optimal service commission contract design of ota to

Research ArticleOptimal Service Commission Contract Design of OTA to CreateO2O Model by Cooperation with TTA underAsymmetric Information

Pingping Shi1 Yaogang Hu 2 and Yongfeng Wang1

1School of Management Chongqing University of Technology Chongqing 400054 China2School of Electrical and Electronic Engineering Chongqing University of Technology Chongqing 400054 China

Correspondence should be addressed to Yaogang Hu 569934301qqcom

Received 16 March 2021 Revised 19 April 2021 Accepted 21 April 2021 Published 3 May 2021

Academic Editor Ming Bao Cheng

Copyright copy 2021 Pingping Shi et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

is paper studies the service commission contract of an online travel agency (OTA) to integrate the online to offline (O2O)modelby cooperation with a traditional travel agency (TTA) under asymmetric information e principal-agent models are establishedwith symmetric and asymmetric service information respectively Further the impacts of asymmetry information on the revenueof the OTA TTA and the whole O2O model and the properties of optimal commission contract are analyzed e paper notesmanagement implications (1) OTA designs service commission contracts by weighing the fixed payment and service commissioncoefficient for different incentives to TTAs with different serviceabilities and (2) because the existence of asymmetric informationalways leads to the damage of OTArsquos expected revenue OTA should encourage the TTA to disclose private service information

1 Introduction

With the rapid development of Internet and informationtechnology it is a popular and convenient way for touriststo buy more diverse tourism products and services throughonline travel agencies (OTAs) [1ndash3] However onlinetourism product homogenization makes OTAs reduce theservice quality to attract more tourists such as cashbackand low price forcing consumption in the experiencewhich seriously increase touristsrsquo complaints and hinderthe healthy development of online tourism [4ndash6] To im-prove competitive advantage and realize differentiationsome OTAs (eg Ctrip) open up offline stores to achieveonline to offline (O2O) model for providing personal in-formation and advice to tourists and some cooperate withtraditional travel agencies (TTAs) such as Uzai (httpwwwuzaicom) and ZhongXin TTA Lvmama (httpwwwlvmamacom) and JinJiang TTA [4] In the processof cooperation between OTAs and TTAs it is difficult forOTAs to observe TTAsrsquo service information which makesthe cooperation and incentive problem more complicated

erefore how to design the commission contract of TTAsstrategically for OTAs is an urgent problem to implementO2O strategy

e O2O model as a new e-commerce business modelcombines online trading and offline experience and hasbecome an important strategy for the development of en-terprises in recent years [7] With the rapid development ofOTAs the online channel plays a crucial role in tourism andhospitality and the tourism O2O model achieved by thecooperation between OTAs and hotels or airlines is acommon phenomenon erefore there is a growing pop-ularity on the cooperation problem between OTAs andhotels or airlines and they have gotten some effective co-operation strategies [1 8ndash11] However to seek betterbusiness opportunities OTAs have to attach importance tothe offline service to achieve differentiation Although somescientific researchers have demonstrated the importance ofTTAsrsquo advice-offering and OTAsrsquo attributes in travelersrsquobooking little literature in the hospitality and tourism fieldshas studied the service contract problem of cooperationbetween OTAs and TTAs

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 9934726 17 pageshttpsdoiorg10115520219934726

To fill this gap and provide some suggestions for OTAsand TTAsmanagers on establishing the O2Omodel throughservice cooperation this paper proposes a cooperationmodel to describe decision interactions of an OTA and aTTA e OTA and TTA play a principal-agent model inwhich the OTA as the principal player determines theservice commission and the TTA as the agent determinesservice effort In addition in the O2O model because theOTA and TTA are relatively independent TTArsquos privateinformation is difficult to disclose and it often hides its ownservice information to obtain higher revenue By comparingthe case of information symmetry and asymmetry theimpacts of asymmetry information on OTArsquos service con-tract TTArsquos information value and the revenue of OTA andTTA are analyzed

e rest of this paper is organized as follows Section 2reviews the related literature Section 3 describes the servicecooperation between an OTA and a TTA Section 4 analyzesand compares equilibriums of symmetric and asymmetricinformation and further analyzes equilibriums when TTArsquosability is continuous Section 5 presents the results of nu-merical analyses Section 6 concludes this paper by sum-marizing some of the managerial implications obtained

2 Literature Review

According to the purpose of studying the service coopera-tion contract design of OTAs and TTAs this section reviewsthree distinct of literature studies about the O2O modelservice cooperation and cooperation between tourism en-terprises and OTAs

21O2OModel Since Alex Rampel put forward the conceptof the O2O model it has attracted wide attention in aca-demia Although the concept and classification of the O2Omodel are different because of the scholarsrsquo different re-search perspectives it is a business model for the effectiveintegration of online and offline stores [4 7] It is believedthat O2O is a new business model which can discloseproduct information change consumer brand awarenessand mitigate the adverse effects of product uncertainty onconsumersrsquo shopping decisions [12] By using 311 respon-dent data from O2O e-commerce users in Greater JakartaArea Savila et al showed that both multichannel integrationand trust have a significant effect on both customer onlineloyalty and customer offline loyalty that drive customerrepurchase intention [13]

According to the existing research about the O2Omodelit can be divided into two categories one is to purchaseonline and then experience offline which can also be calledonline to offline the other is to experience offline and thenpurchase online which can also be called offline to online[14] In the hospitality and tourism industry becausetourists have to experience tourism products offline theO2O model achieved by the cooperation between OTAs andhotels or airlines is a common phenomenon From theperspective of the supply chain scholars more emphasize thesale cooperation between the OTAs and hotels or airlines

and have gotten some effective cooperation strategies[1 8ndash11] In the retail industry as the major characteristicfeature of a product is tangible it can be touched and triedoute O2Omodel achieved by cooperating with others oropening by oneself [14 15] becomes the main researchdirection Considering that the service cost and effort in-formation of the showroom are asymmetry Jin et al studythe design of the commission contract of offline to onlinewhich is considered that the showroom and online retailerhave impacts on consumersrsquo purchasing decisions respec-tively [15] In practice online sale efforts and offline serviceswill affect the demand for each other erefore it is difficultto highlight the integrity of the online retailer and storecocreating O2O model by splitting online and offlinedemand

Despite the impact of technology and the advent ofonline bookings offline channels play a pivotal role in thetourism supply chain Unlike the general retail product thetourism product of a travel agency cannot be touchedtransported stored or returned e functions of offlinechannels are different between general retailer and travelagency e offline channels of general retailer providetouching trying and even online return service for con-sumers e offline channels of travel agency provide sceneexperience personal and professional information andadvice to meet travelersrsquo demands on a continuous basis bygathering and organizing information [16 17] e O2Omodel of travel agency in this paper is an integration ofonline and offline Both online sale efforts and offline ser-vices will have an impact on the overall demand for the O2Omodel

22 Service Cooperation e existing literature on servicecooperation is mostly concentrated in the supply chain fieldService cooperation not only reduces the competition be-tween direct and retail channels but also improves supplychain revenue [18ndash21] Unit service reward and service costsharing are the main forms of service cooperation To avoidchannel conflict and improve service efficiency Xiao Danand Zhang studied after-sale service cooperation betweendirect channel and retailers in which direct channel pays unitservice commission to retailers [18] In the O2O retailmarket the service cost sharing mechanism is introduced tocoordinate conflicts and achieve a win-win strategy therebyimproving the performance of the entire O2O supply chain[19] Due to the importance of presales services in pur-chasing decisions Zhou et al introduce that the service costsharing contract can effectively stimulate the retailer toimprove his service level while free riding occurred [20]Considering that the retailer provides the same service levelin both channels or not Yang and Zhang compare themanufacturerrsquos optimal profit and retailerrsquos optimal profitunder the condition of different services and the sameservice and provide differentiated services to enable thesystem to achieve the optimal profit [21] However thecompletely symmetrical assumptions in the above literatureare not currently common practice e adverse selectionand moral hazard are always in the cooperation process [22]

