This document summarizes a game theoretic model of airport privatization and charges in a multi-airport system. It involves three stages: 1) Regional authorities decide airport ownership, 2) Airports set charges, 3) An airline sets ticket prices. The model finds: 1) Airline ticket prices satisfy a linear system determined by demand parameters. 2) Airport charges at Nash equilibrium also satisfy a linear system, with a unique solution. 3) The optimal ownership structure depends on relative demand parameters. With equal demand slopes, public ownership maximizes welfare if both external airports are also public and demand parameters are close.

1. 1
Privatization and airport charges in multi airport systems: a
game theoretic approach
V. Psaraki - Kalouptsidi, Th. Pantelidis and I. Pagoni
Faculty of Civil Engineering, National Technical University of Athens, 5 Iroon Polytechniou Street,
Zografou Campus, 15773 Athens, Greece
Corresponding author: vpsaraki@civil.ntua.gr
ABSTRACT
We consider government's decisions on airport privatizations, in a multi-airport setting. In particular,
in a three stage non-cooperative game, first, regional authorities decide on the ownership status of the
airports in their region based on airport and airline revenues, as well as passenger surplus. Then,
airports set charges. Finally, a single airline sets the ticket price per passenger. We show that both the
optimal airline fees and the airport charges at Nash equilibrium satisfy a linear system of equations.
The optimal ownership choice for the multi airport system depends on the relative size of demand
parameters. We derive both analytically as well as in simulations when a given ownership choice
offers the highest regional welfare for the multi airport system. In the case of equal demand slopes, we
find that the welfare of the multi airport region is maximal under public ownership of its airports if the
external connecting airport is also publicly owned and the offset demand parameters are close and lie
in a certain cone. This behavior is consistently observed across price coefficients. On the other hand, if
the external airport is private, mixed public private schemes emerge as welfare maximizers.
Keywords
Airport ownership, multi airport systems, airport charges, airline pricing, non-cooperative games, Nash
equilibrium.

2. 2
1. Introduction
This paper considers passenger traffic services in multi-airport systems under different ownership
options and airport charging policies. We present a game-theoretic model to study the joint
determination of airport and airline fees, as well as the ownership status of airports decided upon by
regional authorities.
The model features three types of agents: airlines which cater to consumers, airports and regional
authorities. Passenger traffic is interregional with flows from a single airport region (country, city) to a
multi-airport region. It is assumed for simplicity that traffic is served by a single carrier that is based
on one of the two regions. The price paid by a passenger consists of the charges at the origin and
destination airports and the fee charged by the carrier. This last price component is decided by the
airline under a profit or revenue maximizing objective. The airport charges are set by the airport
authorities so that the airport’s payoff is maximized. The airports in the multi-airport region offer
complimentary services so that demand for round trips to one of these airports is affected not only by
the ticket price of the trip but also by the price of trips to another airport of this region. Due to
networking effects a connecting airport also affects demand through its charges leading to the so called
gravitation effect (Mantin, 2012). Airport payoffs crucially depend on the ownership form of the
airport. Under private ownership, airports set charges to maximize profits. Under state or public
ownership additional factors are taken into account such as consumer surplus and airline revenues if
the airline is based on the region. The ownership status is decided by the relevant regional authority
that seeks to find the right balance between the interests of the various actors. As the landscape
described above involves different stakeholders that make decisions in pursuance of different
objectives, jointly optimal strategy profiles constitute an interesting object of study and form the focus
of this paper.
A number of important contributions have studied jointly or in isolation the issues of airport ownership
and charging (Morrison, 1987), (Zhang & Zhang, 1997), (Brueckner, 2002), (Zhang & Zhang, 2003),
(Basso & Zhang, 2007), (Oum, Zhang, & Zhang, 1996), (Lin, 2013) for single and multi-airport
regions. As observed in (Neufville & Odoni, 2009) the presence of more than one airport in a
large metropolitan area creates competition for the same air traffic demand. (Bonnefoy, 2007)
points out that airport privatizations in the case of multi-airport systems can be a stimulating
event in the development of future secondary airports that emerge as alternative air
transportation nodes in the region and increase competition between players at the regional
market level.
Pricing and competition between two airports in a multi-airport region were considered in
(Noruzoliaee, Zou, & Zhang, 2015) where joint ownership of two airports under a private
monopoly and under a single public agency were analyzed. (Matsumura & Matsushima, 2012),
(Mantin, 2012) examined airport complementarity, using a game theoretic approach to solve a
game of ownership (government or private) between two players. A popular approach in analytic
pricing models is the vertical approach (Basso & Zhang, 2007) where the airline market is modeled
as an oligopoly taking airport charges as given. In short, airports provide an input to the
airline market which reaches a price or quantity decision and demand is realized.
In this paper we focus on the various forms of ownership that might occur in a multi-airport
region under a game theoretic model that exemplifies both duopoly and monopoly ownership
cases. Airport ownership, airport charges and airline fees are considered as a three stage non
cooperative game. Optimal strategies and Nash equilibria are studied through backward reasoning. The
airline fee is first determined by revenue maximization and is obtained by a parametrized linear system
of equations under a deterministic multivariable linear demand model. Demand parameters include the
matrix of price coefficients, and a vector of offset parameters and that collectively summarize all
other factors that affect passenger demand. At stage two, airport charges are set at Nash equilibrium.
They are found to satisfy a linear system of equations which has a unique solution for almost all

