q )qC d ?
#
qC d ? xWC (x) .
3.3 The Three-Part Tari
This section will address the three-part tari separately from the at-rate and twopart taris. The other two taris could have been treated as special cases to the
three-part tari; however, more direct results for the at-rate and two-part taris
were achieved without the added complications of q^, ^, q, and q found in the threepart tari.
Chapter 3: The Static Problem
41
Begin by dening the three surpluses for the three-part tari. Recall that the
three-part tari is expressed as
R(q) = p + r1 q ; 0 q < q^
?
= p + r1 ? r2 q^ + r2 q ; q q^
Total Consumer Surplus is the dierence between total willingness to pay and the
total revenues paid by all consumers.
CS =
=
=
Z x
0
Z ^
0
Z ^
0
[W ? R]d
[W (q) ? R2(q)]d +
[W (q) ? r2 q]d +
Z x
Z x
^
^
[W (q) ? R1(q)]d
[W (q ) ? r1 q ]d ? px ? (r1 ? r2 )^q^
The components of Producer Surplus include the total revenues along with the costs
due to consumer usage, subscriber set size, and capacity.
PS =
=
Z ^
0
Z ^
0
[R2 (q) ? C1 (q)]d +
[r2 q ? C1(q)]d +
Z x
Z x
^
^
[R1 (q) ? C1 (q)]d ? C2 (x) ? C3(K )
[r1 q ? C1 (q)]d
+ px + (r1 ? r2 )^q^ ? C2(x) ? C3 (K )
Weighted Surplus is a combination of the Producer and Consumer Surpluses, as is
stated below for the three-part tari.
WS = CS + (1 ? )PS
=
Z ^
0Z
+
[W (q ) + (1 ? 2 )r2 q ? (1 ? )C1 (q)]d
x
^
[W (q ) + (1 ? 2 )r1 q ? (1 ? )C1 (q)]d
+ (1 ? 2 )px + (1 ? 2 )(r1 ? r2 )^q^ ? (1 ? )C2 (x) ? (1 ? )C3 (K )
(3.12)
Chapter 3: The Static Problem
42
The static three-part tari problem will maximize Equation 3.12 subject to the constraints in Equation 3.13. They include the denition of congestion as the dierence
between total usage and capacity, the upper limit to capacity, non-negativity constraints, description of the last consumer's zero Consumer Surplus, individual Consumer Surplus maximization FONC, description of consumer ^'s indierence between
R1 and R2, and a non-negativity constraint for Producer Surplus.
K +C ?
Z ^
0
qd ?
Z x
^
qd = 0
K ? K 0
C0
K0
x0
p0
r1 0
r2 0
^ 0
q^ 0
1?x 0
x ? ^ 0
W (q; x; x; C ) ? p ? r1 q(x) = 0
Wq (q; ) = r2 ;
Wq (q; ) = r1
W (q; ^) ? [p + (r1 ? r2 )^q + r2 q] ? [W (q; ^) ? p ? r1 q] = 0
PS =
Z ^
0
[r2 q ? C1 (q)]d +
Z x
^
(3.13a)
(3.13b)
(3.13c)
(3.13d)
[r1 q ? C1 (q)]d
+ px + (r1 ? r2 )^q^ ? C2(x) ? C3(K ) 0
Optimization of WS constrained by the equalities and inequalities in Equation 3.13
is equivalent to maximizing I (p; r1 ; r2 ; x; C; K; ^; q^). Variables 1 through 12 are
Chapter 3: The Static Problem
43
Lagrange multipliers for the non-negativity constraints and 1 through 3 are the
multipliers for the equality constraints.
I = WS + 1 K + C ?
Z ^
0
qd ?
Z x
^
qd + 2 [W (q; x; x; C ) ? p ? r1 q(x)]
+ 3 [W (q; ^) ? (r1 ? r2 )^q ? r2 q ? W (q; ^) + r1 q] + 1 (K ? K ) ? 2 (C )
+ 3 (K ) + 4 (x) + 5 (p) + 6 (r1) + 7 (r2 ) + 8 (^) + 9 (^q) + 10 (1 ? x) (3.14)
Z ^
Z x
+ 11 (x ? ^) + 12 [r2 q ? C1(q)]d + [r1 q ? C1 (q)]d
0
^
+ px + (r1 ? r2 )^q^ ? C2 (x) ? C3(K )
FONC for the maximization of I (p; r1 ; r2 ; x; C; K; ^; q^) require its rst-order partial
derivatives with respect to p, r1 , r2 , x, C , K , ^, q^, 1 , and 2 to be zero. Furthermore,
Kuhn-Tucker conditions must be satised for the non-negativity constraints and their
multipliers.
0 = Ip = Ir1 = Ir2 = I^ = Iq^ = Ix = IC = IK = I1 = I2
0 = 1 (K ? K ) = 2 (C ) = 3(K ) = 4 (x) = 5 (p) = 6(r1 )
= 7 (r2 ) = 8 (^) = 9 (^q) = 10 (1 ? x) = 11 (x ? ^)
= 12
Z ^
0
[r2 q ? C1 (q)]d +
Z x
^
[r1 q ? C1 (q)]d
(3.15a)
(3.15b)
+ px + (r1 ? r2 )^q^ ? C2 (x) ? C3 (K )
K ? K; ?C; K; x; p; r1 ; r2 ; ^; q^; 1 ? x; x ? ^; PS 0
(3.15c)
1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 0
(3.15d)
First solve for the partial derivative of I with respect to the xed price.
Ip = 0 = (1 ? 2 )x ? 2 + 5 + 12 x
(3.16)
The optimality condition from partially dierentiating I with respect to the system
capacity is
IK = 0 = ?(1 ? )C3K + 1 ? 1 + 3 ? 12 C3K .
(3.17)
Chapter 3: The Static Problem
44
Partial derivatives of I with respect to ^ and q^ are
?
?
I^ = 0 = (1 ? ) r2 q(^) ? r1 q (^) ? C1 q(^) + C1 q(^) + (r1 ? r2 )^q
(3.18)
+ 1[?q(^) + q(^)] + 3 [W (q; ^) ? W (q; ^)] + 8 ? 10
+ 12[r2 q(^) ? C1 (q(^)) ? r1 q(^) + C1 (q(^)) + (r1 ? r2 )^q]
Iq^ = 0 = (1 ? 2 )(r1 ? r2 )^ ? 3 (r1 ? r2 ) + 9 + 12 (r1 ? r2 )^ .
Next nd the partial derivative with respect to the rst variable price.
Z x
(3.19)
(1 ? 2 )q + (1 ? ) r1 ? C1q (q) ? 1 1 d
Ir1 = 0 =
Wqq (q)
^
+ (1 ? 2 )^q^ ? 2 q(x) ? 3 q^ + 3 q(^) + 6
Z x
1
+ 12 [q + (r1 ? C1q (q))
Wqq (q) ]d + q^^
^
?
Now nd the partial derivative with respect to the second variable price.
Z ^
Ir2 = 0 =
(1 ? 2 )q + (1 ? ) r2 ? C1q (q) ? 1 1 d
Wqq (q)
0
? (1 ? 2 )^q ^ + 3 q^ ? 3 q(^) + 7
Z ^
1
+ 12 [q + (r2 ? C1q (q))
Wqq (q) ]d ? q^^
0
?
W
Maximizing I over congestion leads to
IC = 0 =
Z ^
0
Z x
?
WC (q) ? (1 ? ) r2 ? C1q (q) ?
?
(q)
1 WqC(
qq q )
W (q ) qC
d
WC (q) ? (1 ? ) r1 ? C1q (q) ? 1 W (q) d
qq
^
(3.20)
h
?
?
?
i
+ 1 + 2 WC q(x); x + 3 WC q(^); ^ ? WC q(^); ^ ? 2
"Z #
Z x
^
WqC (q) W
q
)
qC (
? 12
(r2 ? C1q (q))
Wqq (q) d + ^ [(r1 ? C1q (q)) Wqq (q) d .
0
+
Chapter 3: The Static Problem
45
The next optimality condition requires the partial derivative of I with respect to the
number of subscribers to be zero.
Z ^
W (
?
qx q )
Ix = 0 =
Wx (q) ? (1 ? ) r2 ? C1q (q) ? 1 W (q) d
qq
0
Z x
W (q) ?
+
Wx (q) ? (1 ? ) r1 ? C1q (q) ? 1 Wqx(q) d
qq
^
?
+ (1 ? ) r1 q(x) ? C1 q(x) + p ? C2(x) ? 1q (x)
?
?
h
?
i
?
+ 2 W q(x); x + Wx q(x); x + 3 Wx q(^; ^) ? Wx q(^); ^
+ 4 + 10 ? 11 ? 12
+
Z x
^
"Z ^
0
(3.21)
(r2 ? C1q (q)) Wqx(q) d
Wqq (q)
#
Wqx (q) (r1 ? C1q (q))
Wqq (q) d ? r1 q(x) + C1 (q(x)) ? p + C2x(x)
The nal three optimality conditions (Kuhn-Tucker conditions listed in Equations
3.15b through 3.15d will not be repeated) merely recover the equality constraints.
I1 = 0 = K + C ?
Z ^
qd ?
Z x
qd
(3.22)
I2 = 0 = W (q; x; x; C ) ? p ? r1 q(x)
(3.23)
I3 = 0 = W (q ; ^) ? (r1 ? r2 )^q ? r2 q ? W (q; ^) + r1 q
(3.24)
0
^
Proposition 3.6. The at-rate tari p will decrease over x if negative externality
eects outweigh those of the positive externality for the last consumer.
Proof: A larger contribution by the negative externalities (including index eects)
than by the positive externality for the last consumer implies that
Wx(q(x); x) W (q(x); x) + WC dC
dx
=x
.
Chapter 3: The Static Problem
46
Equation 3.23 gives
I2 = 0 = W (q; x; x; C ) ? p ? rq(x) .
(3.25)
Rearranging terms and dierentiating with respect to x yields
dp = d W (q; x; x; C ) ? rq(x)
dx dx
dq(x) dq(x)
dC
= Wq
dx + W + Wx + WC dx ? r dx =
=x
dC
W + Wx + WC dx =x
.
(3.26)
Hence,
dp 0 .
dx
Proposition 3.7. The marginal value of relaxing the constraint in Equation 3.13a
is related to the marginal cost of capacity and the marginal value of relaxing the
constraint restricting capacity to be smaller than the maximum available capacity.
Proof: From Equation 3.17,
0 = ?(1 ? )C3K + 1 ? 1 + 3 ? 12 (C3K ) .
The Lagrange multiplier for Equation 3.13a, 1 , has the interpretation of being the
marginal increase in the objective function due to a relaxation in the corresponding
equality constraint. Since the nontrivial case of K > 0 is assumed, 3 must be zero by
the Kuhn-Tucker condition for non-negative K . When Producer Surplus is optimized
( = 0), 12 is zero so the above expression becomes 1 = C3K + 1 . If some other
Weighted Surplus is desired ( 2 (0; 1]), the marginal value of relaxing Equation 3.13a
is equal to the marginal value of relaxing the constraint restricting capacity to be
smaller than the maximum available capacity plus the marginal cost of capacity times
Chapter 3: The Static Problem
47
the sum of (1 ? ) plus the marginal value of relaxing the non-negativity constraint
for Producer Surplus. Eectively,
1 = (1 ? )C3K + 12 C3K + 1 .
Proposition 3.8. The marginal value of relaxing the constraint in Equation 3.13b is
related to the total number of subscribers, if the at rate p is positive.
Proof: From Equation 3.16,
0 = (1 ? 2 )x ? 2 + 5 + 12 x .
The Lagrange multiplier for Equation 3.13b, 2, has the interpretation of being the
marginal increase in the objective function due to a relaxation in the corresponding
equality constraint. Since p is positive, 5 must be zero by the Kuhn-Tucker condition
for non-negative p. When Producer Surplus is optimized ( = 0), 12 is zero so
the above expression becomes 2 = x. If some other Weighted Surplus is desired
( 2 (0; 1]), the marginal value of relaxing Equation 3.13b is equal to the total number
of subscribers times the sum of (1 ? 2 ) plus the marginal value of relaxing the nonnegativity constraint for Producer Surplus. In other words,
2 = (1 ? 2 ) + 12 x .
Proposition 3.9. The marginal value of relaxing the constraint in Equation 3.13d
is related to the total number of subscribers, if the switching usage q^ is positive.
Proof: From Equation 3.19,
0 = (1 ? 2 )(r1 ? r2 )^ ? 3 (r1 ? r2 ) + 9 + 12 (r1 ? r2 )^ .
(3.27)
The Lagrange multiplier for Equation 3.13d, 3 , has the interpretation of being the
marginal increase in the objective function due to a relaxation in the corresponding
equality constraint. Since q^ is positive, 9 must be zero by the Kuhn-Tucker condition
Chapter 3: The Static Problem
48
for non-negative q^. When Producer Surplus is optimized ( = 0), 12 is zero so
the above expression becomes 3 = ^. If some other Weighted Surplus is desired
( 2 (0; 1]), the marginal value of relaxing Equation 3.13d is equal to the ambivalent
consumer's index, ^, times the sum of (1 ? 2 ) plus the marginal value of relaxing the
non-negativity constraint for Producer Surplus. In other words,
3 = (1 ? 2 ) + 12 ^ .
Proposition 3.10. Assume the maximum available capacity K is not fully utilized
and Producer Surplus is the quantity to be maximized ( = 0). If an increase in
congestion leads to a positive change in the sum of Weighted Surplus and the three
equality constraints multiplied by their respective Lagrange multipliers, the system will
operate optimally at critical congestion, or C = 0.
Proof: Capacity does not equal the maximum available capacity.
K 6= K
Since the Kuhn-Tucker condition requires that
1 K ? K = 0 ,
1 must be zero. Setting = 0 in Equation 3.20 leads to
0=
Z ^
?
r2 ? C1q (q)
0Z x ?
^
? 1 qC d +
r1 ? C1q (q) ? 1 qC d + 1 +
?
h
?
?
12
"Z ^
0
?(r2 ? C1q (q))qC d +
Z x
^
(3.28)
i
2 WC q(x); x + 3 WC q(^); ^ ? WC q(^); ^ ? 2 ?
#
[?(r1 ? C1q (q))qC (q) d .
