AIRLINE NETWORK STRUCTURE
WITH ECONOMIES OF FREQUENCY
Emine YETISKUL1, Kakuya MATSUSHIMA2 and Kiyoshi KOBAYASHI3
1 Institute of Transportation, University of California, Berkeley
(115 Mc Laughlin Hall, UC, Berkeley, CA, 94720, USA)
E-mail: eyetiskul@berkeley.edu
2 Dept. of Urban Management, Graduate School of Engineering, Kyoto University
(Nishikyo-ku, Kyoto, 615-8540, JAPAN)
E-mail: kakuya@ psa.mbox.media.kyoto-u.ac.jp
3 Dept. of Urban Management, Graduate School of Engineering, Kyoto University
(Nishikyo-ku, Kyoto, 615-8540, JAPAN)
E-mail: kkoba@psa.mbox.media.kyoto-u.ac.jp
Recently, the operational and market structure of the airline transport industry has been changing from
the hub-and-spoke networks to point-to-point systems of frequent services. As travelers’ preferences
are affected from time as well as from actual fares, direct frequent services increase flight availability
for consumers. In this research, the economy of scale which occurs as a result of the increase in
consumers’ flexible time is called “economies of frequency”. This paper provides a simple comparison
analysis between two alternative networks for a monopolist airline on the profit levels. Additionally,
the effect of network choice on social welfare is analyzed.
1
1. Introduction
After the deregulation of the aviation industry in the 1980s, airlines made major changes in their network systems.
Almost all of the carriers in North America began to offer connecting services via their hub cities by developing
hub-and-spoke (hereafter HS) operations because HS networks are designed to decrease the costs of airlines by
feeding traffic from spokes. However, the access to the market for a new entrepreneur airline is possible by
developing new business model in which airline operates a point-to-point network type (hereafter PP), providing
frequent, direct flights. The driving force of the model is to charge lower ticket fares. To exploit from lower fares,
the entrepreneur eliminates all kinds of costs superfluous to a cheap as well as reliable flight. Therefore, the new
airlines such as Southwest Airlines in US and Ryanair in Europe adopt this specific operational strategy in recent
years. Even though the de-hub phenomenon have grown rapidly, it is seen in the regional and domestic markets of
North America and Europe. If the entire aviation market is focused, it is observed that HS network is still the main,
current network type.
In order to examine the predominance of HS networks in the aviation markets a large empirical and theoretical
body of literature has developed.1 “Economies of density” has been found as the major reason of HS network
choice. As a result of carrying passengers of different origins and destinations by the same flight, a carrier
increases the average number of passengers per flight, which brings the utilization of the aircrafts with high
loading capacities and centralized maintenance and thereby reduces the costs. The other economic efficiency of
HS networks is the easiness in the expansion of the system, because a direct connection between one more city
and the hub automatically results in the completion of all connections in the network, stimulating an increasing
benefit to the network scale in the total market.
The airlines, providing services in well-developed HS networks, discouraged potential competitors to enter the
market due to economies of operating HS networks, characterized natural monopolies and charged higher prices
in the direct passengers of the itineraries between the spoke and hub airports.2 In addition to inefficiency in
competition and fare premiums at hub cities, HS networks increase inconvenience and time costs not only for the
indirect passengers traveling between spokes, but also for the direct passengers because as much as the network
expands, the level of congestion at hub airports increases. The greater the HS networks are, the more congested
hub airports and the more complex operating systems are, causing an increase in the costs of machinery as well as
labor. Hence, the level of economic efficiency in using hub systems is important. As a result, the PP network type,
providing a carrier to offer cheap flight services due to the utilization of standard machinery and simple schedules
came to appear in last decade.
On the other hand, the research that analyze the impacts of the entry of carriers on competition have developed
and they show enough econometric evidence indicating that fares are decreasing and frequency and traffic levels
are increasing on routes they operate.3 Besides, the studies scrutinizing the sources of competitive advantages of
the entrants over major network carriers have emerged. According to the US DOT (1996), the principal and
fundamental advantage of these carriers is their lower unit operating costs. Yetiskul et al (2005) highlight the
comparative advantages of PP networks over HS ones by thick market externality in airline network structure.
1
The theoretical papers include Brueckner and Spiller (1991), Starr and Stinchcombe (1992), Hendricks, Piccione, and
Tan (1995, 1997 and 1999), Berechman and Shy (1998), Oum, Zhang, and Zhang (1995), Zhang (1996) and Shy
(2001). The empirical ones are Caves, Christensen, and Tretheway (1984), Brueckner, Dyer, and Spiller (1992),
Brueckner and Spiller (1994) and Morrison and Winston (1995).
2
See Berry (1990, 1992), Borenstein (1989, 1990, 1992) and Berry, Carnall and Spiller (1997).
3
The research includes Cohas, Belobaba, and Robert (1995), Dresner, Lin, and Windle (1996), U.S. DOT (1996),
Windle and Dresner (1999) and (Volwes 2001).
2
Briefly, the majority of both theoretical and empirical studies about new entrant phenomenon are pertaining to
their low-cost business and management issues while none of them highlight the mechanism of flight frequency in
direct services. Therefore, we utilize a concept viz. “economies of frequency, described in the following
paragraph.
When a consumer has a tied schedule, to allocate time for a business activity, takes place in another city, bounds
her due to the need for the adjustment of the times of other scheduled activities. If the time adjustment is
impossible, it costs her to cancel the activity and thereby, the trip. Hence, consumers are looking for time
flexibility in their choices. Additionally, the possibility of a consumer to take a trip depends on her schedule on
both legs so the frequency increase affects the consumers’ preferences on both legs as a whole. Time flexibility on
the outbound and return legs results in an increasing benefit to the airline. In the case that the entrant offers more
frequent, direct flights, the consumer utilizes from the flexibility in arranging her schedule and thick market
thickness works in the network choice of the airline. Matsushima and Kobayashi (2005) propose a market
equilibrium model to investigate the structure of a taxi spot market and the externality effects.
In the model, the impact of frequency on demand is captured with an external expression that incorporates the
duration of travel and the frequency of flights. As both are related with network configuration, the network choice
