GraSPP-DP-E-11-001 and ITPU-DP-E-11-001

An Analysis on Airport-Airline Vertical
Relationships with Risk Sharing Contracts underAsymmetric Information Structures
Katsuya Hihara
June 2011

GraSPP Discussion Paper E-11-001

ITPU Discussion Paper E-11-001

GraSPP-DP-E-11-001 and ITPU-DP-E-11-001

An Analysis on Airport-Airline Vertical
Relationships with Risk Sharing Contracts underAsymmetric Information Structures
Katsuya Hihara
June 2011

Professor, Graduate School of Public Policy
The University of Tokyo
7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan
Phone: +81-3-5841-1710
Fax: +81-3-5841-7877
hihara@pp.u-tokyo.ac.jp

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the author.

An Analysis on Airport-Airline Vertical
Relationships with Risk Sharing Contracts underAsymmetric Information Structures
Katsuya Hihara ∗2011/06/10 1830 JST

Abstract
We analyze the double moral hazard problem at the joint venture typeairport-airline vertical relationship, where two parties both contribute efforts
to the joint venture but neither of them can see the other’s efforts. With
the continuous-time stochastic dynamic programming model, we show thatby the de-centralized utility maximizations of two parties under very strict
conditions, i.e., optimal efforts’ cost being negligible and their risk averse
parameters both asymptotically approaching to zero, the vertical contract,which is linear function of the final state with the slope being the productmade by their productivity difference and uncertainty (diffusion rate) levelindex, could be agreed as the optimal sharing rule.
If both parties productivities are same, or the diffusion rate of the underlying process is unity, optimal linear sharing rule do not depend on the final
state. If their conditions not dependent on final state are symmetric as well,then risk sharing disappears completely. In numerical examples, we illustrate
the complex impact of uncertainty increase and end-of-period load factor improvement on the optimal sharing rule, and the relatively simple impact on
total utilities level.

Key words
Airport-Airline Vertical Relationship, Load Factor Guarantee Mechanism, Double
Hidden Action/Moral Hazard, Risk Sharing, Stochastic Dynamic Programming
JEL Classification Numbers:
C61,C73,D82,D86,L14,L93∗ Corresponding

author; Tel:+81-3-5841-1710, Fax:+81-3-5841-7877, E-mail address:
hihara@pp.u-tokyo.ac.jp, GraSPP, University of Tokyo, 7-3-1 Hongo Bunkyo-ku Tokyo, Japan
Preliminary Draft. Do not quote without written permission of the author.

11 Introduction
Faced with revenue and profit level fluctuations, some airports (local governments)
and airlines serving the airports are forming vertical contractual relationship toshare their risk and stabilize their financial conditions so that air transport services
by those airlines to/from the airports could be created or kept.
Hihara (2008) and Hihara (2010) reports that Noto Airport, one of rural airports inJapan, agreed on contract with one airline group to share the demand fluctuation
risk so as to derive the commitment of airline’s service to the airport. They usedload factor as a key indicator. So the mechanism of the contract is called load factor
guarantee mechanism. Also Hihara (2010) analyzed by using incomplete contract
framework to show that when such vertical risk sharing contract satisfy properconditions, such contract can overcome the under-effort problem and improve theutilities levels of both parties.

2 Past Literature
Risk sharing study dates back to Borch (1962). He showed that under pure risk
sharing situation without any efforts, the optimal sharing rule stipulates that theratio of marginal utility of two participating parties in different state are constantand the results hold for any probability distribution.
The risk sharing is also the subject of moral hazard analysis, since the principal
is paying to the agent based on the realization of uncertain state. The essence ofrisk sharing is to share the outcome depend on the realization of contingencies bypaying from one party to the other(s) based on some rule. The moral hazard case,the effort of the agent is involved and the principal is optimizing his utilities bycontrolling payment and also agent’s efforts indirectly through the payment structure.
There are two approaches to model the uncertainty in the moral hazard situation.
First is that random variable with some probability distribution is assigned to thestate itself. This approach dates back to Mirrlees( for example, Mirrlees (1979))
and Rogerson (1985) prove that under this settings, the first order approach is permissible if and only if monotone likelihood ratio condition (MLRC) and the convexity of distribution function condition (CDFC) are satisfied for the probability
function with the condition of the agent’s action.

2

But remember the first approach is not a dynamic approach. The random variable
is assigned to state contingency and the maximization is about all the possible outcomes at one time only.
The second approach is that the difference of the state is modeled by the stochastic
differential equation with the usual Wiener process. This is pioneered by Holmstrom and Milgrom (1987). They showed that if the agent has great action space,the optimal reward contract is simple linear function of the final state (But they did
not prove the sufficient condition).
Then Schaettler and Sung (1993) proved the general cases with necessary and sufficient conditions in the second approach case. Also Schaettler and Sung (1997)
studied the connection between discrete models and continuous model in the second approach case.
Mueller (1996) and Mueller (1998), based on these studies, showed that if the principal can see the agent action, in the first best situation according to his words, thelinear optimal contract is structured by the risk aversion parameters of both principal and agent.
These literatures are about the traditional moral hazard situation where one principal pay to one agent for his efforts that the principal cannot observe. The efforts
are performed only by agent.
If both principal and agent are making efforts, which cannot be observed fromthe other party, the situation is so-called double moral hazard problem. In thefirst approach case, some authors are analyzing the double moral hazard problems.
The recent examples are Bhattacharyya and Lafontaine (1995) and Kim and Wang
(1998a). To our knowledge, there are no literatures yet to model the double moral
hazard situation between airport and airline using the stochastic differential equation.
In the air transport studies, there are a number of studies on vertical relationshipbetween airport and airline. Recent examples are Oum and Fu (2009), Barbot(2009), Feng et al. (2010),Zhang et al. (2010). The airport-airline vertical relationship could contain some double moral hazard problems, since one cannot directly
observe the other s efforts in the relationship.
To our knowledge, there are no literatures yet to model the double moral hazard
3

situation between airport and airline, in the air transport economics field, using thestochastic differential equation.
In this study, we use the second approach (stochastic differential equation with
Weiner process) to analyze the double hidden action/moral hazard problems in
airport-airline vertical relationships.

3 Double Hidden Action/ Moral Hazard Situation
As stated in Hihara (2010), airport(=AP) and airline(=AL) are engaging in a jointventure type project, in which both AP and AL are independently making efforts
to provide air transport service to the passengers using the air route to/from the airport. We believe air transport service at one airport cannot be provided by airline
or airport alone. Only the combination of airline side and airport side can provide
the service to passengers / cargo service users.
In this sense, airport’s service and airline’s service are closely connected with eachother and they are not just a buddle of two different services. For example, if airport make more effort to improve the quality of service, this affects the quality ofairline and enhance the contribution of airline’s service to the joint-venture typeproject.
Usually in moral hazard problem, only one party’s efforts cannot be seen from theother party. But in the case of airport-airline relationship, neither of the parties’
efforts can be seen from the other parties. Hence the double moral hazard situation.
Here we try to use the stochastic differential equation approach to model the double
moral hazard situation. Namely neither airport nor airline can see the efforts of theother side. The contingent variable is according to the usual stochastic differential
equation as in the single moral hazard studies.

4 The Model
The model is based on the single moral hazard situation of Schaettler and Sung
(1993, 1997) with modification to the AP-AL joint-venture type project situation.
As a usual situation, two parties (in our case, airport and airline, in simple moral
4

hazard case, principal and agent) agree at time 0 on a certain contract characterized by a salary S among them which is payable at time 1 and satisfy the parties
reservation utility constraint. The salary or sharing rule S = S(X) is random via dependence on the outcome of a stochastic process X, the common observable among
the parties.
Here we use the outcome or situation variable X(t) for load factor during the contracting period. The effort of airport is uP and that of airline is uL .

