1
Environmental Taxation in Airline Markets 
 
 
 
 
Fredrik Carlsson 
 
 
 
Working Papers in Economics no 24 
May 2000 
Department of Economics 
Göteborg University 
 
 
 
 
Abstract 
Over the last two decades many airline markets have been deregulated, resulting in increased competition 
and use of different types of networks. At the same time there has been an intense discussion on 
environmental taxation of airline traffic. It is likely that an optimal environmental tax and the effects of a 
tax differ between different types of aviation markets. In this paper we derive optimal environmental 
taxes for different types of airline markets. The first type of market is a multiproduct monopoly airline 
operating either a point-to-point network or a hub-and-spoke network. The optimal tax is shown to be 
similar in construction to an optimal tax for a monopolist. We also compare the environmental impact of 
the two types of networks. Given no differences in marginal damages between airports we find that an 
airline will always choose the network with the highest environmental damages. The second type of 
market we investigate is a multiproduct duopoly, where two airlines compete in both passengers and 
flights. The formulation of the optimal tax is similar to the optimal tax of a single product oligopoly. 
However, we also show that it is, because of strategic effects, difficult to determine the effects of the tax 
on the number of flights. 
 
Key words: Environmental taxation, Multiproduct duopoly, Aviation, Networks. 
 
JEL classification: D62, L13. 
 
 
 
Department of Economics         
Göteborg University        
Box 640    
SE-40530 Göteborg          
Sweden            
 
Tel: +46 31 7734174          
Fax: +46 31 7731043 
e-mail: Fredrik.Carlsson@economics.gu.se     
 
 2
1. Introduction1 
In this paper we discuss the issue of optimal environmental taxation for different types 
of aviation markets. In the standard perfect competition model the optimal prescription 
is a tax equal to marginal external damages, a so-called Pigouvian tax. However, airline 
markets has two properties that affect the optimal environmental tax: imperfect 
competition and network effects. Most airline markets consists of only a few actors and 
in many cases only one or two airlines operate on a particular route. Network effects 
occur because aviation markets (connections) in many cases are related on the demand 
and/or cost side. One interesting development of airline markets has been the increased 
use of hub-and-spoke operations (see e.g. Borenstein 1992). The main explanation for 
the formation of hub-and-spoke networks is probably economies of traffic density (e.g. 
Caves et al. 1984), but there could also be positive effects on the demand side since 
hubbing can result in more frequent flights to a larger number of cities (market 
presence). 
The environmental impact from the aviation sector depends on the number of flights, 
types of aircraft engines that are used, and the location of the airports. In this paper we 
take technology as given, even though this is probably an equally important factor for 
environmental improvements. Instead we focus on the number of flights as the 
environmental impact. In the model presented an airline has two choice variables, 
number of passengers and number of flights. We follow the standard approach (see e.g. 
DeVany 1975, Schmalensee 1977) and assume that demand is increasing in capacity 
(number of flights) since delay costs are decreasing in capacity. Consequently an airline 
has incentives to increase the number of flights in order to increase demand for air 
travel. 
In the first part of the paper we elaborate on the model presented in Nero and Black 
(1998). They analyse a monopoly airline and the differences in environmental impact 
between a point-to-point network and hub-and-spoke network. Here we extend this 
model to cover non-symmetric equilibrium, thereby allowing for different effects on the 
number of flights at different connections. We derive optimal environmental taxes for 
the two types of networks, and compare the environmental impacts of the two networks, 
                                               
1 The author would like to thank Gardner Brown, Olof Johansson-Stenman, Åsa Löfgren and participants 
at seminars in Gothenburg for useful comments. Financial support from the Bank of Sweden 
Tercentenary Foundation is gratefully acknowledged. 
 3
by comparing the number of flights. In the second part of the paper we extend the 
discussion to a market with two airlines. The airlines make decisions both about the 
number of passengers and the number flights, which means that the airlines are 
multiproduct duopolists. The two "products", passengers and flights, are related through 
the demand function; where market demand depends on the aggregate number of 
passengers and flights. Given this model we derive the optimal environmental tax and 
the comparative statics of the tax.  
 
2. General outline of the model 
In order to derive useful result we will work with a rather simple model. For any city 
pair ij the cost function for an airline is given by: 
 
ijijijij FtbcQC )( ++= , (1) 
 
where ijQ  is the number of passengers, ijF  the number of flights, c is the marginal 
passenger cost, b is the marginal flight cost, net of any environmental tax, and ijt  is the 
environmental tax. The cost function is the same as in Nero and Black (1998),2 and 
admittedly the assumption of separability and the absence of economies of scope are 
restrictive.3 The demand on city pair ij is a function of price, number of flights and a 
market-specific demand shift parameter, ijW . We assume the following inverse demand: 
 
s-bse-eae W=W= ijijijijijijij QFQFP /1//1 . (2) 
 
Throughout the paper we restrict the absolute value of the price-elasticity, e, to be 
larger than unity, and the flight elasticity, a , to be lower than unity.4 For the inverse 
                                               
