GraSPP-DP-E-10-001 and ITPU-DP-E-10-001

Analysis on Airport-Airline Relationship with
Risk Sharing Contract

Katsuya Hihara
November 2010

GraSPP Discussion Paper E-10-001

ITPU Discussion Paper E-10-001

GraSPP-DP-E-10-001 and ITPU-DP-E-10-001

Analysis on Airport-Airline Relationship with
Risk Sharing Contract

Katsuya Hihara
November 2010

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

GraSPP Discussion Papers can be downloaded without charge from:http://www.pp.u-tokyo.ac.jp/
Discussion Papers are a series of manuscripts in their draft form. They are not intended
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Discussion Papers may not be reproduced or distributed without the written consent of
the author.

Analysis on Airport-Airline Relationship with
Risk Sharing Contract
Katsuya Hihara ∗2010/11/20 2300 JST

Abstract
As an example of airport-airline vertical contractual relationships,’Load Factor Guarantee Mechanism’ contract, actually agreed at Noto
Airport in Japan, stipulates that the airline agrees to serve the airport
and the airport agrees to pay to (or receive from) the airline based on thediﬀerence between the target load factor set at the start of the period in
the contract and a realized one. Airports are often local governments andsuch payments are controversial since tax money is on the line.
From airport side, such contract ensures the commitment of airline
to serve the airport as well as the sharing of up-side proﬁt in high loadfactor case. From airline side, such contract serves not only as a risk mitigating tool to compensate down-side loss of revenue but as an incentive
device to extract airport eﬀorts to avoid large payment hence overcoming
its under-eﬀorts seen typically after start of the service at the airport.
To reﬂect these structures, we revised Hart and Moore hold up model toinclude more interdependent and joint-venture type relationship between
airport and airline. With the revised stochastic model, we show such contract can overcome under-eﬀort problem and restore utilities loss under
modiﬁed Hart and Moore ﬁrst best condition. So contractual payment
from public airport authority could be justiﬁed by such eﬃciency gains.
Finally we illustrate, by numerical examples, the under-eﬀorts, its recoveries and utilities loss restorations by contract with impacts of expectedstate of nature improvements, uncertainty increases and ”risk aversion”
enhancements1 .

key words
Airport-Airline Vertical Relationship, Load Factor Guarantee Mechanism,
Hold-Up Problem, Joint-Venture Type Trade, Eﬃcient Pure Risk Sharing
JEL Classiﬁcation Numbers:
C70,D81,D86,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
1 This paper is a draft material. Do not quote without a written consent of the author.

1

1

Introduction

Airports and airlines are facing revenue and proﬁt ﬂuctuations under increasingly volatile and unstable business environments. Some of them, especially in
local area with low demands or in secondary airports for start-up carriers suchas LCC entering into the area, are forming vertical contractual relationship toshare their risk and stabilize their ﬁnancial conditions so that air transport services by those airlines at the airports could be newly introduced or be kept tocarry on.
An airport and an airline at one airport are, by nature of the business, jointly
making a business project at the airport. Airport are providing airline with airport related service in exchange for landing fee and at the same time airline areproviding air transport service to the airport, with or without incentive moneyfrom the airport. The two services are not separable in a sense that each service
could not exist without the other2 .
Also they are in a strategic complementary relationship, where one side’s eﬀort
could improve not only its own but also the other side’s contribution to thevalue of the joint project. For example, airport’s eﬀort on improving airport
services increase the attractiveness of the airport and help airline’s independenteﬀort to bring more passengers, hence both ending up in enjoying more revenues. Therefore further eﬀorts arising from such contractual relationship in
addition to such interdependent relationship could have the potential to significantly enhance the values of the project both sides are participating in.
We try to model this type of interdependent vertical contractual relationshipbetween airport and airline from the stand-point of incomplete contract frameworks in contract theory. Speciﬁcally the under-eﬀort problem, or hold-up problem, of the participants, entailing utilities loss and its restoration by contract
are our main interest. We also try to analyze the risk sharing nature of thecontract from the eﬃcient pure risk sharing standpoint.
Our plan of the rest of this paper is as follows. In next section, we brieﬂy discuss
the past relevant literature. In section 3, we explain the main contents of the
Load Factor Guarantee Mechanism contract, which is actually agreed andstill binding at Noto airport in Japan. In section 4, we set up a revised Hart andMoore model and explain its characteristics and speciﬁc features. We show that
under the interdependent, joint-venture type relationship there are under-eﬀort
problem as compared with the ﬁrst best situation.
In section 5, we show that the contract of Load Factor Guarantee Mechanism
can overcome the under-eﬀorts problem and restore entailing utilities loss underrevised ﬁrst best conditions of Hart and Moore framework.
2 Passengers in this case is not directly involved in the trade, since we are focusing on thebilateral relationship between airport and airline from contract theory point of view.

2

In section 6, we use concrete numerical examples to illustrate the under-eﬀort
problems, entailing utilities loss and its restoration of the lost utilities by showing the impact of several factor perturbations, namely improved prospect ofstate of nature, uncertainty increase, risk aversion enhancement of parties
on such under-eﬀort levels and utilities loss restoration.

2

Past Literature

Because of the fundamental importance of airport-airline relationship, there are
a number of literatures in air transport economics that studied the relationshipfrom various standpoints. For example, the recent study from the stand point ofconsumer welfare analyses is Barbot (2009) and Oum and Fu (2009). Oum andFu (2009) also studies from the view-points of market competition among many
airport-airline vertical relationships and its eﬀects on networking and pricing.
Zhang et al. (2010) is the latest and unique research about contractual revenue
sharing between airport and airline, and its impacts on pricing and routes.
On the other hand, a lot of literatures are available for hold-up problem, incomplete contact, and other surrounding contract theory frameworks ranging
from Hart and Moore (1988), Noldeke and Schmidt (1995), Chung (1991),
Aghion et al. (1994), Edlin and Reichelstein (1996), Tirole (1999), Itoh (2003),
Bolton and Dewatripont (2005), Itoh and Morita (2009) to Kanemoto (1990),
Kanemoto and MacLeod (1992), Gul (2001), Segal (2002), Pitchford and Snyder(2004) and Iyer and Schoar (2008).
But so far it is very diﬃcult for us to ﬁnd any study on airport-airline vertical relationship from the stand point of hold-up problem under the incomplete
contract theory frameworks. We presume one of the reasons for this situation isthe airport-airline relationship is more interdependent and complex than simple
buyer-seller relationships.
But the basic structure of under-eﬀorts problem and the need to mitigate itwith some proper contract could well apply to airport and airline relationshipin our view. To reﬂect these interdependent vertical relationships and capture
the basic structure in the context of incomplete contract framework is what wetry to pursue in this paper.

33

Noto Airport Load Factor Guarantee Mechanism

Noto airport is a small local airport (only one 2,000m runway) opened in 2003
at Noto peninsula in Ishikawa-Prefecture, central-northern part of Japan. The
airport owned by Ishikawa prefecture promises to pay (or receive) contingent
payment based on the diﬀerence between a realization of load factor θ and thetarget load factor set at the start of the period by contract, in exchange for airline’s commitment of providing air transport service at the airport. The contact
is called ”Noto Airport Load Factor Guarantee Mechanism.”With this contract, there are 2 scheduled air transport services a day between
Noto AP and Tokyo-Haneda AP (530 km away) by ANA group (mostly by A320
with 166 seats). Ticket prices are from about 20, 000 yen (160$) to less than
10, 000 yen (80$).
The basic structure of contractual payments (payment from airport to airline
is positive amount) in the ﬁrst four years from 2003 to 2007 are in Figure 1 3 .
The calculation is done after the end of each one year period and is basically;
Payment amount = parameter ×(target load factor - realized load factor).
The parameter is diﬀerent over several divided ranges of entire load factor possibilities. So the relationship between payment p(θ) and load factor (θ) in thecontract is piece-wise linear functions. Price ceiling is kept at 2 × 108 yen andprice ﬂoor is set at −2 × 108 yen from the 2nd year on. More detail of themechanism and its numerical analysis on its payoﬀ structure is in Hihara (2008)
and Hihara (2010).
(Figure 1 Noto contract’s 4 year contract payoﬀ structure)
From airport perspective, such contract ensures the commitment of airline toserve the airport, while it is also a tool to share up-side proﬁt in high load factor case. From airline side, such contract serves not only as a risk mitigating
tool to compensate down-side loss of revenue but also as an incentive device toextract more airport eﬀorts of trying to avoid large payment hence overcoming
its under-eﬀorts problem seen typically after start of the service at the airport4 .
Notice also that the contract has multiple functions. In addition to ensuringcommitment and incentive device, it has a mechanism of risk sharing between
3 From the ﬁfth year on, the contract is same as that of the fourth year until September
2010.
4 If the eﬀort is hidden from the other, namely under the asymmetric information framework, this is called moral hazard or hidden action problem in agency frameworks. Here wefollow the usual hold up problem framework and assume the eﬀorts are seen from each other,i.e., under symmetric information structure. But the eﬀorts are not veriﬁable to the third
parties.