2 Mathematical Problems in Engineering

23 Cooperation between Tourism Enterprises and OTAsClearly the OTAs are an extremely important part of thetourism supply chain recognized as an online distributionchannel for hotels and airlines A larger number of scholarshave emerged to pay attention to the issue of the cooperationproblem between hotels airlines and OTAs such as pricingstrategy cooperation form and coordination strategy[1 8ndash11] Koo et al point out that the airlines are less likelyto use OTA platforms if the airlines have a large loyalconsumer base or if the OTA platform is highly competitive[8] Ling et al and Guo et al study the optimal pricingstrategy for tourism hotels when they operate their onlinechannel by cooperation with an OTA [1 9] Considering theoverbooking of hotels Dong and Ling study the pricing andoverbooking strategies of a hotel in the context of cooper-ation with multiple OTAs and analyze how these strategiesinfluence the cooperation process [10] e commonly usedldquofirst come first serverdquo form puts hotels in a disadvantageousposition especially when the OTAs have much more marketattractiveness than the hotels Xu et al propose a new formnamed ldquosetting Online-Exclusive-Roomsrdquo for a hotel tocollaborate with a third-party website on room bookingservice [11]

To get a new business opportunity the TTAs are in-creasingly aware of the need to open up online marketsrsquocooperation with OTAs because of the low volume of visitsand lack of e-commence operating experience of self-builtwebsite [3 4] erefore a few scholars have providedsuggestions on the cooperation between TTA and OTABased on the resource-based view Shi and Long analyze thecomplementary resources input decisions of OTAs andTTAs cocreating the O2O model which do not propose aclear form of cooperation [3] In a supplemental study Longand Shi study the optimal pricing strategies of a tour op-erator and an OTA when they achieve the O2O businessmodel through online sale and offline service cooperation todevelop the advantage of complementary resources [4]

However no study has addressed the problem of anOTA that wants to achieve its O2O model by cooperatingwith a TTA with private information In this situation theOTA has little information and experience regarding offlineservice and knows little about how to design a servicecontract to disclosure TTArsquos real information To enrich thescientific literature and provide some suggestions to OTA onhow to pursue offline service this paper studies the optimalservice contract of OTAs under asymmetric informationthrough the analysis of a simple O2Omodel composed of anOTA and a TTA

3 Problem Description and Assumptions

In this paper the O2O model combining both trading andoffline service achieved by service cooperation of an OTAand a TTA is considered herein Table 1 summarizes themain notation and its definitions used in this paper

e OTA makes some effort to induce customers tomake reservations through its website such as rankingposition According to the literature [1] c (eO) is a convexincreasing function with dc(eO)deO gt 0 d

2c(eT)de2T gt 0

and c(eO) 0 e sale cost of OTA is c(eO) ηOe2O2where ηO is the OTArsquos sale cost coefficient which is widelyused in the cost management literature [23ndash26]

While the TTA provides offline services for tourismproducts such as the scene experience and personalized andprofessional travel advice the service effort is the cost ofhiring salespersons by TTA According to the literature[4 25 26] a strictly convex service function c (eT) is used todepict TTArsquos unit cost of service effort c(eT) ηTe2T2where ηT is the TTArsquos service cost coefficientdc(eT)deT gt 0 and d2c(eT)de2T gt 0

e demand for O2O model q is influenced by OTArsquossaleability sO sale effort eO TTArsquos serviceability sT serviceeffort eT market scale θ and market random factors ξ

q f sO eO( 1113857 + g sT eT( 1113857 + u(θ) + ξ (1)

where fsOgt 0 fsO

sO le 0 feOgt 0 feOeO

le 0 gsTgt 0 gsTsT

le 0geTgt 0 geTeT

le 0 uθ gt 0 uθθ le 0 and ξ follows a normaldistribution ξ isin N(0 σ2)

In this paper according to the literature [27] let f (eO)

sOeO g(eT) sTeT and u (θ) θ us the demand for O2Omodel is q sOeO + sTeT + θ + ξ Assuming that the revenue ofO2O model is proportional to its demand the revenue ofO2O model can be simplified into Π(q) sOeO+

sTeT + θ + ξ Following the studies of Jin et al [15] and Heet al [28] OTA provides service commission (a b) tomotivate TTA to make service effort where a representsfixed payment and b denotes service commission coefficientor the rate of revenue sharing erefore given (a b) theservice commission that OTA pays for TTA isT(q) a + bΠ(q)

Based on the above assumptions the OTArsquos revenuefunction is obtained as follows

ΠO Π(q) minus T(q) minus c eO( 1113857 (1 minus b) sOeO + sTeT + θ + ξ( 1113857

minus a minusηOe

2O

2

(2)

Assuming that OTA is risk-neutral and its utility is equalto expected revenue the OTArsquos expected revenue is

EΠO (1 minus b) sOeO + sTeT + θ( 1113857 minus a minusηOe

2O

2 (3)

e TTArsquos revenue is

ΠT T(q) minus c eT( 1113857 a + b sOeO + sTeT + θ + ξ( 1113857 minusηTe

2T

2

(4)

Assuming that TTA is risk-neutral risk aversion one formof the utility function dominant in both theoretical and appliedwork in areas of decision theory and finance is the exponentialutility function Following the literature [29]UT(ΠT) minus eminus rΠT where r denotes TTArsquos risk aversion co-efficient and rgt 0 r 0 and rlt 0 are risk aversion risk neu-trality and risk preference respectively By certaintyequivalencemethod the revenue of TTA is obtained as follows

Mathematical Problems in Engineering 3

EΠT a + b sOeO + sTeT + θ( 1113857 minusηTe

2T

2minus

rσ2b2

2 (5)

Due to the uncertainty of the O2O model market de-mand and equation (5) TTArsquos risk cost is rσ2b22

4 Equilibrium Solutions and Analysis

is paper mainly studies the service cooperation contractdesign of OTA creating O2O model by cooperating with TTAunder symmetric and asymmetric information Further thecontract model with continuous type of TTArsquos serviceabilityunder information asymmetry is researched Here the su-perscripts ldquoNrdquo ldquoSrdquo and ldquoCrdquo represent the case of symmetricinformation the discrete type and continuous type of TTArsquosservice capability under information asymmetry respectively

In the process of establishing service cooperation be-tween an OTA and a TTA the serviceability and effort areTTArsquos private information which OTA is difficult to observeSimilar to the study of Li et al [30] it is assumed that thereare two possibilities for TTArsquos serviceability high service-ability sTH and low serviceability sTL and sTHgt sTL

41 Equilibrium of Symmetric Information In the case ofsymmetric information the OTA fully knows the state ofTTArsquos serviceability sTi (iH L) e OTA designs a servicecontract (aN

i bNi ) and sale effort eN

Oi when the type of TTArsquosserviceability is i en the expected revenue of OTAEΠO(aN

i bNi ) and TTA EΠTi(aN

i bNi ) and the optimization

problem P1 is

maxaN

ibN

ieN

Oi

EΠO 1 minus bNi1113872 1113873 sOe

NOi + sTie

NTi + θ1113872 1113873 minus a

Ni minus

ηOeN2Oi

21113888 1113889

st(IR minus i)aNi + b

Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(6)

where

eNTi argmax

eT

aNi + b

Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2⎛⎝ ⎞⎠ (7)

In the above optimization problem P1 the OTA max-imizes its revenue Inequality (IR-i) is individual constraintsassuring that the TTA will join the service cooperationbecause of exceeding the reservation revenue ΠT Equation

(7) denotes the TTA optimizes service effort by maximizingits revenue e optimal decisions are found out by solvingthe above optimization problem P1 using the backwardinduction method

Table 1 Notation and definitions

Notation DefinitionssO Sale ability of OTAeO Sale effort of OTAC (eO) Sale cost of OTAηO Sale cost coefficient of OTAsT Serviceability of TTAeT Service effort of TTAηT Service cost coefficient of TTAC (eT) Unit cost of service effort of TTAQ e demand for O2O modelθ Market scaleξ Market random factorsT (q) Service commission that OTA pays for TTAA Fixed paymentB Service commission coefficient or the rate of revenue sharingΠ (q) ΠO ΠT e revenue of O2O model OTA and TTA respectivelyEΠO EΠT e expected revenue of OTA and TTA respectivelyUT (EΠT) e utility function of TTAR TTArsquos risk aversion coefficientI e state of TTArsquos serviceability iH LsTi TTArsquos serviceability iH Lρ e probability that the OTA believes that the TTArsquos serviceability is at a high stateNote that the superscript lowastdenotes the optimal solutions