2 Their model in turn builds on the models by DeVany (1975) and Schmalensee (1977). 
3 The assumption is made for analytical convenience, but it should be noted that the comparison between 
the two types of network could be affected by changing the functional form of the cost function. 
4 The restrictions of the elasticities have some support by empirical findings. Summarising major survey 
results on price elasticities Oum et al. (1992) finds that most studies show values of the price elasticity 
between 0.8 and 2.0. However, it should be noted that some studies indicate that business passengers are 
less price elastic, with values around 1.0 (see for example Oum et al. 1986 and Oum et al. 1993). DeVany 
(1972) estimate the flight elasticity for domestic US flights to around 1.2, while Morrison and Winston 
(1986) find flight elasticities for business passengers to be around 0.2 and for leisure travel roughly 0.05. 
Berechman and de Wit (1996), using European data, estimate the flight elasticity to 0.7 for business 
passengers and 0.3 for leisure passengers. 
 4
demand function the absolute value of the inverse price-elasticity is given by s, where 
1<s , and the flight-elasticity is given by b  times e.  
For the monopoly model we assume a network with three cities: h, 1, 2, where city h 
is a potential hub. In the point-to-point network the airline operates the network with 
traffic on all city-pairs, i.e. there are three direct connections h1, h2, and 12. In the hub-
and-spoke network, where city h is the hub airport, the airline operates the network with 
traffic on connection h1 and h2, and passengers travelling between city 1 and 2 are 
funnelled through the hub h. For the oligopoly model we only consider competition on 
one connection. 
The regulator has only one instrument available: the environmental tax. The goal for 
the regulator is to set an optimal tax in the sense that the tax maximises social welfare, 
and the regulator determines the taxes before the airlines make their decisions. It should 
be noted that we do not model the possible effect of the tax on investments in cleaner 
technology; an effect which could affect the environmental impact and the optimal tax. 
 
3. Monopoly network market and environmental taxation 
In this section we derive optimal taxes for the point-to-point network and the hub-and-
spoke network. Let { }12,2,1 hhf =  denote the set of all city-pairs and let { }2,1 hhk =  
denote the set of city-pairs with a direct connection in a hub-and-spoke network. 
 
Point-to-point network 
We assume that an airline maximises its profits with respect to passengers and flights, 
and that the profit for an airline operating a point-to-point network is: 
 
å
Î
+--=p
fij
ijijijij
P FtbQcP )()( . (3) 
 
First order conditions and the equilibrium levels of passengers and flights are therefore: 
 
0)( =
¶
¶
+-=
¶
p¶
ij
ij
ij
ij
ij
P
Q
Q
P
cP
Q
 è  s-bsWs-
= /1)(
)1( ijij
ij F
cQ , 
(4) 
 
and, substituting in for ijQ  and using the fact that sb=a / : 
 5
 
0=--
¶
¶
=
¶
p¶
ij
ij
ij
ij
ij
P
tb
F
P
Q
F
è  a-
s-
s-+
s-
Wb
= 1
1
/1
)(
)1)((
)
1
(
ij
ij
ij tb
cc
F . 
(5) 
 
We assume that the regulator maximises the unweighted sum of consumer- and 
producer surplus net of damage costs. The objective function for the regulator is 
therefore: 
 
å òò +--=
ij
ijijijijij
Q F
ij FtFDCdQdFFQPW
ij ij
)(),(
0 0
, 
(6) 
 
where ijD  is the external damage cost, which is a function only of flights; we assume 
that external damages are strictly increasing in ijF . The regulator sets three 
environmental taxes, one for each connection. Differentiating the regulators objective 
function with respect to the environmental tax ijt , substituting in for b  from the first 
order condition in (5), and solving for ijt  we have: 
 
dt
dF
dt
dF
F
PQ
dt
dQ
cP
F
F
D
t
ij
ij
ij
ijij
ij
ij
ij
ij
ij
ij
)()(
1
1
1
b-
s-
+-
+b
-
¶
¶
= . 
(7) 
 
The optimal environmental tax thus consists of two parts, the marginal damage cost of 
flights (which would correspond to the standard Pigouvian tax), and a correction part 
due to the monopoly situation at the connection. This formulation of the optimal 
environmental tax problem resembles the optimal environmental tax of a single product 
monopolist (Barnett 1980). The difference is that the second part of the tax expression 
consists of two welfare effects: the effect on output and the effect on flights. 
Substituting in the price- and flight elasticity and using the fact that 
1-e
e= cPij , the 
optimal tax can be written: 
 
 6
dt
dF
dt
dF
F
cQ
dt
dQ
cc
F
F
D
t
ij
ij
ij
ij
ijij
ij
ij
ij
)()( 1
111 e
a-
-e
e
-e
e+-
-e
e
e+a
e
-
¶
¶
= . 
(8) 
 
Differentiating the second part of the tax expression it can be shown that, for a given 
change in output and flights, the second part is decreasing in both the price- and flight 
elasticity. The intuition behind this is the same as in the monopoly case; the more elastic 
demand is the smaller is the welfare loss associated with the reduction in output and 
flights. In order to determine the sign of the second part of the tax expression we need to 
know the sign of the tax effects on passengers and flights. Since: 
 
0
))(1(
1 <
+a-
-= ij
ijij
ij F
tbdt
dF
 and 0<
a
=
ij
ij
ij
ij
ij
ij
dt
dF
F
Q
dt
dQ
 
(9) 
 
the second part of the optimal tax is non-negative.5 Consequently the optimal tax is 
lower than the corresponding Pigouvian tax. The reason for this is that the tax has a 
negative effect on output and flights, and since the monopolist is already supplying a 
less than optimal level there is a negative welfare effect of this reduction. The 
magnitude of the second part depends on the elasticities and the magnitude of the effects 
on output and flights. 
 