4

airport and airline in a mitigating way of the latter’s revenue ﬂuctuation. Whena realized load factor is low at the route of the airport, the former pays thecompensating money speciﬁed in the contract depending on the realization ofcontingent variable. When, on the other hand, a realized load factor is high,the latter pays the ”reward” money to the former.
The Noto contract could be seen as one of many cases seen across the world,such as in the case of airport’s trying to keep the low demand route at the
local airport or in the case of a secondary airport’s paying incentive payment
for start-up LCCs. With the incentive money from airport, airline then become
willing to serve the airport. The payoﬀ structure of Noto case, however, notonly covers the downside of low load factor case, where airport pays to airline,
but also the upside of high load factor case, where airline pays to airport.
Table 1 is the summary information about the Noto case for the ﬁrst four years.
In the table, we have 1) target load factors that Noto airport and the airline
agree on in the contract, 2) model load factors (estimated load factors) thatis predicted based on the econometric model about the following year’s loadfactor in Hihara (2008)5 , 3) actual load factors are those that actually realized,
4) actual number of passengers, and 5) actual contractual payments.
(Table 1 Noto Case Summary Information about here)
In the ﬁrst three years, airport did a lot of promotion eﬀorts and received moneyfrom airline. This betrayed most people s expectations that airport will end
up with paying a lot of money to airline. From the fourth year on, there is nomoney payment because the actual load factors are in the no payment range
around target load factors set in the contract6 .
From Table 1, we can calculate the diﬀerence between the estimated load factors
and the realized actual load factors and corresponding passenger number diﬀerence, for example. This could be one measure of the eﬀect the Noto contract hason the project value of the Noto airport. The corresponding passenger number
increases are estimated to bring about as much as 4.5% more revenue. This
revenue increase could be attributed at least partly to the contract extracting
more eﬀorts from both sides.
In the next section, we try to model these interdependent relationship of the twoparties in the context of incomplete contract and hold up problem framework.

5 In Hihara (2008), the domestic passenger air transport market load factors in Japan from1950 s to 1990 s are analyzed as time series data and the structure of (ARIMA(1,1,4)) isestimated. Based on the model, the following years load factors are predicted by normal
distribution. The estimated load factor is the mean of the predicted normal distribution.
6 This range is the ﬂat line part in the middle of p(θ) piece-wise linear function of the Figure
1. Such ﬂat area appeared in the third year and widened in the fourth year.

54

The Model

Under the incomplete contract model settings, airport (=AP) and airline (=AL)
can be thought to be in a bilateral dual ”trading” relationships with relationspeciﬁc eﬀorts (or investments). That is AP is buying AL s air transport service
at AP for incentive/compensating money, and on the other hand, AL is buying
AP s airport service for landing fees. At the same time both AP and AL makerelation-speciﬁc eﬀorts to increase the number of passengers on air service at
the airports, hence to enhance the value of the air service project there.

4.1

Settings

Both airport and airline are risk neutral.
In our analysis, we basically use the canonical settings in Hart and Moore (1988),
where airline has only two choices, either to provide service (one) or not to provide it (zero). Service quantity problem in-between zero and one is not relevant
here.
The time line along which we assume for both parties to transact with eachother is Figure 2 and Figure 3. We also assume time discounting is not relevant
as in the usual settings. Eﬀorts are assumed to be speciﬁc to the relationshipand its cost is to be sunk before the realization of the state of nature. Renegotiations are costless and can attain ex post eﬃcient eﬀort levels7 .
(Figure 2 and Figure 3 time line around here)
In Figure 2, there is no contract at t = 0. When air transport service is provided
at AP, AP and AL each make independent eﬀorts from t = 1 to t = 2 before
the realization of θ. After θ is realized at t = 2, AP and AL renegotiate onshare of project value and the price p(θ) as long as it is eﬃcient. We assume therenegotiation result is decided according to Nash bargaining solution. The payment is paid at t = 3 based on the results. They anticipate these mechanisms
after t = 2 and make eﬀort based on expectations between t = 1 and t = 2. If
no air transport service is provided, no value is realized. Also, if the realizedθ does not justify the air service provision, AP and AL have to tear down therelationship and clean up the transactions up to the point.
7 We assume the costless renegotiation after the contingency realization and the ex posteﬃciency so that we focus on the ex ante ineﬃciencies. This is because we think ex ante inefﬁciencies are very relevant to our problem and also this type of focus on ex ante ineﬃciencies
is the usual settings of incomplete contract frameworks.

6

Figure 3 is about the situation with contract. AP and AL agree on a contract
at t = 0. By contract AL makes commitment to serve at AP. Before the realization of θ, AP and AL make independent eﬀorts. Between t = 2 and t = 3 they
renegotiate on p based on the realization of θ. These are according to Hart andMoore (1988)settings.
However, the AP-AL relationship has its own typical characteristics so wechange the Hart-Moore model accordingly. The biggest change is the jointventure type value structure, which we will describe extensively in the next
subsection.
Basic idea of joint-venture type value structure is AP and AL are both buyer
and seller at the same time. If AL serves AP, AL ”buys” airport service from
AP for landing fee but at the same time, AP pays incentive money to AL to”buy” air transport service at the airport.
If AL thinks the value from service at the airport is lucrative enough without
any incentive payment, then AL is selling air transport service for zero price.
But if the service is not so lucrative, then AP may think the incentive payment
to buy AL’s service is necessary to secure AL’s commitment to service AP and tokeep the accompanying business revenues, such as parking fee from passengers
and concession revenues of land side operations. In that case, AP is ”buying”
as a buyer the air transport service for positive price from AL in addition toselling airport service for landing fee as a seller.
With this modiﬁcation, we can model the situation in a local airport to pay
incentive money to LCCs beyond the discount of landing fee, just like the Noto
case 8 . Also with this model, we can set such structures in which the valuationof AL’s side to serve at the airport is not so lucrative although serving there
meets the minimum participation utility level (as we explain later, this could
be negative) and the incentive money from AP to AL plays an important rolein AL’s decision to actually serve AP.
As signiﬁcant as joint venture type structure is payment aggregation. AP and
AL ”trade” with or without such a contingent payment contract in a real world.
Here our model covers not only the situations without such contract but also
the situations with contract. In a sense, payment in our model is aggregatedversion of all the payment transactions of the real world AP-AL relationship. So
8 Noto airport does have landing charge. But the frequency is only two services a day. Also
landing fee is not dependent on θ but only on MTOW of aircraft in use, which means thefee is basically constant. In addition, landing fee is discounted to one third and its amountis at most about 10% of maximum contingent payment in the contract. So for the sake ofsimplicity in this study we assume the zero landing fee to stay focus only on the contingent
variable payment p(θ) without loss of generality. The analysis is unaﬀected and valid evenwith the consideration of landing fee.

7the payment in our model between AP and AL are only through one payment(aggregate of all payments, from landing fee to incentive payment, if any, inboth directions between AP and AL) 9 .
The direction of net payment from AP to AL is set to be positive and that of
the other direction is negative. With or without such a contract, the payment
is calculated at the end of one business period based on the valuation including
the result from contingent realization of the period.
We further assume that if the valuations of both sides, after the realization ofstate of nature, result in the situation that no trade is eﬃcient, then AP and
AL must dissolve the contractual relationship and have to pay for the cost of
tearing down the relationship. For the sake of simplicity, this is assumed towipe out all the values and ﬁnal valuations for both AP and AL in this case areassumed to be zero. This assumption is necessary for our model to be consistent
with typical incomplete contract settings.

4.2

Joint-Venture Type Contract Model

Here we formally introduce joint-venture type relationship. By joint-venture
type, we mean that AP as well as AL is providing (or selling) service (forexample, AP = providing runway and passenger terminal, AL = providing airtransport service), while AP as well as AL are making relation-speciﬁc eﬀorts
to increase the value of joint venture and to decrease the service providing cost(for example, AP = renovating terminal for more eﬃciency, AL = introducing
smaller and fuel-eﬃcient new type aircraft). AL as well as AP is independently
generating revenues and incurring costs out of the joint-venture.