of a monopolist determines the demand. Therefore, a network economy composed of three cities is considered.
There are three possible city-pair markets where potential passengers originate in one city and terminate in the
other. If the three cities in the network are connected symmetrically with each other by direct services as shown in
Figure 1, the network type is called PP. On the other hand, if the route between city A and city C is via city B
while city-pair market AB and BC are connected by direct services, the network type is HS and city B is the hub
city.
FIGURE 1
The airline in HS network exploits from economy of density and economy of network size as by operating larger
aircraft. However, the flight time becomes longer for the connecting passengers. Additionally, operating a larger
aircraft and increasing the service quality to match the disutility that arises from extra travel time of a connecting
flight and extra waiting time at hub airports, cause an increase in fixed and variable costs for the airline. On the
other hand, if the airline offers services in a PP network, flight time of the connecting passengers is shorter and
scheduling flexibility increases so the airline utilizes from economy of frequency. However, the airline has to
ensure the services on many routes. Consequently, the inquiry of which network type is superior to the other one
is ambiguous because each network type has competitive advantages and employs different kind of economy of
scale.
In this research, distinguishing between two network types, a PP and a HS, we analyze the network model of an
airline and the comparison of the ticket fares, frequency and profit levels of the monopoly airline under two
alternative networks are pointed out. This paper is organized as follows: After introducing the model for a
monopoly airline in a PP network choice in Section 2, we carry out the similar analysis in a HS one in Section 3.
Considering one-way trip demand, the solutions for the profit maximization problem are figured out in Section 4
and the impact of thick market externality and economy of frequency with numerical examples are also examined
in Section 4. Concluding remarks follow in Section 5. The proof of the proposition is included in the appendix.
3
2. PP network and market equilibrium
2.1.
Assumptions
The monopoly airline offers direct services in three city-pair markets, AB, BC, and AC in which consumers,
residing in one city want to take two-way trips. As it is assumed that the size and characteristics of the residents of
each city in the network are identical, travel demand is symmetric on both directions in each city-pair market as
well as in three of them. The number of flights, offered by the airline in the time interval [0,2π) from city A to city
C is equal to n. Besides, we ignore any distance differences between cities in the network of three cities and
assume that the duration of a nonstop travel between any city pairs is identical and it is shown as f, (0<f<2π).
Each potential passenger has a scheduled activity in the destination city C in a time segment (,) and for each
consumer, there is a start time for her activity, denoted by θ. Assuming that the distribution of activity start times
is continuous and uniform in infinitely time, we focus on the potential passengers who are addressed in a circular
time interval [0,2π) where the end joins to the beginning. Besides, we assume that the number of this
corresponding group is equal to M. On the other hand, these consumers have to return to city A after staying for a
time in the destination city until the end of their activities because they have another activities in the origin city A,
which are scheduled before taking the outbound trips to the destination city C. Normalizing time duration of the
activity in the destination city for a each consumer to a fixed value and letting θ' denote the end time of the
activity, we can suppose that the distribution of θ' is uniform and in another circular time interval [2π,4π).
In addition to consumer heterogeneity in the start times of the activities in the destination cities, we assume that
consumers differ in their gross utilities, derived from taking trips to the destination city C. w denotes the
consumer-specific gross utility and has a uniform distribution with support 0, ŵ . Due to the heterogeneity in
the gross utilities, the fare policy of the airline affects the trip demand.
2.2.
Passengers
The travel demand for the outbound trip is generated as follows. A consumer, identified by θ has to complete
another activity in the origin city A before taking the trip to city C. We suppose that she can not leave city A
before the time θA=θ-s. Here, s denotes the time interval from the activity before leaving the origin city until after
arriving in the destination city, for the corresponding consumer. Hereafter, s is called as “inter-activity time”
which can be also defined as the possible time to take outbound trip. On the other hand, the travel demand for the
return trip is similar to that of the outbound trip. Our representative consumer, identified by θ has to return to her
origin city A before the time θ'A=θ'+ s' where θ' denotes her end time of the activity in the destination city C and
s' denotes her inter-activity time that arises from the interval, beginning from θ' and ending at the start time of
next activity in the city A. The circular time interval is illustrated in Figure 2. We assume that inter-activity times
on the outbound as well as return legs are consumer specific times and distributed uniformly in an interval 0, ŝ
and the inter-activity times on both leg for each consumer are independent so a two-way trip exists only if both
times are long enough to complete/reach the activities in the origin city, otherwise there is no trip.
FIGURE 2
The airline company offers n flights on each direction in each city-pair market and the intervals (i.e., the
headways) between departure times of flights are the same. The flights, originating from city A to city C are
indexed by i and t(n)=(t1,…..., tn) denoting the set of arrival times of the flights i. The headway is equal to 2π/n.
Then, the arrival time of each flight that originates from city A and terminates in city C is
4
ti  i  1
2
n
i  1,...,n 
.
(1)
Besides, we assume that the potential passenger who has to arrive in the city C before the start time of the activity
θ chooses the flight belonging to the smaller value of time, spent in the destination city. Similarly, that consumer
chooses the flight for her return trip that leaves the destination city as early as after the activity ends. The
representative potential passenger has s on her outbound and s' on her return trip. While the condition that she
takes the flight, arriving at city C at time 0 can be written as
f   s ,
(2a)
the condition on her return trip is
f     s .
(2b)
As the left sides of both conditions involve the actual flight time as well as the activity start/end time, (2a) and
(2b) guarantee the condition of the demand for a flight, sˆ  f . The indirect utility of a potential passenger on
the outbound leg is given by
Y  w  p
U w, p   
 