3. 3
parameter values. Regional welfare is explored at stage one. Parameter constraints ensuring that all
possible combinations of ownership choices become optimal are derived using a combination of
theoretical analysis and symbolic calculations. In the case of equal demand slopes, we find that the
welfare of the multi airport region is maximal under public ownership of its airports if the external
connecting airport is also publicly owned and the offset demand parameters and are close and lie
in the cone . This behavior is consistently observed across price coefficients. If the
external airport is private, mixed public private schemes emerge as welfare maximizers.
The rest of the paper is organized as follows: section 2 gives a description of the main model, the
network topology and the ingredients of the three stage game. Section 3 presents the solution to the
non-cooperative game. In section 4 we study in detail the case of symmetric demand slopes.
Simulations and symbolic calculations are employed to demonstrate passenger traffic, air ticket price
and welfare at equilibrium. Conclusions are given in section 5 while detailed derivations are
summarized in the appendices.
2. Model description
In this section the basic model configuration is given. The relevant stakeholders, their decisions and
payoffs are defined. The interactions are captured by a three stage static game and the resulting Nash
equilibria.
2.1 Network
We consider air traffic between two geographical regions (countries, cities, etc) as in Figure 1. Region
0 is served by a single airport. A system of two airports is available in region 1. For simplicity we
assume that air traffic between the two regions is carried out by a single carrier. The relevant network
topology is illustrated in Figure 1. Flights between airports are represented by the edges of the graph.
The airports in the multi airport region are not connected; travel within this region is served by other
means of transport. The carrier is based in one of the two regions. Let d be a binary variable that flags
the registration location of the airline; d=0 means that the carrier is based on region 0, while d=1
means that the carrier is based on region 1.
2.2 Game primitives and stages
The interactions and decisions of the relevant stakeholders are formulated as a three stage game. The
basic constituents are summarized in Table 1.
Stage Agents Decisions Payoffs
1 Regional authorities Airport ownership Total regional
welfare
2 Airports Airport charges Airport payoff
(revenues or
welfare)
3 Airlines Passenger volume Airline profit
Table 1. Basic constituents of the game formulation.

4. 4
Figure1. Network configuration of traffic flow between a single airport region and a multi airport
region.
During the first stage the ownership structure of the airports in each region is simultaneously decided.
The two players (decision makers) are the regional authorities (government, state, municipality, etc).
To simplify the exposition we assume the regional authority of the single airport region makes a binary
decision: retain the public nature of the airport or employ some form of a privatization scheme. Thus
the action set is {0,1} where 0 denotes public ownership and 1 stands for any form of privatization. On
the other hand, the multi airport region decides on the ownership of all its airports. The action set takes
four values: 00, 01, 10, 11 for every possible combination of ownership. Each region r picks an action
ur to maximize the region’s total welfare. The total welfare of region r, Wr(ur, u~r) consists of the
profits of airports located in region r, the profits of the airline if the latter is based on region r and the
consumer surplus, namely the aggregate utility of passengers traveling to and from region r. These
terms are made precise later on. Note that the total welfare function depends on the airport ownership
decision taken by the authorities of region r as well as the ownership decisions taken by the other
region.
In the second stage and after the ownership status is decided, airports in each region set airport
charges. The decision makers are now the airports and the airport charges form the decision variables.
In contrast to stage 1, the game is no longer finite with the common action set being the set of positive
real numbers. The payoff of each privately owned airport is taken here to be the revenues collected
from all flights served. In contrast, the payoff of a publicly owned airport is given by the total welfare
of the region it serves. In particular, if the two airports in the multi airport region are publicly owned,
they share the same payoff function and thus lead to a so called team coordination situation.
During the last stage, the air carrier becomes the single active player. It determines airport fees on the
two routes so that total revenues are maximized (costs are again ignored for simplicity). The total
Airport
0
Airport
1
Airport
2
Single airport
region
Multi airport
region
Q1
Q2