Hub-and-spoke network 
Now the airline operates a hub-and-spoke network, where city h is the hub. The airline’s 
cost function for city pair hj is: 
 
hjhjhjhj FtbcQcQC )(12 +++= , khj Î . (10) 
 
The inverse demand for connecting passengers, i.e. passengers travelling between city 1 
and city 2 through the hub, is assumed to have the following form: 
 
                                               
5 The expressions in the parentheses in equation (8) are non-negative since these are the marginal 
willingness to pay minus the marginal cost for passengers and flights. 
 7
s-bsW= 12121212 QFP ; { }2112 ,min hh FFF = . (11) 
 
The reason for this assumption is that we want to catch the effect of "the weakest link". 
Even if the number of flights are high at one connection, connecting passengers also 
have to rely on the number of flights at the other connection. In order to make the model 
simple we therefore assume that only flights on the connection with the lowest number 
of flights affect demand.6 For the other markets we assume the same inverse demand 
function as before. The profit function for the airline can now be written: 
 
åå
ÎÎ
-=p
khj
hj
fij
ijijij
H CQQP )( . (12) 
 
Again the airline maximises its profit with respect to passengers and flights. First order 
conditions and the equilibrium number of passengers are: 
 
0)( =
¶
¶
+- hj
hj
hj
hj QQ
P
cP è s-bsWs-
= /1)(
)1( hjhj
hj F
cQ , 
0)2( 12
12
12
12 =¶
¶+- Q
Q
PcP è s-bsWs-
= /1
1212
12 )( )1(
2
F
cQ . 
(13) 
 
Note that hj
hj
QQ
W
W= s- 12/112 2  if 12FFhj = , and in a symmetric equilibrium, where 
12W=W hj , we have that hjQQ s-= /112 2 . Thus, in a symmetric equilibrium the share of 
connecting passengers depends only on the price elasticity, while in the nonsymmetric 
equilibrium the share also depends on the demand shift parameters. The first order 
conditions for the number of flights is: 
 
01212 =--¶
¶+
¶
¶
hj
hjhj
hj
hj tbF
PQ
F
P
Q  if hihj FF £ , and 
0=--
¶
¶
hj
hj
hj
hj tbF
P
Q  if khihjhihjFF hihj ι> ,;; . 
(14) 
 
 8
Assuming that hihj FF £ , and using the equilibrium solution for the passengers, the first 
order condition in (14) yields the following equilibrium level of flights: 
 
a-
s-s-
s-+
W+W
s-
b
= 1
112
/1/1
)(
)1)((
)22()
1
(
hj
hj
H
hj tb
cc
F  = Phj
hj
Fa-e-
W
W+ 1
1
12 )( 221 . 
(15) 
 
Consequently, the difference in the number of flights between the point-to-point 
network and the hub-and-spoke network depends on the price- and flight elasticities and 
on the demand shift parameters. If Hhi
H
hj FF <  then, for a given tax, the number of flights 
on connection hi is the same as in the point-to-point network: 
 
a-
s-
s-+
s-
Wb
= 1
1
/1
)(
)1)((
)
1
(
hi
hi
H
hi tb
cc
F = PhiF . 
(16) 
 
Now the regulator maximises the following objective function: 
 
hjhjhjhj
khj
hj
fij
Q F
ij FtFDCdQdFFQPW
ij ij
+--= åå òò
ÎÎ
)(),(
0 0
, 
(17) 
 
by setting two different environmental taxes 1ht  and 2ht . The difference between the 
hub-and-spoke network and the point-to-point network is the demand from the 
connecting passengers. Whether this affects the tax expression or not depends on 
whether the number of flights at the connection is higher or lower than the number of 
flights at the other connection. The model is such that for the connection with the 
highest number of flights, the airline has no incentive to increase the flights in order to 
increase the demand for the connecting passengers (there is still an effect for passengers 
travelling directly on that connection). Consequently, for that connection the optimal tax 
will not be a function of the connecting passengers. We proceed assuming that 
hihj FF < , which means that the optimal tax for connection hi  is the same as in the 
                                                                                                                                         
6 This is also a difference from the model in Nero and Black (1998) where the share of connecting 
passengers is exogenous. 
 9
point-to-point case. For connection hj , differentiating the regulators objective function 
with respect to the tax hjt , using the fist order conditions in (14) and solving for the tax: 
 
hj
hj
hj
hj
hjgj
gjqj
hjhj
hj
hj
hj
hj
hj
hj
hj
hj
dt
dF
dt
dF
PQ
Fdt
dQcP
F
dt
dQ
cP
F
F
D
t
å
Î
b-
s-
+-
+b
+-
+b
-
¶
¶
= 12,
12
12 )()()( 1
12
11
 
(18) 
 
Again, the optimal tax consists of two parts, but the second part now involves the 
effects on demand in two markets; passengers travelling directly between airport h and 
j, and passengers travelling between airport i and j through the hub airport. As in the 
point-to-point case we can substitute in the price- and flight elasticities, and show that 
the tax expression is decreasing in both the price- and flight elasticity, for a given 
change in output and flights. The comparative statics are also easy to establish; both the 
number of passengers and flights are decreasing in the tax. Consequently the second 
part of the tax expression is positive. Compared to the point-to-point network, the 
second part of the tax expression now consists of two additional positive expressions 
stemming from the effect on connecting passengers. The tax is therefore reduced even 
more compared to the corresponding Pigouvian tax at the connection with the least 
number of flights. For the other connection, the tax is the same as in the point-to-point 
network. 
 
4. Environmental effects in the two networks 
We can make a direct comparison of the environmental effects of the two types of 
networks, at least in a simplified fashion, by comparing the number of flights in the two 
networks. Thus, in the comparison we rule out any differences in marginal damages 
between airports and any differences in distance between the cities. This is probably not 
the case in reality, where the marginal damage could be higher at the hub airport. This 
mainly due to congestion which is a negative externality, but congestion can also 
increase other external effects such as noise and local emissions (see Carlsson 1999). It 
is also likely that there are differences in distance between the cities. We distinguish 
between two cases: (i) non-symmetric equilibrium with different demand-shift 
 10
parameters and different capacity and (ii) symmetric equilibrium. The comparison 
between the networks is made on the premise that the tax is the same for both networks 
(alternatively that there are no environmental taxes). Note that the difference in the 
number of flights between the two networks depends on output- and flight elasticities 
and the demand shift parameters. From (5) we also have that: 
 
( ) PhjhjP FF a-WW= 111212 . (19) 
 
Symmetric equilibrium 
We begin with the symmetric equilibrium where the demand shift parameters are 
identical; consequently the difference in flights between the networks depends only on 
the price- and flight elasticities. The following propositions are modified versions of the 
propositions given in Nero and Black (1998). The difference is that in our model the 
share of connecting passengers in the hub-and-spoke networks depends on the price 
elasticity, while in Nero and Black (1998) this share is exogenous.  
 