4.1 Settings
For notation and settings we basically follow those of Schaettler and Sung (1993)
with modification to airport and airline relationship and important simplification
along with Mueller (1996).
As usual, at time 0, airport and airline agree on a sharing rule: S : C → R which
specifies a payment between them at time 1. The sharing rule may depend on astochastic load factor process X defined on the interval [0, 1], which is publicly
observable.

4.2 Double Hidden Action/Moral Hazard Model for De-centralized
Decision Making Environment
Here we construct a de-centralized utility maximization model. That is airport andairline are performing independent utility maximizations subject to the other’s being maximizing its own utilities while sharing the same stochastic control processby joint control production function. In this case, the double moral hazard problem
becomes as follows1 ;1 In other model of past literature, principal maximizes the total of both principal’s utility andagent’s. Then principal maximizes the total utility by choosing sharing rule, agent’s effort and principal’s effort. The notion that principal may not observe its own effort might to some extent lackreality. In our view, the assumption that the principal cannot see the agent’s effort but can see its own
effort might have more reality. See the problem (3.3) of Kim and Wang (1998a), for example.

5UP = − exp{−R{PX(1) + FP − S(X) −UL = − exp{−r{LX(1) + FL + S(X) −dX(t) = f (uP , uL )dt + σ dB

∫ 1
0cP (uP )dt}}(1)cL (uL )dt}}

(2)

∫ 1
0

(3)

These are assumptions we make in our model. They are consistent with those in
Schaettler and Sung (1993) and Mueller (1996), with the revision to adjust double
moral hazard situation. We only stipulate the revised aspects to airport-airline vertical relationship contract.
t ∈ R[0, 1]: time during the period the contract between airport and airline coversΩ: space C ∈ R[0, 1] of continuous functions on interval [0, 1] with value in R
Wt : coordinated process on Ω, i.e., Wt (ω ) = ω (t) for ω ∈ Ω
Ft0 : the filtration generated by W until time t
P:Weiner measure on (Ω,Wt0 )If we define Ft to be the augmentation of Ft0 by all null sets of F10 , then the filtration Ft is continuous and the coordinated process (Wt , Ft ) is a Weiner process
on the probability space, (Ω, F1 , P).
X(t): the load factor of the air transport routes to/from airport at time t during thecontract period
UP : the utility of airport
UL : the utility of airline
R ∈ R+ : risk averse parameter of airportr ∈ R+ : risk averse parameter of airline
P: parameter to connect load factor at the year end to the revenues of airport
L: parameter to connect load factor at the year end to the revenues of airline
FP : the portion of revenues, if any, from the source that does not have any connection with load factor, such as fixed income from leasing contracts of terminalbuilding
FL : the portion of revenues, if any, from the source that does not have any connection with load factor, such as fixed income from advertisement fee of aircraft body
painting
S(X): the sharing rule as a payment, from airport to airline (+) and from airline to
airport (-), as a function of load factor, X
uP : the effort of airport (Efforts are represented by the class U of all Ft -predictableprocess uP in some control set U ∈ R. )uL : the effort of airline (Efforts are represented by the class U of all Ft -predictable6

process uL in some control set U ∈ R. )cP : the cost of effort of airport. It is assumed to be convex function (c0P > 0, c00P ≥ 0)
and not directly dependent on time t or load factor X(t).
cL : the cost of effort of airline. It is assumed to be convex function (c0L > 0, c00L ≥ 0)
and not directly dependent on time t or load factor X(t).
f (uP , uL ): the production function converting efforts of airport and airline to theoutcome of load factor. It is assumed to be concave ( f 0 > 0, f 00 ≤ 0) and not directly dependent on time t or load factor X(t).
σ : diffusion rate. Here we assume constant diffusion rate.
B: one-dimensional Brownian motion
W P (t): the certainty equivalence of airport at time t (defined in the appendix)
W L (t): the certainty equivalence of airline at time t (defined in the appendix)
Problem A
max E[− exp{−R{PX(1) + FP − S (X) −P

SP (X),uP ,uL∫ 1
0

cP (uP )dt}}]

(4)

s.t.
dX(t) = f (uP , uL )dt + σ dBuL ∈ arg max E[− exp{−r{LX(1) + FL + SP (X) −ûLmax E[− exp{−r{LX(1) + FL + SL (X) −SL (X),uP ,uL

∫ 1
0

(3)

∫ 1
0

cL (ûL )dt}}]cL (uL )dt}}]

(5)
(6)

s.t.
dX(t) = f (uP , uL )dt + σ dB
uP ∈ arg max E[− exp{−R{PX(1) + FP − SL (X) −
ûP

(3)

∫ 1
0

cP (ûP )dt}}](7)

Notice that there are two moral hazard problems in Problem A. One is that AP isprincipal and AL is agent. The other is the opposite. Also note that both AP and
AL make efforts and have own revenues independently. This is the main points ofour joint-venture type relationship between AP and AL. What makes our double
hidden action/moral hazard model unique is that AP and AL make its own maximizations independently, namely de-centralized maximizations, in contrast withthe centralized maximization such as in Kim and Wang (1998b)2 .
Assumption 1
2 In the usual hidden action/moral hazard problem in our settings, we would have the following

problem.

7

In f (uP , uL ), uP and uL are additively separable and they have no interaction witheach other. This implies as follows.

∂ 2 f (uP , uL )=0
∂ uP ∂ uL(8)Assumption 2
The production function f (uP , uL ) and cost functions, cP (uP ) cL (uL ) are in themax E[− exp{−R{PX(1) + FP − S(X)}}]

S(X),uLs.t.
dX(t) = f (uP , uL )dt + σ dBuL ∈ arg max E[− exp{−r{S(X) −

∫ 1

ûL

0

cL (ûL )dt}}]

Notice that only the agent, AL, makes efforts and only the principal has the revenue.
Also if we were in the centralized maximization setting of double moral hazard as in Kim and Wang
(1998a) and we moved from non-dynamic to dynamic setting, then we would have the following
problem.

max E[PX(1) + FP − S(X) −∫ 1

S(X),uP ,uL+ λ E[− exp{−r{S(X) −cP (ûP )dt]

0

∫ 1
0

cL (ûL )dt}}]s.t.
dX(t) = f (uP , uL )dt + σ dB
uP ∈ arg max E[PX(1) + FP − S(X) −ûPuL ∈ arg max E[− exp{−r{S(X) −
ûL

∫ 1

∫ 1
0

0

cP (ûP )dt]cL (ûL )dt}}]

Like our joint-venture type, both parties make independent efforts (uP , uL ) with cost functions. But
only AP has independent revenues(PX(1) + FP ) and AP is maximizing the aggregation of AP’s andAL’s expected utilities.
The structure that AP is maximizing its effort under the condition of its effort being satisfying IC(Incentive Constrain) in Kim and Wang (1998b) is somewhat unnatural to us. IC means that AP’seffort is assumed to be unobservable from AP while AL’s effort is assumed to be unobservable fromAP, hence the ’Double Moral Hazard’ in their model.
In our de-centralized model, AP and AL make independent maximizations for their own expectedutility, while what one party cannot observe is only the other party’s effort, not its own effort, as canbe seen in Problem A.