if condition (2a) and (2b) are satisfied
otherwise
(3)
where Y is a fixed amount of income for a potential passenger and p is the price of one-way ticket. Trip utility for
each consumer can be found from the specific gross utility w minus the price. This utility function says that a
potential passenger can take a flight only her inter-activity time is long enough to arrive in the destination city
after completing the activity in the origin city; otherwise her net utility is negative. Even though the trip utility is
conditional to the time components of the trip, the function doesn't contain the costs borne by actual flight time
and rescheduling time. However, the impact of them on the demand is captured externally. Introducing the
interactivity time for each potential passenger, the effect of the flight and rescheduling times on the demand is
characterized in a way where the net utility is conditional. Then, the utility condition is written as
U w, p   Y .
(4)
To outline the role of time components, we solve the expression that shows the ratio of the passengers for whom
inter-activity times are sufficiently long to take the trips after the activities in the origin city end and before the
activities in the destination city start. That is
Pn, f , sˆ  
1

 sˆ  f   .
ˆ
s
n
(5)
When the impact of ticket fares on demand is not considered, this key expression gives the patronage of the flights
for outbound trips and 0, ŝ , the return trip expression, Pn, f , sˆ  is the same as (5) because the interactivity
times for the return trips are also distributed uniformly with the support 0, ŝ . Then the probability of consumers
of which both inter-activity times are large enough to travel on both legs is found by taking the square of (5).
Additionally, the potential passengers are specified according to their gross utilities, the overall two-way trip
demand for the flights on one leg can be expressed as
5
X n, p   M Pn, f , sˆ  Probw  p  0 
2
M 