Theorem 1 In the case of symmetric information given thetype of TTArsquos serviceability as i the OTArsquos optimal servicecontract (aNlowast

i bNlowasti ) sale effort eNlowast

Oi and the TTArsquos optimalservice effort eNlowast

Ti are

bNlowasti

s2Ti

s2Ti + ηTrσ2

aNlowasti ΠT minus

s2O

ηO

+ θ1113888 1113889bNlowasti +

12

rσ2 minuss2Ti

ηT

1113888 1113889 bNlowasti1113872 1113873

2

eNlowastOi

sO

ηO

eNlowastTi

s3Ti

ηT s2Ti + ηTrσ21113872 1113873

sTi

ηT

bNlowasti

(8)

eorem 1 shows that (1) with the TTArsquos service costcoefficient ηT market uncertainty σ2 and risk aversion coef-ficient r increasing the service commission coefficient bNlowast

i

decreases (2) when the market scale θ and OTArsquos sale abilitydecreasing sO or OTArsquos sale cost coefficient ηO increasing theOTArsquos fixed payment increases aNlowast

i so as to encourage TTA toparticipate in cooperation for cocreating the O2O model Inaddition sO and ηO only affect OTArsquos fixed payment aNlowast

i buthave no effect on service commission coefficient bNlowast

i (3) theOTArsquos online sales effort eNlowast

Oi is only related to its own saleability

sO and cost coefficient ηO and is directly proportional to sO andinversely proportional to ηO (4) with the service commissioncoefficient bNlowast

i and TTArsquos serviceability sTi increasing or TTArsquosservice cost coefficient ηT decreasing the TTArsquos offline serviceeffort eNlowast

Ti increases

42 Equilibrium of Asymmetric Information In the case ofasymmetric information the TTA surely knows which of thetwo serviceability states will occur while the OTA has only asubjective assessment about the likelihood of the two ser-viceability states Let ρ be the probability that the OTAbelieves that the TTArsquos serviceability is at a high state and 1-ρbe the probability of the low state Due to its parsimony andtractability for analysis this type of asymmetric informationhas been commonly employed in supply chain contracting[15 27 30] In this paper the OTA is an uninformed partythat acts as a principal and the TTA is an agent that holdsprivate information about its serviceability e servicecontract design problem is investigated under asymmetricserviceability informatione goal of the OTA is to design amenu of service contracts so as to maximize its expectedprofit based on the revelation principle

When the state of TTArsquos serviceability is i and OTA givesservice contract (aS

i bSi ) and sale effort eS

Oi the revenue ofOTA and TTA is EΠO(aS

i bSi ) and EΠTi(aS

i bSi ) respectively

and then the optimization problem P2 is

maxaS

HbS

HeS

OHaS

LbS

LeS

OL

EΠO ρ 1 minus bSH1113872 1113873 sOe

SOH + sTHe

STH + θ1113872 1113873 minus a

SH minus

ηOeS2OH

21113888 1113889 +1113896

(1 minus ρ) 1 minus bSL1113872 1113873 sOe

SOL + sTHe

STL + θ1113872 1113873 minus a

SL minus

ηOeS2OL

21113888 11138891113897

st(IC minus H)aSH + b

SH sOe

SOH + sTHe

STH + θ1113872 1113873 minus

ηTeS2TH

2minus

rσ2b2H2ge

aSL + b

SL sOe

SOS + sTHe

LTH + θ1113872 1113873 minus

ηTeL2TH

2minus

rσ2b2L2

(IC minus L)aSL + b

SL sOe

SOL + sTLe

STL + θ1113872 1113873 minus

ηTeS2TL

2minus

rσ2b2L2ge

aSH + b

SH sOe

SOH + sTLe

HTL + θ1113872 1113873 minus

ηTeH2TL

2minus

rσ2b2H2

(IR minus H)aSH + b

SH sOe

SOH + sTHe

STH + θ1113872 1113873 minus

ηTeS2TH

2minus

rσ2b2H2ge ΠT

(IR minus L)aSL + b

SL sOe

SOL + sTLe

STL + θ1113872 1113873 minus

ηTeS2TL

2minus

rσ2b2L2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)


where

eSTH argmax

e

aSH + b

SH sOe

SOH + sTHe

STH + θ1113872 1113873 minus

ηTeS2TH

2minus

rσ2b2H2

1113888 1113889 (10)

eSTL argmax

e

aSL + b

SL sOe

SOL + sTLe

STL + θ1113872 1113873 minus

ηTeS2TL

2minus

rσ2b2L2

1113888 1113889 (11)

In the above optimization problem P2 the expectedrevenue of OTA EΠO is the sum of revenue under thecooperation of serviceability states of TTA Inequalities (IC-i) are incentive compatibility constraints assuring that theTTA does not pretend to choose the other serviceabilitystate where eL

TH(eHTL) indicates the optimal service effort

level of TTA under service reward contract (aSL bS

L)((aS

H bSH)) Inequalities (IR-i) are individual rationality

constraints assuring that the TTA will join to cocreate O2Omodel because of exceeding the reservation revenue ΠT Equations (10) and (11) denote the TTA optimizes its service

effort eNTi by maximizing its revenue according to OTArsquos

service cooperation contract (aSi bS

i ) and sale effort eSOi e

optimal decisions are found out by solving the above op-timization problem P2 using the backward inductionmethod

Theorem 2 Under asymmetric information the OTA re-veals TTArsquos serviceability by designing separation contractse OTArsquos service commission contract optimal sale effortand TTArsquos service effort are

aSlowastH ΠT minus

s2O

ηO

+ θ1113888 1113889bSlowastH +

12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastH1113872 1113873

2+

12ηT

s2TH minus s

2TL1113872 1113873 b

SlowastL1113872 1113873

2

bSlowastH

s2TH

s2TH + ηTrσ2

aSlowastL ΠT minus

s2O

ηO

+ θ1113888 1113889bSlowastL +

12

rσ2 minuss2TL

ηT

1113888 1113889 bSlowastL1113872 1113873

2

bSlowastL

(1 minus ρ)s2TL

s2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ2

eSlowastOH e

SlowastOL

sO

ηO

eSlowastTH

s3TH

ηT s2TH + ηTrσ21113872 1113873

eSlowastTL

(1 minus ρ)s3TL

s2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873ηT

(12)

eorem 2 shows the following

(1) When r⟶ +infin bSlowastH bSlowast

L 0 and aSlowastH aSlowast

L

ΠT can be obtained at is when TTA has no risktolerance its revenue could only be equal to its reser-vation revenue When r⟶ 0 bSlowast

H 1 andbSlowast

L (11 + ρ(s2TH minus s2TL(1 minus ρ)s2TL))lt 1 at iswhen the TTA tends to be risk-neutral TTA with highserviceability will obtain the total revenue and if the

difference between the two types of TTArsquos serviceabilityor the probability of TTA with high serviceability isgreater the OTArsquos service commission coefficient forTTA with low serviceability is lower When rgt0 withthe increase of TTArsquos risk aversion coefficient theservice commission coefficient decreases correspond-ingly indicating that risk aversion can offset the in-centive effect of the service commission coefficient


(2) According to (zbSlowastH zsTH)ge 0(zbSlowast

L zsTL)ge 0 theservice commission coefficient will increase with theincreasing TTArsquos serviceability

(3) e service effort of the TTA with low serviceabilityhas no effort on the service commission coefficientthe TTA with high serviceability obtains Howeverthe difference between service effort of TTA withhigh and low service will affect the service com-mission coefficient TTA with low serviceability ob-tains and the greater the difference is the smaller theservice commission coefficient is

(4) According to (zbSlowastH zρ) 0(zbSlowast

L zρ)le 0 theprobability ρ has no effect on service commissioncoefficient that the TTA with high serviceabilityobtains However with the increase of ρ the servicecommission coefficient that the TTA with highserviceability obtains decreases

(5) e relationship between service commission coef-ficient and TTArsquos service cost coefficient ηT thedegree of market uncertainty σ2 the degree of riskaversion r and the relationship between fixed pay-ment and OTArsquos saleability and sale cost coefficientare the same under symmetric information which isomitted here

Corollary 1 For TTA with different serviceability the op-timal service commission contract parameters have the re-lationships bSlowast

H ge bSlowastL and aSlowast

H le aSlowastL

Corollary 1 shows that the motivation purpose of OTAservice contract is different for TTA with different service-abilityismeans that theOTA should balance themotivationcooperation and the incentive of improving service effort forTTA when designing service commission contract If the TTAis of a low serviceability type the OTA will give TTA a largerfixed payment to ensure TTArsquos initiative in cocreating the O2Omodel If the TTA is of a high serviceability state the OTA willgive a larger service commission coefficient And according tobSlowast

H minus bSlowastL ((s2TH minus s2TL) (ρs2TH + (1 minus ρ) ηTrσ2)

(s2TH +ηTrσ2)(s2TL minus 2ρs2TL +ρs2TH + (1 minus ρ)ηTrσ2))ge 0 thegreater the difference between service efforts of TTA with highand low service the bigger the difference of service commissioncoefficient