8following relationships. For every p ∈ R, the function
H(p, uP , uL ) ≡ p f (uP , uL ) + cP (uP ) + cL (uL )is convex in both uP and uL , and has stationary points in both uP and uL .
By Assumption 2, we are sure the first order approach (Theorem 4.1 in Schaettler
and Sung (1993)) is always valid for the moral hazard problem. By Theorem 4.2 alladmissible control uP and uL are implemented by the other party using the sharing
function S, which is derived by Theorem 4.1 for each control uP or uL .
This is because every admissible control is implementable under specific conditions as proven by these theorems in Schaettler and Sung (1993), and we do nothave to worry about the situation where we are maximizing over larger sets of control possibilities than actually implementable resulting in one party being unable
to implement admissible control, since that control is not optimal within the actualsets of controls.
Proposition 1
Under Assumption 1 and Assumption 2 above, the independent optimal sharing
rules in the joint-venture type relationship between AP and AL, both being risk
averse, under the de-centralized utility maximizations stated in Problem A, are linear of the final outcome. In this case, part of the drift terms and the diffusion term
are weighted by the difference of productivity ratios between the two parities.
Proof (sketch) of Proposition 1
Here we describe only the sketch of the proof. The complete proof is in appendix.
Under Assumption 1, function f (uP , uL ) can be treated completely separately foru p and uL in applying Theorem 3.1, 4.1, and 4.2 for necessary condition and Theorem 5.1 for sufficient conditions in Schaettler and Sung (1993), since there is nointeracting relationship between uP and uL , implied by the equation (8).
With Assumption 2, all admissible controls are implementable, by Theorem 4.2,
through the derived sharing rule from Theorem 4.1. So the first order approach ofusing the derived sharing rule from Theorem 4.1 is justifiable.
So, for example, in solving equation(5) we can apply these theorems only with respect to uL , while in solving equation (4), we can apply these theorems only withrespect to uP .
9

Under this assumption, we can solve these problems by Theorem 3.1 and Theorem 4.1 in Schaettler and Sung (1993). First we use these theorems by solving the
maximization problem of equations of (5) and (7) in the conditions part of Problem A. Then with the results, we use again these two theorems to solve the main
maximization problem of equations of (4) and (6).
Notice also that all admissible controls, including optimal control u∗L of the maximization problem of equations of (5) for example, are implementable with Theorem 4.2 under Assumption 2. In this case, with Theorem 5.1, the derived optimal
control u∗L and its entailing sharing rule SP (X), which are derived by Theorem 3.1
and Theorem 4.1 in maximizing equations of (5), are indeed the optimal control
and sharing rule from the other party’s maximization (4).
Then we can make optimization of (4) by choosing optimal control of u∗P and its
entailing optimal control SP∗ (X) by applying Theorem 3.1 nod 4.1 to this maximization again.
Solving the maximization of (6) subject to (3) and (7) have the same processes.
By doing algebra in all the derivations above, we get the following.
1
SP∗ (X) =PX(1) − LX(1) + {FP − FL − (W P (0) − W L (0)) − (P − L)X(0)}
2
∫ 1
∫ 1
R
−
cP (u∗P )dt −(P − L)2 σ 2 dt
2 0
0
−

∫ 1
0

(P − L)σ dB(9)1SL∗ (X) =PX(1) − LX(1) + {FP − FL − (W P (0) − W L (0)) − (P − L)X(0)}
2
∫ 1
∫ 1
r
+
cL (u∗L )dt +(L − P)2 σ 2 dt
2 0
0
∫ 1

+
0

(L − P)σ dB(10)

If the integrations in these equations are executed, we have the followings.

10

1SP∗ (X) =(P − L)(1 − σ )X(1) + {FP − FL − (W P (0) − W L (0)) + (P − L)X(0)}2R− cP (u∗P ) − (P − L)2 σ 2(11)2
1SL∗ (X) =(P − L)(1 − σ )X(1) + {FP − FL − (W P (0) − W L (0)) + (P − L)X(0)}
2
r
2 2
∗
+ cL (uL ) + (L − P) σ
(12)
2
In this way, we can clearly see that the derived optimal sharing rules are linear
function of the final state, which is the load factor at the end of the contract period,X(1).
In deriving these results, we have the following relationships.

c0 P (u∗P )

=P

(13)

c0 L (u∗L )

=L

(14)

∂ f (u∗P ,uL )∂ uP∂ f (uP ,u∗L )∂ uL

Observe that P can be considered the productivity ratios consisting of ratio of
marginal cost to marginal production. The same could be considered for L in theabove.
By seeing the results in the equations(9), (10), (13) and (14), part of the drift terms
and the diffusion term are weighted by the difference of productivity ratios between
the two parities.
These prove the Proposition 13 .
3 Notice that in our model AP and AL are both risk averse and the optimal sharing rules arelinear in the final outcome in dynamic problem setting. This contrasts with the other double hidden
action/moral hazard models studies. In Kim and Wang (1998a), for example, only agent is risk
averse and principal is risk neutral and their findings are that the optimal contracts do not approach
the linear contract as agent s risk aversion approach zero in non-dynamic, one time setting. In
Romano(1994) and Bhattachryya and Lafontaine(1995), both principal and agent are risk neutral
and they show that there always exists a linear contract, among other possible optimal contracts, that
implements a second-best outcome in non-dynamic, one time setting.

11

By putting these results back in the utility functions of (1) and (2) for each party
and making arrangements, then we get the following.

1
UP∗ = − exp{−R{LX(1) + {FP + FL + W P (0) − W L (0) + (P − L)X(0)}
2
∫
∫ 1
R 1
2 2
+
(P − L) σ dt + (P − L)σ dB}}
(15)
2 0
0
1
UL∗ = − exp{−r{PX(1) + {FP + FL + W L (0) − W P (0) + (L − P)X(0)}
2
∫ 1
∫ 1
r
(L − P)2 σ 2 dt + (L − P)σ dB}}
+
(16)
2 0
0
In this form, we can see the sharing rule more clearly. The actual effort cost compensation part has already disappeared, because of the nature of actual compensation for the actual expense.
We recall that PX(1) + FP and LX(1) + FL are independent revenue terms for airport and airline, the load-factor dependent part and the part independent of loadfactor respectively. So at the optimal controls, the utility functions consist of therevenue part at the end of the period dependent on the level of the other party,
namely LX(1) for UP for example.
12 {FP + FL } is an averaged part of revenue independent of load factor X(t) common
to both parties. W P (0) − W L (0) + (P − L)X(0) could be thought as adjustment

parts from the level of the other side’s, i.e., for UP the adjustment parts are thedifference between W P and W L measured from the level from W L , for example.
The diffusion part (the part related to dB) and the drift part (the part related to dt) are the risk sharing parts, since the former is the compensation error part for thisrisky reward with zero expectation (E[dB] = 0), and the latter is the risk premiumfor the risky project. The diffusion part is the compensation error in that the reward is based on the observable outcome information, in our model load factor,
X(t), rather than efforts, uP or uL .
For the last terms of equations of (15) and (16), the term in parentheses, (P − L)or (L − P), are constant. So the last term in equations of (15) and (16) are linear
12equation of final results, X(1).
The drift term and diffusion term are both weighted by (P − L), which are productivity ratio between the two parties by equations (13) and (14) in addition tovolatility(σ ) or variance(σ 2 ) of the uncertainty.
If we do the (stochastic) integrations in the equations (15) and (16), we get thefollowings.