5. 5
revenues of the air carrier do not explicitly depend on the ownership structure of the airports but they
directly depend on the airport fees which constitute a share of their fixed cost structure.
3. Game solution and Nash equilibrium
In this section the solution of the three stage game is obtained. Using a backward approach, the
behavior of airlines is first analyzed and their optimal choices are determined. Then the best responses
of airports are found and the Nash equilibrium of airport charges is computed. Finally the best
responses of regions are considered and the ownership profile leading to Nash equilibrium is specified.
3.1 Air travel demand
Air travel demand is expressed in the form of round trips between airports 0 and 1 and 0 and 2. Recall
that no air traffic is observed between airports 1 and 2. Let Q1 denote passenger demand for round trips
between airports 0 and 1. Similarly Q2 denotes demand between airports 0 and 2. We consider the
following linear demand model:
(1.1)
(1.2)
where denotes the full ticket price paid by a passenger on a round trip between airport 0 and i, i=1,
2. Thus demand falls with price P1 and rises with price P2. The positive coefficients bij designate the
slope of the demand curve and reflect how passenger demand responds to price changes (such as
regional population, income, or the price of substitutes such as rail). The positive offset parameters ai,
reflect all factors that affect demand other than price. The above equations are written in matrix form
as
(2)
where
(3.1)
(3.2)
(4)
and T denotes matrix transpose. The full ticket price comprises the fees charged by airports per
passenger and the carrier fee Pi
c
:
(5)
The above is written in matrix form as
(6)
where

6. 6
(7)
(8.1)
(8.2)
3.2 Optimal carrier fees and revenues
The carrier seeks to determine the price vector Pc
to maximize its total revenues
(9)
We substitute (3) into (1) to obtain
(10)
Thus the revenue function becomes
(11)
This is a quadratic function and its maximum value (if it exists) obtains when the fee satisfies
(12)
The above linear system of equations has a unique solution provided the determinant of the associated
matrix B+BT
is nonzero. We strengthen this condition and require that the determinant is positive:
(13)
Under this assumption the matrix B+BT
is positive definite and the unique solution of (12) maximizes
carrier revenues. If we substitute (12) into (6) we find the equilibrium price and demand. Indeed, let
(14)
Then
(15)
(16)
We represent the matrix B by its columns:
(17)
so that the transpose becomes

7. 7
(18)
Then the two passenger volumes are
(19.1)
(19.2)
Finally the optimized carrier revenues are
(20)
3.3 Airport charges and Nash equilibrium
At stage two, airports set airport charges to maximize payoffs. Airport payoffs depend on ownership
status; they comprise total welfare or revenues only. The optimum charge of each airport and the
corresponding best response depend on the remaining airport charges. A Nash equilibrium at stage 2 is
a vector of airport charges that jointly optimizes all airport payoffs. The airport payoff is quadratic in
its charge variable and linear with respect to the charges of the remaining airports. Consequently, Nash
equilibria satisfy a linear system of equations. Under general conditions, such pure Nash equilibria
charges exist and are uniquely determined. We further explore and exemplify these intuitive arguments
below.
Consider the revenues of airport 1. These depend on the airport’s own charges F1 as well as on the
charges of the remaining airports. We make this dependence explicit and thus prepare the ground for
the natural emergence of the notion of Nash equilibrium. More precisely, the revenues of airport 1 are
given as follows:
(21.1)
or
(21.2)
A similar expression holds for airport 2:
(22)
Airport 0 collects revenues from air traffic spanned by both airports of region 1:

8. 8
(23)
The best response of each airport results when charges are set to maximize payoffs. Payoffs in turn,
depend on ownership status. Private airports maximize revenues, public airports maximize welfare.
Total welfare consists of the airport revenues, the consumer surplus of passengers it serves and the
revenues of the carrier if the latter is based in the same country. Let da denote the ownership status of
airport a: da=1 if airport is private and 0 if it is state owned. The payoff of airport a is thus given by
(24.1)
or
(24.2)
The consumer surplus reflects the gains offered to passengers by paying a ticket price that is lower
than the highest price they would be willing to pay. Under a linear demand model, passengers traveling
between airports 0 and 1 perceive the highest price as the value obtained by setting Q1=0 in eq. (1) and
solving for P1:
(25)
Then the corresponding surplus for passengers traveling between 0 and 1 is
(26.1)
Substituting (25) into (26.1) gives
(26.2)
Similarly, passengers traveling from 0 to 2 gain
(27)
The passenger surplus for airport 0 is
(28)
Based on the above concepts, it is shown in the Appendix that the airport charges of the three airports
at Nash equilibrium satisfy the following linear system of equations
(29)
where the 3 matrix and the 3 vector are given below.

9. 9
(30)
denotes the element of the matrix ,
(31)
and
(32)
The determinant of N is nonzero almost everywhere. Therefore the Nash equilibrium is unique for
almost all values of the elements of B. The proof is given in Appendix A.
3.3 Demand equilibrium and determination of airport charges
In this subsection we prove that the equilibrium quantities demanded vector satisfy a linear system of
equations which almost surely has a unique solution. A testable condition for invertibility is presented.
Finally we demonstrate that the airport charges can be uniquely determined by equilibrium demand.
We consider the demand vector Q evaluated at the Nash equilibrium airport charges discussed above.
Then Q satisfies the following linear system of equations.
(33)
where
and
(34)

10. 10
The determinant of R is nonzero for almost all values of parameters.
The airport charges are determined from equilibrium demand via the following expressions.
(35)
(36)
(37)
Note that the airport charges in the multi airport system are proportional to passenger demands over
the routes served. The charge of the external airport is a linear combination of the two demand values
at equilibrium. The above claims are established in Appendix B.
3.4 Regional welfare
At stage 1 the two regional authorities decide on airport ownership. The decision of the regional
authority 0 is represented by the binary variable while that of region 1 by the binary variables
and . Both authorities base their decisions on welfare maximization. The welfare of region 0 is the
sum of the airport revenues, the surplus of passengers traveling from and to region 0 and the airline
revenues if the latter is based in region 0. Suppose that the airline is based on region 0. We find it
convenient to express welfare in terms of the demand vector. Based on the above findings we have
(38)
The notation emphasizes that the decision variables of region 1 are . The best response of region
1 for a given action of the external airport results when takes its maximum value
with respect to the four ownership options obtained by the combinations of Each of the
contours of in the demand space forms an ellipse. The welfare of region 0 is a full
quadratic given by
+ (39)
where the expression for airline revenues is obtained when equations (14)-(20) are taken into account.
In the sequel we focus on the welfare of the multi airport system solely.

11. 11
4. Results and simulations in the case of symmetric demand slopes
In this section we illustrate the results of section 3 by carrying out detailed calculations and
simulations in the case of a symmetric demand slope matrix. We calculate equilibrium prices,
passenger volumes as well as respective regional welfare. The smaller number of associated
parameters enables us to gain better insight and enhanced visualization, making interpretation of the
results easier.
The case of symmetric demand slopes results from the general linear demand model (1) when
and . Then the matrix B is symmetric and Negative definiteness of amounts
to
Furthermore
, L=1/4,
γ expresses the ratio of rates of change in demand caused by incremental variations of price on the two
routes. The parameters of the linear system determining the demand vector is given by
(40)
where
(41)
The airport charges are
(42)
(43)
(44)
Next we determine the airline fees on the two routes. Using the basic relations (1) and (2) we find