UAP (b, θ) =vAP (b, θ) − cAP (b, θ) − p(θ) − φ(b)(1)

UAL (s, θ) =p(θ) + vAL (s, θ) − cAL (s, θ) − ψ(s)(2)

b: eﬀort of airport (=AP s eﬀort)b ∈ [0, b̄] ∈ R
s: eﬀort of airline (=AL s eﬀort)s ∈ [0, s̄] ∈ R
θ : random variable corresponding to the state of nature (in Noto case, Load
9 In the case of LCC and local airport owned by local government, for example, the landing
fee is often exempted in the ﬁrst place and that case it is irrelevant. For bigger hub airport, in
our model payment is more like a aggregate payment over all payment between AP and AL,
including landing fee, check-in counter lease payment, incentive payment from AP, discounton hanger payment and so on.

8Factor of the Noto Haneda route)θ ∈ [0, 1] ≡ Θ ∈ R(b,s): observable by AP and AL but not veriﬁable by a third party(e.g., national government including Ministry of Transport or court)
p: price of the reward for the air transport service from airport to airline (minus
value means airport is receiving money from airline)p∈R
q: the Noto=Haneda airport passenger air service quantity levelq ∈ {0, 1}
vAP (b, θ): AP’s valuation function of the air transport service at the airportcAP (b, θ): AP’s valuation function of the airport service (relationship speciﬁc
contribution function to project by its eﬀort to decrease the trade cost of the
airport service to be provided
vAL (s, θ): AL’s valuation of the airport service at the airportcAL (s, θ): AL’s valuation function of the air transport service at the airport(relationship speciﬁc contribution function to project by its eﬀort to decrease
the trade cost of the air transport service to be provided)
000vAP > 0, vAP> 0, vAP< 0 are assumed for each element of (b,θ).
0cAP > 0, −cAP > 0, −c00AP < 0 are also assumed for each element of (b,θ).
000
vAL > 0, vAL> 0, vAL< 0 are assumed for each element of (s,θ).
0cAL > 0, −cAL > 0, −c00AL < 0 are also assumed for each element of (s,θ).

vAP − cAL > 0 and vAL − cAP > 0 are the two conditions for the trades within
the joint-venture relationship. This implies
vAP − cAP + vAL − cAL > 0.
vAP − cAP + vAL − cAL is assumed to be strictly concave function bounded
above.
vAP − cAP > 0 orvAL − cAL > 0 are also assumed.10
φ(b): AP’s direct eﬀort costs
φ(s): AL’s direct eﬀort costs
φ > 0, φ0 > 0, φ00 > 0 andψ > 0, ψ 0 > 0, ψ 00 > 0 are assumed.

In usual settings, buyer s utility consists of only the valuation of the projectand valuation of cost reduction part is not present. Likewise seller s utility
consists only of valuation of cost reduction part and the valuation of the project
10 Notice that v
AP ≥ cAL ≥ vAL ≥ cAP , for example, does not satisfy the condition ofvAL − cAL > 0 but do satisfy the two conditions for the trades in the joint-venture, namely
vAP − cAL > 0 and vAL − cAP > 0.

9

is not present. In our joint-venture settings, both AP and AL are both buyers
and sellers and their utilities have both the valuation of the project part andvaluation of cost reduction part11 .
In the joint-venture type contract model, the service from AP to AL (e.g., airport
service) and the service from AL to AP (e.g., air transport service) are noteconomically separable. Each service cannot exist without the existence of theother. So even though both AP and AL have both ”buyer = service receiver”
and ”seller = service provider” sides but these services and sides cannot be
separated.
With these modiﬁcations, we have two more degrees of freedom, namely vALand cAP in the footnote case, to model the situation realistically. For example,
if AL s opportunity cost, or its valuation, of the air transport service, cAL ,is high as compared with the airport s value to AL, vAL (since AL can use
its resources in other very high yield route to other airports), then AL s total
net valuation in the joint project of the AP (Noto airport) could be most likely
negative vAL − cAL < 0 in low load factor case as depicted in Figure 4. This
loss can be covered by the contract payment. This is shown in Figure 4 thatp(θ) ≥ cAL − vAL in low θ area.
(Figure 4

about here)

This could be a real situation where the AL, with better potential opportunity
to other routes to diﬀerent APs, is reluctant to serve a local AP but the incentive
money from the AP plays decisive role in actually serving there.
On the other hand, if AP s valuation of the air transport service, vAP (suchas those from land side revenues or tourism revenues) is quite high as comparedwith its airport service valuation (opportunity cost), cAP (since there is no othergood candidate for serving the AP), then the trade conditions, vAP − cAL > 0
and vAL − cAP > 0, would more likely to hold. And the incentive money p(θ)
could be very high in low load factor case because of high vAP .
Joint-venture type modiﬁcation is not just to tolerate negative p(θ) in price
11 If we literally follow Hart and Moore (1988), the model would be as follows assuming AP
is buyer and AL is seller.

UAP (b, θ) =vAP (b, θ) − p(θ) − φ(b)UAL (s, θ) =p(θ) − cAL (s, θ) − ψ(s)In other joint venture type contract settings, both buyer and seller could share one combined
value and one combined cost function like in Kim and Wang (1998). In our case, it is natural
AP or AL each has its own valuation function and cost reduction function, since AP and AL
have its own independent revenue sources (AP: landing fee and AL: passenger revenue, forexample). They are generated and accounted independently though they usually cannot exist
without the existence of the other in the joint venture project of the air transport service at
the airport (routes).

10

agglomeration of both directions between AP and AL 12 . But our model canmore realistically and directly describe the reasons and hidden dynamics forsuch p(θ) in AP-AL relationships.
It tolerates, for instance, high positive p(θ) because AP s valuation of airtransport service (vAP ) is quite high because tourism revenue might be quite
valuable for AP even in low load factor case. It also tolerates quite negativep(θ) in high load factor case because in such a case, AL s valuation of serving
the AP (vAL ) is quite high and hence more room to share the proﬁt with AP
through contract payment. In this case, vAL − cAL >> 0 ⇔ cAL − vAL << 0.
And this is shown in Figure 4 that p(θ) ≥ cAL − vAL in high θ area.
Contract of Noto airport load factor guarantee mechanism can be thought as
speciﬁc performance contract with payment directly contingent on realizationof state variable in exchange for the contractual commitment of providing airtransport service at the Noto airport.
Noto AP Load Factor Guarantee Mechanism Contract
C{(p, q) = (p(θ), 1)}(3)

In Hart and Moore (1988) or Noldeke and Schmidt (1995), they are thinking ofa contract, which consists of binary choice of quantity of service (either zero orone) and its corresponding price.
In the case of Noto, by the speciﬁc performance contract, AL are committed toprovide air transport service (q = 1) at Noto airport in exchange for paymentp = p(θ). Notice that AL has no choice of not providing service. Rather theyreceive p(θ) from AP or renegotiate on price, depending on the realization of θand concrete contents of the utility functions of the parties 13 .
Also we can think p(θ) = p1 − p0 . In this case, p0 is set to zero. To make thissetting, we adjust, without loss of generality, all the other relevant values in our
settings. Notice that either one of the parties utility level can be negative fromthe assumptions above. So the minimum utility level also could be negative.
This can happen when, for example, the whole operation of the AL’s all networks are making proﬁt in aggregation, but the local airport routes or start-up
routes of the AL (or LCCs) could be in red ink at least for a while. This is oneof the main purposes of our model.
12 p could be negative in usual settings, too. See, for example, Chung (1991) or Noldeke andSchmidt (1995).
13 With the above assumptions, trade is always eﬃcient. So dissolving the contractual
relationship, which occur when the trade is not eﬃcient after the realization of θ, is not
relevant here.

11

In the speciﬁc contract above, it speciﬁes that AL provides air transport service
at the airport and AP rewards AL by paying p(θ) in aggregation.
Noto contract in equation (3), however, unlike the usual speciﬁc performance
contract with constant price level, payment p is a function of θ and changeable
amount of payment to be made from AP to stabilize the AL’s revenue ﬂuctuations, since low θ makes the p positive, meaning AP is paying to AL with little
revenue, and high θ makes the p negative, meaning AL is paying to AP out oflarger revenue.
In this since, the speciﬁc performance contract is not just the commitment of
AL to provide service at the airport in exchange for ﬁxed amount of reward, but
the AL’s commitment in exchange for risk sharing in a counterbalancing way by
AP. Note that the p(θ) could be negative. So the AL receives money from AP
when the load factor is below the target load factor (beyond the special range
of no payment).