UP∗ = − exp{−R{{L + (P − L)σ }X(1)1+ {FP + FL + W P (0) − W L (0) + (P − L)(1 − 2σ )X(0)}2R+ (P − L)2 σ 2 }}(17)
2UL∗ = − exp{−r{{P + (L − P)σ }X(1)1+ {FP + FL + W L (0) − W P (0) + (L − P)(1 − 2σ )X(0)}2r+ (L − P)2 σ 2 }}(18)
2
In this form too, we can confirm that the optimal sharing rules under the decentralized maximizations stated in Problem A are linear function of the final outcome.
Also from the equations (15) and (16) with equations (13) and (14), we can confirm
that the drift and diffusion terms are weighted by the difference of productivity ratios between the two parities.

4.3

The Agreeable Sharing Rule

The derived SP∗ and SL∗ above are not identical. So it is not possible for airport andairline to actually agree on a sharing rule with this difference remained. We needsome kind of framework to structure the agreement of two parties. So we introduce
the following Assumption 3.
Assumption 3
13

In so far as for every number ε ∈ R+ , there exist numbers, δ1 ∈ R+ , aP ∈ R,
δ2 ∈ R+ and aL ∈ R such that
|SL∗ (r) − SP∗ (R)| < ε and 0 < |R − aP | < δ1 and 0 < |r − aL | < δ2 ,(namely, |SL∗ (r) − SP∗ (R)| → 0 as R → aP and r → aL ),airport and airline can agree on a single sharing rule S∗ , which is defined as follows.
SP∗ (R) → S∗ as R → aP
SL∗ (r) → S∗ as r → aL
We call this S∗ as an agreeable sharing rule.
Under Assumption 3, we have the framework for airport and airline to reach a single agreement out of the de-centralized utility maximizations in Problem A above.
That is, if their risk aversions approach to some fixed numbers asymptotically andthe difference of |SP∗ (R) − SL∗ (r)| approaches to zero with them, then the two parties can reach a risk sharing rule contract which is optimal to both parties under thede-centralized utility maximizations in Problem A.
Next is the assumption about the cost differences between airport and airline in theequations (11) and (12).
Assumption 4
We further assume that the optimal efforts of both airport and airline are costlessor negligibly minuscule. This means as follows.
cP (u∗P ) = cL (u∗L ) = 0 (cP (u∗P ) ≈ 0, cL (u∗L ) ≈ 0)

(19)

Assumption 4 is a stringent assumption. By Assumption 2 and assumptions on cP ,cL and f (uP , uL ), we have unique optimal effort levels, u∗P and u∗L . The costs ofboth parties’ optimal effort levels are negligible by Assumption 4. Here we are nottrying to state this assumption is plausible. But rather we try to clarify that in ourde-centralized maximization model, the cost terms are very important to satisfy
Assumption 3. As can be easily checked from the equations (11) and (12), onlywith Assumption 4, we have a possibility for the agreeable sharing rule definedin Assumption 3. In other words, unless Assumption 4 is satisfied, there is nochance for the two parties to agree on a single contract in our model even with the14agreement framework of Assumption 3. In other words, unless Assumption 4 issatisfied, there is no chance for the two parties to agree on a single contract in ourmodel.
Proposition 2
If both parties’ risk aversion parameters (R and r) approach asymptotically to zerounder Assumption 3 and Assumption 4, airport and airline can agree on a sharing
rule, which is derived as optimal sharing rules out of the de-centralized utility
maximizations in Problem A and is the linear function of the final state of loadfactor. In this case, the agreeable optimal risk sharing rule S∗ is as follows.
S∗ = (P − L)(1 − σ )X(1)1+ {FP − FL − (W P (0) − W L (0)) + (P − L)X(0)}2(20)

Proof of Proposition 2
When both parties’ risk aversion parameters asymptotically approach to zero, thismeans that R → 0 and r → 0 (aP ≡ 0 and aL ≡ 0 in Assumption 3)4 . With Assumption 4 and from equations (11) and (12), we can easily derive S∗ as in the equation(20).
With the result, |SL∗ (r) − SP∗ (R)| → 0. Hence with Assumption 3, airport and airline can agree on S∗ as an optimal risk sharing rule since SP∗ and SL∗ are optimal
for airport and airline respectively.
Q.E.D.
This result is consistent to a certain degree with Romano (1994) and Bhattacharyya
and Lafontaine (1995). Although in a different modeling framework for double
hidden action/moral hazard problem (i.e., non-dynamic setting), they all show that
there always exists a linear contract, which implements the second best (i.e., under
double moral hazard) effort levels when both principal and agent are risk neutral.
However, our result contrasts Kim and Wang (1998a). They show that in nondynamic setting, the optimal non-linear unique sharing rule under double moral
4 Notice that risk parameters (R and r) being approaching asymptotically zero is not necessarilyequivalent to AP and AL becoming risk neutral. Risk aversion parameters being zero (R = r = 0) in
CARA utility functions (U = −exp(−iy), i = {R, r}) means utility functions are all constant (U =−1), which is a special, more strict case of risk neutrality in general.

15

hazard with risk neutral principal and risk averse agent does not approach to thelinear contract as the agent s risk aversion approaches to zero.
In our case, Proposition 2 shows that in dynamic setting, two parties can agree
on an optimal linear contract under double moral hazard in our decentralized utility maximizations if both parties’ risk aversion parameters asymptotically become
zero and Assumption 4 is satisfied.5
The equation (20) shows that the slope of the linear contract is the product of thetwo parts. One is (P − L), which is the difference of productivity of both parties,
which can be seen from the equations (13) and (14)6 . The other is the uncertainty
level measured as (1 − σ ). The slope is the procuct of the two parts.
This means that if both parties productivities are equal, P − L = 0, or if the diffusion rate is unity, 1− σ = 0, then the slope is zero. So both parties are equal-footing
in productivity, then the compensation part dependent on the realized final outcomeof the optimal sharing rule disappears.
Also if the uncertainty in the project indicated by the diffusion rate of the Weiner
process in our model is unity, then the compensation error part related to the realized final outcome, −(P − L)σ X(1), of the optimal sharing rule just cancel out
the productivity difference compensation part elated to the realized final outcome,(P − L)X(1), resulting in the zero slope of the linear function of realized final outcome.
In both cases, the remaining optimal sharing rule is just the utility level adjustments
consisting only of parts not related to the realized final outcome X(1).
In other cases, the sign of the slope of the term related the realized final outcome
depends on the signs of the two pars, the productivity differences (P − L) and uncertainty level measured as (1− σ ). If (P−L) > 0 and (1− σ ) > 0, and, (P−L) < 0
and (1− σ ) < 0, the slope is plus. On the other hand, ff (P−L) > 0 and (1− σ ) < 0,and, (P − L) < 0 and (1 − σ ) > 0, the slope is minus.
5 On the other hand, Kim and Wang (1998a) showed that under double moral hazard situation, on

their model, the optimal sharing rule between risk neutral principal and risk averse agent is not linear
contract and that linear contract is not a limiting contract as agent’s risk aversion approaches to zero.
In our case, the optimal contracts are linear contract and as both parties’ risk aversions asymptoticallyapproach to zero, they can agree on a limiting optimal linear contract.
6 In non-dynamic double moral hazard setting for risk neutral principal and agent case, the slopeof the linear optimal sharing rule is the agent s productivity.

16

As can be seen from these explanations, the slope of the linear function of theagreeable optimal sharing rule is the product made by the productivity difference(P − L) and the uncertainty level index (1 − σ ).
Based on the contents of Proposition 1 and Proposition 2, we can construct Proposition 3.
Proposition 3
If we assume perfect symmetry, i.e., everything being same between the two contracting parties, in addition to Assumption 1 through 4, the risk sharing rule disappears completely in the double moral hazard situation in this setting.
Proof of Proposition 3
Under the perfect symmetry, everything between two parties is same. This meansthat among other things
P = L,
R = r,FP = FLW P (0) = W L (0).
With these put in the equation (20), we have the following resultsS∗ ≡ 0Hence the risk sharing rule disappears completely with the additional perfect symmetry assumption.
Q.E.D.