Because the objective is to maximize Producer Surplus, 12 equals zero by the KuhnTucker condition for non-negative Producer Surplus. Furthermore, for = 0, 1 =
Chapter 3: The Static Problem
49
C3K by Proposition 3.7, 2 = x by Proposition 3.8, and 3 = ^ by Proposition 3.9.
Note that 3 has been set to zero because the nontrivial case of K 6= 0 has been
assumed. The above expression then becomes
2 =
Z ^
?
r2 ? C1q (q) ? C3K qC d +
0Z x ?
r1 ? C1q (q) ? C3K qC d + C3K
^
?
h
?
+
i
?
xWC q(x); x + ^ WC q(^); ^ ? WC q(^); ^ .
Finally, because an increase in congestion leads to a positive change in the sum
of Weighted Surplus and the three equality constraints multiplied by their respective
Lagrange multipliers,
Z ^
?
r2 ? C1q (q) ? C3K qC d +
0Z x ?
r1 ? C1q (q) ? C3K qC d + C3K
^
h
?
?
?
+
i
xWC q(x); x + ^ WC q(^); ^ ? WC q(^); ^ 0
and so 2 > 0. Hence, critical congestion (C = 0) is optimal by the Kuhn-Tucker
condition for non-positive congestion.
Proposition 3.11. Assume that there is a positive value for relaxing the constraint
restricting capacity to be less than the maximum available capacity, that is, 1 > 0.
Furthermore, assume that Producer Surplus is the quantity to be maximized ( = 0).
If congestion is optimally set below critical congestion, 1 is equal to the change in
the sum of Weighted Surplus and the three equality constraints multiplied by their
respective Lagrange multipliers due to an increase in congestion, all divided by 1
minus the change in total consumption by all users due to a change in congestion.
Proof: Congestion is set below critical congestion.
C0
Since the Kuhn-Tucker condition requires that
2 C = 0
Chapter 3: The Static Problem
50
2 must be zero. Setting = 0 in Equation 3.20 leads to
0=
Z ^
?
r2 ? C1q (q)
0Z x ?
? 1 qC d +
r1 ? C1q (q) ? 1 qC d + 1 +
^
?
h
?
?
i
2 WC q (x); x + 3 WC q(^); ^ ? WC q(^); ^ ?
12
"Z ^
?(r2 ? C1q (q))qC d +
0
Z x
^
#
[?(r1 ? C1q (q))qC (q) d .
Because the objective is to maximize Producer Surplus, 12 equals zero by the KuhnTucker condition for non-negative Producer Surplus. Furthermore, for = 0, 1 =
C3K by Proposition 3.7, 2 = x by Proposition 3.8, and 3 = ^ by Proposition 3.9.
Note that 3 has been set to zero because the nontrivial case of K 6= 0 has been
assumed. Rearranging the above expression then leads to
"
Z ^
?
?
1
r2 ? C1q (q) ? C3K qC d +
1 = R ^
R
1 ? 0 qC d ? ^x qC d 0
Z x ?
r1 ? C1q (q) ? C3K qC d + C3K +
^
?
h
?
?
#
i
xWC q(x); x + ^ WC q(^); ^ ? WC q(^); ^
.
3.4 The Flat-Rate/Two-Part Tari
This section will discuss the at-rate/two-part tari which was described in detail
in the last chapter. Begin by dening the three surpluses for the at-rate/two-part
tari. Recalling that this revenue function is expressed as
R(q) = p ; 0 q < q^
?
= p + r q ? q^ ; q q^ ,
Chapter 3: The Static Problem
51
Total Consumer Surplus is the dierence between total willingness to pay and the
total revenues paid by all consumers.
Z x
CS =
0
Z ^
=
0
[W ? R]d
[W ? (p + r(q ? q^)]d +
Z ^
=
0
Wd +
Z ~
^
W (^q)d +
Z ~
Z x
~
^
[W (^q) ? p]d +
Z x
~
Wd ? px + rq^^ ? r
[W ? p]d
Z ^
0
qd
The components of Producer Surplus include the total revenues taken in from all consumers along with the costs due to consumer usage, subscriber set size, and capacity.
PS =
=
Z x
Z ^
[(p + r(q ? q^) ? C1 (q)]d +
0Z
+
=
[R(q) ? C1 (q)]d ? C2 (x) ? C3(K )
0
Z ^
0
x
~
Z ~
^
[p ? C1 (^q)]d
[p ? C1(~q)]d ? C2(x) ? C3 (K )
[rq ? C1(q)]d ?
? C2 (x) ? C3 (K )
Z x
^
C1(~q)d + px ? rq^^ ? (~ ? ^)C1(^q)
Weighted Surplus is a combination of the Producer and Consumer Surpluses, stated
below for the two-part tari.
WS = CS + (1 ? )PS
=
Z ^
[W + (1 ? 2 )rq ? (1 ? )C1 (q)]d +
0Z
+
x
~
Z ~
^
[W (^q )]d
[W (~q ) ? (1 ? )C1 (~q)]d + (1 ? 2 )px ? (1 ? 2 )rq^^
(3.29)
? (1 ? )(~ ? ^)C1(^q) ? (1 ? )C2 (x) ? (1 ? )C3 (K )
The static two-part tari problem will maximize Equation 3.29 subject to the constraints in Equation 3.30. They include the denition of congestion as the dierence
between total usage and capacity, upper limit for capacity, non-negativity constraints,
Chapter 3: The Static Problem
52
description of the last consumer's zero Consumer Surplus, individual Consumer Surplus maximization FONC, denitions of the two middle edge consumers, and a nonnegativity constraint for Producer Surplus.
K +C ?
Z ^
q~d ? q^(~ ? ^) = 0
(3.30a)
K ? K 0
C0
K0
x0
p0
r0
^ 0
~ 0
x ? ~ 0
~ ? ^ 0
W (~q; x; x; C ) ? p = 0
Wq (q; ; x; C ) = r; Wq (~q; ; x; C ) = 0
q(^) ? q^ = 0;
q(~) ? q^ = 0
(3.30b)
(3.30c)
(3.30d)
PS =
Z ^
0
0
qd ?
Z x
~
[rq ? C1 (q)]d ?
Z x
^
C1 (~q)d + px ? rq^^
? (~ ? ^)C1 (^q) ? C2 (x) ? C3(K )
(3.30e)
Optimization of WS constrained by the equalities and inequalities in Equation 3.30 is
equivalent to maximizing I (p; r; x; C; K; ~; ^). Variables 1 through 1 2 are Lagrange
multipliers for the non-negativity constraints and 1 through 3 are the multipliers
for the equality constraints.
Chapter 3: The Static Problem
I = WS + 1 K + C ?
53
Z ^
0
qd ?
Z x
~
q~d ? q^(~ ? ^)
+ 2[W (~q ; x; x; C ) ? p] + 3 [q(~) ? q^] + 1 (K ? K ) ? 2 (C )
+ 3(K ) + 4 (x) + 5 (p) + 6(r) + 7 (^) + 8 (~) + 9 (x ? ~)
+ 10 (~ ? ^) + 11
Z ^
0
[rq ? C1 (q)]d ?
Z x
^
C1 (~q)d + px ? rq^^
? (~ ? ^)C1 (^q) ? C2 (x) ? C3(K )
FONC for the maximization of I (p; r; x; C; K; ~; ^) require its rst-order partial
derivatives with respect to p; r; x; C; K; 1 ; and 2 to be zero. Furthermore, KuhnTucker conditions must be satised for the non-negativity constraints and their multipliers.
0 = Ip = Ir = Ix = IC = IK = I~ = I^ = I1 = I2 = I3
0 =1 (K ? K ) = 2 (C ) = 3 (K ) = 4 (x) = 5 (p) = 6 (r)
=7 (~) = 8 (^) = 9 (x ? ~) = 10 (~ ? ^)
=11
Z ^
0
[rq ? C1(q)]d ?
Z x
^
C1(~q)d + px ? rq^^ ?
(3.31a)
(3.31b)
(~ ? ^)C1(^q) ? C2(x) ? C3 (K )
K ? K; ?C; K; x; p; r; ~; ^; x ? ~; ~ ? ^; PS 0
1 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 0
(3.31c)
(3.31d)
First solve for the partial derivative with respect to the xed price.
Ip = 0 = (1 ? 2 )x ? 2 + 5 + 12 x
(3.32)
And solve for the partial derivative with respect to capacity.
IK = 0 = ?(1 ? )C3K + 1 ? 1 + 3 ? 12 C3K
Next nd the partial derivative with respect to the variable price.
(3.33)
Chapter 3: The Static Problem
Ir = 0 =
54
Z ^
1
(1 ? 2 )q + ((1 ? )(r ? C1q ) ? 1 )
Wqq d
0
Z ~
1
+
Wqq (^q; ^) ^
W (^q )d ? (1 ? 2 )r^ ? (1 ? )(~ ? ^)C1q (^q)
? 1 (~ ? ^) ? 3 ? (1 ? 2 )^q ^ + 6 + 12
1
? W (^
r^ + (~ ? ^)C1q (^q) ? q^^
qq q; ^)
h
i
Z ^ h
0
i
(
r
?
C
1q )
q+ W
d
qq
The partial derivative of I with respect to ~ is
W
q
;
~)
q (~
I~ = 0 = 3 W (~q; ~) ? 8 ? 9 + 10 .
qq
The partial derivative of I with respect to ~ is
Z ~
W
q
;
^)
q (^
I^ = 0 = ? W (^q; ^) W (^q)d ? (1 ? 2 )r^ ? (1 ? )(~ ? ^)C1q (^q)
qq
^
q; ^) hr^ + (~ ? ^)C (^q)i .
q (^
? 1 (~ ? ^) ? 3 + 7 ? 9 + 12 W
1q
W (^q; ^)
qq
Maximizing I over congestion leads to
IC = 0 =
+
Z ^ 0
Z ~
^
qC
WC ? ((1 ? )(r ? C1q ) ? 1 ) W
Wqq d
WC (^q)d +
Z x
~
WC (~q) + (1 ?
W
(~q)
)C1q (~q) + 1 WqC(~
qq q)
Z ~
W
q
;
^
)
qC (^
? W (^q; ^) W (^q )d ? (1 ? 2 )r^ ? (1 ? )(~ ? ^)C1q (^q)
qq
^
q; ~) ? qC (~
? 1 (~ ? ^) ? 3 + 1 + 2 WC (q(x); x) ? 3 W
Wqq (~q; ~) 2
Z ^h
Z x
i
W
W (~q)
qC
+ 12 ?
(r ? C1q )
d
+ C1q (~q) qC d
Wqq
Wqq (~q)
0
~
h
i
W (^q; ^) r^ + (~ ? ^)C (^q) .
+ qC
1q
W (^q; ^)
qq
d
Chapter 3: The Static Problem
55
The next optimality condition requires the partial derivative of I with respect to the
number of subscribers to be zero.
Z ^
W
qx
Ix = 0 =
Wx ? ((1 ? )(r ? C1q ) ? 1 ) W d
qq
0
Z ~
Z x
W (~
qx q )
+ Wx (^q)d +
Wx (~q) + (1 ? )C1q (~q) + 1 W (~q) d
qq
^
~
Z ~
q; ^) W (^q)d ? (1 ? 2 )r^ ? (1 ? )(~ ? ^)C (^q)
qx (^
?W
1q
Wqq (^q; ^) ^
? 1 (~ ? ^) ? 3 + (1 ? )p ? (1 ? )C2 (x) ? 1q(x) ? (1 ? )C1x(x)
+ 2 W (q(x); x) + Wx (q(x); x) ? 3 Wqx (~q; ~) + 4 + 10 ? 11
Wqq (~q; ~)
Z ^h
Z x
i
+ 12 ?
(r ? C1q ) Wqx d + C1q (~q) Wqx(~q) d
Wqq
Wqq (~q)
0
~
h
i
W
q
;
^
)
qx (^
+ p ? C2 (x) ? C1x(x) +
Wqq (^q; ^) r^ + (~ ? ^)C1q (^q)
The nal three optimality conditions (Kuhn-Tucker conditions listed in Equations
3.31b through 3.31d will not be repeated) merely recover the equality constraints.
h
i
I1 = K + C ?
Z ^
0
qd
I2 = W (~q ; x; x; C ) ? p = 0
I3 = q(~) ? q^ = 0
3.5 The Nonlinear Tari
3.5.1 Problem Formulation
Begin by dening the three surpluses for the nonlinear tari. Since the nonlinear
tari is expressed as R(q), total Consumer Surplus is the dierence between total
willingness to pay and the total revenues paid by all consumers. Consumer Surplus is
Chapter 3: The Static Problem
56
the dierence between revenues and costs. Weighted Surplus is a convex combination
of the other two surpluses.
CS =
PS =
Z x
Z0
x
0
[W ? R(q)]d
[R(q) ? C1 (q)]d ? C2 (x) ? C3 (K )
WS = CS + (1 ? )PS
=
Z x
0
[W + (1 ? 2 )R(q) ? (1 ? )C1 (q)]d ?
(1 ? )C2 (x) ? (1 ? )C3 (K )
This is a Calculus of Variations problem whose solution includes the optimal subscriber set size, optimal congestion level, and optimal usage patterns for all consumers.
From these, we can then easily determine the optimal tari.
Maximize
Rx
0 [W (q ( ); ; x; C ) + (1 ? 2 )R(q ( )) ? (1 ? )C1 (q ( ))]d ?
(1 ? )C2 (x) ? (1 ? )C3 (K )
Subject To
(3.34)
Z x
K + C ? q()d = 0
0
K ? K 0
C0
K0
x0
q() 0 ; 2 [0; x]
W (q(x); x; x; C ) ? R(q(x)) = 0
Wq (q(); ; x; C ) ? Rq (q()) = 0 ; 2 [0; x]
The objective function, Equation 3.34, could now be optimized over all system
variables: the congestion level C , subscriber consumption q(), and tari function
Chapter 3: The Static Problem
57
R(q). However, an interdependence between R(q) and q() in Equation 3.36c forces
us to eliminate one of the two variables to ensure an independent set of controls. Oren,
Smith, and Wilson [1982] introduced a simple method of doing this by eliminating
R(q) from the objective function altogether with Equations 3.36b and 3.36c. To
do so, integrate by parts the portion of the objective function containing R(q) :
Rx
0 (1 ? 2 )R(q )d .