43 Comparison of Results under Symmetric and AsymmetricInformation

431 Validity of Separation Service Contract By intro-ducing the results of eorem 2 into incentive compatibilityconstraints (IC-i) and comparing the TTArsquos revenue of TTAfalse and truthfulness reporting serviceability it is obtainedas

EΠTH aSlowastH b

SlowastH1113872 1113873 minus EΠTH a

SlowastL b

SlowastL1113872 1113873 0

EΠTL aSlowastL b

SlowastL1113872 1113873 minus EΠTL a

SlowastH b

SlowastH1113872 1113873

bSlowastH minus b

SlowastL1113872 1113873 b

SlowastH + b

SlowastL1113872 1113873 sTH + sTL( 1113857

2ηT

gt 0

(13)

When the TTA with low serviceability pretends to be theTTA with high serviceability the revenue is strictly lower thanthat when TTA truthfully reports serviceability(EΠTL(aSlowast

H bSlowastH )ltEΠTL(aSlowast

L bSlowastL )) erefore the TTA

with low serviceability has no motivation to disguise the TTAwith high serviceability In addition if the TTA is of a highserviceability type its false report will not affect its revenue(EΠTH(aSlowast

H bSlowastH ) EΠTH(aSlowast


with high serviceability has nomotivation to disguise TTAwithlow serviceability It can be seen that separation of servicecontract has the characteristics of ldquoself-selectionrdquo in the OTAand TTA cocreating O2Omodelat is the TTAwith high orlow serviceability needs to choose the service contract corre-sponding to its serviceability type

Corollary 2 By comparing the expected revenue and thereservation revenue with different serviceability types it isobtained as

EΠT sTL( 1113857 ΠT

EΠT sTH( 1113857 ΠT +s2TH minus s

2TL1113872 1113873

2ηT

bSlowast 2L

(14)

Corollary 2 shows that when the TTA is of a low ser-viceability type TTArsquos revenue is reservation revenue afteraccepting the service commission contract However whenthe revenue of TTA with high serviceability is greater thanreservation revenue the difference ((s2TH minus s2TL)2ηT)bSlowast 2

L

shows that in order to obtain private information of TTArsquosserviceability the OTA needs to pay TTA informationsharing fees If the OTA does not pay the information feeTTA may lie about its serviceability and damage the OTArsquosexpected revenue

432 e Impact of Information Asymmetric on ServiceContract and Expected Revenue According to eorems 1and 2 by comparing and analyzing the service contractTTArsquos optimal service effort and the revenues of thepartners under symmetric and asymmetric informationconditions eorems 3 and 4 are obtained

Theorem 3 e optimal service contract sale effort andTTA optimal service effort under symmetry and asymmetryinformation have the following relationships


(1) aSlowastH ge aNlowast

H bNlowastH bSlowast

H

(2) aSlowastL ge aNlowast

L bSlowastL le bNlowast

L

(3) eNlowastTH eSlowast

THeSlowastTL lt eNlowast

TL

(4) eNlowastOi eSlowast

Oi

Ineorem 3 (1) and (2) show that the OTA should pay ahigher fixed payment to mobilize TTA to participate in coc-reating the O2O model under asymmetric information If theTTA is of a low serviceability type theOTA lacksmotivation toencourage TTA to provide higher service effort and the servicecommission coefficient is ldquodistorted downwardrdquo If the TTA isof a high serviceability type the OTA will provide a higherservice commission coefficient to encourage the TTA toprovide higher service effort under asymmetrical informationHere the TTA can obtain additional information rent byldquoupward distortedrdquo fixed paymentis also means that due tothe existence of asymmetric information the service contractdesign of OTA needs to adjust the fixed payment and servicecommission coefficient to ensure that TTA can participate incooperation and choose the appropriate level of service effortsthat are beneficial to OTA From (3) the service effort of TTAwith the high serviceability is the same in symmetrical andasymmetrical information and the service effort of TTA withthe low serviceability is lower in asymmetrical informationthan that in symmetrical information erefore the OTA isdominant in encouraging low serviceability type TTA toparticipate in cooperation which will also provide incentives toparticipate in cooperation and service efforts to TTA with highserviceability state (4) shows that the OTArsquos optimal sale effortis the same in symmetrical and asymmetrical informationisis straightforward because the OTA knows its own saleabilityinformation

Theorem 4 ere is the following relationship betweenOTArsquos and TTArsquos expected revenue under symmetry andasymmetry information

(1) EΠSlowastOi leEΠNlowast

Oi

(2) EΠSlowastTH geEΠNlowast

TH

(3) EΠSlowastTL EΠNlowast

TL

eorem 4 shows that the information disadvantage ofOTA is detrimental to its expected revenue the informationadvantage of TTA does not necessarily generate informationrent When the TTA is the high serviceability type theincrease of fixed payment cost and constant service com-mission coefficient make OTA pay higher service com-mission erefore the TTA with high serviceability can getadditional information rent from the increased fixed pay-ment When the TTA is of a low serviceability type theasymmetric information makes the service commissioncoefficient tilt downward and fixed payment cost rise andthe rise of fixed payment can make up for the loss of TTAcaused by the decline of service commission coefficienterefore the expected revenue of TTA with low service-ability remains unchanged

44 Strategic Analysis When TTArsquos Service Ability IsContinuous Assuming that TTAsrsquo service capability iscontinuously distributed and satisfied sT isin H [sT sT]F(sT) and f(sT) are TTArsquos cumulative distribution anddistribution density function respectively and f(sT)gt 0

Due to the fact that ξ is not related to sT the expectedrevenue of risk-neutral OTA can be obtained as

EΠO sT( 1113857 1113946sT

sT

1 minus bC

sT( 11138571113872 1113873 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873 minus a

CsT( 1113857 minus

ηOeC2O sT( 1113857

21113896 1113897f sT( 1113857dsT (15)

e expected revenue of TTA is

EΠT sT( 1113857 aC

sT( 1113857 + bC

sT( 1113857 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873

minusηTe

C2T sT( 1113857

2minus

rσ2bC2sT( 1113857

2

(16)Firstly determine the service effort level of TTA eC

T(sT) thatis forallsT isinH eC

T(sT) argmaxeEΠT(sT) (sCTbC(sT)ηT)

Bringing eCT(sT) to equation (16) it is obtained as

EΠT sT( 1113857 aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857

+12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857

(17)

Furthermore when sT is continuous the optimizationproblem P3 is

maxaCbC

1113946sT

sT

1 minus bC

sT( 11138571113872 1113873 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873 minus a

CsT( 1113857 minus

ηOeC2O sT( 1113857

21113896 1113897f sT( 1113857dsT

st(IC)bCmiddot

sT( 1113857ge 0

(18)

(IC)aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857 +s

C2T

ηT

minus rσ21113888 1113889bCmiddot

sT( 1113857 0 (19)


(IR)aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge ΠT (20)

where eCT(sT) (sC

TbC(sT)ηT) Theorem 5 When (ddsT)(1 minus F(sT)sTf(sT))lt 0

aClowast

sT( 1113857 ΠT minuss2O

ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT

minus rσ21113888 1113889bClowast2

sT( 1113857 + 1113946sT

sT

ωηT

bC2

(ω)dω

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

eClowastO

sO

ηO

eClowastT sT( 1113857

sCTb

ClowastsT( 1113857

ηT

(21)

eorem 5 shows that when the serviceability of TTA iscontinuous (ddsT)(1 minus F(sT)sTf(sT))lt 0 is the precon-dition for the existence of a service commission separationcontract At this time bClowast(sT) is strictly increasing and allTTA can choose appropriate service contracts corre-sponding to its serviceability type

Corollary 3 EΠT(sT) minus ΠT 1113938sT

sT(ωηT)bClowast2(ω)dω

Corollary 3 shows that apart from the TTA with theworst serviceability all other types of TTA can get strictinformation rent With the increase of TTArsquos serviceabilitythe information rent and the excepted revenue of TTAincrease which is consistent with the conclusion whenTTArsquos serviceability is discrete

5 Numerical and Sensitivity Analysis

e impacts of the risk aversion coefficient serviceabilityand asymmetric information on OTA service contract andboth TTA and OTArsquos revenue are analyzed so as to getmore management implications Considering that thereferences on cooperation between the TTA and OTA areless and numerical examples are usually hypothetical thevalues of parameters are assigned according to Jin et al[15] A selected set of parameters is as follows sO 03ηO 01 ηT 03 sTH 05 sTL 05 ρ 05 θ 0 σ2 4and ΠT 5 In order to study the impacts of TTArsquos riskaversion coefficient r and serviceability sTi on the servicecommission and revenue of both TTA and OTA take sTias the horizontal axis that is when sTH 05 andsTL isin (02 05) or when sTL 05 and sTH isin (05 1) fixedpayment service commission coefficient and revenueunder symmetric information and asymmetric informa-tion are plotted respectively with r 01 r 05 andr 09 e results of numerical examples are summarizedin Figures 1ndash3