5 Numerical Illustration
5.1 Concrete Settings
Here we illustrate some of the structures we used in the preceding chapters. First
we show the concrete results for modeling X(t) for load factor during the contract
period.

17

We assume the following concrete structures of functions7 ;f (uP , uL ) = uP + uL
k P 2 P2
uP −
2
2kP
L2
kL
cL (uL ) = uL 2 −2
2kL
dX(t) = (uP + uL )dt + σ dBcP (uP ) =We further assumeX(0) = 0.6886,
σ = 0.056,
uP = uL = 0.005 or 0.025.
The 5 sample paths of X(t) for both uP = uL = 0.025 (left graph) and 0.005 (rightgraph) cases are shown in the Figure 1.
(Figure 1 is about here.)
Also we simulated the X(1) results with 10,000 iterations. The simulated results
are the Figure 2.
(Figure 2 is about here.)
The mean and the standard deviation of the simulated X(1) for uP = uL = 0.005 is;mean µ̂ = 0.6411
standard deviation σ̂ = 0.0266.
The mean and the standard deviation of the simulated X(1) for uP = uL = 0.025 is;mean µ̂ = 0.7013
standard deviation σ̂ = 0.0194.
7 The cost function could be negative in our settings like Schaettler and Sung (1993) and Kim
and Wang (1998a). This could be explained if both parties are receiving some financial aid fromnational government, for example. This aid could be lump sum or volume based for the purposeof cost alleviation. In our example, we assume lump some financial aid to reduce the two parties’
costs. This could be the case, if local government, which owns and leases land for both airline (officepremises) and airport (parking space) surrounding area of the airport, decreases its lease premium by
fixed amount, the costs of airline and airport decrease accordingly by the same lump sum amount.

18

We can see the improvement of final load factor at the end of the period with more
efforts of both parties even if the initial load factor is the same.

5.2 Optimal Levels
By the assumptions and these numbers with equations (13) and (14), we can derive
the optimal efforts levels and optimal sharing rules.

P
kP
L
u∗L =
kL

u∗P =

(21)
(22)

Notice that these solutions satisfy Assumption 4. Namely, we have( )k P P 2 P2
−
=0
2 kP
2kP
( )
kL L 2 L2∗−= 0.
cL (uL ) =2 kL
2kL
cP (u∗P ) =With the result of the optimal effort levels, we have optimal sharing rules. These
are the equations (11) and (12) with concrete parameter values.

1SP∗ (X) = (P − L)(1 − σ )X(1) + {FP − FL − (W P (0) − W L (0)) + (P − L)X(0)}2R− (P − L)2 σ 2(23)21SL∗ (X) = (P − L)(1 − σ )X(1) + {FP − FL − (W P (0) − W L (0)) + (P − L)X(0)}2r+ (L − P)2 σ 2(24)
2
If both parties’ risk aversions R and r asymptotically approach to 0, both canagree, based on Assumption 3, on the optimal sharing rule of the equation (20)19with concrete parameter values as follows.

S∗ = (P − L)(1 − σ )X(1)1+ {FP − FL − (W P (0) − W L (0)) + (P − L)X(0)}2(25)

In Figure 3, we show that |SL∗ − SP∗ | → 0 as R → 0 and r → 0 under the following
numerical settings.
P = 0.055; kP = 11u∗P = kPP = 0.005
L = 0.011; kL = 2.2u∗L = kLL = 0.005
X(1) ∼ N(0.6411, 0.02662 )W P (0) = 1.1; W L (0) = 1
FP = 11; FL = 10
X(0) = 0.668(Figure 3 is about here.)This means airport and airline can agree on an optimal sharing contract S∗ withconcrete paremater values as their risk aversion parameters asymptotically becomezero.

5.3 Diffusion Rate Effects
As the equation (20) and (25) indicate, the optimal agreeable contract , which islinear function of the final state, has the slope of the product of both parties productivity difference P − L and uncertainty (diffusion rate) level index 1 − σ .
The productivity difference part of the slope is the part to level the productivity difference dependent on the final state through the sharing rule. By this, the optimal
agreeable sharing rule adjusts the revenue differences between two parties generated by the productivity differences even if they share the common final outcome.
The uncertainty (diffusion rate) level index part of the slope is the part to adjust
20

the compensation error after adjusting the productivity differences. Compensation
error adjustment is necessary, since the sharing is based on the uncertain outcome
not on the efforts actually performed. In our model, the productivity difference
adjustment part has the same term P − L in the opposite sign. So the compensation
error part in the slope is the form of1 − σ , not just σ .
If the diffusion is zero, then no uncertainty entails. Only the productivity differences matter. So the slope is just the productivity difference adjustment, P − L. If
the diffusion is unity σ = 1 in our model, then the compensation error adjustment
is just the same as the countering productivity difference. So the slope is zero. In
this case, the optimal sharing rule has no part with the final outcome X(1) . Only
the fixed income part FP and FL , and the initial condition parts are involved in the
optimal sharing rule. Therefore the optimal agreeable sharing rule has only theattributes of the two parties that are not dependent on the final outcome X(1).
In Figure 4, we illustrate the optimal sharing contract S∗ linear in X(1), the loadfactor at the end of the contract period with the impact of the diffusion rate σ .
(Figure 4 is about here.)
As diffusion rate σ increases, the slope of optimal contract S∗ decreases. This
mean that as the volatility of Brownian motion increases, the compensation error
necessary part in the optimal sharing rule becomes large. If the compensation error
overwhelms the positive impact of load factor X(1) impact, then the slope of thelinear sharing rule becomes negative.
More specifically, if diffusion rate σ is unity, then the slope is zero. This meansthat the compensation error part just negate the productivity difference adjustmentin the slope, if σ = 1. In this case, the optimal sharing rule has only the parts that
do not depend on the load factor at the end X(1), as we explained above.
If diffusion rate σ is less than 1, the effect of load factor X(1) on the sharing ruleis greater than the effect of compensation error, which is the opposite sign of theeffect of load factor, resulting from the diffusion rate σ . So the slope of S∗ (X(1))
is positive. The contradicting effect between that with X(1) and that with σ can beseen from the equation (9) and (10). But on the other hand, if the diffusion rate σ
is greater than 1, the negative compensation error effect overwhelms the positiveeffect of load factor improvement, hence the negative slope of the optimal sharing
contract.

21

Under the agreeable sharing rule, we have the optimal utility functions.

UP∗ (X(1), σ ) = − exp{−R{{L + (P − L)σ }X(1)1+ {FP + FL + W P (0) − W L (0) + (P − L)(1 − 2σ )X(0)}2(26)
UL∗ (X(1), σ ) = − exp{−r{{P + (L − P)σ }X(1)1+ {FP + FL + W L (0) − W P (0) + (L − P)(1 − 2σ )X(0)}2(27)
In Figure 5, we show the visualized utility level with the final outcome of loadfactor X(1) and the diffusion rate σ , setting R = r = 10−4 .
(Figure 5 is about here.)
As we can see from the left graph of Figure 5, the impact of increased diffusion
rate on U P∗ is relatively simple. As diffusion rate increases, so does the impactP∗σ)of X(1), i.e., the slope ∂ U ∂ (X(1),. The slope is always positive. The intercept
X(1)P∗
U (0, σ ), on the other hand, decreases as σ increases.
In the right graph of Figure 5, the impact on U L∗ is relatively complex in ourL∗σ)numerical example. As σ increases, the impact of X(1), i.e., the slope ∂ U ∂ (X(1),,X(1)
is decreasing. If σ ≥ 1.25, the slope becomes negative and continue to decrease.
The intercept U L∗ (0, σ ), on the other hand, increases as σ increases.
Now we turn to the total utilities level for both AP and AL. With the above results,
we can gauge the impact of increased uncertainty and the end-of-period load factor
improvement on the total utilities level. To specify these impacts, we can calculate
the derivatives of the total utilities with respect to σ and X(1).