R
R
Selecting u = R(q) so that du = Rq q d and v = dv = d = ,
this leads to
Z x
0
(1 ? 2 )R(q)d = (1 ? 2 )
[R(q)]x
0
?
Z x
= (1 ? 2 ) xR(q)j=x ?
0Z
Rq q d
x
0
Rq q d .
Now use Equation 3.36b and Equation 3.36c to replace R(q)j=x and Rq with W (q)j=x
and Wq , respectively, so that
Z x
0
(1 ? 2 )R(q)d = (1 ? 2 ) xW (q)j=x ?
Z x
0
Wq q d .
Substitution of this result into the objective function yields:
Z x
Z0
0
x
[W + (1 ? 2 )R ? (1 ? )C1 ]d ? (1 ? )C2 (x) ? (1 ? )C3 (K ) =
[W ? (1 ? 2 )Wq q ? (1 ? )C1 ]d ? (1 ? )C2 (x) ?
(1 ? )C3 (K ) + (1 ? 2 )W (q)j=x x .
And the static problem becomes the following.
Maximize
Rx
0 [W
? (1 ? 2 )Wq q ? (1 ? )C1 ]d ? (1 ? )C2 (x) ?
(1 ? )C3 (K ) + (1 ? 2 )W (q)j=x x
Chapter 3: The Static Problem
Subject To
58
Z x
K + C ? q()d = 0
0
K ? K 0
C0
K0
x0
q() 0 ; 2 [0; x]
W (q(x); x; x; C ) ? R(q(x)) = 0
Wq (q(); ; x; C ) ? Rq (q()) = 0 ; 2 [0; x]
(3.36a)
(3.36b)
(3.36c)
Note that the last two constraints have already been satised through the elimination
of R and Rq from the objective function; subsequent derivations do not explicitly refer
to them. As a result, we can now express an equivalent representation of the modied
optimization problem by introducing the Hamiltonian function, F (; Y; Y ).
Call the Lagrange multiplier for the rst equation . Depict the last ve constraints as 1 through 5 , respectively, and call the associated Lagrange multipliers
1 through 5 . The only multiplier which varies over is 5 . The system variables
can be grouped into the vector
Y = [y1; y2 ; y3]T = [q; C; K ]T , where Y = [q ; C ; K ]T = [q ; 0; 0]T . Furthermore,
Y = fq; Y g, and Y = fY2 ; Y3g.
Let us represent the integrand of the objective function as
f (; x; Y; Y ) = W ? (1 ? 2 )Wq q ? (1 ? )C1 (q)
and dene the salvage function, S , to be
S (x; Y j=x ) = ?(1 ? )C2 (x) ? (1 ? )C3 (K ) + (1 ? 2 )W j=x x +
(K + C ) + 1 (K ? K ) ? 2 (C ) + 3 (K ) + 4 (x) .
If we assume the solution will be nontrivial, 0 = 1, and the Hamiltonian function
Chapter 3: The Static Problem
59
can be dened as
F (; x; Y; Y ) =0 f (; x; Y; Y ) ? q + 5q
=[W ? (1 ? 2 )Wq q ? (1 ? )C1 (q)] + (5 ? )q .
(3.37)
Consequently, the equivalent optimization problem becomes the following:
Maximize
I (x; Y ) =
Z x
0
F (; x; Y; Y )d + S (x; Y j=x) .
(3.38)
3.5.2 Necessary Conditions
Although the problem of maximizing Equation 3.38 over the vector Y appears to
correspond to the format for a standard Calculus of Variations problem, the presence
of the subscriber set size x in both the integrand as well as upper limit of integration
requires an explicit derivation of the Euler and transversality conditions for this particular problem. See Kamien and Schwartz [1991] for treatments of dierent problems
with similar development.
Dene the variation H to be the dierence between the optimal system variable
vector, Y , and the comparison vector, Y . For any constant a, the value of I (x; Y ) as
a function of a is
g(a) = I (x + ax; Y + aH )
=
Z x+ax
F (; x + ax; Y + aH; Y + aH )d +
S (x + ax; Y j=x + aY j=x) .
0
Proposition 3.12. g(a) will reach its maximum at a = 0 and g0 (a = 0) = 0.
Proof: Since Y has been designated as the optimal system variable vector, g(a)
must reach its maximum at a = 0. Therefore, by the rst order necessary condition
of maximization, g0 (a) equals zero at a = 0.
Chapter 3: The Static Problem
60
First take the derivative of g(a) with respect to a.
Z x+ax
@ [F (; x + ax; Y + aH; Y + aH )]d + F j
0
g (a) =
=x+ax x +
@a
0
Sx(x + ax; Y j=x + aY j=x )x +
SYT (x + ax; Y j=x + aY j=x )Y j=x
=
Z x+ax
[Fx x + FYT H + FYT H ]d + F j=x+ax x +
0
Sx(x + ax; Y j=x + aY j=x )x +
SYT (x + ax; Y j=x + aY j=x )Y j=x
Next, set g0 (a) equal to zero at a = 0.
g0 (0) = 0Z
x
=
[Fx x + FYT H + FYT H ]d + F j=xx + Sx(x; Y j=x )x +
0
SYT (x; Y j=x)Y j=x
Integration of FYT H by parts leads to
Z x"
#
dFYT
T
0 =
Fx x + FY H ? d H d + [FYT H ]j=x ? [FYT H ]j=0 +
0
[F j=x + Sx (x; Y j=x)]x + SYT (x; Y j=x)]Y j=x .
Use the following approximation from Kamien and Schwartz [1991]:
hi(x) yi (x) ? yi (x)x , i = 1.
(3.39)
Furthermore, since the initial point, = 0 is xed,
hi (0) = yi(0) , i = 2, 3.
Rearrange terms a bit to get the following:
#
Z x
Z x"
T
dF
0 = Fxxd +
FYT ? dY Hd + FYT j=x H j=x ?
0
0
FYT j=0 H j=0 + SYT (x; Y j=x )Y j=x + [F j=x + Sx(x; Y j=x)]x .
Dene H such that H = fH1 ; H g; and H = fH2 ; H3 g.
(3.40)
Chapter 3: The Static Problem
61
Proposition 3.13. If I is maximized over C and K then Euler Condition - 1 is true
for optimal solutions.
Euler Condition - 1
dFY 0 =
FY ? d d + SY (x; Y j=x )
(3.41)
0
Proof: Note that the C and K components of H do not vary over since C and
K themselves are not functions of . Furthermore, Equation 3.40 must be zero for
any value of H2 or H3 . The coecients for H2 and H3 must therefore be zero. The
components of Equation 3.40 which contain H2 and H3 must by themselves equal zero
and are
Z x
Z x"
#
dFYT T
0=
FY ? d Hd + FYT j=x H j=x ? FYT j=0 H j=0 +
0
SYT (x; Y j=x)Y j=x .
Finally, use the fact that FY = 0 and factor out H to get
Z x"
T #
dF
Y
FYT ? d d + SYT (x; Y j=x ) .
0=
0
Proposition 3.14. The following Euler and transversality conditions must be true
if I is optimized over q and x.
EulerCondition ? 2
For 2 [0; x],
dF
Fq ? dq = 0 :
(3.42)
Fq j=x + Sq (x; Y j=x) = 0
(3.43)
TransversalityConditions
If qj=x is free,
Chapter 3: The Static Problem
62
If qj=0 is free,
Fq = 0
If x is free,
Z x
0
(3.44)
Fxd + F j=x + Sx (x; Y j=x) ? Fq j=x q j=x
(3.45)
Proof: Further substitutions and rearranging of Equation 3.40 to nd the optimality conditions for q and x give
Z x
dFq 0 =
Fq ? d H1 d ? Fq j=0 qj=0 +
0
[Fq j=x + Sq (x; Y j=x)]qj=x +
Z x
0
Fxd + F j=x + Sx(x; Y j=x) ? Fq j=x q (x) x .
(3.46)
Because Equation 3.46 must be zero for any value of H , qj=0 , qj=x , or x, the
coecient of each must be zero. Setting each coecient equal to zero yields the Euler-2
and transversality conditions.
3.5.3 Static Nonlinear Solution
We can now solve the necessary conditions in Equations 3.41 through 3.45 for the
optimization problem at the end of Section 3.5.1.
The Euler Conditions require the following:
0 = Fq ?
0 =
Z x
0
dFq
d
dFC FC ? d d + SC (x; Y j=x)
Chapter 3: The Static Problem
0 =
63
Z x
0
dFK FK ? d d + SK (x; Y j=x) .
The rst requirement can be solved as follows.
dF
0 =Fq ? q
d
@
= f[W ? (1 ? 2 )Wq q ? (1 ? )C1(q)] + (5 ? )qg ?
@q
d f @ [W ? (1 ? 2 )W q ? (1 ? )C (q)] + ( ? )q g
q 1
5
d @q
=Wq ? (1 ? 2 )Wqq q ? (1 ? )C1q + (5 ? ) +
(1 ? 2 )[Wq + (Wqq q + Wq )]
0 =(1 ? 2 )Wq + (1 ? )[Wq ? C1q ] + 5 ? (3.47)
The second requirement is solved below.
Z x
dFC FC ? d d + SC (x; Y j=x )
0=
0
Z x
= [FC ] d + SC (x; Y j=x)
=
0
Z x"
0
@ f[W ? (1 ? 2 )W q ? (1 ? )C (q)] +
q 1
@C
#
(5 ? )qg d + (1 ? 2 )WC j=x x + ? 2
0=
Z x
0
[WC ? (1 ? 2 )WqC q ] d + (1 ? 2 )WC j=x x + ? 2
(3.48)
Chapter 3: The Static Problem
64
Solve the third requirement as follows.
Z x
dFK 0=
FK ? d d + SK (x; Y j=x)
0
Z x
= [FK ] d + SK (x; Y j=x)
=
0
Z x"
0
@ f[W ? (1 ? 2 )W q ? (1 ? )C (q)] +
q 1
@K
#
(5 ? )qg d ? (1 ? )C3K + ? 1 + 3
0 = ? (1 ? )C3K + ? 1 + 3
(3.49)
Now solve for the transversality conditions for q and x. In order for q(x) to be
free, we require:
0 = Fq j=x + Sq (x; Y j=x )
= [?(1 ? 2 )Wq ]j=x + (1 ? 2 )Wq j=x x
= 0.
(3.50)
Likewise, Equation 3.43 is required for q(0) to be free.
0 = Fq j=0
= [?(1 ? 2 )Wq ]j=0
= 0
(3.51)
Chapter 3: The Static Problem
65
The nal tranvsersality condition, Equation 3.45, results from allowing the subscriber set size, x, to be free.
Z x
Fxd + F j=x + Sx(x; Y j=x ) ? Fq j=xq j=x
x @ W ? (1 ? 2 )Wq q ? (1 ? )C1 (q) + (5 ? )q d +
=
@x
0
W ? (1 ? 2 )Wq q ? (1 ? )C1 (q) + (5 ? )q +
0 =
Z0
=x
@ ?(1 ? )C (x) ? (1 ? )C (K ) + (1 ? 2 )W j x +
2
3
=x
@x
(K + C ) + 1 (K ? K ) ? 2 (C ) + 3 (K ) + 4 (x) ? Fq q j=x
Z x
[Wx ? (1 ? 2 )Wqx q ]d + W ? (1 ? 2 )Wq q ?
(1 ? )C1 + (5 ? )q ? (1 ? )C2x +
(1 ? 2 )[W + x(W + Wx )] + 4 + (1 ? 2 )Wq q =x
=
0
=x
Z x
[Wx ? (1 ? 2 )Wqx q ]d + (1 ? )[W ? C1 ? C2x ]j=x +
(1 ? 2 )x[W + Wx]j=x + (5 (x) ? )q(x) + 4
0 =
0
(3.52)
From Equations 3.47, 3.48, 3.49, and 3.52, we can nally state the Euler equations
and transversality conditions necessary for an optimal solution to the static problem.
Static Euler Equations
0 = (1 ? 2 )Wq + (1 ? )[Wq ? C1q ] + 5 ? , 2 [0; x]
0=
Z x
0
(3.53a)
[WC ? (1 ? 2 )WqC q ] d + (1 ? 2 )WC j=x x + ? 2
(3.53b)
0 = ?(1 ? )C3K + ? 1 + 3
(3.53c)
Static Transversality Equation
If x is free,
0=
Rx
0 [Wx ? (1 ? 2 )Wqx q ]d + (1 ? )[W
? C1 ? C2x ]j=x +
(1 ? 2 )x[W + Wx ]j=x + (5(x) ? )q(x) + 4 .
(3.54)
Chapter 3: The Static Problem
66
Static Kuhn-Tucker Equations
1 [K ? K ] = 0
K ? K 0
2 [C ] = 0
C0
3 [K ] = 0
K0
4 [x] = 0
x0
5 [q()] = 0 ; 2 [0; x]
q() 0 ; 2 [0; x]
Static Integral Equation
K +C ?
Z x
0
q()d = 0
(3.55a)
(3.55b)
(3.55c)
(3.55d)
(3.55e)
(3.55f)
(3.55g)
(3.55h)
(3.55i)
(3.55j)
(3.56)
3.5.4 Results
Proposition 3.15. The marginal value of relaxing the constraint in Equation 3.36a
is related to the marginal cost of capacity and the marginal value of relaxing the
constraint restricting capacity to be smaller than the maximum available capacity.
Proof: From Equation 3.53c,
0 = ?(1 ? )C3K + ? 1 + 3 .
The Lagrange multiplier for Equation 3.36a, , has the interpretation of being the
marginal increase in the objective function due to a relaxation in the corresponding
equality constraint. Since the nontrivial case of K > 0 is assumed, 3 must be zero by
the Kuhn-Tucker condition for non-negative K . When Producer Surplus is optimized
the above expression becomes = C3K + 1. If some other Weighted Surplus is desired
( 2 (0; 1]), the marginal value of relaxing Equation 3.36a is equal to the marginal
Chapter 3: The Static Problem
67
value of relaxing the constraint restricting capacity to be smaller than the maximum
available capacity plus (1 ? ) times the marginal cost of capacity. In other words,
= (1 ? )C3K + 1 .