51 Analysis of the Parameters of Service CommissionContract Figure 1 shows that (1) when TTArsquos serviceabilitysTi increases fixed payment of aSlowast

i and aNlowasti always de-

creases but aSlowasti gt aNlowast

i is is straightforward because OTAneeds to give TTA more fixed payment to stimulate itsinitiative to cooperate and create an O2O model underasymmetric information (2) when r takes different valuesthe influence of asymmetric information on fixed paymentincreases with the increase of the difference between TTArsquosserviceability which indicates that the smaller the differencebetween TTArsquos different serviceability the smaller the in-fluence of asymmetric information on fixed payment (3) nomatter the serviceability type of TTA is aSlowast

i and aNlowasti in-

crease and the distance between fixed payment curves de-creases with the increase in r indicating that r can alleviatethe impact of asymmetric information on fixed payment

Figure 2 shows that (1) when TTArsquos serviceability sTiincreases service commission coefficient of bSlowast

i and bNlowasti

always increases and bSlowastH bNlowast

H and bSlowastL lt bNlowast

L is in-dicates that the existence of asymmetric information doesnot affect the service commission coefficient of OTA paid forTTA with high serviceability but it will reduce the servicecommission coefficient of OTA paid for TTA with lowserviceability (2) no matter the serviceability type of TTA isbSlowast

i and bNlowasti decrease with the increase in r indicating that r

can offset the incentive effect of service commission coef-ficient on TTA (3) Figure 2(b) shows the distance betweenservice commission coefficient curves decreases with r or sTLincreases which indicates that r can alleviate the impact ofasymmetric information on service commission coefficientand the smaller the difference between TTArsquos differentserviceability the smaller the influence of asymmetric in-formation on service commission coefficient

52 Revenue Analysis of Both TTA and OTA Figure 3 showsthat (1) no matter the serviceability type of TTA is there isΔEΠSNlowast

Oi lt 0 indicating that the existence of asymmetric


information always leads to loss of OTA expected revenueΔEΠSNlowast

TH gt 0 and ΔEΠSNlowastTL 0 show that only TTA with high

serviceability can obtain additional information rent ΔEΠSNlowastTH

and ΔEΠSNlowastOH are symmetric with respect to the 0 value curve

which shows that in the O2O model composed by TTA andOTA the existence of asymmetric information transfers part of

revenue from OTA to TTA with high service capability (2)when sTH increases ΔEΠSNlowast

Oi firstly decreases and then in-creases e changing trend of ΔEΠSNlowast

OH depends on whetherOTArsquos revenue from the improvement of TTArsquos serviceabilitycan compensate for the cost of information value (3) |ΔEΠSNlowast

Oi |

decreases with r increases which shows that r can alleviate the

5

45

aH

STH

4

35

305 06 07 08 09 1

r = 01r = 05r = 09

aHN lowast

aHN lowast

aHS lowast

aHSlowast

aHNlowast

aHS lowast

(a)

aLN lowast

aLN lowast

aLN lowast

aLS lowast

aLSlowast

aLS lowast

5

48

aL

STL

46

44

4202 03 04 05

r = 01r = 05r = 09

(b)

Figure 1 Impacts of r and sTi on fixed payment (a) sTL 05 (b) sTH 05

1

06

08

bH

STH

04

02

005 06 07 08 09 1

r = 01r = 05r = 09

bH = bHN lowast S lowast


bH = bHN lowast Slowast

(a)

bLNlowast

bLN lowast

bLNlowast

bLS lowast

bLS lowast

bLS lowast

08

06

bL

STL

04

02

002 03 04 05

r = 01r = 05r = 09

(b)

Figure 2 Impacts of r and sTi on service compensation coefficient (a) sTL 05 (b) sTH 05


adverse effect of OTArsquos information disadvantage ΔEΠSNlowastTH

decreases with r increases which indicates that r can inhibit theinformation advantage of TTA with high serviceability

6 Conclusion and Future Research

e rationality of the OTA service commission contract is animportant factor to ensure the efficient operation of the O2Omodel in the establishment of service cooperation betweenOTA and TTA Considering the serviceability informationasymmetry the OTA designs reasonable service commissioncontracts to differentiate TTA serviceability and motivateTTA to improve service effort level Some suggestions forestablishing a cooperation contract are provided

(1) When an OTA has low saleability or high sale costthe OTA will increase the fixed payment to the TTAto support the cocreation of the O2O model eincentive effect of the service commission coefficientcan be offset by the TTArsquos service effort cost coef-ficient risk aversion coefficient and the degree ofmarket uncertainty

(2) When the type of TTArsquos serviceability is continuousthe precondition of separation contract is(ddsT)(1 minus F(sT)sTf(sT))lt 0 At this time theTTA can choose the appropriate contract corre-sponding to its serviceability type

(3) In cocreating the O2O model the existence ofasymmetric information always results in the loss ofOTArsquos expected revenue Except for TTA with thelowest serviceability all other types of TTA canobtain strict information rent e stronger the TTAserviceability is the more information rent it obtainsand the greater its expected revenue is

(4) e OTA designs service contracts by weighing thedifferent incentives of fixed payments and service

commission coefficient to the TTAWhen the TTA isa high serviceability type the OTA sets a high servicecommission coefficient to encourage the TTA tomake more service When the TTA is of a lowserviceability type the OTA sets a larger fixedpayment stimulating its enthusiasm for participat-ing in cocreating the O2O model

Due to some of its basic assumptions there are stillsome limitations of this paper Firstly in this paper theservice commission contract is considered as a linearform In future research it can be extended by otherservice cooperation forms for example service cost-sharing contract Secondly this study only considers thatan OTA establishes service cooperation with a TTA ormultiple TTAs e next step is to extend this model inwhich two or more OTAs achieve the O2O model byservice cooperation with multiple TTAs irdly theoptimal service commission contract is studied from ashort-term perspective However in some cases long-term cooperation is more conducive to the stable devel-opment of both For example He et al studied sustainabletourism by using evolutionary game models [31 32]Although it is more complex and challenging servicecooperation could be discussed from a long-term per-spective Finally our paper mainly utilizes an analyticalapproach us another future research direction is toconduct empirical research to validate our analyticalfindings

Appendix

Proof of TTArsquos revenue

Assuming that the random variable x follows normal dis-tribution xsimN (m n2) its utility function are the same asTTArsquos U (x) -e-rx and its expected utility is

01

0

005

∆EprodH

STH

ndash005

ndash0105 06 07 08 09 1

r = 01r = 05r = 09

∆EprodOHSN lowast

∆EprodOHSN lowast∆EprodTH

SN lowast

∆EprodTHSNlowast

∆EprodTHSN lowast

∆EprodOHSN lowast

(a)

0

times10ndash3

∆EprodL

STL

ndash5

ndash10

ndash1502 03 04 05

r = 01r = 05r = 09

∆EprodTHSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

(b)

Figure 3 Impacts of r and sTi on ΔEΠSNlowastTi and ΔEΠSNlowast

Oi (a) sTL 05 (b) sTH 05


EU(x) 1113946+infin

minus infinminus e

minus rx 12πn

radic eminus (xminus m)22n2( )dx minus e

minus r mminus rn22( )

(A1)

According to certainty equivalence method EUU (CE)minus eminus r(CE) minus eminus r(mminus rn22) can be obtained and also CE m minus

(rn22) can be get According to ξ isin N(0 σ2) EΠT a + b

(sOeO + sTeT + θ) minus (ηTe2T2) Var(ΠT) b2σ2 e TTArsquosrevenue ΠT is regarded as a random variable x its certaintyequivalent revenue is a + b(sOeO + sTeT + θ)minus

(ηTe2T2) minus (rσ2b22)

Proof of eorem A1 e optimization problem P1 issolved by using reverse induction and K-T method eprocess is solved in two steps

① e TTArsquos response function is eNTi (sTib

Ni ηT)

with a given service contract (aNi bN

i ) and sale efforteN

Oi② Solving OTArsquos optimal service contract and sale

effort

Plugging eNTi back to optimization problem P1 it is easy

to know that the OTArsquos expected revenue EΠO is linearfunction of aN

i and a joint concave of bNi and eN

Oi ereforethere are corner solution aN

i and interior solution bNi and eN

Oi

under the constraint equation (IR) at is there exists aunique optimal solution e Lagrange function is con-structed as