4.3

Speciﬁc Features of the Model

Note that p could be negative. The reward price in negative means AP is receiving money from AL rather than paying to AL.
In the standard setting of incomplete contact frameworks, not only both parties’ eﬀorts but also state contingent variable are unveriﬁable. Therefore thepayment is usually not directly contingent on state variable.
In our model, however, the state variable of load factor is oﬃcially announced by
the Ministry of Land, Infrastructure, Transport and Tourism. So θ in our model
is veriﬁable. Hence the contract can specify the target load factor and payment
is able to be calculated directly from load factor (state variable); p = p(θ). That
was exactly what the airport and airline did in Noto case.
AP and AL still cannot make contract that directly speciﬁes the target of efforts, since they are unveriﬁable.
By introducing p = p(θ), AP is assuming part of AL’s revenue ﬂuctuation risk
and is stabilizing AL’s revenue ﬂuctuation by paying money to AL when θ islow and hence AL’s revenue is low or by receiving the money when θ is highand hence AL’s revenue is high.
Note also that, unlike usual incomplete contract settings, UAL has vAL (s, θ) as
one of its elements in addition to −cAL (s, θ). This is because in the AL and AP
relationship, AL has its own revenue from the air transport service as well as
the reward revenue from AP.
12

In the usual incomplete contract settings, sellers (or employees or service providers)have only one revenue source from the transaction (with or without contract).
That is the reward revenue from buyers (or employers or service receivers) represented by p.
In the case of AP and AL relationship, AL always has its own revenue from the
air transport service. Oﬀ course AP also has its own revenues, such as landing
fees, concession revenues from the ground side operations at the airports andparking revenues, out of such air transport services by AL. It could be said
that AP and AL ”share” the entire revenue from the joint-venture air transport
service project at the airport.
We are assuming that the two diﬀerent revenue sources can be additively separable in the utility function of AP and AL. This means that AL’s two elements(vAL (s, θ) and −cAL (s, θ))of utility function, for example, are linearly separable.
Accordingly, our model in equation (2) captures the unique features of AP-AL’
joint-venture type relationship. This type of ”double structure” on the bothsides of a contract, just like that of AP and AL, is, to our knowledge, neverformally set up, although this kind of relationship is, we presume, no t so rare
in a real business world 14 .
In addition, AP and AL have other types of close relationship. In their value
and cost contribution functions, based on the above assumptions, AP and AL
are in so called ”strategic inter-complementary relationship,” where AP’s eﬀort
level enhance the eﬀect of AL’s same level of eﬀorts and vise versa.
We deﬁne the project value, V (b, s), as follows (noticing we already assumedvAP (b, θ) − cAP (b, θ) + vAL (s, θ) − cAL (s, θ) > 0),
V (b, s, θ) ≡vAP (b, θ) − cAP (b, θ) + vAL (s, θ) − cAL (s, θ)V (b, s) ≡Eθ [V (b, s, θ) | b, s].

(4)
(5)

We can show, with the assumptions on diﬀerentiability, that,∂ 2 V (b, s)
≥ 0.
∂b∂s

(6)

For a reference, if V (b, s) is, contrary to our assumptions here, not diﬀerentiable,
14 For example, in franchising contracts of retail convenience store, franchisees (convenience
stores) are paying royalty fees to franchisers (brand holding/managing companies) and on theother hand, franchisers are paying incentive or risk mitigating money to franchisees. Anotherexample could be tenants and land owners in commercial shopping mall projects and they
could be in the same kind of joint-venture type situation. Land owners are receiving rents
from tenants but they could oﬀer some incentive/ risk neutralizing payment mechanism to(potential) tenants to ensure their commitments in the project.

13the ”increasing diﬀerence” condition is as follows ;V (b, s) − V (b0 , s) ≥V (b, s0 ) − V (b0 , s0 )
0

0

(7)
0

0

∀{b, b , s, s } ∈ {b > b , s > s }This property is called ”increasing diﬀerences,” or, in game theory terms, ”strategic inter-complementarity.” Under the property, if one party increases marginal
eﬀort, it does not decrease the marginal contribution by the other party’s
marginal eﬀort to increase the value of the project, in which both are jointly
participating.
Our model of equations, (1) and (2) above, captures, through assumptions, thisstrategic inter-complementary relationships between AP and AL.

4.4

First BestIn the ﬁrst best, both parties jointly maximize the total expected utility level.
maxEθ [UAL + UAP ]{b,s}= max Eθ [vAP (b, θ) − cAP (b, θ) + vAL (s, θ) − cAL (s, θ)] − φ(b) − ψ(s){b,s}= max V (b, s) − φ(b) − ψ(s){b,s}The ﬁrst best level of eﬀorts(b∗ , s∗ ) are characterized by FOCs:
0Eθ [vAP(b∗ , θ) − c0AP (b∗ )] = φ0 (b∗ )(8)0Eθ [vAL(s∗ , θ) − c0AL (s∗ , θ)] = ψ 0 (s∗ )(9)In the above,
∗

∗

∗

0
0
vAP
(b∗ , θ) = ∂vAP∂b(b ,θ) , c0AP (b∗ ) = ∂cAP∂b(b ,θ) , vAL(s∗ , θ) = ∂vAL∂s(s ,θ) , c0AL (s∗ ) =∗∗∗∂cAL (s ,θ))), φ0 (b∗ ) = ∂φ(band ψ 0 (s∗ ) = ∂ψ(s
∂s
∂b
∂s .

Note that the ﬁrst best eﬀort level for AL s∗ is increased from the ﬁrst best
situation in usual incomplete contract settings. This is because AL has its own
revenue source (i.e., sales from the operation), hence its own valuation function
vAL·s (s∗ , θ) > 0 in its utility function. This increase the marginal stake to bebalanced by the marginal eﬀort costs in equation (9) resulting in the higher s∗ .
Here, we further assume that the objective function of the maximization problems is strictly concave and that the problem has interior solution for simplicity.
14

Therefore the ﬁst best eﬀorts level (b∗ , s∗ ) exists, it is unique and 0 < b∗ , 0 < s∗ .

4.5

Hold Up Problem

Assumption 1 - Process
If both parties cannot agree on any ex ante contract, we assume, according toHart and Moore (1988) and Edlin and Reichelstein (1996), that ﬁrst AP and AL
independently make eﬀorts, wait for the realization of load factor, θ, make renegotiation according to generalized Nash bargaining solution framework about
what part of realized value are gained by AP or AL, and at the same time priceis also decided according to the bargaining solution15 .
Based on this prospect of the transaction, AP and AL make ex ante utility
maximization after taking expectation over the contingency. Also if both parties’ renegotiation results in valuation and price that cannot support providing
service in the ﬁrst place, they have to dissolve the relationship, if any, withsigniﬁcant cost as much as wiping out the valuation until then.
In the generalized Nash bargaining solution, we can assume AP’s share isα(0 < α < 1) portion of and AL’s share is (1 − α) of the total value fromthe realized contingency, which is as follows16 ;huUAP(b, s) ≡ α{vAP (b, θ) − cAP (b, θ) + vAL (s, θ) − cAL (s, θ)}{θ∈Θ} − φ(b)huUAL(b, s) ≡ (1 − α){vAP (b, θ) − cAP (b, θ) + vAL (s, θ) − cAL (s, θ)}{θ∈Θ} − ψ(s)Anticipating these processes ahead, AL and AP are ex ante maximizing theirutilities by expectations over their eﬀort level;hu(b, s) =Eθ [α{vAP (b, θ) − cAP (b, θ) + vAP (s, θ) − cAL (s, θ)}] − φ(b)ŪAP=αV (b, s) − φ(b)huŪAL(b, s) =Eθ [(1 − α){vAP (b, θ) − cAP (b, θ) + vAL (s, θ) − cAL (s, θ)}] − ψ(s)}=(1 − α)V (b, s) − ψ(s)So the maximization is as follows;hu
max{ŪAP(b, s)} = max{Eθ [α{vAP (b, θ) − cAP (b, θ) + vAP (s, θ) − cAL (s, θ)}]bb