Proposition 1a: The number of flights at the leg airport is higher under the point-to-
point network than under the hub-and-spoke network if 5.022 >- e-a- . 
 
Proof: From the equilibrium levels of flights in (5) and (15), and using (19) and the fact 
that ijW=W 12 , we have that proposition 1a is true if PP FF a-e-+> 1
1
)221(2 . 
 
This means that when the flight elasticity is low and the price elasticity is high the 
number of flights is higher at the leg-airports under the point-to-point network 
compared to the hub-and-spoke network. When 0®a  this is always true, and when 
1®e  this is never true. From the proposition we can calculate critical levels of the 
elasticities, i.e. values where the number of flights are the same in the two networks. For 
example, when the flight elasticity is 0.5 then the price elasticity would have to be 
higher than 2.27 in order for this to be the case. 
 
 11
Proposition 1b: The total number of flights at the hub airport is always higher under the 
hub-and-spoke network. 
 
Proof: From the equilibrium levels of flights in (5) and (15), and using the fact that 
ijW=W 12 , we have that proposition 1b is true if PP FF 2)221(2 1
1
a-e-+< . This implies 
that the proposition is true if 1)221( 1
1
>+ a-e- , and since 1>a  this is always true. 
  
Proposition 1b is not surprising, but what is interesting is under what conditions the 
difference in number of flights is small or large. This follows directly from the 
proposition; there is a lower difference in number of flights when the flight elasticity is 
low and the price elasticity is high. In that case demand does not increase that much due 
to the increase in number of flights, and demand decreases much due to the increase in 
prices. 
 
Proposition 1c: The total number of flights is higher under the point-to-point network if 
125.1 11 >- e-a- . 
 
Proof: From the equilibrium levels of flights in (5) and (15), and using (19) and the fact 
that ijW=W 12 , we have that proposition 1c is true if PP FF 4)221(6 1
1
a-e-+> . 
 
Consequently, the total number of flights is higher in the point-to-point network when 
the flight elasticity is low and the price elasticity is high. If we assume no differences in 
marginal damages between airports, we could also conclude that total external damages 
are higher in a point-to-point network when the flight elasticity is low and the price 
elasticity is high.7 However, the elasticities will also determine the airline's choice of 
network. For the airline's choice of network we have the following proposition. 
 
                                               
7 The flight elasticity would have to be rather low in order for this to be the case. For example if the flight 
elasticity is 0.3, then the price elasticity would have to be at least 5.3 in order for this to be the case, while 
if flight elasticity is 0.1, then the price elasticity would have to be at least 2.51. 
 12
Proposition 1d: It is optimal for the airline to operate a hub-and-spoke network when 
e-a- -> 11 25.11 . 
 
Proof: See Proof 1 in the Appendix. 
 
The interesting aspect is that this condition is the opposite of the condition in 
Proposition 1c. This means that the airline will choose the network with the largest 
number of flights. Given no differences in marginal damage between flights, this also 
means that the airline will choose the network with the highest environmental damages. 
 
Nonsymmetric equilibrium 
It is easy to extend the propositions to the nonsymmetric case, although the exact 
interpretation of them is more complicated. Suppose that hihjFF hihj ¹< ; in the hub-
and-spoke network. A crucial difference between this case and the symmetric case is 
then that the number of flights is the same at the other leg airport hi . In this extreme 
case, the only effect on this connection is an increased load factor. We can now 
establish the following propositions. 
 
Proposition 2a: The total number of flights and the number of flights at leg airport j is 
higher under the point-to-point network if 0211 1
1
1211
1
12 )()( >
W
W+-
W
W+=Y a-e-a-
hjhj
. 
 
Proof: From the equilibrium levels of flights in (5) and (15), and using (19) we have that 
Proposition 2a is true if: 0221)1( 1
1
121
1
12 )()( >
W
W+-
W
W+ a-e-a- Phj
hj
P
hj
hj
FF , since Hhi
P
hi FF = . 
 
The difference between the symmetric and non-symmetric equilibrium is the demand-
shift parameters. However, as in the symmetric case, the elasticities and the demand 
shift parameters will affect the airline's choice of network. We therefore have the 
following proposition. 
 
 13
Proposition 2b: It is optimal for the airline to operate a hub-and-spoke network when 
0211 1
1
1211
1
12 )()( <
W
W+-
W
W+=Y a-e-a-
hjhj
. 
 
Proof: The proof is similar to the proof of proposition 1d (using the result of proposition 
2a). 
 
Again, an airline will choose the network with the largest number of flights and, given 
the assumptions about the damage function, the highest environmental damages. In 
Table 1 we present the level of the price elasticity where Y  is approximately equal to 
zero (i.e. where total traffic and profits are equal between the networks) for different 
levels of the demand shift parameter and the flight elasticity.8 We then see that when 
12W  is larger than hjW , total traffic is higher under a point-to-point network for most 
cases.9 Furthermore, by looking at the case when the demand shift parameters are equal 
it is also easy to make a comparison with the symmetric equilibrium. Not surprisingly 
the critical level of the price elasticity is lower in this case, and the reason for this is of 
course that in the nonsymmetric case the flights from airport hi are the same in both 
networks. 
 