Proposition 3.16. Assume the maximum available capacity K is not fully utilized
and Producer Surplus is the quantity to be maximized ( = 0). If an increase in
congestion leads to a positive change in the objective function, the system will operate
optimally at critical congestion, or C = 0.
Proof: Capacity does not equal the maximum available capacity.
K 6= K
Since the Kuhn-Tucker condition requires that
1 K ? K = 0 ,
1 must be zero. Setting = 0 in Equation 3.53b leads to
0=
Z x
?WqC q d + WC j=xx + ? 2 .
0
(3.57)
For = 0, 1 = C3K by Proposition 3.15. Note that 3 has been set to zero because
the nontrivial case of K 6= 0 has been assumed. The above expression then becomes
Z x
2 =
0
?WqC q d + WC j=x x + C3K .
Finally, because an increase in congestion leads to a positive change in the objective
function,
Z x
0
?WqC q d + WC j=xx + C3K 0 ,
and so 2 > 0. Hence, critical congestion (C = 0) is optimal by the Kuhn-Tucker
condition for non-positive congestion.
Chapter 3: The Static Problem
68
Proposition 3.17. Assume that there is a positive value for relaxing the constraint
restricting capacity to be less than the maximum available capacity, that is, 1 > 0.
Furthermore, assume that Producer Surplus is the quantity to be maximized ( = 0).
If congestion is optimally set below critical congestion, 1 is equal to the change in
the objective function due to an increase in congestion.
Proof: Congestion is set below critical congestion.
C0
Since the Kuhn-Tucker condition requires that
2 C = 0
2 must be zero. Setting = 0 in Equation 3.53b leads to
0=
Z x
0
?WqC q d + WC j=x x + .
For = 0, 1 = C3K + 1 by Proposition 3.15. Rearranging the above expression then
leads to
1 =
Z x
0
WqC q d ? WC j=x x ? C3K .
Proposition 3.18. If 6= 1 and the rst ( = 0) consumer's marginal willingness to
pay is greater than marginal cost, capacity is optimal at its maximum possible value.
Proof: From Equation 3.53a,
0 = (1 ? 2 )Wq + (1 ? )[Wq ? C1q ] + 5 ? , 2 [0; x] .
In particular, this must be true for = 0, so that
0 = (1 ? )[Wq ? C1q ]j=0 + 5 (0) ? .
Assume a non-trivial solution in which at least one consumer (who will be = 0)
will choose to subscribe to the service. Therefore, q(0) 6= 0 and by the Kuhn-Tucker
condition, 5 (0) = 0. After some rearranging, the above equation then becomes
= (1 ? )[Wq ? C1q ]j=0 .
Chapter 3: The Static Problem
69
From Proposition 3.15,
= (1 ? )C3K + 1
= (1 ? )[Wq ? C1q ]j=0
so that
1 = (1 ? ) [Wq ? C1q ]j=0 ? C3K .
Therefore, because Consumer Surplus is not being maximized ( 6= 1) and because
the rst consumer's marginal willingness to pay is greater than marginal cost, this
simplies to 1 6= 0. Hence, from the Kuhn-Tucker condition, K must equal its
maximum possible value, K .
Chapter 4
The Dynamic Problem
4.1 Subscriber Set Growth Function
The dynamic optimization problem is a more complicated version of the static problem. We are now concerned with a more realistic setting in which imperfect information and market lags aect product adoption.
This chapter will adapt the static problems introduced in the last chapter to a
dynamic framework by denoting the time variable, t, where appropriate. The dynamic
problem will maximize the integral of discounted Weighted Surplus over the time
interval [t1 ; t2 ]. Discounting is implemented by multiplying Weighted Surplus at any
t 2 [t1 ; t2 ] by e?it , where i is the discount rate; in eect, all surpluses are discounted
back to time t1 dollars. In addition, the capacity cost is now incurred due to a change
in capacity, or as a function of the derivative of capacity, Kt.
Market lag will be modeled as in Dhebar [1983] in which the subscriber set size
x has a growth rate determined by the current subscriber set size and last potential
subscriber, dened to be d . Variable d is the last consumer with index greater than
x who believes that the subscriber set will eventually be larger than x. His perception
of the eventual subscriber set size is a weighted sum of x and d , or d + (1 ? )x,
where 2 [0; 1].
As described in Dhebar [1983], the weight could be aected by any number of
70
Chapter 4: The Dynamic Problem
71
factors forming consumer awareness: advertising, word-of-mouth, media reports, etc.
The case = 0 corresponds to a subscriber set who is myopic; in other words, consumers do not make predictions about the future. At the other end of the spectrum,
= 1 describes consumers who have perfect foresight, perhaps as a result of intensive
advertising promotions and numerous accurate market studies.
The subscriber set growth rate thus depends on both the current number of subscribers and last potential subscriber.
Assumption 4.1. The number of subscribers will grow at a rate xt dened by
xt = G(d ; x)
where G is dierentiable over d and x.
Since d is the last consumer who would subscribe believing that the subscriber
set size will be d + (1 ? )x, his Consumer Surplus will be zero.
W (q(d (t); t); d (t); [d (t) + (1 ? )x(t)]; C (t)) ?
R(t; q(t; d (t))) = 0 ; t 2 [t1 ; t2 ]
As with all consumers x, d will optimally choose his consumption quantity
by setting marginal willingness to pay equal to marginal tari.
Wq (q(d(t); t); d (t); [d (t) + (1 ? )x(t)]; C (t)) ?
Rq (q(d (t); t); t) = 0 ; t 2 [t1 ; t2 ]
4.2 The Two-Part Tari
Recall from Chapter 3 the expression for Weighted Surplus where the revenue function
is the two-part tari.
WS = CS + (1 ? )PS
=
Z x
0
[W (q; ; x; C ) ? (1 ? )C1 (q) + (1 ? 2 )rq]d
+ (1 ? 2 )px ? (1 ? )C2 (x) ? (1 ? )C3 (K )
Chapter 4: The Dynamic Problem
72
Adapt this Weighted Surplus for the dynamic problem discussed in the last section.
WS =
Z x(t) 0
W (q(; t); ; x(t); C (t)) ? (1 ? )C1 (q(; t)) +
(1 ? 2 )r(t)q(; t) d + (1 ? 2 )p(t)x(t)
(4.1)
? (1 ? )C2 (x(t)) ? (1 ? )C3 (Kt(t))
The dynamic problem will maximize discounted Weighted Surplus, subject to the
constraints from the static two-part tari problem in addition to those listed in the
last section.
R
Maximize tt12 e?it WSdt
Subject To
xt (t) ? G(x(t); d (t)) = 0
Z x(t)
(4.2a)
K (t) + C (t) ?
q(; t)d = 0
(4.2b)
0
K ? K (t) 0
C (t) 0
K (t) 0
p(t) 0
r(t) 0
W (q(d (t); t); d (t); d (t) + (1 ? )x(t); x(t); C (t))
?p(t) ? r(t)q(d(t); t) = 0
(4.2c)
W (q(x; t); x(t); x(t); C (t)) ? p(t) ? r(t)q(x; t) = 0
(4.2d)
Wq (q(d (t); t); d (t) + (1 ? )x(t); x(t); C (t)) ? r(t) = 0
(4.2e)
Wq (q(; t); ; x(t); C (t)) ? r(t) = 0 , 2 [0; x]
(4.2f)
Dene Y (t) = fy1 ; Y2; y3 ; Y4 g and Y (t) = fY2 ; y3; Y4 g. Let y1 = x and dene
Y2t =fy21t ; y22t ; y23t ; y24t ; y25t g
=fp; r; C; Kt ; d g .
Chapter 4: The Dynamic Problem
73
Also let y3t be the multiplier for Equation 4.2a and Y4 = fy41t ; y42t ; y43t t g such
that y41t is the multiplier for Equation 4.2b, y42t is the multiplier for Equation 4.2c,
and y43t is the multiplier for Equation 4.2d. Finally, dene = f1 ; 2 ; 3 ; 4 ; g where
the four components of are the Lagrange multipliers for Inequalities 4.2, 4.2, 4.2,
and 4.2, respectively.
Using these variables and multipliers, reformulate the maximization problem as
the equivalent function F .
F = e?it WS + y3t y1t ? G(y1; y25t ) + y41t 1 + y42t 2 + y43t 3
+ 1 1 + 2 2 + 3 3 + 4
4
(4.3)
This problem can now be solved as a Kuhn-Tucker Reformulation to a Calculus of
Variations problem with equality and inequality constraints as discussed in Gregory
and Lin [1992]. The optimality conditions for this problem are summarized below.
Euler Equation: dtd FYt = FY
Kuhn-Tucker Conditions:
1. T (t; y1 ; Y2t ) = 0
2. (t; y1 ; Y2t ) 0
?
Transversality Condition: = FYt t2 ; Y (t2 ); Yt(t2 )
Proposition 4.1. Fy t (t) = 0 for optimal solutions.
Proof: By the Euler Condition:
21
dF =F
dt y21t y21
=0 .
Hence Fy21t is a constant over t. Since the Transversality Condition requires that
?
FYt t2 ; Y (t2 ); Yt(t2 ) = 0, Fy21t (t2 ; Y (t2); Yt (t2 ) = 0.Fy21t = 0 follows.
Chapter 4: The Dynamic Problem
74
Using Proposition 4.1 and similar arguments for Fyit , i = 22, 23, 25, 3, 41, 42,
and 43, solve for the optimality conditions required by the Euler Equation.
dF =F
y1
dt y1t
d [y ] = e?it Z x hW ? (1 ? )(r ? C ) Wqx id + (1 ? )W (q(x); x)
x
1q
dt 3t
Wqq
0
? (1 ? )C1 (q(x)) ? (1 ? )C2x (x) ? y3t Gy1 +
hZ x W
i
h
i
qx
d
?
q
(x) + y42t Wx(q(d ); d )
(4.4)
y41t
0 Wqq
h
i
+ y43t W (q(x); x) + Wx (q(x); x)
(4.5)
0 = Fy21t
= e?it (1 ? 2 )x ? y42t ? y43t + 3
0 = Fy22t
Z x
(1 ? )(r ? C1q ) 1 + (1 ? 2 )q d ? y41t
Wqq
0
? y42t q(d ) ? y43t q(x) + 4
= e?it
(4.6)
Z x
0
1 d
Wqq
0 = Fy23t
Z x
W
qC
qC ?(1 ? )(r ? C1q ) W + WC d + y41t 1 + W
d
qq
0
0 Wqq
+ y42t WC (q(d ); d ) + y43t WC (q(x); x) ? 2
= e?it
Z x
dF =F
y24
dt y24t
d e?it (1 ? )C (K ) = y ? 3K t
41t
1
dt
0 = (1 ? )ie?it C3K (Kt ) + y41t ? 1
0 = Fy25t
= ?y3t Gy25t + y42t W (d )
(4.7)
(4.8)
Chapter 4: The Dynamic Problem
75
0 = Fy3t
= y1t ? G(y1; y25t )
The Euler Condition for Y4 merely recovers Equations 4.2b, 4.2c, and 4.2d.
The Dynamic Two-Part optimality conditions are summarized below. Results
from the Euler Condition are restated and then followed by a list of the Kuhn-Tucker
Conditions.
Chapter 4: The Dynamic Problem
76
Dynamic Two-Part Euler Equations
dF =F
y1
dt y1t
d [y ] = e?it Z xhW ? (1 ? )(r ? C ) Wqx id + (1 ? )W (q(x); x)
x
1q
dt 3t
W
0
qq
? (1 ? )C1 (q(x)) ? (1 ? )C2x (x) ? y3t Gy +
hZ x W
i
h
i
qx
y41t
W d ? q(x) + y42t Wx(q(d ); d )
1
0
h
qq
(4.9a)
i
+ y43t W (q(x); x) + Wx(q(x); x) , t 2 [t1 ; t2 ]
0 = e?it (1 ? 2 )x ? y42t ? y43t + 3 , t 2 [t1 ; t2 ]
Z x
1
1 d
0=e
(1 ? )(r ? C1q )
+ (1 ? 2 )q d ? y41t
Wqq
0
0 Wqq
? y42t q(d ) ? y43t q(x) + 4 , t 2 [t1 ; t2 ]
Z x
Z x
W
qC
qC ?
it
0=e
?(1 ? )(r ? C1q ) W + WC d + y41t 1 + W
d
qq
0
0 Wqq
+ y42t WC (q(d); d ) + y43t WC (q(x); x) ? 2 , t 2 [t1 ; t2 ]
0 = (1 ? )ie?it C3K (Kt) + y41t ? 1 , t 2 [t1 ; t2 ]
0 = ?y3t Gy25t + y42t W (d ) , t 2 [t1 ; t2 ]
0 = y1 ? G(y1; y25t ) , t 2 [t1 ; t2 ]
Z x
?it 0 = K (t) + C (t) ?
Z x(t)
0
q(; t)d , t 2 [t1 ; t2 ]
0 = W (q(d (t); t); d (t); d (t) + (1 ? )x(t); x(t); C (t))
?p(t) ? r(t)q(d(t); t) , t 2 [t1 ; t2 ]
0 = W (q(x; t); x(t); x(t); C (t)) ? p(t) ? r(t)q(x; t) , t 2 [t1 ; t2 ]
(4.9b)
(4.9c)
(4.9d)
(4.9e)
(4.9f)
(4.9g)
(4.9h)
(4.9i)
(4.9j)
Chapter 4: The Dynamic Problem
77
Dynamic Two-Part Kuhn-Tucker Conditions
1 K ? K = 0 ; t 2 [t1 ; t2 ]
K ? K 0 ; t 2 [t1 ; t2 ]
2 C = 0 ; t 2 [t1 ; t2 ]
C 0 ; t 2 [t1 ; t2 ]
3 p = 0 ; t 2 [t1 ; t2 ]
p 0 ; t 2 [t1 ; t2 ]
4 r = 0 ; t 2 [t1 ; t2 ]
r 0 ; t 2 [t1 ; t2 ]
(4.10a)
(4.10b)
(4.10c)
(4.10d)
(4.10e)
(4.10f)
(4.10g)
(4.10h)
4.3 The Nonlinear Tari
Recall from Chapter 3 the expression for Weighted Surplus where the revenue function
is the nonlinear tari.