ˆ  p
 sˆ  f   w
2
ˆ
sˆ w
n
(6)
2

where Prob(.) defines the probability of passengers of whom net utilities, derived from taking trips by purchasing
a ticket are non-negative. As it is observed from (6), the second power of Pn, f , sˆ  is included in finding the
flight demand for one leg. Therefore, the increase in the number of flights on outbound leg has an effect on the
demand of return leg and the expression (6) shows the externality of market thickness, arises from the positive
interaction of the increase in both legs.
2.3.
Firm Behavior and Market Equilibrium
The monopolist maximizes his profit according to two-way trip demand and optimizes the number of flights, n
and a ticket fare, p. We suppose that there is no restriction in the capacity of aircrafts. In the case that the airline
operates the flights in a PP network, airline’s fixed cost per flight on each connection between two cities is equal
to dp. The fixed cost per flight consists of maintenance cost, landing fee and services provided to the aircraft while
it is on the ground as well as salaries of the flight crew so the size of aircraft and from where the services are
provided affect the cost. As the size of an aircraft, served under a PP/HS network, landing fees and the services of
hub/periphery airport, etc is different from the other, we analyze the relation between flight demand and fixed cost
by distinguishing the airline fixed cost between two alternative network types. On the other hand, passenger cost,
associated with services such as providing food and drink on board and passenger baggage is denoted as cp in PP
network. As the demands for two-way trips are symmetric in both directions of AC city-pair market, the profits,
earned by the carrier from operating the flights on one direction in AC market can be written as
 n, p   2 p  c p X  p, n   nd p .
(7)
Additionally, there is symmetry between the demands of each city-pair market in the three-city-model so the
monopolist, offering n flights in each direction and setting the ticket fare, p, faces the following maximization
problem
max n, p   3 n, p .
(8)
n, p
Substituting (6) and (7) into (8), we can write the profit as a function of p and n. After taking the first derivative
with respect to p, the optimum price p* for one-leg of the two-way trip is found as
p* 
1
w  c p  .
2
(9)
After finding the optimum price, we can rewrite the total profit function. To find the optimal n*, the first
derivative is equalized into zero. Then, it gives the following condition
 p sˆ  f n     d p n3
p 
ˆ  c p 2
M w
.
(10)
ˆ
sˆ 2 w
The left-hand-side and the right-hand-side of the condition (10) are illustrated in Figure 3, where the S-shaped
curve and the line represent the RHS and LHS of the expression, respectively. To hold the condition sˆ  f ,
6
LHS must be positive, deducting that the line is upward sloping. The curve and the line intersect at three points in
which two are positive while one is negative. When the second-order condition is checked, the solution, marked
with a point in Figure 3 is found. The way outlined here is essentially the way presented in Brueckner (2004).
FIGURE 3
3. HS network and market equilibrium
3.1.
Passengers
In this section the market equilibrium model is formulated in the case that the monopolist operates flights only on
the AB and BC routes and transfers the passengers of AC market via city B as its hub so the passengers’ behavior
in AB and BC city-pair markets of the HS network is the same as that in the markets of the PP network but the
behavior of AC market passengers can be different because of traveling via hub. Therefore, we focus on the
potential passengers, residing in city A and having scheduled activities in city C in a circular time interval [0,2π].
The monopolist offers n flights on each of two connections in the same time interval. The headways between
the flights are same and equal to 2 / n . Under HS network, we distinguish between two different types of price:
p and q , the price set for the direct and connecting market passengers, respectively. In the connecting market
of HS, the duration of the trip is equal to two actual flight times (2f). It is assumed that the timetable coordination
at the hub airport is perfect so the layover time for connecting passengers is zero.
As well as other assumptions under the HS network, the distributions of activity start and end times of the
potential passengers are the same as those under the PP case. The only difference is that a potential passenger of
the connecting market involves additional time that arises from flying on both connections so her inter-activity
times have to be large enough to cover the duration of the total travel time, 2f and these conditions can be written
as
2 f   s ,
(11a)
2 f     s .
(11b)
The indirect utility of a connecting market passenger that can actually take the trip is given by
Y  w  q
U w, q   
 