22UA∗ = UP∗ +UL∗
∂ UA∗
= UP∗ [−R{{L + (P − L)σ }}] +UL∗ [−r{{L + (P − L)σ }}] > 0∂ X(1)∂ UA∗= UP∗ (−R){(P − L)(X(1) − X(0)) +UL∗ (−r){(L − P)(X(1) − X(0))} ≈ 0∂σNotice that since R ≈ 0 and r ≈ 0, we have the last results.
Figure 6 is the graph for UA∗ = UP∗ + UL∗ with X(1) and σ . The slope of UA∗ withrespect to X(1) is positive and the slope with respect to σ is almost zero.
(Figure 6 is about here.)
The impact of uncertainty increase is almost zero with some perturbations, since
we already assuming that the risk averse parameters approach asymptotically tozero. The impact of the final outcome improvement, on the other hand, has a positive effedct on the total utilities level with some perturbations.
Because of risk averse parameters approaching to zero, the conflicting effects causedby diffusion rate changes on each party’s utility through the optimal agreeable sharing rule as depicted in Figure 5 are almost smoothed out in the total utilities level
as depicted in Figure 6. With some perturbations, σ changes have almost no effecton the total utilities level in Figure 6.

6 Concluding Remarks
We show that, under the double hidden action/moral hazard situation of de-centralized
utility maximizations based on the stochastic load factor process under additively
separable efforts assumption and convexity assumption about the relationship between production and cost function, the optimal sharing structures are the linear
function of the final state for both parties. In this case, the drift and diffusion terms
are weighted by the difference of productivity between the two parities.
If we further assume that the costs of optimal effort are negligible and also assume
both parties’ risk aversion parameters approach asymptotically to zero, then we23

show that both parties can agree on a single optimal contract, which is also a linear
function of final load factor.
Our finding is to some degree consistent with some of double hidden action/ moral
hazard situation analysis of preceding literatures. Namely, if both parties are risk
neutral, the linear function of the end-state is among the optimal sharing rules.
However, our result contrasts with Kim and Wang (1998a). They show, in nondynamic setting, the optimal non-linear unique sharing rule under double moral
hazard with risk neutral principal and risk averse agent does not approach to thelinear contract as the agent s risk aversion approaches to zero.
We find that in dynamic setting, two parties can agree on an optimal linear contract
under double moral hazard in our decentralized utility maximizations if both parties risk aversion parameters asymptotically become zero and optimal effort are
negligible, as we alread stated.
The optimal agreeable contract, which is linear function of the final state, has theslope of the product of both parties productivity difference and uncertainty (diffusion rate) level index.
If the productivities are same between the two parties in our setting, there is no needto adjust their productivity difference by the optimal sharing rule. So the slope iszero. In this case, the optimal sharing rule has no part with the final outcome.
If the diffusion rate is unity in our model, then the compensation error adjustment
is just the same as the countering productivity difference adjustment. So the slope
is zero. In this case, the optimal sharing rule has no part with the final outcome.
In addition, if everything is symmetric, then the risk sharing rule disappears completely.
With numerical examples, we show the sample paths for load factor processes generated according to underlying Weiner process. The improving effects of effort
increases of both parties on the end-of-period load factors are provided. Also weshow the convergence to the agreeable sharing rule as both parties’ risk averse parameters get asymptotically to zero.
We show that the complex effect of increased diffusion rate on the linear functionof the agreeable optimal sharing rule. The impacts on total utilities level are rel24

atively simple. The impacts of diffusion rate increase on total utilities are almost
zero. The increase of final outcome has a positive effect on the total utilities level.
As one example of the next steps, we can look into the other double hidden action /moral hazard model of utility maximizations, which is other than our de-centralized
model. This direction is in line with the past literature. However, in the stochastic
process model, this would require much more mathematical sophistication.
Also we can try to relax the restrictive assumptions on the functions, including f
and cP and cL . For example, we assume they are not function of time t. This would
also require much more rigorous theoretical treatments. Since even in our simple
model with restrictive functions, we can show some results, we should try to balance clarity and reality when pursuing in this direction.
In the context of network industries of routes and airports, another important next
step could be to try to link the abstract efforts in our model to concrete tools of theaviation and airport industries, such as pricing, routing and frequencies, so that we
can more directly see the interaction within the airport and airline vertical relationship in more realistic settings.

7 Acknowledgement
The author expresses gratitude to Professor Naoki Makimoto (University of Tsukuba)
for valuable comments. Any remaining errors are entirely the author s responsibility.
Financial supports from the Airport Environment Improvement Foundations, Kansai International Airport Co., Ltd., Narita International Airport Co., Ltd. and Japan
Airport Terminal Co., Ltd. are gracefully acknowledged.

A Appendix A Proof of Proposition 1
As already stated, under Assumption 1, function f (uP , uL ) can be treated completely separately for u p and uL in applying Theorem 3.1 and 4.1 for necessary
condition and Theorem 5.1 for sufficient conditions in Schaettler and Sung (1993),
since there is no interacting relationship between uP and uL , i.e., equation (8). So
25

for example, in solving equation(5) we can apply these theorems only to uL , while
in solving equation (4) we can apply these theorems only to uP .
We show the proof of the derivation of SP (X) only, since that of SL (X) is basicallysame. Also to avoid the just duplication, we only describe the main part of thederivation in the following equations.

max E[− exp{−R{PX(1) + FP − SP (X) −SP (X),uP ,uL∫ 1
0

cP (uP )dt}}]

(4)

s.t.
dX(t) = f (uP , uL )dt + σ dBuL ∈ arg max E[− exp{−r{LX(1) + FL + SP (X) −
ûL

(3)

∫ 1
0

cL (ûL )dt}}] (5)First, we apply Theorem 3.1 and 4.1 in Schaettler and Sung (1993) to the maximization problem of equation (5) under equation (3) holding uP constant. Noticethat in dX(t) we have also uL .
As usual, we assume the SP (X) has the following structure.
SP (X) = S1P (X) +∫ 10α (s, X)ds +∫ 10β (s, X)dXs∫∫

From equation (5), structure of SP (X) and X(1) = X(0) + 01 f (uP , uL )dt + 01 σ dB
with some arguments suppressed, the utility UL becomes as follows;
UL = − exp{−r(LX(0) + FL + S1P (X)
−
−

∫ 1
0

∫ 1
0

r(L f (uP , uL ) + α + β f (uP , uL ) − cL (uL ))dtr(L + β )σ dB}

By applying Theorem
3.1 in Schaettler and Sung (1993), there exist Ft -adapted∫1
process Vt and ∇Vt , 0 | ∇Vt σ |2 dt < ∞ a.e., such that u∗L ∈ U is optimal control
if and only if

