).
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 . Due to the heterogeneity in the gross utilities, the fare policy of the airline affects the trip demand.

### 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 *θ*_{A}*=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 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)=(t*_{1}*,…..., t*_{n}*)* 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

. (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

, (2a)
the condition on her return trip is

. (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, . The indirect utility of a potential passenger on the outbound leg is given by

(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

. (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

. (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 , the return trip expression, is the same as (5) because the interactivity times for the return trips are also distributed uniformly with the support . 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

(6)
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 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.

### 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 *d*_{p}. 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 *c*_{p} 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

(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

(8)
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

. (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

. (10)
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 , 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 **

## HS network and market equilibrium

### 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 flights on each of two connections in the same time interval. The headways between the flights are same and equal to . Under HS network, we distinguish between two different types of price: and , 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

, (11a)

. (11b)
The indirect utility of a connecting market passenger that can actually take the trip is given by

(12)
Each potential passenger flies if only the utility of taking the trip exceeds his income, *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

. (14)

### Firm Behavior and market equilibrium

Airline’s fixed cost per flight on each connection in a HS network is equal to *d*_{h} and the variable cost per passengers is *c*_{h}. 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

. (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

. (16)
where and 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

. (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 and rearranging the first order yields the condition for frequency

. (18)
If d_{p}=d_{h} 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.

### 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 , it is found that *.
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 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 and , it is found that *.
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.

## Thick market externality and economy of frequency

### 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

(19)
Terms with subscript _{o} are assigned to one-way trip demand. The total one-way demand for the flights on each connection is

(20)
The optimum ticket fare and the frequency level for one-way trip are found as

, . (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

, . (22)
To compare the optimal number of flights between a PP and HS under one-way trip demand,

(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 and* * are assumed, the optimal frequency level in a HS network is 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,

.
respectively. Taking the first and second derivatives of (6) with respect to number of flights yields the followings,

(24a)

(24b)
where . To investigate economy of frequency, expression in (24b) is reduced to

. (25)
In equation (42), the expression 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, . Letting denote the critical frequency level in which the economy of frequency works well, we can write

(26)
As it is observed from (26), the critical frequency increases when consumers become more time sensitive (when 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.

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

(27)

Comparing (26) and (27), under *f>0*, it is clear that

. (28)
Since , 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

(29a)

(29b)

where . 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 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.

### 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 obtain positive net utilities by taking flights, the average consumer gross utilities is found as . Then the aggregate consumer surplus in the PP case can be expressed as

. (30)
As the sum of the consumer surplus and firm’s profit gives the total surplus, the social planner maximizes . 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

, (31)
where

. (32)
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

. (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

, (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.*

### 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*, *c*_{p}*=c*_{h}*=π/10 *and *d*_{p}*=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 *d*_{h} 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 . 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, *d*_{h}*/d*_{p} 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**
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 *d*_{p}*=** d*_{h} and *c*_{p}*=** c*_{h} are assumed. Denoting the difference in profit and social welfare levels between under PP and HS case as and , 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**

## 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 d_{p}=d_{h} and c_{p}=c_{h}, compare (10) and (18) after replacing in (21) with *n*. The RHSs and the first expression in LHSs are same in both. Denoting the second expression in the LHS of (10) and (18) as g_{p}(n) and g_{h}(n), the difference can be written that . In the satisfaction of the condition , the frequency is higher in HS than PP. On the other hand, the condition for positive demand in *AC* city-pair market is . Since both are same, , implying that in HS network, the monopolist offers more flights than in PP network.

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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*.

Figure 3. A point-to-point network and a hub-spoke network

Figure 4. 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*.