if condition (11a) and (11b) are satisfied
otherwise
(12)
Each potential passenger flies if only the utility of taking the trip exceeds his income, Y,
U w, q   Y .
(13)
As, the travel demand for the return trip is similar to that of the outbound trip and the interactivity times of each
consumer are mutually independent, the overall two-way trip demand for the flights of connecting market on each
leg can be expressed as
X n , q  
M 

sˆ  2 f   wˆ  q  .
2 
sˆ wˆ 
n
2
(14)
7
3.2.
Firm Behavior and market equilibrium
Airline’s fixed cost per flight on each connection in a HS network is equal to dh and the variable cost per
passengers is ch. The revenue of the firm under a HS network comes from carrying the passengers of two direct
markets and one connecting market while total fixed cost is identified by the cost of flying only on the AB and BC
markets so HS operator can utilize from economies of density if the fixed cost doesn’t increase under HS network.
However, operating a large scale aircraft as well as providing services from hub airport increases the fixed cost of
the airline under the HS network. Additionally, it is observed that the variable cost per passenger decreases when
the airline chooses PP network due to offering shorter direct trips. Then it is assumed that
dh  d p
ch  c p .
(15)
As a result of the symmetry between both directions in each connection, the profit, earned by the carrier from
operating the flights under a HS network is given by
 n , p, q   4 p  ch X n , p   2q  ch X n , q   2n d h .
(16)
where X n , p  and X n , q  are demands for the flights of the direct and connecting market, respectively.
Taking the first derivatives with respect to the price of the local market and the price of the connecting market, we
find the optimum prices for one leg of the two-way trip in both markets as
p*  q * 
1
wˆ  ch  .
2
(17)
From equation (17), we can say that the monopolist sets the same ticket prices for the passengers of direct and
connecting markets. Additionally, it is the same as the optimum price, set in the PP case if the variable costs under
PP and HS are same. After substituting (17) into (16), we can rewrite the total profit function of the monopolist
under HS network. Taking the first derivative with respect to n and rearranging the first order yields the
condition for frequency


 h  sˆ 
h 

4 
f n     d h n 3
3 

.
(18)
ˆ  ch 
3M w
ˆ
2 sˆ 2 w
2
If dp=dh is assumed, the RHS of the condition is the same as that in (10). However, the slope and the intercepts of
the LHS are different. The diagram of the condition resembles the one shown in Figure 3.
3.3.
Comparison of two alternative networks
In previous sections, the optimum price and frequency solutions of the monopolist under two network structures
are scrutinized without discussing which type of network offers higher fares or frequency or traffic volumes. As
cost heterogeneity in network types is assumed, a change in one of cost parameters alters the levels, causing a
difficultly in identifying the network choice of the monopolist. However, following results can be established:
Proposition 1. Under the assumption ch  c p , it is found that p*  p*  q * .
Proposition 1 implies that the difference between the fares under a PP and HS network arises from the variable
cost differences between operating a HS and PP network so the assumption (15) plays an essential role in the
8
Proposition 1. By this result, the model supports the observations and hypotheses about the entry and success of
low cost carriers that are charging low fares by adopting low-cost strategy. The more the airline succeeds to
decrease the variable cost per passenger, the lower the airline charges the fares. The following result about
frequency levels under both network types can be found by comparing (10) with (18).
Proposition 2. Under the assumption d h  d p and ch  c p , it is found that n*  n * .
Proposition 2 implies that flight frequency is higher in the HS network than in the PP network when the fixed cost
per flight as well as variable cost per passenger when operating a PP network is the same as those when operating
a HS network. Because the left-hand-side of HS network in equation (18) has a higher position than that of PP
network in (10) while the right-had-sides of two conditions are the same (Appendix 1). As cost assumptions of the
first case are the same as in Brueckner (2004), the result, established in Proposition 2, is also same. The reason
that the line of HS network has a higher level than that of PP network is the passenger volumes in each connection.
The fact that higher frequency under HS network on one connection doesn’t yield the result that the total flights,
operated under HS network is more than those under PP one.
4. Thick market externality and economy of frequency
4.1.
The impact of thick market externality
In this section, the set-up of the model is changed to scrutinize the impact of thick market externality on the
monopolist network choice so the optimum price and frequency level in two alternative networks, PP and HS, are
solved for one-way trip demand. In the previous section, the consumers have two time constraints, one of which is
on the outbound leg while the other is on the return leg so the probability that shows the total number of
passengers taking two-way trips is found by multiplying the probabilities defined for both ways. In the following
optimization problem, it is assumed that a potential passenger considers taking a one-way trip so time restriction
runs in only one-way. Therefore, assuming the cases in which the potential passengers have time limitations on
the outbound leg or vis-à-vis, the model is outlined again. The indirect utility of a potential passenger is given by
Y  w  po
U w, po   
 