WS =
Z x
0
[W + (1 ? 2 )R(q) ? (1 ? )C1 (q)]d ? C2 (x) ? C3(K )
(4.11)
Adapt this Weighted Surplus for the dynamic problem as was discussed in Section
4.1.
WS =
Z t2
t1
e?it
hZ x(t)
0
[W (q(; t); ; x(t); C (t)) + (1 ? 2 )R(q(; t)) ?
i
(1 ? )C1 (q(; t))]d ? (1 ? )C2(x(t)) ? (1 ? )C3 (Kt(t)) dt
The dynamic nonlinear tari problem will maximize discounted Weighted Surplus,
subject to the constraints from the static nonlinear tari problem in addition to those
listed in Section 4.1.
R
Maximize tt12 e?it WSdt
Subject To
Chapter 4: The Dynamic Problem
78
xt (t) ? G(x(t); d (t)) = 0
Z x(t)
K (t) + C (t) ?
q(; t)d = 0
0
K ? K (t) 0
C (t) 0
K (t) 0
x(t) 0
q(; t) 0
W (q(d (t); t); d (t); [d (t) + (1 ? )x(t)]; C (t)) ?
R(t; q(t; d (t))) = 0
Wq (q(d(t); t); d (t); [d (t) + (1 ? )x(t)]; C (t)) ?
Rq (q(d (t); t); t) = 0
W (q(x(t); t); x(t); x(t); C (t)) ? R(q(x(t); t)) = 0
Wq (q(x(t); t); x(t); x(t); C (t)) ? Rq (q(x(t); t)) = 0
(4.12a)
As was done with the static nonlinear tari case, integrate out the revenue function, R(q), from the objective function of the dynamic model and solve for the corresponding Euler equations and transversality conditions. To eliminate the revenue
R
function from the objective function, integrate 0x e?it (1 ? 2 )R(t; q)d by parts.
R
R
Select u = R(t; q) so that du = Rq (t; q)q d. Also choose v = dv = d = .
This leads to
Rx
e?it (1 ? 2 )R(t; q)d =
R
e?it (1 ? 2 ) [R(t; q)]x0 ? 0x Rq (t; q)q d
R
= e?it (1 ? 2 ) xR(t; q)j=x ? 0x Rq (t; q)q d
R
= e?it (1 ? 2 ) xW j=x ? 0x Wq q d .
0
(4.13)
Chapter 4: The Dynamic Problem
79
Substitute this result into the objective function. Note that most function arguments are suppressed for ease of exposition.
Z t2
t1
e?it
hZ x 0
W + (1 ? 2 )R ? (1 ? )C1 (q) d ? (1 ? )C2 (x) ?
i
(1 ? )C3 (Kt) dt
=
Z t2
t1
e?it
hZ x 0
W ? (1 ? 2 )Wq q ? (1 ? )C1 (q) d ? (1 ? )C2 (x) ?
i
(1 ? )C3 (Kt) + (1 ? 2 )xW j=x dt
The dynamic problem becomes:
Maximize
R t2 ?it R x
0 [W
t1 e
f
? (1 ? 2 )Wq q ? (1 ? )C1 (q)]d ?
(1 ? )C2 (x) ? (1 ? )C3 (Kt) + (1 ? 2 )xW j=x gdt
(4.14)
xt (t) ? G(x(t); d (t)) = 0
(4.15a)
K (t) + C (t) ?
q(; t)d = 0
0
K ? K (t) 0
C (t) 0
K (t) 0
x(t) 0
q(; t) 0
W (q(d (t); t); d (t); [d (t) + (1 ? )x(t)]; C (t)) ?
R(t; q(t; d (t))) = 0
Wq (q(d(t); t); d (t); [d (t) + (1 ? )x(t)]; C (t)) ?
Rq (q(d (t); t); t) = 0
W (q(x(t); t); x(t); x(t); C (t)) ? R(q(x(t); t)) = 0
Wq (q(x(t); t); x(t); x(t); C (t)) ? Rq (q(x(t); t)) = 0 .
(4.15b)
Z x(t)
(4.15c)
(4.15d)
(4.15e)
(4.15f)
(4.15g)
(4.15h)
(4.15i)
(4.15j)
(4.15k)
Chapter 4: The Dynamic Problem
80
Note that the last two constraints have already been satised via the replacements
for R and Rq in the objective function. As a result, subsequent derivations do not
explicitly refer to them.
Let 1 through 5 represent inequality constraints 4.15c through 4.15g, respectively. Similarly, let 1 and 2 represent equality constraints 4.15h and 4.15i.
Furthermore, make the following representations: Y = [y1; Y2T ; y3 ; Y4T ]T ; y1 = x;
Y2 = [y21; y22 ; y23 ; y24]T ; y21t = C ; y22t = d ; y23 = q ; y23 = q ; y24t = Kt ; and
Y4 = [y41 ; y42 ]T
Whereas y3t represents the multiplier for trajectory equation 4.15a, Y4t is the
vector of multipliers for constraints 4.15h and 4.15i. Call the multiplier associated
with constraint 4.15b . The multipliers associated with 1 through 5 are labeled
1 through 5 , respectively. Dene functions F1 and F2 as follows:
F1 =e?it [W ? (1 ? 2 )Wy y23 ? (1 ? )C1 (y23)] ? ( ? 5 )y23
=F1(t; ; y1 ; y23; y21t ; y23 ; ; 5 )
23
(4.16)
F2 =e?it [?(1 ? )C2 (y1) ? (1 ? )C3 (y24t ) + (1 ? 2 )y1 W j=y ] +
y3t [y1t ? G(y1; y22t )] + (y24 + y21t ) + 1 1 (y24t ) + 2 2(y21t ) +
3 3 (y24t ) + 4 4 (y1) + y41t 1(t; y1 ; y23j=y t ; y21t ; y22t ) +
y42t 2 (t; y1; y23 j=y t ; y21t ; y22t )
=F2(t; y1 ; y23j=y ; y23j=y t ; y24; y1t ; y21t ; y22t ; y24t ; y3t ; y41t ; y42t ; ;
1
22
22
1
22
1 ; 2 ; 3 ; 4 ) .
An equivalent problem is therefore to
Maximize
Z t Z y
f F1 (t; ; y1; y23 ; y21t ; y23 ; ; 5 )d +
2
t1
1
0
F2(t; y1 ; y23j=y ; y23j=y t ; y24; y1t ; y21t ; y22t ; y24t ; y3t ; y41t ; y42t ;
; 1 ; 2 ; 3 ; 4 )gdt .
1
22
(4.17)
Dene the variation H to be the dierence between the optimal system variable vector,
Y , and the comparison vector, Y . For any constant a, the value of Equation 4.17 as
Chapter 4: The Dynamic Problem
81
a function of a is
g(a) =
Z t2 +at2 Z y +ah1
1
0
t1
F1(t; ; Y + aH; Yt + aHt ; Y + aH )d+
F2(t; Y + aH; Yt + aHt )] dt
(4.18)
Recall that Proposition 3.12 showed that g(a) will reach its maximum at a = 0.
Therefore, dierentiate g(a) with respect to a.
g0 (a) =
Z t2 +at2
@
@a
t1
Z y +ah1
1
0
Z y +ah1
1
0
0
=
0
t2
2 t=t2 +at2
@ F d + F j h + @ F dt +
1 =y1 +ah1 1
@a 1
@a 2
2 F1d + F
Z t2 +at2 "Z y +ah1 1
t2
t=t2 +at2
T
T
T
#
F1 j=y +ah h1 + F2TY H + F2TYt Ht dt +
1
"Z
y1 +ah1
0
F1Y H + F1Yt Ht + F1Y H d +
0
t1
F1 d + F2 dt +
F1 d + F
Z t2 +at2 Z y +ah1
1
t1
Z y +ah1
1
1
#
2 F1d + F
t=t2
t2
Chapter 4: The Dynamic Problem
82
Proposition 3.12 also states that g0 (a) equals zero at a = 0, so set g0 (0) = 0.
g0 (0) =0
=
Z t2 "Z y
1
0
t1
[F1TY H + F1TYt Ht + F1TY H ]d + F1j=y1 h1 +
#
F2TY H + F2TYt Ht dt +
=
Z t2 "Z y h
1
y1
0
#
2 F1d + F
t=t2
t2
i
F1y h1 + F1y h23 + F1y t h21t + F1y h23 d +
1
0
t1
"Z
23
21
23
F1j=y h1 + F2y h1 + F2y j=y h23 j=y + F2y j=y t h23 j=y t +
F2y h24 + F2y t h1t + F2y t h21t + F2y t h22t + F2y t h24t +
1
1
24
1
23
1
21
1
#
22
F2y t h3t + F2y t h41t + F2y t h42t dt +
3
41
23
22
24
"Z
42
22
y1
0
F1 d + F
#
2 t=t2
t2
The following approximations and integrations by parts will help in deriving the
necessary Euler and Transversality Conditions for the dynamic nonlinear tari problem. The two approximations to hi (t2) and h23 j=y1 are similar to ones derived in
Kamien and Schwartz [1991].
hi (t2) yi (t2) ? yit (t2)t2 ; i = 1; 21; 22; 24; 3; 41; 42; 43
h23 j=y y23 j=y ? y23 j=y h1
1
1
Z t2 Z y
1
t1
0
1
F1y h23 d =
23
=
Z t1
t2
Z t1
[[F1y23 h23 ]0 ?
y1
Z y
1
0
@ F1y
h23 @23 d]dt
[[F1y23 h23 ]j=y1 ? [F1y23 h23]j=0 ?
@ F1y
h23 @23 d]dt
0
t2
Z y
1
Chapter 4: The Dynamic Problem
Z t2 Z y
1
t1
0
F1y t h21t ddt =
21
83
Z t2
t1
=[h21
h21t
Z y
1
0
Z t2
t1
Z y
1
F1y t ddt
21
0
F1y t d]tt ?
Z y @ F
1y t
2
1
21
h21[
Z y
1
1
0
d
+ [F1y21t ]j=y1 y1t ]dt
@t
21
[F1y21t ]jt=t2 d ?
Z t2
Z y @ F
1
1y21t
h21[
d
+ [F1y21t ]j=y1 y1t ]dt
@t
t1
0
=h21 (t2)
For i = 1; 21; 22; 24; 3; 41; 42
Z t2
t1
0
@ F2
hi @tyit dt
t1
Z t2 @ F
2yit
=[F2yit hi]jt=t2 ? hi
@t dt .
t1
F2yit hit dt =[F
t
2yit hi ]t21
?
Z t2
Substitute results from the two approximations along with the derived integrations
by parts into the expression for g0 (0) = 0.
Chapter 4: The Dynamic Problem
g0 (0) =0
Z t2
@F
F1y d + F1j=y + F2y ? @t2y t ]dt ?
0
Z y @ F
@ F2y t
1y t
d
+ F1y t j=y y1t +
h21 [
@t
@t ]dt ?
0
Z t Z y
@ F1y
@ F2y t
h22 [ @t ]dt + [ h23[F1y ? @ ]d +
h1[
t1
Z t2
t1
Z t2
t1
84
Z y
1
1
1
1
1
1
21
21
21
2
22
t1
1
1
0
23
23
h23j=y1 [F1y23 ]j=y1 ? h23j=0 [F1y23 ]j=0 + h23 j=y1 [F1y23 ]j=y1 +
Z t2
@ F2
h23j=y22 t [F2y23 ]j=y22 t ]dt + h24[F2y24 ? @ty24t ]dt ?
t1
Z t2
@ F2
h3 [ @ty3t ]dt ?
t1
Z t2
Z t2
@ F2y41t
@ F2
h41 [ @t ]dt ? h42 [ @ty42t ]dt +
t1
t1
Z y (t2 )
1
0
[F1y21t ]jt=t2 d[y21 (t2 ) ? y21 t (t2)t2 ] +
[F2y1t ]jt=t2 [y1 (t2 ) ? y1t (t2 )t2 ]
[F2y21t ]jt=t2 [y21(t2 ) ? y21 t (t2)t2 ] +
[F2y22t ]jt=t2 [y22(t2 ) ? y22 t (t2)t2 ] +
[F2y24t ]jt=t2 [y24(t2 ) ? y24 t (t2)t2 ] +
[F2y3t ]jt=t2 [y3 (t2 ) ? y3t (t2 )t2 ] +
[F2y41t ]jt=t2 [y41(t2 ) ? y41 t (t2)t2 ] +
[F2y42t ]jt=t2 [y42(t2 ) ? y42 t (t2)t2 ] +
[
Z y
1
0
F1 d + F2]jt=t t2
2
Chapter 4: The Dynamic Problem
85
Rearranging terms, we get
g0 (0) @F
F1y d + F1 j=y + F2y ? @t2y t ?
t
0
[y23 F1y ]j=y ? [y23 F2y ]j=y ]dt ?
Z t
Z y @ F
@ F2
1y t
d
+ F1y t j=y y1t + y t ]dt ?
h21[
@t
@t
t
0
Z t
Z
Z
t
y
@ F1y
@ F2
h22[ @ty t ]dt + [ h23[F1y ? @ ]d ?
Z t2
1
Z y
1
h1[
1
1
1
1
23
2
1
1
23
1
21
21
1
21
1
2
2
22
t1
t1
1
23
23
0
[h23 F1y23 ]j=0 + [h23 F2y23 ]j=y22 t ]dt +
Z t2
Z t2
@ F2
@ F2y24t
h24[F2y24 ? @t ]dt ? h3[ @ty3t ]dt ?
t1
t1
Z t2
Z
t
2
@ F2
@ F2
h41[ @ty41t ]dt ? h42 [ @ty42t ]dt +
t1
t1
[F2y1t y1 + [F2y21t +
Z y
1
0
(4.19)
F1y t d]y21 + F2y t y22 +
21
22
F2y t y24 + F2y t y3 + F2y t y41 + F2y t y42 ]jt=t +
24
3
41
[F1y23 + F2y23 ]j=y1 y23 j=y1 + f
Z y
1
Z y
1
[
0
0
2
42
F1d + F2 ? [F2y t ]y1t ?