L aNi b

Ni e

NOi1113872 1113873 1 minus b

Ni1113872 1113873 sOe

NOi + sTie

NTi + θ1113872 1113873 minus a

Ni minus

ηOeN2Oi

21113888 1113889

+ χ aNi + b

Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2minus ΠT

⎛⎝ ⎞⎠

(A2)

Let (zLzaNi ) (zLzbN

i ) (zLzeNOi) 0 (zLzχ)ge

0χ ge 0 andχ(zLzχ) 0e only set of solutions can be obtained

(aNlowasti bNlowast

i eNlowastOi )


① e TTArsquos response function can be obtainedeS

Ti (sTibSi ηT) with a given service contract (aS

i bSi )

and sale effort eSOi

② Solving OTArsquos optimal service contract and saleeffort

Plugging eSTi back to optimization problem P2 it is easy

to know that the OTArsquos expected revenue EΠO is linearfunction of aS

i and a joint concave of bSi and eS

Oi ereforethere are corner solution aS

i and interior solution bSi and eS

Oi

under the constraint equation (IR-i) It is easy to prove that(IC-H) and (IR-L) are tight constraints so the above op-timization problem P2 can be rewritten as follows P2prime

maxaS

HbS

HeS

OHaS

LbS

LeS

OL


SOH +

bSHs

2TH

ηT

+ θ1113888 1113889 minus aSH minus

ηOeS2OH

21113888 1113889 +(1 minus ρ) times1113896

1 minus bSL1113872 1113873 sOe

SOL +

bSLs

2TL

ηT

+ θ1113888 1113889 minus aSL minus

ηOeS2OL

21113888 11138891113897


SH sO( e

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2b2H2ge

aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTH1113872 1113873

2

2ηT

minusrσ2b2L2

(IR minus L)aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2b2L2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A3)


e Lagrange function is constructed as

L aH bH eOH aL bL eOL χ1 χ2( 1113857 ρ 1 minus bSH1113872 1113873 sOe

SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ)times

middot 1 minus bSL1113872 1113873 sOe

SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 1113889 + χ1 a

SL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2bS2

L

2minus ΠR

⎛⎝ ⎞⎠

+ χ2 aSH + b

SH sOe

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2bS2

H

2minus a

SL minus b

SL sOe

SOL + θ1113872 1113873 minus

bSLsTH1113872 1113873

2

2ηT

+rσ2bS2

L

2⎛⎝ ⎞⎠

(A4)

e Kuhn-Tucker condition is obtained as

zL

zaSi

zL

zbSi

zL

zeSOi

0zL

zχ1ge 0

zL

zχ2ge 0 (A5)

where χ1 ge 0 and χ1(zLzχ1) 0 χ2 ge 0 and χ2(zLzχ2) 0

In χ1 1 χ2 ρaSlowastH bSlowast

H eSlowastOHa

SlowastL bSlowast

L and eSlowastOL can be

obtained

Proof of Corollary A1

bSlowastH minus b

SlowastL

s2TH minus s

2TL1113872 1113873 ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

s2TH + ηTrσ21113872 1113873 s

2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

ge 0

aSlowastH minus a

SlowastL

s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 11138731113888 1113889 b

SlowastL minus b

SlowastH1113872 1113873

(A6)

When rle 0 it is easy to get aSlowastH lt aSlowast

L When rgt 0 let

G(r) s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 1113873

s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

bSlowastL + b

SlowastH1113872 1113873 minus

12

rσ2 bSlowastL + b

SlowastH1113872 1113873 (A7)

where (s2TH2ηT)(bSlowastL + bSlowast

H ) is about r monotonicallydecreasing

12

rσ2 bSlowastL + b

SlowastH1113872 1113873

12

rσ2s2TH

s2TH + ηTrσ2

+(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs


1113888 1113889

1

2ηT

s2TH + s

2TL minus

s4TH

s2TH + ηTrσ2

minus(1 minus ρ)s

2TL s

2TL minus 2ρs

2TL + ρs

2TH1113872 1113873

s2TL minus 2ρs

2TL + ρs


⎛⎝ ⎞⎠

(A8)


(12)rσ2(bSlowastL + bSlowast

H ) is about r monotonically increasingerefore G(r) is about r monotonically decreasing

which maximizes r⟶ 0 at and minimizes at r⟶ +infin

G(+infin) s2O

ηO

+ θ1113888 1113889

G(0) s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

1 +(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs

2TH

1113888 1113889

(A9)

erefore G(r)gt 0 and aSlowast

L ge aSlowast

H only when r⟶ +infinaSlowast

L aSlowastH

Proof of eorem A3 (1)aSlowast

H minus aSlowast

L (12ηT)(s2TH minus s2TL)(bSlowast

L )2 ge 0(2) bSlowast

L minus bNlowast

L (ρs2TL(s2TL minus s2TH)(s2TLminus 2ρs2TL+ ρs2TH +

(1 minus ρ)ηTrσ2)(s2TL + ηTrσ2))le 0

aSlowast

L minus aNlowast

L bNlowast

L minus bSlowast

L1113872 1113873s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TL

ηT

1113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889 (A10)

e process of solving value of (aSlowast

L minus aNlowast

L ) is similar tothe proof of Corollary 1

e proofs of (3) and (4) of theorem are easy so they areomitted

Proof of eorem A4

(1) ΔEΠSNlowast

OH EΠSlowast

OH minus EΠNlowast

OH (12ηT)(s2TL minus s2TH)

(bSlowast

L )2 le 0

ΔEΠSNlowast

OL EΠSlowast

OL minus EΠNlowast

OL bSlowast

L minus bNlowast

L1113872 1113873s2TL

ηT

minus12

s2TL

ηT

+ rσ21113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889

1ηT

bSlowast

L minus bNlowast

L1113872 1113873 s2TL minus

s2TL

2bNlowastL

bNlowast

L + bSlowast

L1113872 11138731113888 1113889 s2TL

2ηT

bSlowast

L minus bNlowast

L1113872 1113873 1 minusb

Slowast

L

bNlowast

L

⎛⎝ ⎞⎠

minuss2TL + ηTrσ21113872 1113873

2ηT

bSlowast

L minus bNlowast

L1113872 11138732le 0

(A11)

(2) ΔEΠSNlowast

TH EΠSlowast

TH minus EΠNlowast

TH (12ηT)(s2TH minus s2TL)

(bSlowast

L )2 ge 0

ΔEΠSNlowast

TH EΠSlowast

TL minus EΠNlowast

TL 0 (A12)

Proof of eorem A5 According to the display principlewhen analyzing the validity of continuous variables theanalysis restriction can be displayed a(sT

harr) b(sTharr

)1113966 1113967 directlyerefore forall(sT sT

harr) isin H there is

aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC

sTharr

1113872 1113873 + sOeCO sTharr

1113872 1113873 + θ1113872 1113873bC

sTharr

1113872 1113873 +12

s2T

ηT

minus rσ21113888 1113889bC2

sTharr

1113872 1113873

(A13)


According to In equation (A13) there is

aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC

sTprime( 1113857 + sOe

CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sTprime( 1113857

(A14)

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

sprime2T

ηT

minus rσ2⎛⎝ ⎞⎠bC2

sTprime( 1113857ge

aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

sprime2T

ηT


sT( 1113857

(A15)

In equations (A14) and (A15) are added and simplifiedas

sT minus sTprime( 1113857 b

CsT( 1113857 minus b

CsTprime( 11138571113872 1113873ge 0 (A16)

So bC(sT) is no decreasing which shows bC(sT) is dif-ferentiable everywhere in the interval [sT sT] and

bCmiddot

(sT)ge 0 In addition the first-order condition for sT is

obtained aCmiddot

(sTharr

) + sOeCO

middot

(sT)bC(sT) + (sOeCO(sT) + θ)bC

middot

(sT) + ((s2TηT) minus rσ2)bCmiddot

(sT) 0 from In equation (A13)According to the direct display mechanism forallsC

T isin H

there is

aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857

+s2T

ηT


sT( 1113857 0

(A17)

In equations (18) and (19) are incentive compatibilityconstraints to ensure that the TTA has no motivation todeviate from the service reward contract provided by OTAIn equation (20) is the constraint condition for the TTA toparticipate in cooperation and cannot be lower than itsreserved revenue

Derivatives of equation (17) and further simplifiedaccording to equation (19)

EΠT

middot

sT( 1113857 sT

ηT

bC2

sT( 1113857 (A18)