Terms with subscript
connection is
o
if condition (2a) is satisfied
(19)
otherwise
are assigned to one-way trip demand. The total one-way demand for the flights on each
X no , po   M
M
 
 sˆ  f  w
ˆ  po 
ˆ
sˆw
no 
(20)
The optimum ticket fare and the frequency level for one-way trip are found as
po 
*
1
wˆ  c p ,
2
no* 
wˆ  c 
p
2
2M
.
ˆ sˆd p
w
(21)
Then we optimize the price and frequency, set by the monopolist in a HS network under one-way trip
demand and write both as
po  qo 
*
*
1
wˆ  ch  ,
2
no* 
9
wˆ  ch 
2
3M
.
ˆ sˆd h
w
(22)
To compare the optimal number of flights between a PP and HS under one-way trip demand,
ˆ  cp
w
no*

ˆ  ch
w
no*
2d h
3d p
(23)
is written. It can be seen that the higher frequency is offered when operating either a HS network or PP one under
one-way demand because it depends only on the cost parameters. Note that there is no impact of time on the
frequency level differences between two alternative networks. Therefore, the actual flight time doesn’t affect the
frequency choice of the monopolist. An observation of the market equilibrium analysis under one-way trip
demand generates the same results as those, established in Proposition 1 and 2. If d h  d p and ch  c p are
assumed, the optimal frequency level in a HS network is
4.2.
3 / 2 times that level in a PP one.
Economy of frequency
In this section, the relation between the flight demand and frequency is analyzed. To hold positive flight demand,
(6) and (14), the following frequency conditions in a PP network and HS one must be satisfied,
n

sˆ  f
n
,

sˆ  2 f
.
respectively. Taking the first and second derivatives of (6) with respect to number of flights yields the followings,
X n   

 2 sˆ  f     0
n
n 
n
(24a)
 2 X n   
3 
 3  2sˆ  f  

2
n
n 
n 
(24b)



ˆ  p * / sˆ 2 w
ˆ . To investigate economy of frequency, expression in (24b) is reduced to
where   2M w

 2 X n   0

n 2
 0

if
if
3
 sˆ  f
2n
.
3
 sˆ  f
2n
(25)
In equation (42), the expression sˆ  f shows the upper bound value among net interactivity times (the time after
the actual flight time is subtracted from the gross interactivity times) and the expression 3π/2n is the critical
interval in which the airline has to arrange his flight schedule. When the demand of business travelers’ augments
in the market, implying the decrease in the inter-activity times of the potential passengers; the airline has to
increase the number of flights to satisfy the condition, 3 2n  sˆ  f . Letting n o denote the critical frequency
level in which the economy of frequency works well, we can write
no 
3
ˆ
2s  f
(26)

As it is observed from (26), the critical frequency increases when consumers become more time sensitive (when
ŝ decreases). Even though, the market size M has no direct effect on the economy of frequency, the expression
(24a) as well as (24b) increases when M rises, causing higher frequency.
10
On the other hand, taking the first and second derivatives of the flight demand in the connecting market of a HS
network (14) with respect to frequency, we can find the critical frequency level as
no 
3
2sˆ  2 f
(27)