Table 1. Critical levels of the price elasticity. 
 
Demand shift parameters, 
hjW
W 12  Flight elasticity, a  Value of price elasticity, e, 
where 0»Y  
0.5 0.1 / 0.25 / 0.5 / 0.75 1.29 / 1.81 / 3.08 / 6.01 
0.75 0.1 / 0.25 / 0.5 / 0.75 1.24 / 1.65 / 2.59 / 4.40 
1 0.1 / 0.25 / 0.5 / 0.75 1.21 / 1.55 / 2.27 / 3.40 
1.5 0.1 / 0.25 / 0.5 / 0.75 1.17 / 1.43 / 1.90 / 2.40 
2 0.1 / 0.25 / 0.5 / 0.75 1.14 / 1.35 / 1.69 / 1.96 
 
Finally, for the hub airport, it is easy to establish a similar condition as for the 
symmetric case, but where the difference in flights between the networks is low when 
12W  is large compared to hjW . 
                                               
8 Note that Y is always increasing in the price elasticity, e. 
9 Given that city h is the hub airport, it is more likely that the demand shift parameter hjW  is larger than 
12W , since these parameters in a sense measure the economic activity (income) in the different cities. 
 14
5. Duopoly market and environmental taxation 
After deregulation, airlines now face competition on many routes. One interesting 
problem is then how an optimal environmental tax should be designed under 
competition. We apply a Cournot duopoly model where each airline simultaneously 
maximises its profit with respect to passengers and flights. Previous research on 
environmental taxation in duopoly models has focused on single product oligopolies 
(see e.g. Carlsson 2000, Simpson 1995). The problem with an analysis of multiproduct 
oligopolies is especially to determine conditions for stability of the equilibrium, and to 
derive the comparative statics. Here we use the conditions for stability derived by Zhang 
and Zhang (1996). There have been some papers on multiproduct oligopolies in the case 
of aviation markets (Brueckner and Spiller 1991 and Oum et al. 1995). However, in 
those papers the multiproduct nature of the model is that an airline operates a network 
where travel on each city-pair market is seen as a single product, and where each 
product (market) is possibly related through the cost function. In our model, the two 
products, passengers and flights, are related through the demand function. The inverse 
market demand is a function of total number of passengers, 21 qqQ += , where iq  is 
airline i’s number of passengers, and total number of flights, 21 ffF += , where if  is 
airline i’s number of flights. Consequently, passengers do not differentiate between the 
two airlines’ flights; they only care about total number of flights. The inverse market 
demand function is therefore s-bsW= QFP , and the profit for airline i is: 
 
[ ] iiii ftbqcFQP )(),( q+--=p ; 2,1=i . (20) 
 
In order to allow for differences between the airlines we impose an exogenous aircraft 
engine technology, which in turn affects the emissions from a particular flight. 
Emissions from a flight are equal to ii fq , and if 21 q=q  then the airlines use the same 
technology. Each airline maximises its profit with respect to iq  and if , given its rivals' 
choice of these variables, and the choice of iq  and if  is made simultaneously. First 
order conditions are therefore: 
 
 15
0)( =
¶
¶+- i
i
q
q
PcP  and 0=q--
¶
¶
i
i
i tbf
Pq . 
(21) 
 
We assume that the regulator cannot differentiate the tax between the airlines, and that 
he maximises the following objective function: 
 
åòò -q+-=
i
iii
Q F
EDftCdzdxxzPW )(),(
0 0
, 
(22) 
 
by setting the environmental tax, t, where å q=
i
ii fE . Differentiating the objective 
function with respect to the tax, substituting in for b  from the first order conditions in 
(21) and solving for t we have: 
 
011
2
1
2
1
)()(
=
q
b-
s-
+-
+b-
¶
¶=
å
å
=
=
i
i
i
i
i
i
i
dt
df
dt
df
F
qQP
dt
dq
cPF
E
Dt . 
(23) 
 
The resulting optimal tax is similar to an optimal tax for a single product duopoly 
(Carlsson 2000, Simpson 1995). The difference from a single product duopoly is that 
there are welfare effects from effects on both passengers and flights. However, the main 
problem is to determine the sign of the tax effect on the number of passengers and 
flights. In the single product oligopoly case, output of both firms must not necessarily 
be decreasing in the tax since the tax can shift production between firms. However, 
under the condition that marginal costs are increasing in the tax industry output will be 
decreasing in the tax (see Carlsson 2000). As we will see it is difficult to determine how 
the tax will affect the number of flights in this model. This implies that we do not know 
if the optimal tax is lower or higher than the marginal damage of flights. 
When determining the sign of the comparative statics we will use the stability 
condition for multiproduct oligopolies derived by Zhang and Zhang (1996). Let vector 
iX  denote firm i's passengers and flights, and let )( jii XRX =  denote firm i's reaction 
function. A sufficient condition for stability of the equilibrium point is then that for 
some matrix norm × , 
 16
 
1
2
1
2
2
1
<
¶
¶
¶
¶
pX
R
X
R  and 1
2
2
1
1
2
<
¶
¶
¶
¶
pX
R
X
R ; ¥= ,2,1p . 
(24) 
 
This condition implies that the magnitude of the eigenvalues of the matrices 
)( 1
2
2
1
X
R
X
R
¶
¶
¶
¶  and )( 2
1
1
2
X
R
X
R
¶
¶
¶
¶  are less than unity (for a proof of the condition see Zhang 
and Zhang 1996).10 We will assume that the equilibrium is stable, and hence impose this 
condition on the reaction functions. Furthermore, note that the profit function has the 
following properties: 
 