Bring EΠT

middot

(sT) into equation (15) the above optimi-zation problem P3 can be rewritten as follows P3prime

maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2

sT( 1113857 minus EΠT sT( 1113857 minusηOe

C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A19)

st in equation (18) and (A18) EΠT(sT)ge ΠT

Let EΠT(sT )ge ΠT the solution of integral equation(A18) is obtained

EΠT sT( 1113857 1113946sT

sT

ωηT

bC2

(ω)dω + ΠT (A20)

According to 1113938sT

sT[1113938

sT

sT(ωηT)bC2(ω)dω]f(sT)dsT

1113938sT

sT[1113938

sT

sTf(ω)dω] (sTηT)bC2(sT)dsT 1113938

sT

sT(1 minus F(sT)f

(sT))(sTηT)bC2(sT)f(sT)dsT

en the objective function of P3 is

maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2

sT( 1113857 minus1 minus F sT( 1113857

f sT( 1113857

sT

ηT

bC2

sT( 1113857 minus ΠT minusηOe

C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A21)


e optimal solution is

eClowastO

sO

ηO

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

(A22)

Where bClowast(sT ) (11 + (ηTrσ2sT ) + (2 sT f(sT )))bClowast(sT) (11 + (ηTrσ2sT

2))

When sT sT according to equation (16) and EΠT(sT )

ΠT aClowast(sT ) can be obtained as

aClowast

sT1113872 1113873 ΠT minuss2O

ηO

+ θ1113888 1113889bClowast

sT1113872 1113873 minus12

sT

2

ηT

minus rσ2⎛⎜⎜⎜⎜⎝ ⎞⎟⎟⎟⎟⎠bClowast2

sT1113872 1113873

(A23)

Bring bClowast(sCT) into equation (A20) EΠT(sT) can be

obtained as

EΠT sT( 1113857 1113946sT

sT

ωηT

bClowast2

(ω)dω + ΠT (A24)

Bring equation (A24) into equation (17) aClowast(sT) can beobtained as

aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857

+ 1113946sT

sT

ωηT

bC2

(ω)dω

(A25)

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

is work was supported by the Research Fund for theScientific Research Foundation of Chongqing University ofTechnology and the Humanities and Social Sciences Re-search Project of Chongqing Education Commission(2020CJR06)

References

[1] L Ling X Guo and C Yang ldquoOpening the online mar-ketplace an examination of hotel pricing and travel agencyon-line distribution of roomsrdquo Tourism Management vol 45no 1 pp 234ndash243 2014

[2] E H C Wu R Law and B Jiang ldquoPredicting browsers andpurchasers of hotel websitesrdquo Cornell Hospitality Quarterlyvol 54 no 1 pp 38ndash48 2013

[3] P Shi and Y Long ldquoA research on complementary resourcesinput decision of the travel agencies co-creating the O2Omodelrdquo Tourism Science vol 31 no 2 pp 55ndash68 2017

[4] Y Long and P Shi ldquoPricing strategies of tour operator andonline travel agency based on cooperation to achieve O2Omodelrdquo Tourism Management vol 62 pp 302ndash311 2017

[5] X Guo X Zheng L Ling and C Yang ldquoOnline coopetitionbetween hotels and online travel agencies from the per-spective of cash back after stayrdquo Tourism Management Per-spectives vol 12 no 12 pp 104ndash112 2014

[6] P Shi and Y Long ldquoService quality decision-making in O2Otourism supply chains considering sharing service costsrdquoTourism Tribune vol 33 no 11 pp 90ndash100 2018

[7] C Lu and S Liu ldquoCultural tourism O2O business modelinnovation-A case study of CTriprdquo Journal of ElectronicCommerce in Organizations vol 14 no 2 pp 16ndash31 2016

[8] B Koo B Mantin and P OrsquoConnor ldquoOnline distribution ofairline tickets should airlines adopt a single or a multi-channel approachrdquo Tourism Management vol 32 no 1pp 69ndash74 2011

[9] X Guo L Ling Y Dong and L Liang ldquoCooperation contractin tourism supply chains the optimal pricing strategy ofhotels for cooperative third party strategic websitesrdquo Annalsof Tourism Research vol 41 pp 20ndash41 2013

[10] Y Dong and L Ling ldquoHotel overbooking and cooperationwith third-party websitesrdquo Sustainability vol 7 no 9pp 11696ndash11712 2015

[11] L Xu P He and Z Hua ldquoA new form for a hotel to collaboratewith a third-party website setting online-exclusive-roomsrdquo AsiaPacific Journal of Tourism Research vol 20 no 6 pp 1ndash12 2014

[12] D Bell S Gallino and A Moreno ldquoShowrooms and in-formation provision in omni-channel retailrdquo Productionand Operations Management vol 24 no 3 pp 360ndash3622015

[13] I D Savila R N Wathoni and A S Santoso ldquoe role ofmultichannel integration trust and offline-to-online cus-tomer loyalty towards repurchase intention an empiricalstudy in online-to-offline (O2O) e-commercerdquo ProcediaComputer Science vol 161 pp 859ndash866 2019

[14] B Swoboda and AWinters ldquoEffects of themost useful offline-online and online-offline channel integration services forconsumersrdquo Decision Support Systems vol 145 Article ID113522 2021

[15] L Jin X Zhang B Dan and S Li ldquoCommission contractdesign in ldquooffline evaluation online purchaserdquo (O2O) supplychain in the presence of cross-sellingrdquo Chinese Journal ofManagement Science vol 25 no 11 pp 33ndash46 2017

[16] J K Dong W G Kim and S H Jin ldquoA perceptual mappingof online travel agencies and preference attributesrdquo TourismManagement vol 28 no 2 pp 591ndash603 2007

[17] A H Walle ldquoTourism and the internetrdquo Journal of TravelResearch vol 35 no 1 pp 72ndash77 1996

[18] J Xiao B Dan and XM Zhang ldquoService cooperation pricingstrategy between manufactures and retailers in dual-channelsupply chainrdquo Systems Engineering-eory amp Practice vol 30no 12 pp 2203ndash2211 2011

[19] Q Xu G Fu and D Fan ldquoService sharing profit mode andcoordination mechanism in the Online-to-Offline retailmarketrdquo Economic Modelling vol 91 pp 659ndash669 2020

[20] Y-W Zhou J Guo and W Zhou ldquoPricingservice strategiesfor a dual-channel supply chain with free riding and service-cost sharingrdquo International Journal of Production Economicsvol 196 pp 198ndash210 2018


[21] Q Yang and M Zhang ldquoService cooperation incentivemechanism in a dual-channel supply chain under servicedifferentiationrdquo American Journal of Industrial and BusinessManagement vol 05 no 04 pp 226ndash234 2015

[22] L Zou ldquoreat-based incentive mechanisms under moralhazard and adverse selectionrdquo Journal of Comparative Eco-nomics vol 16 no 1 pp 47ndash74 1992

[23] F Chen ldquoSalesforce incentives market information andproductioninventory planningrdquo Management Sciencevol 51 no 1 pp 60ndash75 2005

[24] S-H Kim M A Cohen and S Netessine ldquoPerformancecontracting in after-sales service supply chainsrdquoManagementScience vol 53 no 12 pp 1843ndash1858 2007

[25] Y He X Zhao L Zhao and J He ldquoCoordinating a supplychain with effort and price dependent stochastic demandrdquoApplied Mathematical Modelling vol 33 no 6 pp 2777ndash2790 2009

[26] P He Y He C Shi et al ldquoCost-sharing contract design in alow-carbon service supply chainrdquo Computers amp IndustrialEngineering vol 139 Article ID 106160 2020

[27] J Cao C J Yang P Li and G Zhou ldquoDesign of supply chainlinear shared-saving contract with asymmetric informationrdquoJournal of Management Sciences in China vol 12 no 2pp 19ndash30 2009

[28] Y He H Huang and D Li ldquoInventory and pricing decisionsfor a dual-channel supply chain with deteriorating productsrdquoOperational Research vol 20 no 3 pp 1461ndash1503 2020

[29] G Xie W Yue S Wang and K K Lai ldquoQuality investmentand price decision in a risk-averse supply chainrdquo EuropeanJournal of Operational Research vol 214 no 2 pp 403ndash4102011

[30] Z Li S M Gilbert and G Lai ldquoSupplier encroachment underasymmetric informationrdquoManagement Science vol 60 no 2pp 449ndash462 2013