− φ(b)}
15 This assumption is, as already stated, not because we think this is realistic but becauseby the renegotiation the ex post eﬃciency is guaranteed so that we can focus on the ex ante
ineﬃciencies, which are very relevant in our model.
16 The price p in the Nash bargaining case is set to satisfy the relationship; v
AP − cAP − p =α(vAP − cAP + vAL − cAL )15

humax{ŪAL(b, s)} = max{Eθ [(1−α){vAP (b, θ)−cAP (b, θ)+vAL (s, θ)−cAL (s, θ)}]ss

− ψ(s)}
This utility maximizing results in eﬀorts level (b∗∗ , s∗∗ ), which are obtained by
the FOCs;
0αEθ [vAP(b∗∗ , θ) − c0AP (b∗∗ , θ)] = φ0 (b∗∗ )0(1 − α)Eθ [{vAL(s∗∗ , θ) − c0AL (s∗∗ , θ)}] = ψ 0 (s∗∗ )

(10)
(11)

On the other hand, we think about the eﬀort making game between AP andhuhu
AL with their payoﬀ functions being ŪAP(b, s) and ŪAL(b, s) respectively. Nashhu huequilibrium (b , s ) in this game is deﬁned as follows;hu
hu hu hu(b, shu )(b , s ) ≥ ŪAPŪAP∀b ∈ [0, b̄](12)hu hu huhu hu
ŪAL(b , s ) ≥ ŪAL(b , s)∀s ∈ [0, s̄]

(13)

hu
hu
If we further assume ŪAP(b, s) and ŪAL(b, s) are strictly concave function andits maximization problem has interior solution, there is a unique solution. In
this case, Nash equilibrium (bhu , shu ), if any, satisﬁes the above ﬁrst order condition in equations (10) and (11).

Proposition 1 Hold Up Problem
If the process follows Assumption 1, then there exists Nash Equilibrium (bhu , shu )that is less than the ﬁrst best levels of eﬀort;(bhu , shu ) < (b∗ , s∗ ).

(14)

Proof of Proposition 1
The proof is a little complicated so we put it in Appendix A.
So without any contractual agreement between AL and AP in our settings, there
is an under-eﬀort problem by both sides. This means their welfare levels are
not eﬃcient in comparison with the ﬁrs best level.

165First Best Possibility by Price-Contingent-Variable
Speciﬁc Performance Contract

Hart and Moore are thinking of a contract, which consists of binary choice ofquantity of service ( either zero or one) and its corresponding price. We describesuch a contract as { C{(p0 , q0 ) = (p0 , 0); (p1 , q1 ) = (p0 +k, 1)}(k : constant) }. In
their settings, p1 = p0 + k. Also they specify the ﬁrst best conditions for speciﬁc
performance contract C{(p, q) = (p1 , 1)} in proposition 1 and proposition 3 inHart and Moore (1988).

5.1

First Best Condition in Hart and Moore (1988)

As speciﬁed in the Proposition 1 and Proposition 3 in Hart and Moore (1988),
if there exists some constant k(= p1 − p0 ) for which the condition v(b, θ) ≥k ≥ c(s, θ)) holds for all θ, b and s, then the speciﬁc performance contract {C(p, q) = (p1 (= p0 + k), 1) } achieves the ﬁrst best eﬀort levels.17
This means that if the above conditions hold, speciﬁc performance contract of
doing trade with price p1 = p0 + k (C(p, q) = (p1 (= p0 + k), 1)) always guarantee
the ﬁrst best level of eﬀorts on both sides.
Here we modiﬁed the original conditions in Hart and Moore (1988) to suit thecontext of our settings. First AL is by contract already committed to provide
service and no choice of not providing it. In this sense the contract is speciﬁc
performance contract with no room for choice (But AL can dissolve the contract
afterwards).
Second, there is only one contingent variable with common distribution function for both parties. In Hart and Moore (1988) settings, each party has its
own independent probability distribution. Therefore, as to the probability distribution, our settings are one special case (both parties share a single common
probability distribution) of Hart and Moore (1988).
But the same is that the renegotiation by non-veriﬁable message sending gameto change the price is assumed to be possible. In their case contingency realization results in trade or no trade case with entailing price renegotiation results.
In our case, doing trade is always eﬃcient by assumption and parties renegotiate
price after the contingency realization.
The price resulting from renegotiation is not always satisfying the ﬁrst best
17 The message game in this case is by using non-veriﬁable messages. Also although there
are 3 other cases Proposition 3 in Hart and Moore (1988), the three cases are not relevantin our speciﬁc contract case. This is because valuation vAP and vAL cAP and cAL are allcontingent on b, s, θ.

17conditions. Next we cover under what conditions the price after renegotiation
always achieves the ﬁrst best eﬀorts level.

5.2First Best Condition of AP-AL Speciﬁc PerformanceContract

In our settings about the speciﬁc performance contract of Noto case in equation(3), we can adjust Hart and Moore’s ﬁrst best conditions above so as to reﬂect
the special relationship between AP and AL, the Joint-Venture Type contract.
Proposition 2
If the condition vAP (b, θ) − cAP (b, θ) ≥ p(θ) ≥ cAL (s, θ) − vAL (s, θ) holds for all
θ, then the speciﬁc performance contract with risk sharing function { C(p, q) =C(p(θ), 1) } achieves the ﬁrst best eﬀort levels.18
Proof of Proposition 2
vAP (b, θ)−cAP (b, θ) and cAL (s, θ)−vAL (s, θ) have the same functional properties
as v and c in Hart and Moore (1988) by assumptions. We can think of vAP (b, θ)−cAP (b, θ) as v̆AP (b, θ) and cAL (s, θ) − vAL (s, θ) as c̆AL (s, θ). So the Proposition
2 with v̆AP and c̆AL states the same conditions with Proposition 1 and 3 inHart and Moore (1988). Then the contents of Region (2) in APPENDIX A in
Hart and Moore (1988) holds for the v̆AP and c̆AL , although in our case thecontingency is only one common θ.
In this non-veriﬁable message sending game, the ﬁnal outcome is to trade at
price p(θ). So speciﬁc performance contract { C(p, q) = C(p(θ), 1) } always
corresponds to the ﬁnal outcome of the game under the settings.
AP and AL under the speciﬁc performance contract { C(p, q) = C(p(θ), 1) }in this setting are individually maximizing each utility under the condition ofalways paying p(θ) for air transport service that is committed to be provided
by the contract.
c=Eθ [vAP (b, θ) − cAP (b, θ) − p(θ)] − φ(b)
ŪAP
c
ŪAL
=Eθ [p(θ) + vAL (s, θ) − cAL (s, θ)] − ψ(s)18 Here again we assume the message game by using non-veriﬁable messages in the renegotiation stage just as in Hart and Moore (1988).

18

So the maximization is as follows.
cmax{ŪAP} = max{Eθ [vAP (b, θ) − cAP (b, θ) − p(θ)] − φ(b)}bbcmax{ŪAL } = max{Eθ [p(θ) + vAL (s, θ) − cAL (s, θ)] − ψ(s)}ssThis utility maximization results in the eﬀort levels (bc , sc ), which are obtainedby the FOCs;
0Eθ [vAP(bc , θ) − c0AP (bc , θ)] = φ0 (bc )(15)0Eθ [vAL(sc , θ) − c0AL (sc , θ)] = ψ 0 (sc )(16)These FOCs are exactly the same with the FOCs in the ﬁrst best case, namely
equations (8) and (9).
Hence,(bc , sc ) = (b∗ , s∗ )(17)

Therefore the speciﬁc performance contract { C(p, q) = C(p(θ), 1) } achieves theﬁrst best levels of eﬀort.
Q.E.D.
Notice that in our joint-venture type contract, because the v̆ and c̆ have twoelements, vAP , −cAP and cAL , −vAL respectively. So cAL − vAL can be negative by assumptions. Therefore emerges the possibility that Noto type zero-sum
payment structure contract achieves the ﬁrst best eﬀort level even thought thepayment p(θ) could go into the deep negative range.
These two relevant conditions are illustrated in the Figure 4. The actual contractual piece-wise linear price schedule in the fourth year contract of Noto caseis indicated by the piece-wise linear line p(θ).
As you can see in the Figure 4, the reward price function p(θ) is satisfying theﬁrst best condition for the speciﬁc performance contract to achieve the ﬁrst
best level in Proposition 2. The line of p = p(θ) lies within the upper red linevAP − cAP , which is the net valuation of AP and lower decreasing blue linecAL − vAL , which is the negative of the net valuation of AL, as indicated in the
Proposition 2.
Especially in our model, we can see the situation where, like in the case of AP’s

19trying to keep the low demand route at the local airport or in the case of a secondary AP’s paying incentive payment for start-up LCCs, AL’s net valuationvAL − cAL of the project is negative in a bad state of nature ( low θ) but by
signing a the contract with the incentive payment p(θ), the adjusted valuation is
non-negative (vAL −cAL +p(θ) ≥ 0) even in the bad state of nature, while at thesame the contract overcomes the under-eﬀort problem with ﬁrst best conditions
holding.