Comparing (26) and (27), under f>0, it is clear that
n o  no .
(28)
Since n o  n o , a HS operator has to offer higher level of frequency than a PP operator to utilize from the economy
of frequency. As the actual flight time in the connecting market is longer, the impact of a flight increase on
passenger time flexibility is less in the HS case and due there is a time window, increasing frequency doesn’t
work well. Therefore, higher frequency isn’t the only source of the frequency economies so the airline under a HS
must offer more flights than under a PP to exploit from the economies.
To analyze the relation between the flight demand and frequency under one-way trip demand, the following first
and second derivatives of the flight demand (20) in the PP case are given by
X no 

 o2  0
no
no
(29a)
 2 X no 
2
  3o  0
2
no
no
(29b)
where o  2M w
ˆ  p *  / sˆ 2 w
ˆ  . The second expression (29b) as well as the second derivative of the demand
in the HS case doesn’t involve the actual flight time f and interactivity time ŝ so the economy of frequency
doesn’t exist. This result shows that the economy of frequency works when the time flexibilities on both legs are
considered.
4.3.
Social welfare analysis
In this section, we analyze the first-best allocation, which maximizes social welfare. While the monopolist
maximizes his profit level, the social planner focus on the maximization of consumer benefits minus airline costs.
To derive the social welfare differences between two alternative network types, aggregate consumer surplus and
profit level of the monopolist is specified for each type. Consumer welfare is the sum of the individual utilities
derived by the passengers. As the potential passenger who are located at some gross trip utilities
 p, wˆ 
obtain
positive net utilities by taking flights, the average consumer gross utilities is found as w
ˆ  p  / 2 . Then the
aggregate consumer surplus in the PP case can be expressed as
6M
CS p n,p   2
sˆ
   wˆ  p  wˆ  p


 p.
 sˆ  f   

ˆ  2
n  w


2
(30)
As the sum of the consumer surplus and firm’s profit gives the total surplus, the social planner maximizes

 

W n  CS p n, p*   n, p* . The first-order condition for the choice of p yields marginal cost pricing in the
first-best allocation. Then we can write the welfare function in the PP case as
11
~
2
3 p 

W n  
 sˆ  f    3nd p ,
2 
n
(31)
ˆ  c p 2
2M w
p 
 2 p .
ˆ
sˆ 2 w
(32)
where
~
Taking the first derivative of (31) with respect to n and rearranging the terms, the first order condition for the
flight frequency in the PP case is given as
 p sˆ  f n     d p n 3 .
~
(33)
The form of (33) under social welfare is the same as that in (10). However, it differs from traffic level which is
two times that in (10). Therefore, the comparison of frequency levels between the profit and social welfare
maximization results in
n **  n * ,
(34)
showing that the frequency and thereby traffic level under social optimum is higher than under monopoly in PP
case. The same result is found in HS case so the discussion has established the following
Proposition 3. The comparison of frequencies between under the monopoly and social welfare solutions shows
that the monopolist in each network choice offers lower frequency than the social planner optimizes.
4.4.
Numerical Example
To compare the PP solutions with HS ones and to reconfirm the results established above, numerical examples are
illustrated. Given a set of parameters as M=100, w
ˆ  π / 2, sˆ  π 6 , cp=ch=π/10 and dp=1, the HS and PP
solutions are found. As the variable cost per passenger is given by the same parameter in both network types, the
ticket fares are also same from (9) and (17). In the first example, the comparison between two network types as a
result of an increase in actual flight time and the fixed cost dh are illustrated in Figure 4 in which the lower and
upper lines show the boundaries between profit and frequency levels, respectively. While the former one is
determined by setting equilibrium profit    , the latter one is from setting equilibrium flight frequency
n *  n * . It can be seen from the diagram that the profitability of the HS network decreases when either the actual
flight time or the ratio between two fixed costs, dh/dp increase. If the ratio continues to increase, frequency is
higher in the PP than the HS network.
FIGURE 4
FIGURE 5
In the second example, the effect of market size, M, on the frequency and profit levels under both network types is
examined. Holding the actual travel time equal to π/100, the solutions are given in Figure 5. To analyze the effect
of thick market externality on the monopolist’s network choice, with the same parameter set the HS and PP
solutions under one-way trip demand are as before specified in Figure 6 and 7. The aim of calculating these
numerical examples is to compare the regions between under two-way trip demand and one-way trip demand
cases so comparison of regions in the Figures results that the monopolist network choice is inclined toward the PP
case under two-way demand.
FIGURE 6
FIGURE 7
12
Finally, we compare the network choices of the monopolist and social planner with a numerical example. The
comparison under both regimes will be held on in the case that dp= dh and cp= ch are assumed. Denoting the
difference in profit and social welfare levels between under PP and HS case as      and   W  W ,
respectively, we focus on Δ and Γ as functions of fixed cost. The welfare difference is affected from a change in d
value twice as large as that the profit difference is affected. Δ and Γ as functions of d are illustrated in Figure 8 in
which Δ intersects the horizontal axis at a value, smaller than that Γ intersects. Denoting these values as d* and
d**, respectively, we conclude that at the values below d*, both the monopolist and social planner prefer the PP
network and at the values above d**, both prefer the HS network. However, between d* and d**, the monopolist
prefers the HS network while the planner favors the PP one. Hence, over this range of fixed cost values, the
choice of the profit maximizing monopolist toward HS network is inefficient.
FIGURE 8
5. Conclusion
This paper is established to model the success of low-cost airlines maintained over larger, major ones in a
theoretical base. While the major airlines offer indirect connections via hub city and benefit from economies of
density and the fall in total fixed costs in HS networks, new entrants develop operating strategies with PP network
systems. Offering direct as well as more frequent services in the PP networks provide competitive advantages so
new entrants have increased their shares and thereby their profit levels in the air transport markets. Hence,
distinguishing a simple network model between two basic network types, a PP network and a HS network, we
examine the network choice of a monopolist airline.
As the impact of time costs on the consumers’ preferences in taking trips is large enough not to be neglected, in
the model, we investigate whether economies of scale arise as a result of the increase in scheduling flexibility.
Thus, we introduce inter-activity times for consumers and characterize an external expression that embodies the
duration of travel and the frequency of flights to define market thickness. As the actual travel time and the number
of flights, set by the airlines are based on the network configuration, the demand for flights under each network
type is different. Thus, the airline offering direct services in a PP network and acquiring more flights confer
external benefits on the consumers. However, the solutions and discussion in this paper is based on monopoly
case. A further analysis on the duopoly case can be conducted. Schipper (2001) examines the frequency choice in
air transport oligopoly markets but not network choice of the airlines.
Appendix 1
Proof of Proposition 2: Using dp=dh and cp=ch, compare (10) and (18) after replacing n in (21) with n. The
ˆ
RHSs and the first expression M w2 ch 
ˆ
2 sˆ w
2
in LHSs are same in both. Denoting the second expression in the
LHS of (10) and (18) as gp(n) and gh(n), the difference can be written that g h(n)  g p(n)  s  2 f n-π . In the
satisfaction of the condition s  2 f - π n  0 , the frequency is higher in HS than PP. On the other hand, the
condition for positive demand in AC city-pair market is n 