26∇Vt { f (uP , u∗L ) − rσ 2 (L + β )}1
+ Vt {−r(L + β ) f (uP , u∗L ) − rα + rcL (u∗L ) + r2 σ 2 (L + β )2 }2≡ max {∇Vt { f (uP , uL ) − rσ 2 (L + β )}uL inU1+ Vt {−r(L + β ) f (uP , uL ) − rα + rcL (uL ) + r2 σ 2 (L + β )2 }2for almost all (t, ω ) ∈ [01] × Ω. Furthermore, the value process Vt has an Ito differential of the form
Vt = V0 −∫ t
0[∇Vs { f (uP , u∗L ) − rσ 2 (L + β )}+ Vt {−r(L + β ) f (uP , u∗L ) − rα + rcL (u∗L )
∫ t
1 2 2
2
+ r σ (L + β ) }]ds + ∇Vs dXs .
20

ˆ t are value functions of stochastic process as defined in Appendix
Vt , ∇Vt and ∇V
A in Schaettler and Sung (1993).
ˆ t ≡ ∇Vt − Vt r(L + β ), we can get the cL related part as follows;
Using ∇V
ˆ t f (uP , uL ) + Vt rcL (uL )∇V
FOC for maximization for this, we have the following;ˆ tc0 (u∗ )∇V= − ∂ fL(u ,uL ∗ )
P L
rVt
∂ uL

As usual, we define the following certainty equivalent value process Wt as follows;1
Wt = − ln(−Vt )rWith some arrangements, we get the following equation.
dWt = cL (u∗L )dt +ˆ t )2ˆ tr ( ∇V∇V
σ 2 dt +
σ dB
2 rVt
rVt
− (α dt + β dX) − LdX
27

So we get SP (X) as follows.
SP (X) = −(W1L − W0L ) + S1P (X) − L(X(1) − X(0))
)2
(
∫ 1
∫
r 1 c0L (u∗L )∗σ 2 dt+cL (uL )dt +
∗)
∂
f
(u
,u
P
L
2
0
0
∫ 1

+
0

∂ uL
0
cL (u∗L )σ dB∂ f (uP ,u∗L )∂ uLHere we use Wt L instead of Wt for notational clarity in the following further derivation of SP (X).
By the Theorem 5.1 in Schaettler and Sung (1993), this SP (X) satisfies the sufficient condition of maximizing the equation (4) with holding uP as a constant.
Rearranging the terms and defining newly F P (X), α 0 and β 0 , we get the following.
SP (X) = − (W1L − W0L ) + S1P (X) − L(X(1) − X(0))
(
)2
)
(
∫ 1
∫ 1
0 (u∗ )
0 (u∗ )ccr
L L
+ [cL (u∗L ) +σ 2 − ∂ fL(u ,uL ∗ ) f (uP , u∗L )]dt
P L
2 0 ∂ f (uP ,u∗L )0∂ uL0cL (u∗L )∗ dX
0 ∂ f (uP ,uL )
∂ uL
∫ 1
0

∫ 1

+
≡ F (X) +P∫ 1
0

α 0 dt +

0

∂ uL

β dX

(A.1)
(A.2)

F P (X) ≡ −(W1L − W0L ) + S1P (X) − L(X(1) − X(0))
Now we turn to the maximization of the equation (4) by choosing uP , which is now
a controlling variable rather than a constant. We put SP (X) in the equation (4),rearrange terms and have the following equation.

28

UP = − exp{−R(PX(0) + FP − F P (X))
∫ 1

+
0

R{(β 0 − P) f (uP , u∗L ) + cP (uP ) + α 0 }dt
∫ 1

+
0

R(β 0 − P)σ dB}

We apply Theorem 3.1 again here. Then there exist Vt P and ∇Vt P ,dt < ∞ a.e., such that u∗P ∈ U is optimal control if and only if∫1

0 | ∇Vt

P σ |2∇Vt P { f (u∗P , u∗L ) + Rσ 2 (β 0 − P)}1+ Vt P {R{(β 0 − P) f (u∗P , u∗L ) + α 0 + cP (u∗P )} + R2 σ 2 (β 0 − P)2 }2≡ max {∇Vt P { f (uP , u∗L ) − Rσ 2 (β 0 − P)}
uP inU1+ Vt P {R{(β 0 − P) f (uP , u∗L ) + α 0 + cP (uP )} + R2 σ 2 (β 0 − P)2 }2
for almost all (t, ω ) ∈ [0, 1] × Ω. Furthermore, the value process Vt P has an Ito
differential of the form
Vt P = V0P −∫ t
0[∇VsP { f (u∗P , u∗L ) + Rσ 2 (β 0 − P)}+ Vt P {R{(β 0 − P) f (u∗P , u∗L ) + α 0 + cP (u∗P )}∫ t
1+ R2 σ 2 (β 0 − P)2 }]ds + ∇VsP dXs ].
20

We can construct the following.
0∗ˆ P ≡ ∇V P + RV P (β 0 − P) − RV P c L (uL∗)∇Vtttt ∂ f (u ,u )P

∂ uLWith this, we get the uP related part as follows;ˆ P f (uP , u∗ ) + V P RcP (uP )∇Vt
Lt
FOC for maximization for this, we have the following;ˆ P∇Vc0 (u∗ )t= − ∂ fP(u∗ ,uP∗ )
P L
RVt
∂ uP

29

L

Here, we define the following certainty equivalent value process Wt P as follows;
1
Wt P = − ln(−Vt P )R
With some arrangements, we get the following equation.
ˆ P
ˆ P
R ∇V
∇V
dWt P = cP (u∗P )dt + { t + (β 0 − P)}2 σ 2 dt − { t P + (β 0 − P)}σ dB
2 RVt
RVt
0
− (α dt + β 0 dX) − PdX∫∫

By integrating and using SP (X) = F P (X) + 01 α 0 dt + 01 β 0 dX, we have as follows.
SP (X) =W1P − W0P + F P (X) + P(X(1) − X(0))
(
)2
∫ 1
∫ 1
0 (u∗ )
0 (u∗ )
c
c
R
L L
− P P
σ 2 dt−cP (u∗P )dt −2 0 ∂ f (u∗P ,u∗L ) ∂ f (u∗P ,u∗L )
0
∂ uL
∂ uP
)
(
∫ 1
c0L (u∗L )c0P (u∗P )
+
−
σ dB
∗
∗
∂ f (u ,u )∂ f (u∗ ,u∗ )
0

P

∂ uL

L

P

∂ uP

L

Notice that the terminal conditions are that LX(1) + FL + S1P (X) = W L (1) andPX(1)+FP −{W P (1)−W P (0)+F P (X)+P(X(1)−X(0))} = W P (1). With these
conditions and by expanding F P (X), we get the following.

1
SP (X) =PX(1) − LX(1) + {FP − FL − (W P (0) − W L (0)) − (P − L)X(0)}
2
(
)2
∫ 1
∫
R 1 c0 P (u∗P )c0 L (u∗L )∗−cP (uP )dt −−σ 2 dt2 0 ∂ f (u∗P ,u∗L ) ∂ f (u∗P ,u∗L )
0
∂u
∂ uL
(
)P
∫ 1
0
∗
0
∗
c L (u )c P (uP )(A.3)−− ∂ f (u∗ ,uL∗ ) σ dB
∂ f (u∗ ,u∗ )
0

P

∂ uP

L

P

∂ uL

L

By the same process, we get the following.

30

1
SL (X) =PX(1) − LX(1) + {FP − FL − (W P (0) − W L (0)) − (P − L)X(0)}
2
)2
(
∫ 1
∫
r 1 c0 L (u∗L )c0 P (u∗P )∗+cL (uL )dt +σ 2 dt∗ ,u∗ ) − ∂ f (u∗ ,u∗ )
∂
f
(u
P
L
P
L
2
0
0
∂ uL
∂ uP
(
)
∫ 1
0
∗
0
∗
c L (uL )c P (u )+− ∂ f (u∗ ,uP∗ ) σ dB
(A.4)
∂ f (u∗ ,u∗ )
0

P

∂ uL

L

P

∂ uP

L

Putting these back in the utility functions of UP and UL , the resulting utility functions at the optima are the followings.