1
F1y t d + F2y t ]y21 t ? [F2y t ]y22 t ? [F2y t ]y24 t ?
21
21
22
24
[F2y3t ]y3t ? [F2y41t ]y41 t ? [F2y42t ]y42 t gjt=t2 t2 .
From Equation 4.3, we can nally state the Euler equations and transversality conditions below.
Chapter 4: The Dynamic Problem
86
Dynamic Euler Equations
@F
F1y d + F1 j=y + F2y ? @t2y t ? [y23 F1y ]j=y ?
0
[y23 F2y ]j=y = 0 ; t 2 [t1 ; t2 ]
Z y
@ F2y t
@ F1y t
d
+ [F1y t ]j=y y1t +
@t
@t = 0 ; t 2 [t1 ; t2 ]
0
@ F2y t
@t = 0 ; 2 [0; y1 (t)] ; t 2 [t1 ; t2 ]
@ F1
F1y ? @y = 0 ; t 2 [t1 ; t2 ] ; 2 [0; y1 ]
Z y
1
1
1
1
1
1
23
1
23
21
21
1
21
22
23
23
F1y = 0 ; t 2 [t1 ; t2 ] ; = 0
F2y = 0 ; t 2 [t1 ; t2 ] ; = y22 t
@F
F2y ? @t2y t = 0 ; t 2 [t1 ; t2 ]
@ F2y t
@t = 0 ; t 2 [t1 ; t2 ]
@ F2y t
@t = 0 ; t 2 [t1 ; t2 ]
@ F2y t
@t = 0 ; t 2 [t1 ; t2 ]
23
23
24
24
1
(4.20a)
(4.20b)
(4.20c)
(4.20d)
(4.20e)
(4.20f)
(4.20g)
3
(4.20h)
41
(4.20i)
42
(4.20j)
Dynamic Integral Equation
Z y1
0
y23 d ? y24t ? y21t = 0 ; t 2 [t1 ; t2 ]
(4.21)
Chapter 4: The Dynamic Problem
87
Dynamic Kuhn-Tucker Conditions
1 [y24t ? K ] = 0 ; t 2 [t1 ; t2 ]
y24t ? K 0 ; t 2 [t1 ; t2 ]
2 [y21t ] = 0 ; t 2 [t1 ; t2 ]
y21t 0 ; t 2 [t1 ; t2 ]
3 [y24t ] = 0 ; t 2 [t1 ; t2 ]
?y24t 0 ; t 2 [t1 ; t2 ]
4 [y1] = 0 ; t 2 [t1 ; t2 ]
?y1 0 ; t 2 [t1 ; t2 ]
5 [y23] = 0 ; 2 [0; y1 (t)] ; t 2 [t1 ; t2 ]
?y23 0 ; 2 [0; y1 (t)] ; t 2 [t1 ; t2 ]
(4.22a)
(4.22b)
(4.22c)
(4.22d)
(4.22e)
(4.22f)
(4.22g)
(4.22h)
(4.22i)
(4.22j)
Dynamic Transversality Equations
"Z
y1 (t2 )
0
[F1y23
0=f
Z y
1
0
[F2y1t ]jt=t2 = 0 , if y1(t2 ) free
#
F1y t d + F2y
21
21t
(4.23a)
jt=t = 0 , if y21 (t2) free
(4.23b)
2
[F2y22t ]jt=t2 = 0 , if y22 (t2) free
+ F2y23 ]j=y1 = 0 , if y23j=y1 free, t 2 [t1 ; t2 ]:
[F2y24t ]jt=t2 = 0 , if y24 (t2) free
[F2y3t ]jt=t2 = 0 , if y3(t2 ) free
[F2y41t ]jt=t2 = 0 , if y41 (t2) free
[F2y42t ]jt=t2 = 0 , if y42 (t2) free
F1 d + F2 ? F2y t
1
y
1t ? [
Z y
1
0
(4.23c)
(4.23d)
(4.23e)
(4.23f)
(4.23g)
(4.23h)
F1y t d + F2y t ]y21 t ? [F2y t ]y22 t ?
22
21
21
F2y t y24 t ? F2y t y3t ? F2y t y41 t ? F2y t y42 t gjt=t if t2 free
24
3
41
42
2
(4.23i)
Chapter 5
Examples
5.1 Overview
In this chapter, analytical results from previous chapters are applied to optimize
static and dynamic examples. First order necessary conditions are derived for different revenue functions, maximum capacity levels, capacity costs, and congestion
circumstances. For both the static and dynamic cases, is chosen to be 0 so that
Producer Surplus, or prot, is maximized.
The FONC are then numerically solved using Mathematica T M . In particular,
the Runge-Kutta algorithm [Wolfram, 1988] is applied to dierential equations encountered in the static case for the nonlinear tari and in the dynamic case for the
at-rate tari.
5.2 Static Examples
5.2.1 Assumptions
The willingness to pay function used in the following static examples was derived
from a similar model proposed in Oren, Smith and Wilson [1982]. In that paper, each
consumer was assumed to have a saturation, or maximum desirable, consumption
level, Q(; x). Here, consumers are assumed to also be aected by congestion such
88
Chapter 5: Examples
89
that their saturation consumption is given by
2
Q(; x; C ) = 2Tx(2 ? x)(1 ? )(1 + CT 2 )
where the parameter T describes some expected average consumption level per consumer. Setting C = 0 retrieves the saturation consumption function in the nocongestion scenario of Oren, Smith and Wilson [1982].
Q(; x; C ) = 2Tx(2 ? x)(1 ? )
The willingness to pay function can then be dened as
2
W (q; ; x; C ) = 2wo q ? Q(w;ox;q C )
2
W (q; ; x; C ) = 2wo q ? 2x(2 ? x)(1wo?q T)(T 2 + C 2)
where wo is the average consumer willingness to pay for one unit of consumption.
Producer costs take the following forms.
C1 (q) = 0
C2(x) = k0 x
C3 (K ) = k1 K
The cost attributed to each subscriber's individual consumption level is assumed to be
negligible. In addition, the subscriber size and capacity costs are proportional to the
number of subscribers and capacity level, respectively. In these static examples, k0 =
wo T ; in other words, the subscriber set size sensitive cost is about 20% of the average
5
consumer's willingness to pay for the average consumption amount. Parameter k1 is
set to either 0 for a negligible capacity cost or w4o if the cost of capacity is about a
quarter of the willingness to pay of the average consumer.
Separate results are obtained for the at-rate, two-part, three-part, and nonlinear
taris. Furthermore, each tari is optimized for increasing values of the maximum
capacity parameter, K .
Chapter 5: Examples
90
5.2.2 Numerical Results
Table 5.1 summarizes numerical results for the static problem with no congestion
eects when K = T , k0 = T 5wo , k1 = w4o , and = 0. In this particular example,
consumer willingness to pay is assumed to not be aected by congestion. Because
customers are still aected by the positive externality, their subscriber set will settle at
an equilibrium value uninhibited by negative externality eects. In comparison with
other results in which consumers are aected by congestion, however, the resulting
Consumer and Producer Surpluses will not be as high. In the case with congestion,
consumers are willing to pay more for less congestion, so producers can raise more
revenues. Furthermore, consumers will also have higher CS since their congestion
problem is improved and because producers are inecient at extracting surplus from
consumers with multi-part taris. On the other hand, consumers who are unaected
by congestion are not willing to pay more for improved congestion, so both parties
do worse than in the congestion case.
Variable x decreases from 0.4795 for the at-rate tari to 0.8406 for the nonlinear tari as illustrated in Figure 5.1. Fixed rate p increases from 0:7590wo T to
0:0020wo T with the nonlinear tari. Revenue functions are compared in Figure 5.3.
Consumer Surplus dips from the at-rate result of 0:1676wo T to the nonlinear value
of 0:1226wo T . Producer Surplus, which is also the Weighted Surplus since = 0,
rises from the nonlinear result of 0:1351wo T to the nonlinear optimum of 0:3273wo T .
Table 5.2 summarizes numerical results for the static problem with congestion
eects when K = 2:5T , k0 = T w5 o , k1 = w4o , and = 0. Variable x decreases from
0.4526 for the at-rate tari to 0.8870 for the nonlinear tari, while xed rate p
increases from 1:95880wo T to 0:0011wo T with the nonlinear tari as illustrated in
Figure 5.4. Revenue functions are compared in Figure 5.6. The optimum congestion
level is ?1:2469T for the at-rate tari and ?1:3319T for the nonlinear tari. Consumer Surplus dips from the at-rate result of 0:3665wo T to the nonlinear value of
0:1864wo T . Producer Surplus, which is equivalent to Weighted Surplus since = 0,
rises from the nonlinear result of 0:1711wo T to the nonlinear optimum of 0:7854wo T .
In both Tables 5.1 and 5.2, note the considerable jump in Producer Surplus from
Chapter 5: Examples
q
x
p
r1
r2
C
K
or q
q
Total
Usage
Consumer
Surplus
Producer
Surplus
91
Flat Rate
0:4795
0:7590w T
{
{
0
o
T
Two-Part
0:7115
0:1681w T
0:8726w
{
0
o
o
T
Three-Part
0:7977
0:04796w T
1:2969w
0:5990w
0
o
o
o
T
1:4581(1 ? )T 1:0336(1 ? )T 1:3437(1 ? )T
{
{
0:6743(1 ? )T
0:5315T
0:4738T
0:5371T
Nonlinear
0:8406
0:0020w T
(2q ?
o
0:8933q1:5
T 0:5
{
0
T
0:5662T
T
0:1475w
T
0:1314w
T
0:1226w
T
0:1351w
T
0:2723w
T
0:3104w
T
0:3273w
T
o
o
o
o
o
o
1:7055(?1 + )2T
{
0:1676w
o
)w
o
o
Table 5.1: Flat-Rate, Two-Part, Three-Part, and Nonlinear Results if K = T , k0 =
T wo , k = wo , and = 0, without Congestion Eects
1
5
4
0.9
x
p
0.6
0.3
0
Flat-Rate Two-Part Three-Part Nonlinear
Figure 5.1: x and p for Dierent Taris, without Congestion Eects
Chapter 5: Examples
92
CS
PS
0.3
0.2
0.1
0
Flat-Rate Two-Part Three-Part Nonlinear
Figure 5.2: CS and PS for Dierent Taris, without Congestion Eects
Three-Part, R1
2.5
Two-Part
Three-Part, R2
Nonlinear
Flat-Rate
2
1.5
1
0.5
0.5
1
1.5
2
q
Figure 5.3: Optimal Taris, without Congestion Eects
Chapter 5: Examples
93
1.8
x
p
1.2
0.6
0
Flat-Rate Two-Part Three-Part Nonlinear
Figure 5.4: x and p for Dierent Taris, with Congestion Eects
0.8
CS
PS
0.6
0.4
0.2
0
Flat-Rate Two-Part Three-Part Nonlinear
Figure 5.5: CS and PS for Dierent Taris, with Congestion Eects
Chapter 5: Examples
94
Three-Part, R1
Two-Part
Three-Part,
Nonlinear R2
6
5
4
3
Flat-Rate
2
1
1
2
3
4
q
Figure 5.6: Optimal Taris, with Congestion Eects
the at-rate tari to the two-part tari and the smaller increases to the three-part
and then nonlinear taris also shown in Figures 5.2 and 5.5. Adding more revenue
choices (i.e. marginal tari rates) after the two-part tari does not result in marked
improvements in Producer Surplus. Therefore, a more complicated tari structure
may not be a fair trade-o for a small gain in prot. nonlinear optimum of 0:7854wo T .
Figure 5.7 shows the switching capacity, Ks for the at-rate, two-part, three-part,
and nonlinear taris. For K 2 [0; Ks ], both the no-congestion and congestion cases
optimally set K = K . Beyond K = Ks, the no-congestion case will allow K anywhere
between Ks and K ; no additional Producer Surplus is derived from increasing capacity
and, since the cost of capacity is zero, total costs do no change. The congestion case,
however, continues to optimally require K = K since additional capacity increases
Producer Surplus. Table 5.3 lists the optimal system variable values at Ks for the
dierent taris.
Figures 5.8 through 5.11 show the optimal switching capacity, Ks, and attening
capacity, Kf , for K1 = w4o for several taris when = 0. Where K < Kf , K exactly
equals the maximum available capacity. At K < Kf , adding more capacity no longer
Chapter 5: Examples
q
x
p
r1
r2
C
K
95
Flat Rate
Two-Part
Three-Part
Nonlinear
0:4526
0:7479
0:8400
0:8870
1:9588w T
0:2423w T
0:0507w T
0:0011w T
15
{
1:1723w
1:5215w
(2q ? 0 45600 5 )w
{
{
0:9782w
{
?1:2469T
?1:4127T
?1:3562T
?1:3319T
2:5T
2:5T
2:5T
2:5T
3:5785(1 ? )T 2:3221(1 ? )T 2:8270(1 ? )T 3:5092(?1 + )2 T
{
{
1:3239(1 ? )T
{
o
o
o
o
o
o
:
T :
:
q
o
o
or q
q
Total
Usage
Consumer
Surplus
Producer
Surplus
1:2531T
1:0873T
1:1438T
1:1681T
0:3665w
T
0:2688w
T
0:2112w
T
0:1864w
T
0:1711w
T
0:6812w
T
0:7548w
T
0:7854w
T
o
o
o
o
o
o
o
o
Table 5.2: Flat-Rate, Two-Part, Three-Part, and Nonlinear Results if K = K =
2:5T , k0 = T 5wo , k1 = w4o , and = 0, with Congestion Eects
K
with congestion
no congestion
Ks
Ks
K
Figure 5.7: Comparison of Optimal Capacities, = 0, K1 = 0
Chapter 5: Examples
q
x
p
r1
r2
C
K
or q
q
Total
Usage
Consumer
Surplus
Producer
Surplus
96
Flat Rate
Two-Part
Three-Part
Nonlinear
0:5686
0:7320
0:8254
0:8734
0:7022w T
0:1999w T
0:0491w T
0:0013w T
15
{
0:7320w
1:2380w
(2q ? 0 95050 5 )w
{
{
0:4127w
{
0
0
0
0
0:6624T
0:5462T
0:6202T
0:65465T
1:6278(1 ? )T 1:1769(1 ? )T 1:5389(1 ? )T 1:9679(?1 + )2 T
{
{
0:7387(1 ? )T
{
o
o
o
o
o
o
:
T :
:
q
o
o
0:6624T
0:5462T
0:6202T
0:6547T
0:2631w
T
0:1999w
T
0:1759w
T
0:1628w
T
0:2856w
T
0:3997w
T
0:4551w
T
0:4800w
T
o
o
o
o
o
o
o
o
Table 5.3: Flat-Rate, Two-Part, Three-Part, and Nonlinear Results if K = K s,
k0 = T 5wo , k1 = 0, and = 0
increases Producer Surplus, so K remains equal to Kf since the cost of capacity is
non-zero. For K Kf , total usage equals capacity. With the congestion case, this
translates to critical congestion: markets which only have access to a small amount
of capacity will purchase all available capacity and consumers consequently bear the
maximum level of congestion. Producer Surplus for the no-congestion case reaches
its maximum at K = Kf .