0<pi qq ii , 0<p=p
i
ff
i
ff jiii , 0>p=p=p
i
fq
i
fq
i
qf jiiiii , 0<p
i
qf ji , 0=p
i
tqi
, i
i
tfi
q-=p , (25) 
 
where subscripts denote partial derivatives, 2,1, =ji , ji ¹ . We assume that both 
passengers and flights are strategic substitutes, i.e. 0<pi ff ji  and 0<p
i
qq ji .
11 
Differentiating the first order conditions with respect to the tax we have by matrix 
notation: 
 
( ) jijiRRRI
dt
dX i
it
i
ii
j
jt
j
jj
i
j
j
i
i
j
i
¹=PP+PP--= --- ;2,1,;)()()( 111 , 
(26) 
 
where iij
i
iij
i
i
j X
RR PP-=
¶
¶= - 1)( , 
ú
ú
û
ù
ê
ê
ë
é
pp
pp
=P i
ff
i
qf
i
fq
i
qqi
ii
iiii
iiii  and 
ú
ú
û
ù
ê
ê
ë
é
pp
pp
=P i
ff
i
qf
i
fq
i
qqi
ij
jiji
jiji . Let us 
write the derivative matrix of firm i's reaction function as: 
 
( )
ú
ú
û
ù
ê
ê
ë
é
=PP
D
-= i
ff
i
fq
i
qf
i
qqi
ij
i
iii
i
j rr
rr
adjR 1 , 
(27) 
 
                                               
10 See Szidarovszky and Li (2000) for an extended discussion of stability of multiproduct oligopolies. 
They derive stability conditions for a more general case including adaptive expectations. Further, they 
show that the necessary conditions for stability given in Zhang and Zhang (1996) must not necessarily 
hold. However, this does not affect the discussion in this paper. 
11 For a discussion of strategic substitutes and complements see Bulow et al. (1985). 
 17
where iD  is the determinant of iiiP , which is greater than zero by the second order 
condition. It can then be shown that 0, <iffiqq rr , and 0=iqfr . The reason for iqfr  being 
zero is the assumption about the demand function; the passengers do not differentiate 
between the two airlines’ flights. This means that the direct and cross effect of flights on 
marginal profits will be equal. Furthermore, a sufficient condition for 0<ifqr  is that (see 
Proof 2 in the Appendix): 
 
)/(2)/)(1()/( 22 QqQqQq iii s-+ss+sb-b  < 0. (28) 
 
The expression in (28) is negative if either the price- or the flight eleasticity is low; 
when proceeding we will assume that 0<ifqr . The elements of the matrix: 
 
úû
ù
êë
é
dg
a=
ú
ú
û
ù
ê
ê
ë
é
+
=
ii
i
j
ff
i
ff
j
fq
i
ff
j
qq
i
fq
j
qq
i
qqj
i
i
j rrrrrr
rr
RR
00
, 
(29) 
 
can then be shown to be 0,, >dga iii . Note that the matrix jiij RR  is a triangular matrix, 
which means that the eigenvalues of ji
i
j RR  are the entries on the main diagonal. The 
stability condition then implies that the diagonal elements all are less than unity, i.e. that 
121 <qqqqrr  and 21 ffff rr  < 1. A sufficient condition for the diagonal elements to be less than 
unity is that the absolute values of the diagonal elements of the matrices ijR  and 
j
iR  are 
less than unity, i.e. that 1<iqqr  and 1<iffr . These conditions are similar to stability 
conditions for the two markets in isolation, i.e. when the two "markets" are not related 
(Zhang and Zhang 1996). Consequently, we have similar conditions as in the single 
product case, with the difference that we do not have any restrictions on ifqr , apart from 
that it should be negative. Given that the equilibrium is stable, which implies that the 
eigenvalues of the matrices ji
i
j RR  are all less than unity, we have by the Neumann 
lemma (Ortega and Rheinboldt 1970, p. 45) that 1)( -- jiij RRI  exists and that: 
 
 18
( ) úû
ù
êë
é==- å
=¥®
-
ii
ii
k
i
ij
i
i
jk
j
i
i
j dc
ba
RRRRI
0
1 lim)( . 
(30) 
 
This series must converge to a matrix of the same sign as the matrix ji
i
j RR , 
consequently we have that 0,, >iii dca  and 0=ib . Furthermore, since jiij RR  is a 
triangular matrix, we have that aaa == 21  and ddd == 21 . 
Using (27) and (30) we can now calculate the effect of the tax on the number of 
passengers (see Proof 3 in the Appendix): 
 
44 344 21434 21
positive
j
j
qfj
i
qqi
negative
i
i
qfiii jjii
raa
dt
dq
D
pq
-
D
pq
-= .  
(31) 
 
The first part on the right-hand side in (31), representing the own effect, is negative, 
while the second part, the strategic effect, is positive. Consequently, the effect of the tax 
on the individual airline's number of passengers is not determined. However, the effect 
on the total number of passengers is determined: 
 
2
2
2
1
1
1
1
2
2211
)1()1(
D
pq+
-
D
pq+
-= qfqqqfqq
arar
dt
dQ  < 0 if 1<iqqr . 
(32) 
 
The total number of passengers is always decreasing in the tax, which means that at 
least one airline's number of passengers must be decreasing in the tax. Consequently, for 
a symmetric equilibrium both airlines' number of passengers is decreasing in the tax.12 
Using (27) and (30) we can also calculate the effect of the tax on the number of flights 
(see Proof 3 in the Appendix): 
 