1
UP∗ = − exp{−R{LX(1) + {FP + FL + W P (0) − W L (0) + (P − L)X(0)}
2
(
)2
∫ 1
c0 P (u∗P )R
c0 L (u∗L )+σ 2 dt−2 0 ∂ f (u∗P ,u∗L ) ∂ f (u∗P ,u∗L )
∂ uP
∂ uL
)
(
∫ 1
0
∗
0
c P (uP )c L (u∗L )+− ∂ f (u∗ ,u∗ ) σ dB}}
(A.5)
∂ f (u∗ ,u∗ )
0

P

L

∂ uP

P

L

∂ uL

1
UL∗ = − exp{−r{PX(1) + {FP + FL − (W P (0) − W L (0)) − (P − L)X(0)}
2
(
)2
∫ 1
c0 P (u∗P )r
c0 L (u∗L )+σ 2 dt−2 0 ∂ f (u∗P ,u∗L ) ∂ f (u∗P ,u∗L )
∂ uL
∂ uL
(
)
∫ 1
0
∗
0
c L (uL )c P (u∗P )+− ∂ f (u∗ ,u∗ ) σ dB}}
(A.6)
∂ f (u∗ ,u∗ )
0

P

∂ uL

L

P

∂ uP

L

Notice that, as we already mentioned, by Theorem 5.1 in Schaettler and Sung
(1993), the SP (X) and SL (X) above satisfy the sufficient condition of the (relaxed)
maximization problems of UP in the equation (4) and UL in the equation (6).
During the deriving process of u∗L , for example, the airport’s relaxed problem requires uL to satisfy the following equation of dynamic programming according to31

Theorem 5.1 in Schaettler and Sung (1993).

0≡∂ W P (t, X) 1 ∂ 2 W P (t, X) R ( ∂ W P (t, X) )2 2
σ
+
−
∂t
2
∂ X2
2
∂X
∂ W P (t, X)+ max[{ f (uP , uL ) + Rβ 0 σ 2 }dtuL∂X(
R + r 02 2 )− c p (uP ) + cL (uL ) +β σ dt]2(A.7)

From the maximization for u∗L and noticing β 0 = β 0 (t, X), we get the following.
c0 L (u∗ )∂ W P (t, X)= ∂ f (u ,uL∗ )
P L
∂X

(A.8)

∂ uP

The dynamic programming equation above is separable and it is well known how
to solve it. Noticing the terminal condition of W P (1) = LX(1) + 12 {FP + FL +W P (0) − W L (0) + (P − L)X(0)}, we get the following.
W P (t, X) =LX
R+ { L2 σ 2 − β 0 f (uP , u∗L )2
R − r 02 2−
β σ + cL (u∗L ) + cP (uP ))}(t − 1)
2
1
+ {FP + FL + W P (0) − W L (0) + (P − L)X(0)}
2

(A.9)

From this, we can derive the following.

∂ W P (t, X)
c0 L (u∗ )= ∂ f (u ,uL∗ ) = L
P L∂X(A.10)∂ uL

From this result along with the assumptions of convexity of cL and concavity of f
and Assumption 2, the optimal control u∗L is unique.
Also u∗L is constant across the period, namely u∗L (t) ≡ u∗L , since neither cL nor f
depends on time t by assumptions.
32

The same process can be performed for u∗P . Then we get the following.

∂ W L (t, X)
c0 P (u∗ )= ∂ f (u∗ ,uP ) = P
P L∂X(A.11)∂ uP

Therefore the optimal controls, u∗P and u∗L , are unique and constant during the period.
Putting these results back in (A.3) and (A.4), we have the following.

1
SP (X) =PX(1) − LX(1) + {FP − FL − (W P (0) − W L (0)) − (P − L)X(0)}
2
∫ 1
∫ 1
R
−
cP (u∗P )dt −(P − L)2 σ 2 dt
2 0
0
−

∫ 1
0

(P − L)σ dB(A.12)1
SL (X) =PX(1) − LX(1) + {FP − FL − (W P (0) − W L (0)) − (P − L)X(0)}
2
∫ 1
∫ 1
r
(L − P)2 σ 2 dt+
cL (u∗L )dt +
2 0
0
∫ 1

+
0

(L − P)σ dB(A.13)

If the integrations in these equations are executed, we have the followings.
1
SP∗ (X) =(P − L)(1 − σ )X(1) + {FP − FL − (W P (0) − W L (0)) + (P − L)X(0)}2R− cP (u∗P ) − (P − L)2 σ 2(A.14)
2

1
SL∗ (X) =(P − L)(1 − σ )X(1) + {FP − FL − (W P (0) − W L (0)) + (P − L)X(0)}
2
r
2 2
∗
+ cL (uL ) + (L − P) σ
(A.15)
2
33

These are exactly the equations (11) and (12) as indicated.
With these results, we have the UP∗ and UL∗ as follows.

1
UP∗ = − exp{−R{LX(1) + {FP + FL + W P (0) − W L (0) + (P − L)X(0)}
2
∫
∫ 1
R 1
2 2
+
(P − L) σ dt + (P − L)σ dB}}
2 0
0

(15)

1
UL∗ = − exp{−r{PX(1) + {FP + FL + W L (0) − W P (0) + (L − P)X(0)}
2
∫
∫ 1
r 1
2 2
(L − P) σ dt + (L − P)σ dB}}
+
2 0
0

(16)

By doing the integrations of these equations, we have the followings.

UP∗ = − exp{−R{{L + (P − L)σ }X(1)1+ {FP + FL + W P (0) − W L (0) + (P − L)(1 − 2σ )X(0)}2R+ (P − L)2 σ 2 }}(17)
2UL∗ = − exp{−r{{P + (L − P)σ }X(1)1+ {FP + FL + W L (0) − W P (0) + (L − P)(1 − 2σ )X(0)}2r+ (L − P)2 σ 2 }}(18)
2These are exactly the equations (17) and (18) as indicated.
Q.E.D.

34

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36

Figure 1

5 Sample Paths for Load Factor State Variable, X(t)Left graph for u_{P}=u_{L}=0.005, Right graph for u_{P}=u_{L}=0.025

Figure 2

Simulated Paths for Load Factor State Variable, X(t)Left graph for u_{P}=u_{L}=0.005, Right graph for u_{P}=u_{L}=0.025;
each 10
10,000000 iterations

Figure 3 Convergence into a Risk Sharing Contract
L*P
S ,S →Sx 10[|SL - SP | → 0, as R, r → 0]

-6

1.8
1.6
1.4

L

P

|S -S |

1.2
1
0.8
0.6
0.4
0.2
0
1.5

1.5
1

1
0.5
r

0.5
0

0

R

Optimal Contract S* Linear in X(1)withith Diff
Diffusioni R
Ratet ((σ))

Figure 4*

Optimal Linear Contract S (X(1), )
0.52
0.51
0.5
0.49

0.47

S

*

0.48

0.46
0 45
0.45
0.44
0.43
0.42
2
1.5
1
0.5
0


0

0.1

0.2

0.3

0.4
X(1)

0.5

0.6

0.7

0.8

0.9

1

Figure 5 U_P* and U_L* with X(1) and σFigure 6
6. Total Utilities with X(1) & σ