12. 12
(45)
Therefore
(46)
Thus, the demand vector becomes
(47)
and The demand equation (1) leads to the following expression for the full passenger
ticket prices on the two routes
(48)
Next we consider the welfare of the multi airport region 1. We assume for simplicity that the
airline is not based on this region. According to eq. (38) the welfare of region 1 is given by
(49)
In this way, all parameters of interest are expressed as linear functions of Q.
Next we examine welfare maximization of the multi airport system under the condition that the
external airport is state owned. In particular, we characterize the set of parameters than ensure
that state ownership is optimal for the multi airport region. Consider eq.(49) for Notice that
affects the welfare scale but not welfare comparisons. So in the search for maximum welfare
performed below it is disregarded. We substitute the demand vector given by (40) in eq. (49) to obtain
(50)
where
(51)
We look for conditions on the parameters that will secure that the welfare of a joint public ownership
will achieve the biggest welfare benefits. Let us recall the following property from matrix
analysis: if matrices A and B are positive definite, then if and only if
. Therefore, joint state ownership of the multi airports achieves maximum welfare if the
following inequalities hold:
(52)
(53)

13. 13
(54)
The following result is established in Appendix C.
Suppose that the parameters are restricted in the cone
(55)
Then for all values of the ratio γ in the admissible range [0, 1] the welfare of the multi
airport region is maximized when both its airports 1 and 2 are publicly owned.
Parametric conditions ensuring that one of the remaining three ownership schemes lead to maximum
welfare can be studied along similar lines. A notable difference is that welfare optimality is no longer
valid across all values of the parameter γ; different ownership structures may emerge as optimal
choices as the demand slope ratio varies over [0, 1]. This point is further illustrated in the simulations
below. We finally note that the above analysis can be replicated in the case where the external airport
is privately owned.
4.1 Passenger volumes at equilibrium
Next we interpret the above results using symbolic calculations performed in the MATLAB
environment and the symbolic toolbox. We describe the price elasticity matrix B in terms of the
equivalent parameters . The remaining demand parameters are set to fixed
values (10 and 15 respectively) that satisfy the cone condition (55). They, among other things, reflect
capacity constraints on the airports. The parameter γ ranges in the interval [0 1]. The parameter b is
positive.
Figure 2 illustrates passenger volumes on routes 1 and 2 at equilibrium conditions under the
assumption that the external airport 0 is state owned. In a similar fashion, Figure 3 depicts passenger
flows when airport 0 is privately owned. As Figure 2 demonstrates, demand is highest when both
airports in the multi airport system are state owned under public ownership of the external airport. This
behavior is consistently observed for all values of the parameter γ.

14. 14
Figure 2. Demand on routes 1 and 2 under all possible ownership scenario for the multi airport
system. Airport 0 is state owned.
A different pattern emerges in Figure 3 where the external airport is private. For values of γ less than a
certain threshold (γ<1/3), the passengers volume on route 1 is highest when airport 1 is public and
airport 2 is state owned. When γ exceeds the above threshold, passenger demand is maximized when
airports 1 and 2 are state owned.
Figure 3. Demand on routes 1 and 2 under all possible ownership scenario for the multi airport
system. Airport 0 is privately owned.
Passenger volumes are higher on route 2 than route 1. As γ stays smaller than 1/3, passenger volume is
highest when airport 1 is private and airport 2 is public. Once γ>1/3, state ownership of the multi
airport system yields the highest passenger flows on route 2.

15. 15
4.2 Ticket prices at equilibrium
Next we analyze air ticket price over the two routes for various combinations of ownership status and
parameter values. Route 1 is highlighted in Figure 4 under private and public ownership of airport 0. If
the external airport is private and γ lies below a threshold equal to 0.15, the lowest prices are offered
when airport 1 is public and airport 2 is private. Above that threshold, state ownership of both airports
yields the lowest ticket price.
Figure 4. Ticket price on route 1 under all possible ownership scenario for the multi airport system
and both private and public ownership of the external airport.
The plots of ticket price on route 1 when the external airport is state owned are presented in Figure
4(ii). For all values of γ, public ownership of both airports in the multi airport region offers the lowest
prices on route 1.
Analogous conclusions are drawn for route 2.
4.3 Welfare
The four possible ownership structures over the multi airport region 1 under public ownership of the
external airport generate the welfare functions plotted in Figure 7(i) as functions of the parameter γ.
The best choice for the multi airport system in terms of welfare is state ownership consistently for all
values of the parameter γ. This validates the theoretical result described earlier as the chosen
parameters satisfy the sector constraints. The uniformity in terms of γ ceases to exist when the external
airport is private. We observe that for values of γ less than a threshold (approximately 0.41), private
ownership of both airports offers the best welfare for the region. A γ moves to values higher than the
threshold, a public private scheme offers most benefits to the region.