( ) ( )
4 34 214444 34444 21444 3444 21
negative
j
j
qqj
i
ffi
positive
j
j
qf
i
fqi
j
qf
i
qqij
negative
i
i
qfi
i
qqiii jjjjjjiiii
rdrdrccd
dt
df
D
pq
+
D
p+pq
-
D
p-pq
= . 
(33) 
 
                                               
12 Note that the equilibrium is symmetric if the airlines use the same technology, i.e. if 
21 q=q . 
 19
The first part on the right-hand side in (33), representing the own effect, is negative, 
while the sum of the second and third part, the strategic effects, is not determined in 
sign. Consequently, the effect of the tax on the number of flights for an airline is not 
determined.13 The effect on the total number of flights is: 
 
2
2
1
1
2
2
21212
1
1
2
21
1
1
12121
1
)()(
)()(
22222222
11111111
D
+p
-
D
p+p-p
+
D
+qp
-
D
p+p-pq
=
crcrrd
crcrrd
dt
dF
qqqfqqffqffqqq
qqqfqqffqffqqq
 
(34) 
 
If we impose the restriction that 1<ifqr , then total number of flights is decreasing in the 
tax, if 2c  is not sufficiently larger than 1c  (which is true for a symmetric equilibrium). 
Consequently, under these conditions both airlines' number of flights is decreasing in 
the tax under a symmetric equilibrium. This result is in a sense dissatisfying, since it 
implies that we do not know with certainty how the number of flights will be affected 
by the environmental tax. Therefore, the environmental effect of the tax is unclear, and 
we might even have the perverse result of increased flights (emissions) from the 
environmental tax. It should however be noted that we have ruled out economies of 
scope in this mode. This could clearly affect the results 
 
6. Conclusions 
Optimal environmental taxation of airline traffic is complicated by several factors, and 
in this paper we have discussed networks effects and imperfect competition. The 
formulation of the optimal environmental tax is similar to the optimal tax for both a 
monopolist and a duopoly market. The environmental tax depends on both the 
competitive situation and the type of network that the airline operates. For a monopoly 
the optimal tax is always lower than the marginal damages, while for a duopoly the 
result is ambiguous. 
Another interesting result in the paper is that a monopoly airline will choose the type 
of network with the highest number of flights. This implies that if there are no 
                                               
13 However, if we have the extreme case that the competing airline uses a clean technology, i.e. if 
0=q j , then the number of flights is decreasing in the tax. 
 20
differences in marginal damages between flights, an airline will choose the network 
with the higher environmental damages. Of course, in reality, the marginal damages 
differ between different connections, especially in the case of local externalities. One 
implication of adopting a hub-and-spoke network is that the number of flights will 
increase at the hub airport, given the construction of our model. Since there are reasons 
to believe that the external damages are higher at the hub airport (perhaps mainly due to 
the simple fact that the traffic volume is larger), this implies that a regulator, from an 
environmental perspective, should pay attention to the adoption of hub-and-spoke 
network. However, we have ruled out certain effects of a hub-and-spoke system such as 
increased load factors, which could affect the results. This would be a natural extension 
of the model presented here. We also showed that there are significant differences 
between a symmetric and a non-symmetric equilibrium. Especially we presented an 
extreme case where the number of flights on one route was not affected by the 
introduction of a hub-and-spoke network, which of course was in favour for the hub-
and-spoke network. 
Even with the seemingly simple model of two competing airlines as multiproduct 
duopolists, the analysis became rather complicated. We can show that for a symmetric 
equilibrium both airlines' number of passengers is decreasing in the tax, and that the 
total number of passengers is always decreasing in the tax. The total number of flights is 
decreasing in the tax if some additional, reasonable, conditions are imposed. This 
implies that there might be perverse effects of an environmental tax, since at least one 
airlines' number of flights can be increasing in tax. 
The formulation of competing airlines as multiproduct firms is interesting, and a 
natural development of this type of model is to link this with the model of Oum et al. 
(1995), and thus to include several connections, where the connections are linked 
through the cost function. Another interesting development is to introduce adaptive 
expectations and use the stability conditions in Szidarovszky and Li (2000).  
 21
References 
 
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Review 70, 1037-1041. 
Berechman, J. and J. de Wit (1996), An analysis of the effects of European aviation 
deregulation in airline's network structure and choice of a primary West European 
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Borenstein, S. (1992), The evolution of U.S. airline competition, Journal of Economic 
Perspectives 6, 45-73. 
Brueckner, J. and P. Spiller (1991), Competition and mergers in airline networks, 
International Journal of Industrial Organization 9, 323-342. 
Bulow, J., J. Geanakoplos, and P. Klemperer (1985), Multimarket oligopoly: Strategic 
substitutes and complements, Journal of Political Economy 93, 488-511. 
Carlsson, F. (1999), Incentive-based environmental regulation of domestic civil aviation 
in Sweden, Transport Policy 6, 75-82. 
Carlsson, F. (2000), Environmental taxation and strategic commitment in duopoly 
models, Environmental and Resource Economics 15, 243-256. 
Caves, D., L. Christensen and M. Tretheway (1984), Economics of density versus 
economies of scale: Why trunk and local service airline costs differ, Rand Journal of 
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DeVany, A. (1975), The effect of price and entry regulation on airline output, capacity 
and efficiency, Bell Journal of Economics 6, 327-345. 
Morrison, S. and C. Winston (1986), The Economics Effects of Airline Deregulation, 
Brookings Institution, Washington. 
Nero, G. and J. Black (1998), Hub-and-spoke networks and the inclusion of 
environmental costs on airport pricing, Transportation Research D 4, 275-296. 
Ortega, J.M. and W.C. Rheinboldt (1970), Iterative Solutions of Nonlinear Equations in 
Several Variables, Academic Press, New York. 
Oum, T., D. Gillen and S. Noble (1986), Demand for fareclasses and pricing in airline 
markets, Logistics and Transportation Review 22, 195-222. 
Oum, T., A. Zhang and Y. Zhang (1993), Inter-firm rivalry and firm-specific price 
elasticities in deregulated airline markets, Journal of Transport Economics and 
Policy 27, 171-192. 
Oum, T., W.G. Waters II and J-S. Yong (1992), Concepts of price elasticities of 
transport demand and recent empirical estimates, Journal of Transport Economics 
and Policy 26, 164-169. 
Oum, T., A. Zhang and Y. Zhang (1995), Airline network rivalry, Canadian Journal of 
Economics 95, 836-857. 
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Bell Journal of Economics 8, 565-576. 
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Simpson, R. (1995), Optimal pollution taxation in a Cournot duopoly, Environmental 
and Resource Economics 6, 359-369. 
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case, Journal of Mathematical Economics 26, 441-462. 
 22
Appendix 
 