5.3

Utilities Loss Restoration

If the condition for ﬁrst best holds for the speciﬁc performance contract, thenthe utilities level for both AP and AL are brought back from hold up situation
to the ﬁst best situation. Then we can calculate the diﬀerence between thetwo situations and see the utilities loss restoration from hold up (second best)
situation to ﬁrst best situation.
c
c
hu hu
hu hu
ResU ≡ ŪAP(b∗ ) + ŪAL(s∗ ) − [ŪAP(b ) + ŪAL(s )]= v̄AP (b∗ ) − v̄AP (bhu ) − (c̄AP (b∗ ) − c̄AP (bhu ))+ v̄AL (s∗ ) − v̄AL (bhu ) − (c̄AL (s∗ ) − c̄AL (shu ))− (φ(b∗ ) − φ(bhu )) − (ψ(s∗ ) − ψ(shu ))As far as the value of ResU is positive, there is utilities loss restoration. With
our assumptions, economically meaningful setting strongly suggests ResU ≥0, although exact results depend on concrete functional forms and concrete
parameters. In fact, later we show, with concrete numerical examples that
support the ﬁrst best conditions, ResU is actually positive in the relevant range.
The payment from the public entity to private ﬁrm, like government owned local
airport to private airline, is often very controversial because tax payers’ money
is on the line. But the positive value of ResU means that by the contract and its
stipulated payment the lost utilities are recovered. So this eﬃciency gain could
be one of the reasons to justify such use of tax money from economic point ofviews.

6

Numerical Examples

Here we illustrate the theoretical analysis with some concrete models and numbers. We also try to show some relationship between parameter perturbationand the eﬀort levels and utility levels change in our models.

20

To make this as real as possible in addition to keeping as simple as possible,
the price function is using the Noto contract real values, namely, price ceiling2 × 108 yen and price ﬂoor −2 × 108 yen. Henceforth we use simple linear function between the price ceiling and ﬂoor (i.e., p(θ) ≡ 2 × 108 − 4 × 108 θ), whichis indicated by dotted line in Figure 4, rather than piece-wise linear function in
the actual contracts of Noto case, which is also indicated in line of Figure 4 19 .
We assume the functional forms and its parameters as follows.
vAP (b, θ) = B1 − (B1 − A1 )e{−λ1 (θ+b)}cAP (b, θ) = C1 − (C1 − D1 )e{−λ2 (θ+b)}vAL (s, θ) = B2 − (B2 − A2 )e{−λ3 (θ+S)}
cAL (s, θ) = C2 − (C2 − D2 )e{−λ4 (θ+S)}p(θ) = E − F θ, θ ∼ N (µ, σ 2 )φ(b) = eb , ψ(s) = es
A1 = 2.4 × 108 B1 = 4 × 108 C1 = 0.3 × 108 , D1 = 0.1 × 108 , A2 = 0.7 × 108 ,B2 = 2.5 × 108 , C2 = 2.2 × 108 , D2 = 0.1 × 108 , E = 2 × 108 , F = 4 × 108
We further assume that λ1 = λ2 = λ3 = λ4 = λ (> 0) in all the valuation
functions.
Also we use these numbers in the course of our analysis with some variations.
µ = 0.7, σ = 2.38, λ = 3, α = 0.5001
If we use these functions and parameters, we get the utilities for both AP and
AL20 .

19 This is for the sake of maneuverability rather than implying that the eﬃcient contract islinear.
20 The expected utilities are calculated here by using the following approximation, since the0 or 1 are estimated to be at least 6 standard deviation away from the mean value in normal
distribution by the study on the Noto contract in Hihara (2008).

Z 1
0

Ui (j, θ)dF (θ) ≈Z ∞−∞Ui (j, θ)dF (θ)(i, j) = {(AP, b)or(AL, s)}21

ŪAP = Eθ [UAP (b, θ)] = B1 − D1 −(B1 − A1 )e{−λµ−λb+λ2 σ 2}
2−(C1 − D1 )e{−λµ−λb++λ2 σ 2}
2−(E − F µ) − eb
ŪAL = Eθ [UAP (b, θ)] = B2 − D2 −(B2 − A2 )e{−λµ−λs+

λ2 σ 2}
2−(C1 − D1 )e{−λµ−λs++λ2 σ 2}
2+(E − F µ) − esFrom the ﬁrst order conditions of the ﬁrst best situation in ŪAP + ŪAP by b
and s, we can get the system of equations.
(∂(ŪAP (b) + ŪAL (s)))/∂b = 0 and (∂(ŪAP (b) + ŪAL (s)))/∂s = 0
Solving the equations, we get the ﬁrst best eﬀort levels (b∗ , s∗ ).
1(ln(M N1 )(λ + 1)
1
sF B =(ln(M N2 )(λ + 1)bF B =

M = λe{−λµ+λ2 σ 2}2

N1 = B1 − A1 + C1 − D1
N2 = B2 − A2 + C2 − D2
In the no-contract case, we have the following second best (hold up) situations
under Assumption 1.
SBŪAP=α[Eθ [UAP (b, θ)] + Eθ [UAL (s, θ)]] − eb
SBŪAL=(1 − α)[Eθ [UAP (b, θ)] + Eθ [UAL (s, θ)]] − esFrom the ﬁrst order conditions of ŪAP and ŪAL , we have the second best eﬀort
levels (bSB , sSB ) as follows.
1(ln α + ln(M N1 ))(λ + 1)1(ln(1 − α) + ln(M N2 ))sSB =(λ + 1)bSB =22

For the Noto contract to mitigate the under-eﬀort problem and to lead the twoparties to reach the ﬁrst best eﬀort levels, the contract must satisfy the condition in Proposition 2. We can check the condition in our settings here bydrawing the graphs for vAP (θ) − cAP (θ), vAL (θ) − cAL (θ), p(θ) = E − F θ andcAL (θ) − vAL (θ) with θ in our settings.
These graphs are actually in the Figure 4. These graphs are drawn exactly by
the functional forms and parameters we set here. The piece-wise linear line in
Figure 4 is the real price schedule in Noto contract in the fourth year.
For the piece-wise linear price schedule indicated by line in Figure 4, the condition of Proposition 2 under our settings is equivalent to the following.
vAP (0) − cAP (0) = 2.1 × 108 ≥ 2 × 108
cAL (0) − vAL (0) = 1.5 × 108 ≤ 2 × 108
cAL (0.9354) − vAL (0.9354) = −2.16 × 108 ≤ −2 × 108
The point for (θ = 0.9354, P = −2 × 108 ) is the beginning point of the lower
horizontal line part of the piece-wise linear function of the real Noto contract
price schedule. As we can see, all these conditions are met.
Also it is not hard to conﬁrm that the simpliﬁed price schedule p(θ) = E − F θindicated by dotted line in Figure 4 we assume here satisfy these conditions,
too.
Therefore, by Noto speciﬁc performance contract of conﬁrming the commitmentof AL’s service at the airport in exchange for reward money p(θ) can lead AP
and AL to achieve the ﬁrst best eﬀort levels according to Proposition 2.
Now that we see the numerical example for the speciﬁc performance contractsuch as Noto contract satisfy the relevant conditions for the contract to leadboth AP and AL to reach the ﬁrst best eﬀort levels. We look into more detailfor the structure of our numerical model.
First we like to see the diﬀerence between ﬁrst best eﬀort level and the second
best eﬀort level. Notice that bF B , sF B are the function of µ, σ and λ in M
above and bSB and sSB are the function of α in addition to the three. So we
can draw the graphs for bF B and bSB by µ, σ and λ with appropriate numberof α.
(Figure 5 about here)
The Figure 5 left is the graph for the followings. The upper red graph is forﬁrst best case, and the lower black graph is for the second best case.