16. 16
Figure 5. Welfare of the multi airport region under all possible ownership scenario of its airports. (i)
External airport is private, (ii) External airport is state owned.
5.Conclusion
In this paper decisions regarding airport ownership, airport charges and airline fees for multi airport
systems are studied in the context of a three stage game and a multivariable linear demand model. Best
responses and Nash equilibrium decisions are expressed in closed form in terms of the demand
parameters. These expressions are particularly amenable to analysis when they are viewed in terms of
the demand components. The global behavior of important pertinent variables over an admissible
range of parameter values is analyzed using symbolic calculations and simulation. Passenger volumes
ticket prices and regional welfare are illustrated at equilibrium for representative parameter values in
the important case of symmetric demand slopes. Under public ownership of the external airport,
passenger flow peaks when both airports in the multi airport system are state owned. This behavior is
consistently observed for all values of the rate of change of demand ratio which turns out to be the
only necessary elasticity indicator. If the external airport is private, the above uniformity no longer
exists: for values of γ below a certain threshold, mixed private public generate higher passenger
volumes. Similarly, if the external airport is private and γ lies below a threshold, the lowest air ticket
prices are offered under mixed private public schemes. Above that threshold, state ownership of both
airports yields the lowest ticket price.
The results also indicate that in the case of symmetric demand slopes, the welfare of the multi airport
region is maximized under public ownership of its airports provided the external connecting airport is
also publicly owned and the demand parameters and are close and properly confined. This
behavior is consistently observed across price coefficients. If the external airport is private, mixed
public private schemes emerge as welfare maximizers, depending on the magnitude of the cross price
elasticity of demand. The proposed approach is applicable to a variety of settings and can be used to
study the implications of airport privatization schemes. Empirical analysis that will test the theoretical
findings of this paper using real data will be carried out in future work.

17. 17
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18. 18
Appendix A: Calculation of airport charges
In this appendix we provide detailed calculations regarding the airport charges. First we establish that
the Nash equilibrium in airport charges is given by the solution of the linear system (). To this end we
first establish the following expression.
(A1)
Indeed,
Now
(A2)
The formula for L given in (31) is straightforward.
Next we consider the first order condition for revenue maximization of each of the three airports. We
emphasize the dependence on the maximizing charge, ignoring the other charge variables. The
payoff of airport 1 is written as
The above payoff is maximized if the charge is chosen to satisfy the first order condition
Thus
The term in the brackets equals Therefore the first order condition for airport 1 becomes
Equation (1) implies

19. 19
(A3)
Therefore the first order condition becomes
(A4)
where
(A5)
Recall that (see eq. (19.1)
If we substitute the above into (4) and collect terms we obtain
This is the first equation in the linear system for the airport charges.
The first order condition for airport 2 is similar. The partial derivative of the payoff with respect to
gives
(A6)
Thus
(A7)
The second equation in the linear system for the airport charges is established. We finally consider the
first order condition for airport 0. The corresponding payoff is
The partial derivative with respect to gives
Thus
or

20. 20
Next we substitute from (5) and (6) to obtain
The latter expression leads to the third equation in the linear system ( ).
Appendix B: Calculation of demand equilibrium
In this appendix the linear system of equations yielding the demand equilibrium is derived.
Recall expressions (19) () repeated here for convenience
and
We replace ( ) and ( ) into the latter expression to obtain
We substitute the expressions for the airport charges in the first demand equation to obtain
Hence

21. 21
A similar equation holds when demand is considered. Thus
The combination of these two equations leads to the desired linear system of equations given in the
main body of the paper.
Appendix C: Symmetric demand slope and welfare maximization
In this Appendix we prove that if the external airport is state owned and the parameters satisfy
eq. (55), joint state ownership maximizes the welfare of the multi airport region. Consider inequality
(52). After a series of calculations it takes the following equivalent form
(C1)
Under the cone constraint (55), and so that the above is equivalent to
(C2)
The latter is also written as
(C2)
The latter holds for all values of γ in the interval [0, 1].
Next we establish (53). Consider the order reversing matrix
The following identities are established by inspection
and JR(0,0)J=R(0,0). Therefore inequality (53) takes the equivalent form given by (C2) but with
replaced by . The second part of the cone constraint (55), proves the claim.
Finally inequality (54) is similarly established.

22. 22