Proof 1: Proof of Proposition 1d 
Using the equilibrium levels of passengers in (4) and (13), and the equilibrium levels of 
flights in (5), (15) and (16), the reduced forms of the profit functions can be written: 
PP Ftb
b
b-s+=p )(3  
HH Ftb
b
b-s+=p )(2  
Since [ ] PH FF a-e-+= 1 1221  we have that profits are higher in the hub-and-spoke 
network if: 
[ ] PP FtbFtb
b
b-s+>+
b
b-s+ a-e- )(3221)(2 1
1
 
[ ]
2
3221 1
1
>+ a-e-  è  a-e- >+ 11 )
2
3(21  
 
Proof 2: Sign of elements in the derivative matrix of firm i's reaction function 
Firm 1's derivative matrix of the reaction function, equation (27), is: 
( )
ú
ú
û
ù
ê
ê
ë
é
pp
pp
ú
ú
û
ù
ê
ê
ë
é
pp-
p-p
D
-=PP
D
-= 11
11
11
11
11
1
2
2121
2121
1111
1111
11
ffqf
fqqq
qqqf
fqffi
ij
i
iiadjR  
úû
ù
êë
é
dg
ba
D
-=
ú
ú
û
ù
ê
ê
ë
é
pp+pp-pp+pp-
pp-pppp-pp
D
-=
11
11
111111111
11111111
1
1
2
11
2111211121112111
2111211121112111
ffqqfqqfqfqqqqqf
fffqfqffqffqqqffR  
Since 0,,, 1111
21211111
<pppp qfqqqqff  and 011 1111 >p=p fqqf , it follows directly that 0, 11 >ga . 
Further 01 =b  since 11 2111 ffff p=p  and 
11
1121 fqfq
p=p . Finally,  
i
ii qPF
Q
q
PQ
Q
q
PF 21211 )1)(2)1(()1( )( --- b-b-+ss+s-b-=d  
 )( )2)1)((1()1( 2221 -+s-bs-s-bb-=d - Q
q
Q
q
Q
q
FP iii   
The expression in the inner parenthesis can be written: 
)2)1)((()2)(1( 22 -+ss-bs-s-s+b
Q
q
Q
q
Q
q
Q
q
Q
q iiiii  
Q
q
Q
q
Q
q
Q
q
Q
q
Q
q iiiiii s-+ss+bs++sbs-bs-bs+b 2))(1(2)1()(2)( 2222  
Q
q
Q
q
Q
q iii s-+ss+bs-b 2))(1()( 222 .  
This expression is negative, and consequently 01 >d , if s and b  are sufficiently low. 
 
 23
Proof 3: Comparative statics for the duopoly market 
Using (27) and (30) the comparative statics for firm 1 can be written: 
÷
÷
ø
ö
ç
ç
è
æ
P
ú
ú
û
ù
ê
ê
ë
é
pp-
p-p
D
+P
ú
ú
û
ù
ê
ê
ë
é
pp-
p-p
Dú
ú
û
ù
ê
ê
ë
é
úû
ù
êë
é-=
ú
ú
ú
û
ù
ê
ê
ê
ë
é
1
111
11
1
2
222
22
211
11
11
11
1
1
1111
1111
2222
2222
11
t
qqfq
qfff
t
qqfq
qfff
fffq
qfqq
rr
rr
dc
ba
dt
df
dt
dq
Since, úû
ù
êë
é
q-
=
ú
ú
û
ù
ê
ê
ë
é
p
p
=P
1
1
1
1
1
0
1
1
tf
tq
t  and úû
ù
êë
é
q-
=
ú
ú
û
ù
ê
ê
ë
é
p
p
=P
2
2
2
2
2
0
2
2
tf
tq
t , we have: 
÷
÷
ø
ö
ç
ç
è
æ
ú
ú
û
ù
ê
ê
ë
é
pq-
pq
D
+
ú
ú
û
ù
ê
ê
ë
é
pq-
pq
Dú
ú
û
ù
ê
ê
ë
é
úû
ù
êë
é-= 1
1
1
1
12
2
2
2
211
11
11
11
1
11
11
22
22 11
qq
qf
qq
qf
fffq
qfqq
rr
rr
dc
ba
dt
dX  
Finally, since 0, 11 =qfrb , we have: 
ú
ú
û
ù
ê
ê
ë
é
pq-
pq
Dúû
ù
êë
é-
ú
ú
û
ù
ê
ê
ë
é
pq-pq
pq
Dúû
ù
êë
é-= 1
1
1
1
1
11
1
2
2
12
2
1
2
2
1
2
11
1
1
11
11
2222
22
1010
qq
qf
qqffqffq
qfqq
dc
a
rr
r
dc
a
dt
dX  
Solving this yields (31) and (33).