231(ln(M N1 )(λ + 1)
1bSB (µ, σ) =(ln α + ln(M N1 ))(λ + 1)
α = 0.1; λ = 3

bF B (µ, σ) =The Figure 5 right is the graph for the followings. The upper blue graph is forﬁrst best case, and the lower black graph is for the second best case.
1(ln(M N1 )(λ + 1)
1sSB (µ, σ) =(ln(1 − α) + ln(M N1 ))(λ + 1)
α = 0.9; λ = 3

sF B (µ, σ) =(Figure 6 about here )The Figure 6 left is the graph for the followings. The upper red graph is forﬁrst best case, and the lower black graph is for the second best case.
1(ln(M N1 )(λ + 1)
1bSB (µ, λ) =(ln α + ln(M N1 ))(λ + 1)
α = 0.5001; σ = 2.38

bF B (µ, λ) =The Figure 6 right is the graph for the followings. The upper blue graph is forﬁrst best case, and the lower black graph is for the second best case.
1(ln(M N1 )(λ + 1)
1sSB (µ, λ) =(ln(1 − α) + ln(M N1 ))(λ + 1)
α = 0.5001; σ = 2.38

sF B (µ, λ) =The vertical diﬀerences in these graphs indicate the under-eﬀort level. We cansay that the under-eﬀort level situation indicated in lower graphs is overcomeby the contract and AP and AL reach the ﬁrst best eﬀort levels indicated in24

upper graphs in these Figures.
Also we can see the eﬀects of perturbation of µ, σ and λ on the eﬀort level by
seeing the relevant slopes of these graphs. The exact eﬀects of these can beindicated by the derivatives of bF B , sF B and bSB , sSB with respect to each ofµ, σ, λ and α. For the sake of space, we write only some of them here.
−λ
∂bSB
=
<0
∂µ
λ+1
∂bSB
λ2
=
>0
∂σ
λ+1
1
∂bSB
1
=
[(λ + 1){ − µ + σ 2 λ}∂λ(λ + 1)2λ
λ2 σ 2− {ln α + ln λ − λµ ++ ln N1 }]
2
SB
∂b
1
=
>0
∂α
α(λ + 1)
µ increasing means that expected level of state of nature improves. Then toreach the same level of utility, you need less eﬀort. Hence the derivative isnegative.
σ increasing means that the state of nature is more volatile and more uncertain.
The expected level of utility decreases, which means also world for you shrinks
under more uncertain environment. Then to reach the same level of utility, youneed more eﬀorts. Hence the derivative is positive.
α increasing means that you will share more realized value of the project. Since
the stake for the eﬀorts of AP is higher, the eﬀort for the higher stakes is also
higher at the optimal. Hence the positive value of the derivative. α has nothing
to do with the state of nature.
The eﬀect of λ change on b is a little complicated. λ is just like the risk averse
parameter in the CARA utility function in the valuation and cost reduction
functions in our settings. The perturbation of it means the objective valuation
function is more against risk (i.e., low certainty equivalent), although we assume
risk neutral for AP and AL. Its derivative is so complex that by only seeing thederivative we cannot readily tell the sign. From the graphs in Figure 6, the slopeof derivative is ﬁrst positive starting from λ = 0, then turns to be negative andagain turn to be positive between λ = 1.5 and λ = 2. We have to be careful
to pick the number for λ, since the its eﬀect on the level of eﬀorts is rather
big and marginal eﬀect of changing λ is complicated in order to avoid wrong
implications from model analysis.
With the derivation of (bF B , sF B ) and (bSB , sSB ) above, we can put these back25

into ŪAP + ŪAL in both ﬁrst best situation with contract and second best
situation without contract. Then by taking the diﬀerence of the two situations
utility levels ResU , we can capture the utility loss of not having proper contract
and of not reaching the ﬁrst best eﬀort levels on both sides but of being restored
by making a proper contract satisfying the condition of Proposition 2 in our
settings.

SB SB
SB SB
ResU ≡ ŪAP (bF B ) + ŪAL (sF B ) − [ŪAP(b ) + ŪAL(s )]The Figure 7 is the graph for the utilities loss from under-eﬀorts by not having
proper speciﬁc performance contract in our settings that are restored by thecontract to going back to the ﬁrst best eﬀort levels. As you might see from the
Figure 7, ResU is in fact positive over the entire range for all the four choiceparameters, α, µ, σ and λ in our settings. Figure 7 left is the loss explainedby µ and σ. Figure 7 right is the loss explained by µ and λ. The ﬁve diﬀerent
sheets of graphs are for 5 diﬀerent levels of α. α indicates the share balance
between the two in the hold up situation. α = 0.5 is perfectly balanced between
AP and AL. The value near to 0 or 1 indicates less balanced share among AP
and AL.
(Figure 7 about here)
The perturbation eﬀects on the utility loss are more complicated than that on
eﬀort levels, since functions and their derivatives are more complicated. The
eﬀect of µ and σ, however, in our settings cold be said, though the exact eﬀects
are dependent on other parameters, that their eﬀects on utility loss are sameas in the eﬀort levels. The eﬀect of perturbation of λ is also complicated justlike in the eﬀect of λ change on eﬀort levels. We can see the eﬀects in Figure 7
right, although the exact eﬀects are dependent on other parameters just like inthe case of µ and σ.
The eﬀect of α change is the most complicated one. We can see from the twographs in Figure 7 that, roughly speaking, the more balanced the share of value
between AP and AL is, the less utility loss results from under-eﬀorts without
contract. Although the exact eﬀects are dependent on the other parameters,
especially λ, the graph reading means the loss is greater for α = 0.3 or 0.7 than
α around 0.5. The same caution for λ for eﬀort level is valid here for both α aswell as λ.
By seeing the two graphs in Figure 7, we could say that if the factors indicate
good prospect (high µ), low uncertainty (low σ), modest ”risk averse” (middle
λ value) and more balanced share in hold up situation (α more toward 0.5),
then the utility loss is not so severe. On the other hand, if the factors indicate
poor prospect (low µ), high uncertainty (high σ), high ”risk averse” (extreme26λ value) and less balanced share in hold up situation (α more away from 0.5),
then the utility loss could be severe and the restoration by contract of the loss
is more imminently needed.

7

Concluding Remarks

As an example of airport-airline vertical contractual relationships, ’Load Factor Guarantee Mechanism’ contract, actually agreed at Noto Airport in Japan,
stipulates that the airline agrees to serve the airport and the airport agrees to
pay to (or receive from) the airline based on the diﬀerence between the target
load factor set at the start of the period in the contract and a realized one.
Airports are often local governments and such payments are controversial since
tax money is on the line.
From AP perspective, such contract ensures the commitment of AL to serve
AP as well as the sharing of up-side proﬁt in high load factor case. From AL
perspective, such contract serves not only as a risk mitigating tool to compensate down-side loss of revenue but as an incentive device to extract AP eﬀorts
to avoid large payment hence overcoming its under-eﬀorts seen typically after
start of the service at the airport.
In this study, we try to model such AP-AL vertical contractual relationship,
namely joint-venture type relationship, in hold up problem settings under theincomplete contract theory framework. In such relationship, both AP and AL
are buyer and seller at the same time and the relationships are not economicallyseparable. For this, we modiﬁed the model of Hart and Moore (1988) so as tofollow such joint-venture type contract.
With this modiﬁcation, we can model the situation in a local airport to pay
incentive money to LCCs beyond the discount of landing fee. Also we can construct the structures in which the valuation of AL’s side to service at AP is not
so lucrative, although serving there meets the minimum participation utility
level, and the incentive money from AP to AL plays an important role in AL’s
decision to actually serve AP.
We show that, without the contractual commitment of providing air transport
service to the airport, there is an under-eﬀort problem under the usual incomplete contract settings. The speciﬁc performance contracts such as Noto case,however, can achieve the ﬁrst best levels of eﬀorts and restore utilities losses
under the modiﬁed ﬁrst best conditions of Hart and Moore (1988).
We also show, by numerical examples, that if we have good prospect of project,
low uncertainty, modest ”risk averse” and more balanced share in hold up situation, then the utilities loss is not so severe. On the other hand, if we have poor

27

project prospect, high uncertainty, high ”risk averse” and less balanced sharein second best situation, then the utilities loss could be severe and the utilities
loss restoration by such contracts is more imminently needed.
With these results, we could justify at least partly from theoretical point ofview such incentive/risk mitigating payments by vertical contract from local orsecondary airport to LCCs, for example, even when the airport is local government. This is because such contracts can achieve the ﬁrst best eﬃcient eﬀort
levels and restore the ﬁrst best utilities levels. Also the risk mitigating aspectsof zero-sum payment by such contracts could lead to the eﬃcient pure risk sharing.
The next steps would be, among other points, to model the multi-year contract.
Since our model is one year setting, multi-year dynamic model would be more
realistic. Another possibility is to include the eﬀort externalities. In our model
each eﬀorts does not aﬀect the other s valuations. With eﬀort externalities,
AP’s eﬀorts can directly enhance the valuation of AL and vice versa. Also we
could look into more speciﬁc eﬀorts, such as price discount or frequency increase, in the context of network industries. These modiﬁcations could enable
us to model much more interdependent and plausible relationships between AP
and AL.
With such reﬁnements, we have the better possibility to model more closely thecomplex relationship and dynamic negotiations between APs and ALs in thereal world. This could contribute to better public policy analysis on the matter.