. Since both are same, g h (n)  g p n  ,
s 2f
implying that in HS network, the monopolist offers more flights than in PP network.
13
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15
Figure 1. A point-to-point network and a hub-spoke network
Figure 2. Interactivity times of two passengers
Figure 3. The frequency solution under PP network
Figure 4. The effect of f
Figure 5. The effect of M
Figure 6. The effect of f under one-way
Figure 7. The effect of M under one-way
Figure 8. Δ and Γ as functions of fixed cost, d.
16
B
A
B
C
A
PP Network
C
HS Network
Figure 3. A point-to-point network and a hub-spoke network
17
s2
t1
s1
2 0
t2 
f 
2
n
Figure 4. Interactivity times of two passengers
18
LHS
RHS
n
Figure 3. The frequency solution under PP network
19
dh/dp
1.6
dh/dp
1.6
  , n*  n *
1.5
  , n*  n *
1.5
1.4
1.4
   , n*  n *
1.3
1.3
   , n*  n *
1.2
1.2
   , n*  n *
1.1
1.1
f/s
1
0.03
0.04
0.05
   , n*  n *
1
0.06
50
Figure 4. The effect of f
75
100
125
150
Figure 5. The effect of M
20
M
dh/dp
1.6
dh/dp
1.6
   , no*  no*
1.5
1.4
1.5
   , no*  no*
1.4
   , no*  no*
1.3
1.3
1.2
1.2
   , no*  no*
1.1
   , no*  no*
   , no*  no*
1.1
f/s
1
0.03
0.04
0.05
1
0.06
50
Figure 6. The effect of f under one-way
75
100
125
150
M
Figure 7. The effect of M under one-way
21
6
4
2
d
0
0.05
0.1
0.15
0.2
0.25
0.3
  W W
-2
 
Figure 8. Δ and Γ as functions of fixed cost, d.
22