At K = Ks, Producer Surplus once again benets by the addition of capacity for
the congestion case, so capacity is set equal to K . Although consumers who perceive
congestion are willing to pay more for increased capacity, the no-congestion case
maintains capacity at Kf because consumers do not perceive a value for a decrease
in congestion.
Note that Ks is much smaller when K1 = 0 than when K1 = w4o . Increasing
the cost of capacity delays the point at which market players are willing to pay
more, in revenues and capacity costs, to deect the eects of congestion. Also note
that the ratio KKfs decreases with every revenue change (at to two-part, two-part to
three-part, three-part to nonlinear). This signies a tendency once again for market
Chapter 5: Examples
97
K
with congestion
2:2227
no congestion
0:5315
0:5315
2:2227
K
Figure 5.8: Flat-Rate Tari Optimal Capacity Comparison, = 0, K1 = w4o
players to take advantage of capacity increases when the revenue function allows more
exibility in terms of extracting producer or Consumer Surplus. Reduction of the cost
of capacity yields results approaching the situation depicted in Figure 5.7.
Table 5.4 summarizes numerical results for the static problem when K = K S ,
k0 = T 5wo , k1 = w4o , and = 0.
Chapter 5: Examples
98
K
with congestion
1:2176
no congestion
0:4738
0:4738
1:2176
K
Figure 5.9: Two-Part Tari Optimal Capacity Comparison, = 0, K1 = w4o
K
with congestion
1:1879
no congestion
0:5371
0:5371
1:1879
K
Figure 5.10: Three-Part Tari Optimal Capacity Comparison, = 0, K1 = w4o
Chapter 5: Examples
99
K
with congestion
1:1808
no congestion
0:5662
0:5662
K
1:1808
Figure 5.11: Nonlinear Tari Optimal Capacity Comparison, = 0, K1 = w4o
q
x
p
r1
r2
C
K
or q
q
Total
Usage
Consumer
Surplus
Producer
Surplus
Flat Rate
Two-Part
Three-Part
Nonlinear
0:4585
0:7215
0:8043
0:8454
1:7066w T
0:1814w T
0:0481w T
0:0019w T
15
{
0:9910w
1:3746w
(2q ? 0 74900 5 )w
{
{
0:7406w
{
?1:1089T
?0:6222T
?0:5539T
?0:5279T
2:2227T
1:2176T
1:1879T
1:1808T
3:1518(1 ? )T 1:2910(1 ? )T 1:5828(1 ? )T 1:9661(?1 + )2 T
{
{
0:7861(1 ? )T
{
o
o
o
o
o
o
:
:
q
o
T :
o
1:1139T
0:5954T
0:6340T
0:6530T
0:3313w
T
0:1695w
T
0:1402w
T
0:1274w
T
0:1351w
T
0:2723w
T
0:3104w
T
0:3273w
T
o
o
o
o
o
o
o
o
Table 5.4: Flat-Rate, Two-Part, Three-Part, and Nonlinear Results if K = K s,
k0 = T 5wo , k1 = w4o , and = 0
Chapter 5: Examples
100
5.3 Dynamic Examples
5.3.1 Assumptions
Similar to the willingness to pay function from the previous section, the dynamic
form makes allowances for the system variables to change over time.
2
o q (; t)T
W (q(; t); ; x(t); C (t)) = 2wo q(; t) ? 2x(t)(2 ? x(wt))(1
? )(T 2 + C 2 (t))
Furthermore, assume that the subscriber set growth rate, xt , takes the following form:
xt = a(d ? x) ; a > 0 .
Growth of x depends on a scaled dierence of the last potential subscriber, given
the last subscriber is now x, and the current size of the subscriber set, x. If the
last potential subscriber also happens to be the last subscriber, the subscriber set's
growth halts. On the other hand, the bigger the dierence is between the last potential
subscriber and the current last subscriber, the faster x will grow. In these dynamic
examples, a = :2 so that x grows at a rate one-fth the dierence between x and d
per unit of time.
Producer costs take the following forms.
C1 (q) = 0
C2 (x(t)) = k0 x(t)
C3(Kt (t)) = k1 Kt (t)
Note that, in contrast to the static capacity cost, C3 is directly proportional to the
instantaneous change in K , or Kt . As in the previous section, k0 = :2wo T and
k1 = :25wo .
The producer's revenue function is the at-rate tari. The discounting rate, i, is
taken to be 5%.
5.3.2 Numerical Results
Table 5.5 summarizes numerical results for the dynamic problem when k0 = T 5wo ,
k1 = w4o , and = 0. Optimal system variable values are found for maximum capacities
Chapter 5: Examples
K
x(t = 25)
p(t = 25)
C (t = 25)
Total Usage
(t = 25)
Consumer Surplus
(t = 25)
Producer Surplus
(t = 25)
Total Discounted
Consumer Surplus
(t = 0 to t = 25)
Total Discounted
Producer Surplus
(t = 0 to t = 25)
101
0.5
1.0
2.0
0.41
0.43
0.42
0:77w T 0:93w T 1:63w T
?0:07T ?0:45T ?1:06T
o
o
o
0:43T
0:55T
0:94T
4.0
0.43
3:53w T
?1:89T
10.0
0.41
11:29w T
?3:70T
2:11T
6:30T
o
0:11w
o
T
0:15w
T
0:25w
T
0:58w
0:23w
o
T
0:31w
o
T
0:61w
o
T
1:33w
o
T
1:86w
o
T
2:93w
o
2:72w
T
3:71w
o
T
6:94w
o
o
o
o
o
T
1:64w
1:44w
o
T
4:58w
T
7:13w
o
T
15:42w
o
T
T
17:31w
T
48:89w
o
T
o
o
o
o
T
T
Table 5.5: Flat-Rate Dynamic Results for Increasing K when k0 = wo5T , k1 = w4o , and
=0
from 0.5 to 10.0. The largest discounted total Producer Surplus is achieved with
K = 10:0. The subscriber set size, x, hovers around 0.4, p increases from 0.77 to
11.29, and C changes from -0.07 to -3.70 as K is increased from 0.5 to 10.0. Trends
are also illustrated in Figures 5.12 and 5.13.
The No-Congestion Case
The following results are for the case in which congestion has no eect on consumers.
As with the critical congestion case, C = 0, but WC and Wq C are also zero. The
optimal subscriber set function over time is
2
x(t) =0:028482 ? 0:0315141
et + 0:0803223t ? 0:00665977t +
0:000256762t3 ? 3:781310 ?6t4
Figure 5.14 shows x, d , p, Total Usage, and K for the no congestion case from
t = 0 to t = 23. t = 23 is the time at which x attens out. System variables remain
at their t = 23 levels for t 23. In Figure 5.14 and subsequent graphs, wo and T are
Chapter 5: Examples
102
p
9
6
3
0
K = 0:5 K = 1:0 K = 2:0 K = 4:0 K = 10:0
Figure 5.12: Fixed Tari for Varying Maximum Capacity
45
CS
PS
30
15
0
K = 0:5
K = 1:0
K = 2:0
K = 4:0
K = 10:0
Figure 5.13: CS and PS for Varying Maximum Capacity
Chapter 5: Examples
103
p
0.6
d
0.4
0.2
x
K
5
10
15
20
t
Figure 5.14: x, d , p, and K for the No Congestion Case
taken to be one to facilitate the simultaneous plotting of system variables. Note that
x and d converge around t = 23.
Figure 5.15 shows CS, PS, and TS for the no congestion case from t = 0 to t = 23.
Figure 5.16 shows q(; t) for the no congestion case from t = 0 to t = 23.
Chapter 5: Examples
104
TS
0.35
0.3
PS
0.25
0.2
0.15
CS
0.1
0.05
5
10
15
20
t
Figure 5.15: CS, PS, and TS for the No Congestion Case
The Congestion Case
Results for markets which do experience the eects of congestion are shown in the
next several graphs.
First considered is the case where capacity is set to its maximum available value,
K = T . Figure 5.17 shows x, d , p, C , Total Usage, and K for the congestion case
from t = 0 to t = 1:5 when K = K = T . The number of subscribers levels o after t =
1:5 so system variables remain steady beyond that point. Figure 5.18 shows CS, PS,
and TS for the congestion case from t = 0 to t = 1:5 when K = K = T . Figure 5.19
shows q(; t) for the congestion case from t = 0 to t = 1:5 when K = K = T .
Chapter 5: Examples
105
1
q(; t)
20
0.5
15
0
0
10
0.2
0.4
t
5
0.6
0.8
10
Figure 5.16: Consumption Trajectory for the No Congestion Case
Chapter 5: Examples
106
1
K
0.5
0.2
p
Total Usage
x
d
0.4
0.6
0.8
1
1.2
1.4
t
C
-0.5
-1
Figure 5.17: x, d , p, C , Total Usage, and K for the Congestion Case, K = K = T
TS
0.4
PS
0.3
0.2
CS
0.1
0.2
0.4
0.6
0.8
1
1.2
1.4
t
?0:25
PS,TS
Figure 5.18: CS, PS, and TS for the Congestion Case, K = K = T
Chapter 5: Examples
107
1.5
q(; t)
1.5
1
0.5
1
0
0
0.2
0.5
0.4
t
0.6
0.8
10
Figure 5.19: Consumption Trajectory for the Congestion Case, K = K = T
Now consider the case where capacity is set to its maximum available value, K =
4T . Figure 5.20 shows x, d , p, C , Total Usage, and K for the congestion case from
t = 0 to t = 1:6 when K = K = 4T . The number of subscribers levels o after t = 1:6
so system variables remain steady beyond that point.
Figure 5.21 shows CS, PS, and TS for the congestion case from t = 0 to t = 1:6
when K = K = 4T . Figure 5.22 shows q(; t) for the congestion case from t = 0 to
t = 1:6 when K = K = 4T .
Chapter 5: Examples
108
4
p
K
2
Total Usage
d
0.25
0.5
0.75
-2
x
1
1.25
1.5
t
C
-4
Figure 5.20: x, d , p, C , Total Usage, and K for the Congestion Case, K = K = 4T
TS
2
PS
1.5
1
CS
0.5
0.25
0.5
0.75
1
1.25
1.5
t
?1:0
PS,TS
Figure 5.21: CS, PS, and TS for the Congestion Case, K = K = 4T
Chapter 5: Examples
109
6
q(; t)
4
1.5
2
1
0
0
0.2
0.5
0.4
t
0.6
0.8
10
Figure 5.22: Consumption Trajectory for the Congestion Case, K = K = 4T
The nal scenario explores the congestion case when capacity is set to its maximum
available value, K = 10T . Figure 5.23 shows x, d , p, C , Total Usage, and K for the
congestion case from t = 0 to t = 7 when K = K = 10T . The number of subscribers
levels o at t = 7 and system variables remain at their t = 7 levels after that point.
Figure 5.24 shows CS, PS, and TS for the congestion case from t = 0 to t = 7
when K = K = 10T .
Figure 5.25 shows q(; t) for the congestion case from t = 0 to t = 7 when
K = K = 10T .
Chapter 5: Examples
110
p
20
15
10
5
K
Total Usage
1
2
3
x; d
4
5
6
7
t
C
-5
-10
Figure 5.23: x, d , p, C , Total Usage, and K for the Congestion Case, K = K = 10T
TS
6
PS
5
4
3
CS
2
1
1
2
3
4
5
6
7
t
?2:5
PS,TS
Figure 5.24: CS, PS, and TS for the Congestion Case, K = K = 10T
Chapter 5: Examples
q(; t)
111
20
6
10
0
0
4
0.2
0.4
2
t
0.6
0.8
10
Figure 5.25: Consumption Trajectory for the Congestion Case, K = K = 10T
Chapter 6
Conclusions and Extensions
6.1 Conclusions
6.1.1 General Conclusions
This research has provided a exible model which can maximize surplus for a variety of optimizers for dierent consumer patterns and capacity availability. Dierent
regulatory policy and technology variables and their eects on capacity, congestion,
and consumer willingness to pay can be investigated by carrying out a cost-benet
analysis by reiterating the optimization problem for every scenario we wish to study.
General conclusions are drawn from the necessary conditions for the maximization of Weighted Surplus in both static and dynamic frameworks when the revenue
function is the at, two-part, three-part, at/two-part, or nonlinear tari. These conclusions include guidelines for pricing decisions and capacity expansion under specic
conditions as summarized below in Tables 6.1 and 6.2.
In these two tables, \PE<NE?", or \Positive Externality
< Negative External
ity?", asks whether Wx (q(x); x) W (q(x); x)+ WC dC
. In other words, it asks
dx =x
whether the negative externality eects exceed the positive externality eects for the
last consumer. Recall that , K , and C refer to the Weighted Surplus parameter,
system capacity, and congestion, respectively. Finally, \U" means that the quantity
in question is unspecied.