8

Acknowledgement

The author expresses gratitude to Professor Naoki Makimoto (University ofTsukuba), Professor Yoshitsugu Kanemoto (University of Tokyo) and Professor
Anming Zhang (University of British Columbia) for their valuable comments
and questions. 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

First we present the proof of the existence of Nash equilibrium. Then, we show
the inequality (14) holds.

28

Here, to show the existence of Nash equilibrium, we follow the explanation inFundenberg and Tirole (1991) and Itoh (2003) about the existence of purestrategy Nash equilibrium in super-modular game.
V (b, s) has increasing diﬀerence property by assumption. Therefore, it is relahu
hutively easy to see that ŪAP(b, s) and ŪAL(b, s) also satisfy increasing diﬀerence
property. We derive this as follows;huŪAP(b, s) =Eθ [α{vAP (b, θ) − cAP (b, θ) + vAP (s, θ) − cAL (s, θ)}] − φ(b)=αV (b, s) − φ(b)huŪAL(b, s) =Eθ [(1 − α){vAP (b, θ) − cAP (b, θ) + vAL (s, θ) − cAL (s, θ)}] − ψ(s)}=(1 − α)V (b, s) − ψ(s)So for example,hu∂ ŪAP(b, s)
0=Eθ [α{vAP(b, θ) − c0AP (b, θ)}] − φ0 (b)∂bhu∂ 2 ŪAP(b, s)=0∂b∂sHence,hu∂ 2 ŪAP(b, s)
≥0.
∂b∂s

(A.1)

hu
hu
Therefore ŪAP(b, s) and ŪAL(b, s) also satisfy increasing diﬀerence property.

In addition, b and s, which are elements of pure strategies proﬁle S{b ∈ S, s ∈ S}
for AP and AL, are one dimension. S is easily shown, by assumptions, that it
is non-empty and compact. So the game by AP and AL with payoﬀ function
hu
hu
ŪAP
(b, s) and ŪAL(b, s) respectively, are super-modular game. Therefore theset of pure strategy Nash equilibria is non-empty and has greatest and least
points.
Now we show the inequality (14) holds. Here we follow the simple explanationin Itoh (2003) about under-eﬀort level Nash equilibrium.
Suppose the condition (bhu , shu ) < (b∗ , s∗ ) in equation (14) does not hold.
First we assume b∗ ≤ bhu in conﬂict with the condition above.
If s∗ ≤ shu , then29

0 >[V (bhu , shu ) − φ(bhu ) − ψ(shu )] − [V (b∗ , s∗ ) − φ(b∗ ) − ψ(s∗ )]=αV (bhu , shu ) − φ(bhu ) + (1 − α)V (bhu , shu ) − ψ(shu )− [αV (b∗ , s∗ ) − φ(b∗ )] − [(1 − α)V (b∗ , s∗ ) − ψ(s∗ )≥αV (b∗ , shu ) − φ(b∗ ) + (1 − α)V (bhu , s∗ ) − ψ(s∗ )− [αV (b∗ , s∗ ) − φ(b∗ )] − [(1 − α)V (b∗ , s∗ ) − ψ(s∗ )]=α[V (b∗ , shu ) − V (b∗ , s∗ )] + (1 − α)[V (bhu , s∗ ) − V (b∗ , s∗ )]≥0.
The ﬁrst strict inequality is because, by assumption, (b∗ , s∗ ) is the unique efﬁcient eﬀort level. The second inequality comes from (bhu , shu ) is Nash equilibrium. The third inequality is since V (b, s) is, by assumption, monotonicallyincreasing function. Hence contradiction.
If s∗ > shu , then,0 >[V (bhu , s∗ ) − φ(bhu ) − ψ(s∗ )] − [V (b∗ , s∗ ) − φ(b∗ ) − ψ(s∗ )]>[αV (bhu , s∗ ) − φ(bhu )] − [αV (b∗ , s∗ ) − φ(b∗ )]≥[αV (bhu , shu ) − φ(bhu )] − [αV (b∗ , shu ) − φ(b∗ )]≥0.
The ﬁrst strict inequality is because, by assumption, (b∗ , s∗ ) is the unique efﬁcient eﬀort level. The second strict inequality comes from α ∈ (0, 1) andV (b, s) being monotonically increasing function. The third inequality is because
V (b, s) satisﬁes ”increasing diﬀerence” condition. The last inequality comesfrom (bhu , shu ) is Nash equilibrium. Hence contradiction.
By symmetry, the contradiction case for s∗ ≤ shu goes the same way.
Q.E.D.

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32

Figure 1

P(θ) schedule by ‘Noto Contract’ (+: AP→AL)
P(θ)

2.5

x 10

8

8

P(θ)

2.5

1st year

2

1

1

0.5

0.5

θ

0

θ

0
-0.5

-0.5
-1

-1

-1.5

-1.5

-2

-2

-2.5

-2.5
0

0.2

0.4

0.6

0.8

0

1

8

2.5

3rd
year

2
1.5

1.5

P(θ)

x 10

0.2

0.4

0.6

0.8

1

8

x 10

P(θ)

2.5

2

2

2nd
year

1.5
1

x 10

4th year

1.5
1

0.5

θ

0
-0.5

0.5

θ

0
-0.5

-1

-1

-1.5

-1.5

-2

-2

-2.5

0

0.2

0.4

0.6

0.8

1

-2.5

0

0.2

0.4

0.6

0.8

1

Table 1

Noto Case Summary Information
Yr Target
θ(LF)

Model
θ(LF)

Actualθ(LF）# of Actual
Passenger- P(θ)
AP←AL

1

70%

65%

79.5%

151,015973.29

2

63%

63.7%

64.6%

155,623

159.80

3

64%

63.5%

66.5%

160,052

200.00

4

62%

65.0%

65.1%

156,6540

Figure 3

Target θ(LF) is a load factor AP and AL agreed on in the contract.
Model θ(LF) is an estimated load factor based on econometric model in Hihara(2008).
Actual θ(LF) is the load factor that actually realized.

(100,000 ¥)Figure 2

Time Line for No Contract
t0

t1

No Contract

t2

t3

t4

time

p(θ) is paid.
θ is realized.
(possible(Re)negotiation
Efforts (b,s) aredisputes)on p based on
chosen by Airlportrealized θand Airline, if air
Anticipate Nash Bargaining
service is provided.
Solution
Under-efforts by AP&AL inevitable;
Hold-Up Problem

Figure 3

Time Line for Contract (p, q) = (p(θ),1)
t0

t1

Contract
is signed.

Air+
Serviceprovided;
q=1

t2

t3

θ is realized. (Renegotiation p(θ) is paid.
on p based on
Efforts(b,s) are realized θ)chosen by
Airport andAirline.

Anticipate contract
execution

t4

time

(possible
dispute)Figure 4

Condition for Overcoming Under-effort Problem by Contract

4

x 10

8

vAP - cAP
3
vAL- cAL

p(θ)
2

v,c,p

1

0

-1
cAL- vAL

-2

-3

0

0.1

0.2

0.3

0.4

0.5
θ

0.6

0.7

0.8

0.9

1

Figure 5 Under-efforts and Recovery by Contract- 1 (b/s by μ & σ)

1.

The red graph is the first best effort level of AP by contract.

2.

The blue graph is the first best effort level of AL by contract

3.

The black graph is the second best effort level of AP and AL without contract.

Figure 6 Under-efforts and Recovery by Contract-

2(b/s by μ & λ)1.

The red graph is the first best effort level of AP by contract.

2.

The blue graph is the first best effort level of AL by contract

3.

The black graph is the second best effort level of AP and AL without contract.

Figure 7

Utilities Losses from Under-Efforts that are Recovered by Contract(left

graph: Loss by μ & σ;right graph: Loss by μ & λ)The graphs are the difference between first best expected utility level of both AP and AL
restored by contract minus second best expected utility level of AP and AL.
Graphs in colors are;red forα=0.5;

blue forα=0.2;

green forα=0.8;

cyan forα=0.1;

yellow forα=0.9.