112
Chapter 6: Conclusions and Extensions
,
K,
C,
PE<NE?
Conclusion
113
Interpretation
Reference
U,
U,
U,
Yes
The at-rate tari R(q) = p de- The producer has to make service Prop.
creases over x.
more attractive to potential con- 3.1, 3.6.
sumers by lowering prices when externalities aect consumers negatively. Alternatively, observation of
whether p increases or decreases over
time in the dynamic model can signal
the current dominant externality.
U,
U,
U,
U
Relaxing
the
constraint
W (q (x); x; x; C ) ? p ? rq (x) = 0
increases TS by the amount
gained from charging a slightly
higher xed rate.
If we are allowed to increase the xed Prop.
rate p slightly but still maintain the 3.3, 3.8.
same x, we do better by the amount
our TS improves. In eect, surpluses
for consumers and producers change
due to the increased revenue taken in
by producers and paid by consumers.
= 0, The marginal valueR of relaxing
6= K , the constraint C = 0 q()d ? K
U,
is equal to the marginal cost of
U
capacity.
Relaxing the constraint is equivalent Prop.
to one of the following: increasing 3.2, 3.7,
total usage, decreasing capacity, or 3.15.
increasing congestion, while holding
all other variables constant. Hence
the value associated with one of these
changes is merely the value of decreasing capacity costs.
K
x
Table 6.1: Summary of Analytical Conclusions
Chapter 6: Conclusions and Extensions
,
K,
C,
PE<NE?
Conclusion
114
Interpretation
Reference
= 0, The marginal valueR of relaxing
= K , the constraint C = 0 q()d ? K
U,
is equal to the marginal cost of
U
capacity plus the marginal value
of relaxing the constraint K K .
Similar to the previous conclusion Prop.
except that we must also take into 3.2, 3.7,
account the value of relaxing the 3.15.
nonnegativity constraint for K ? K .
= 0, If an increase in congestion leads
6= K , to a positive change in the objecU,
tive function, the system will opU
erate optimally at critical congestion, or C = 0.
Under certain conditions, if we are
maximizing prot and are not using
as much capacity as we could, we
should buy just enough capacity to
meet demand.
K
x
K
U,
U,
U
6= 1,
If the rst ( = 0) consumer's
marginal willingness to pay is
greater than marginal cost, capacity is optimal at its maximum
possible value.
Prop.
3.4,
3.10,
3.16.
The rst subscriber will be the rst Prop.
person to consume more. There- 3.18.
fore, if the producer can recover the
additional marginal cost of capacity through the rst consumer's increased willingness to pay, the producer will want to increase capacity.
Table 6.2: Summary of Analytical Conclusions, Cont.
Chapter 6: Conclusions and Extensions
115
Congestion
Eects?,
,
K
C3 (K )
Observation
U,
Small,
U
Operate at critical congestion (let the system's capacity exactly equal total
usage). If the capacity available to a system is very small but consumers value
service even under constrained capacity, they will use up all available capacity.
No,
Large,
U
Neither producers nor consumers gain from increased capacity beyond a certain level. If consumers are unaected by congestion, they will not be willing
to pay for more capacity (and therefore congestion alleviation) beyond a certain level. Hence, producers will not be able to charge more for more capacity.
Yes,
Large,
C3 (K ) = 0
Set capacity at its maximum possible value. If the cost of capacity is negligible and consumers value a decrease in congestion, buy as much capacity as
possible to allow subscribers to consume as much as possible.
Yes,
U,
Small
Set capacity at its maximum possible value. Maximum capacity is desirable
when consumers are aected by congestion and the cost of capacity is very
small.
Table 6.3: Summary of Observations from Examples
6.1.2 Numerical Example Observations
Numerical analysis of specic examples yielded results which were partially predicted
by the analytical results above. Numerically derived observations are recorded in
Tables 6.3 and 6.4. Recall that K is the maximum capacity available to the system
and C3(K ) is the cost of capacity. \Congestion Eects?" asks whether the potential
subscribing population's willingness to pay depends on congestion. As before, \U"
means that the quantity of interest is unspecied.
6.2 Extensions
Multiple Providers Let us now rethink the restaurant example. The problem it
poses is that, right next door to Restaurant A is Restaurant B. And for that matter,
Restaurant C is just down the street. So what we have is at least an oligopolistic and
Chapter 6: Conclusions and Extensions
116
Congestion
Eects?,
,
K
C3 (K )
Observation
Yes,
Large,
U
CS and PS higher. Since consumers are willing to pay more for less congestion,
producers can raise more revenues. Furthermore, consumers will also have
higher CS since their congestion problem is improved and because producers
are inecient at extracting surplus from consumers with multi-part taris.
No,
Large,
U
CS and PS lower. Consumers are not willing to pay more for improved congestion, so both parties do worse than in the above case.
U,
U,
U
Although multiple-part taris Pareto-dominate taris with fewer parts, CS
and PS may still be comparable. A more complicated tari structure may not
be a fair trade-o for a small gain in surplus.
Table 6.4: Summary of Observations from Examples, Cont.
possibly competitive market situation. You can easily compare the characteristics of
the three or more establishments and make your dining choice. So our optimization
problem now has numerous service providers with a common subscriber pool, yet
separate subscriber set sizes, consumption functions, congestion levels, and, of course,
food prices. Obviously this is a much more complicated problem.
Having said this, I should probably clarify my reasons for using the simpler monopolistic model for our domestic cellular/PCS industry. One, the U.S.'s cellular
markets have been duopolies, not oligopolies or competitive markets. Two, competing cellular service providers have charged very similar prices. And three, even if
the PCS market for one geographical area is licensed out to several parties, both
the House and Senate versions of the Communications Act of 1996 \direct the FCC
to prescribe regulations that permit joint coordinated network planning, design, and
cooperative implementation."
At any rate, one interesting extension of this research would be to allow for multiple service providers who charge dierent rates. Xiu and Sirbu [1995] propose a
simple density demand function which is dependent on the number of subscribers for
Chapter 6: Conclusions and Extensions
117
each of two duopolistic providers. Consumers described by the density function are
assumed to have some predisposition towards one provider or the other or neither,
depending on the number of subscribers of each. Adaptation of this model subject to
the demand function could also assume some distribution of consumption, with either
service provider, over all potential customers and the subsequent congestion eects.
Game theory could then be applied to this model to determine optimal pricing and
capacity planning strategies if consumers can choose to subscribe to service from one
of the two providers.
Distributed Congestion Another possible twist on the problem would be to allow
for a more complicated congestion model which could more accurately approximate
the various congestion sources in the Internet. Positive and negative externalities
both play denite roles in the Internet as the boom in World Wide Web (WWW)
sites encourages on-line subscribership but congestion problems continue to plague
service providers as well as slow down transmission. This would entail accounting for
a more distributed form of congestion which can vary according to the site visited,
the computer from which the transaction is initiated, and the connection between the
two.
We could, for example, represent congestion by the sum of congestion eects felt
at the host computer, the link to some web site, and the WWW site itself. The rst
congestion component would depend on the combined speed of the consumer's home
computer and modem, which could be some function of that person's income index
. The second component could depend on the ratio of the number of users to the
number of possible links. Finally, the last congestion component could depend on
some expected percentage of the total number of users who will link to the site of
interest.
Stochastic Environment Consideration of an uncertain environment where the
available technology and regulatory variables can be described stochastically would
also be an interesting extension of this research. The deterministic representations of
maximum capacity and the constraint dening congestion would thus be replaced by
Chapter 6: Conclusions and Extensions
118
probability distributions for the static problem or else maybe for some xed period(s)
in the dynamic problem. Optimal pricing and capacity decisions would then outlined
based on maximizing expected Weighted Surplus.
Calculus of Variations And nally, the optimality conditions newly derived in this
research extend the reach of the Calculus of Variations in its ability to treat broader
classes of problems. Necessary conditions are presented for univariate problems with
integrand limits which also appear in the integrand. In addition, complete analysis
is provided for the bivariate Calculus of Variations problem with equality, inequality,
and integral constraints.
Appendix A
Appendix
A.1 Regulated Rate of Return
Recall that the basic problem investigated in this thesis is a maximization of a
weighted sum of Consumer Surplus and company prot subject to some constraints:
the denition of congestion;
the description of the subscriber set size;
each subscriber chooses her optimal consumption quantity;
non-negative system variables; and
non-negative Producer Surplus.
An alternate formulation whose importance bears some discussion is the regulated
rate of return problem, in which Consumer Surplus is maximized subject to the
constraint that investors would like to earn a minimum rate of return on capital
investment. If the total investment is represented by capacity costs, the remaining
costs are lumped into what will be called operating costs (OC), and r represents the
rate of return on investment, the problem becomes one in which Consumer Surplus
is maximized subject to the following constraints:
119
Appendix
120
the denition of congestion;
the description of the subscriber set size;
each subscriber chooses her optimal consumption quantity;
non-negative system variables; and
Total Revenue, TR, is not less than the sum of OC and (1 + r) times the
Capacity Cost, C3(K ).
Note that the rst four constraints are the same for each formulation. Call the
vector of these four constraints and the vector of their respective Lagrange multipliers . Let the Lagrange multipliers associated with the last constraint in the two
problems be WS and RR .
Proposition A.1. The Weighted Surplus maximization problem and the regulated
rate of return problem are equivalent if the following is true.
TR ? OC 1 + WS ? RR ? C3 (K ) 1 + WS ? RR (1 + r) ? CS
=
TR ? OC ? C3(K ) ? CS
(A.1)
Proof: The Lagrangian for the WS maximization problem is equal to the sum of
the Weighted Surplus, the Producer Surplus non-negativity constraint times WS, and
the various remaining constraints multiplied by their Lagrange multipliers.
LWS = WS + WS[TR ? OC ? C3(K )] + T The Lagrangian for the regulated rate of return problem is composed of the Consumer
Surplus, the rate of return constraint times RR , and the remaining constraints multiplied by their Lagrange multipliers.
LRR = CS + RR [TR ? OC ? (1 + r)C3(K )] + T Appendix
121
Equivalence of the two problems implies that the Lagrangians LWS and LRR are equal.
Thus,
LWS = LRR
WS + WS [TR ? OC ? C3 (K )] + T = CS + RR [TR ? OC ? (1 + r)C3(K )] + T WS + WS [TR ? OC ? C3 (K )] = CS + RR [TR ? OC ? (1 + r)C3(K )] .
Substitute CS + (1 ? )[TR ? OC ? C3 (K )] for WS, rearrange terms, and solve for
to get
TR ? OC 1 + WS ? RR ? C3(K ) 1 + WS ? RR (1 + r) ? CS
=
.
TR ? OC ? C3 (K ) ? CS
Whereas Proposition A.1 does not explicitly give a value of for any specic rate
of return, it does provide a relationship between and r given the optimal values for
the surpluses, revenues, costs, and Lagrange multipliers. A conclusion can be drawn
about for the simplest case of r = 0.
Proposition A.2. If the two problem formulations are identical when r = 0, then
= 1.
Proof: If r = 0 then
TR ? OC ? (1 + r)C3(K ) = TR ? OC ? C3 (K )
= PS .
For equivalent problems and outcomes, WS must equal RR . Hence, Equation A.1
reduces to = 1.
A.2 Special Case: No Externalities
Consider a special case of the static nonlinear tari problem when the potential
subscriber population is aected by neither positive nor negative externalities. The
absence of the positive externality translates to zero values for Wx and therefore Wqx .
Appendix
122
WC and WqC equaling zero result when congestion has no eect on willingness to
pay. Whereas the willingness to pay function used to be a function of four variables
in W (q(); ; x; C ), it now depends on only two: W (q(); ). The solution to this
problem is the variation on the solution to the general nonlinear tari problem when
Wx = Wqx = WC = WqC = 0. First order necessary conditions therefore reduce to:
Static Euler Equations
0 = (1 ? 2 )Wq + (1 ? )[Wq ? C1q ] + 5 ? , 2 [0; x]
0 = ? 2
0 = ?(1 ? )C3K + ? 1 + 3
(A.2a)
(A.2b)
(A.2c)
Static Transversality Equation
If x is free,
0 = (1 ? )[W ? C1 ? C2x ]j=x + (1 ? 2 )x[W ]j=x + (5 (x) ? )q(x) + 4(A.3)
Static Kuhn-Tucker Equations
1 [K ? K ] = 0
K ? K 0
2 [C ] = 0
C0
3 [K ] = 0
K0
4 [x] = 0
x0
5 [q()] = 0 ; 2 [0; x]
q() 0 ; 2 [0; x]
Static Integral Equation
K +C ?
Z x
0
q()d = 0 .
(A.4a)
(A.4b)
(A.4c)
(A.4d)
(A.4e)
(A.4f)
(A.4g)
(A.4h)
(A.4i)
(A.4j)
(A.5)
Appendix
123
Equation A.2a remains unchanged since it does not contain any partial derivatives
of willingness to pay or marginal willingness to pay with respect to x or C . Individual
consumer usage, q(), can be derived from this condition. Equation A.2c does not
change for the same reasons and yields the marginal value, , of relaxing the constraint
dening congestion. Equation A.2b reduces to equate 2 to . The contribution
R
of 0x Fxd is eliminated in Equation A.3, which can be solved for the number of
subscribers.
Congestion, C , becomes a slack variable in Equation A.5. If the cost of capacity
is non-zero, congestion will be optimal at critical congestion so that capacity exactly
meets total usage. Negligible capacity costs can yield similar results since consumers
do not value decreased congestion. Because capacity costs are most likely non-zero,
congestion will be set to its critical level, and capacity will be determined by Equation A.5.
The Weighted Surplus maximization problem reduces to a system of equations
from which we can easily derive q(), 2 [0; x], x, K , and C and then the optimal
tari. An equivalent problem would nd the optimal consumption levels for all subscribers parametric on subscriber set size, capacity, and congestion. As described
above, capacity and congestion are linked by their Lagrange multipliers and the constraint dening congestion, so the equivalent problem mentioned above would then
nd optimal consumption levels for all subscribers parametric on just subscriber set
size and capacity.
As a result, the elimination of the positive and negative externalities yields a
subscriber set whose growth depends on only the revenue function, capacity, and
consumer dierences.
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