[31] Y He P He F Xu and C Shi ldquoSustainable tourism mod-eling pricing decisions and evolutionarily stable strategies forcompetitive tour operatorsrdquo Tourism Economics vol 25no 5 pp 779ndash799 2019

[32] P He Y He and F Xu ldquoEvolutionary analysis of sustainabletourismrdquoAnnals of Tourism Research vol 69 pp 76ndash89 2018


To fill this gap and provide some suggestions for OTAsand TTAsmanagers on establishing the O2Omodel throughservice cooperation this paper proposes a cooperationmodel to describe decision interactions of an OTA and aTTA e OTA and TTA play a principal-agent model inwhich the OTA as the principal player determines theservice commission and the TTA as the agent determinesservice effort In addition in the O2O model because theOTA and TTA are relatively independent TTArsquos privateinformation is difficult to disclose and it often hides its ownservice information to obtain higher revenue By comparingthe case of information symmetry and asymmetry theimpacts of asymmetry information on OTArsquos service con-tract TTArsquos information value and the revenue of OTA andTTA are analyzed

e rest of this paper is organized as follows Section 2reviews the related literature Section 3 describes the servicecooperation between an OTA and a TTA Section 4 analyzesand compares equilibriums of symmetric and asymmetricinformation and further analyzes equilibriums when TTArsquosability is continuous Section 5 presents the results of nu-merical analyses Section 6 concludes this paper by sum-marizing some of the managerial implications obtained

2 Literature Review

According to the purpose of studying the service coopera-tion contract design of OTAs and TTAs this section reviewsthree distinct of literature studies about the O2O modelservice cooperation and cooperation between tourism en-terprises and OTAs

21O2OModel Since Alex Rampel put forward the conceptof the O2O model it has attracted wide attention in aca-demia Although the concept and classification of the O2Omodel are different because of the scholarsrsquo different re-search perspectives it is a business model for the effectiveintegration of online and offline stores [4 7] It is believedthat O2O is a new business model which can discloseproduct information change consumer brand awarenessand mitigate the adverse effects of product uncertainty onconsumersrsquo shopping decisions [12] By using 311 respon-dent data from O2O e-commerce users in Greater JakartaArea Savila et al showed that both multichannel integrationand trust have a significant effect on both customer onlineloyalty and customer offline loyalty that drive customerrepurchase intention [13]

According to the existing research about the O2Omodelit can be divided into two categories one is to purchaseonline and then experience offline which can also be calledonline to offline the other is to experience offline and thenpurchase online which can also be called offline to online[14] In the hospitality and tourism industry becausetourists have to experience tourism products offline theO2O model achieved by the cooperation between OTAs andhotels or airlines is a common phenomenon From theperspective of the supply chain scholars more emphasize thesale cooperation between the OTAs and hotels or airlines

and have gotten some effective cooperation strategies[1 8ndash11] In the retail industry as the major characteristicfeature of a product is tangible it can be touched and triedoute O2Omodel achieved by cooperating with others oropening by oneself [14 15] becomes the main researchdirection Considering that the service cost and effort in-formation of the showroom are asymmetry Jin et al studythe design of the commission contract of offline to onlinewhich is considered that the showroom and online retailerhave impacts on consumersrsquo purchasing decisions respec-tively [15] In practice online sale efforts and offline serviceswill affect the demand for each other erefore it is difficultto highlight the integrity of the online retailer and storecocreating O2O model by splitting online and offlinedemand

Despite the impact of technology and the advent ofonline bookings offline channels play a pivotal role in thetourism supply chain Unlike the general retail product thetourism product of a travel agency cannot be touchedtransported stored or returned e functions of offlinechannels are different between general retailer and travelagency e offline channels of general retailer providetouching trying and even online return service for con-sumers e offline channels of travel agency provide sceneexperience personal and professional information andadvice to meet travelersrsquo demands on a continuous basis bygathering and organizing information [16 17] e O2Omodel of travel agency in this paper is an integration ofonline and offline Both online sale efforts and offline ser-vices will have an impact on the overall demand for the O2Omodel

22 Service Cooperation e existing literature on servicecooperation is mostly concentrated in the supply chain fieldService cooperation not only reduces the competition be-tween direct and retail channels but also improves supplychain revenue [18ndash21] Unit service reward and service costsharing are the main forms of service cooperation To avoidchannel conflict and improve service efficiency Xiao Danand Zhang studied after-sale service cooperation betweendirect channel and retailers in which direct channel pays unitservice commission to retailers [18] In the O2O retailmarket the service cost sharing mechanism is introduced tocoordinate conflicts and achieve a win-win strategy therebyimproving the performance of the entire O2O supply chain[19] Due to the importance of presales services in pur-chasing decisions Zhou et al introduce that the service costsharing contract can effectively stimulate the retailer toimprove his service level while free riding occurred [20]Considering that the retailer provides the same service levelin both channels or not Yang and Zhang compare themanufacturerrsquos optimal profit and retailerrsquos optimal profitunder the condition of different services and the sameservice and provide differentiated services to enable thesystem to achieve the optimal profit [21] However thecompletely symmetrical assumptions in the above literatureare not currently common practice e adverse selectionand moral hazard are always in the cooperation process [22]








2c(eT)de2T gt 0




q f sO eO( 1113857 + g sT eT( 1113857 + u(θ) + ξ (1)

where fsOgt 0 fsO


le 0 gsTgt 0 gsTsT

le 0geTgt 0 geTeT







minus a minusηOe

2O

2

(2)



2O

2 (3)



2T

2

(4)




2T

2minus

rσ2b2

2 (5)










problem P1 is

maxaN

ibN

ieN

Oi


NOi + sTie

NTi + θ1113872 1113873 minus a

Ni minus

ηOeN2Oi

21113888 1113889


Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(6)

where

eNTi argmax

eT

aNi + b

Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2⎛⎝ ⎞⎠ (7)









Ti are

bNlowasti

s2Ti

s2Ti + ηTrσ2

aNlowasti ΠT minus

s2O

ηO

+ θ1113888 1113889bNlowasti +

12

rσ2 minuss2Ti

ηT

1113888 1113889 bNlowasti1113872 1113873

2

eNlowastOi

sO

ηO

eNlowastTi

s3Ti

ηT s2Ti + ηTrσ21113872 1113873

sTi

ηT

bNlowasti

(8)


i








Ti increases








maxaS

HbS

HeS

OHaS

LbS

LeS

OL


SOH + sTHe

STH + θ1113872 1113873 minus a

SH minus

ηOeS2OH

21113888 1113889 +1113896


SOL + sTHe

STL + θ1113872 1113873 minus a

SL minus

ηOeS2OL

21113888 11138891113897


SH sOe

SOH + sTHe

STH + θ1113872 1113873 minus

ηTeS2TH

2minus

rσ2b2H2ge

aSL + b

SL sOe

SOS + sTHe

LTH + θ1113872 1113873 minus

ηTeL2TH

2minus

rσ2b2L2

(IC minus L)aSL + b

SL sOe

SOL + sTLe

STL + θ1113872 1113873 minus

ηTeS2TL

2minus

rσ2b2L2ge

aSH + b

SH sOe

SOH + sTLe

HTL + θ1113872 1113873 minus

ηTeH2TL

2minus

rσ2b2H2

(IR minus H)aSH + b

SH sOe

SOH + sTHe

STH + θ1113872 1113873 minus

ηTeS2TH

2minus

rσ2b2H2ge ΠT

(IR minus L)aSL + b

SL sOe

SOL + sTLe

STL + θ1113872 1113873 minus

ηTeS2TL

2minus

rσ2b2L2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)


where

eSTH argmax

e

aSH + b

SH sOe

SOH + sTHe

STH + θ1113872 1113873 minus

ηTeS2TH

2minus

rσ2b2H2

1113888 1113889 (10)

eSTL argmax

e

aSL + b

SL sOe

SOL + sTLe

STL + θ1113872 1113873 minus

ηTeS2TL

2minus

rσ2b2L2

1113888 1113889 (11)




L)((aS








aSlowastH ΠT minus

s2O

ηO

+ θ1113888 1113889bSlowastH +

12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastH1113872 1113873

2+

12ηT

s2TH minus s

2TL1113872 1113873 b

SlowastL1113872 1113873

2

bSlowastH

s2TH

s2TH + ηTrσ2

aSlowastL ΠT minus

s2O

ηO

+ θ1113888 1113889bSlowastL +

12

rσ2 minuss2TL

ηT

1113888 1113889 bSlowastL1113872 1113873

2

bSlowastL

(1 minus ρ)s2TL

s2TL minus 2ρs

2TL + ρs


eSlowastOH e

SlowastOL

sO

ηO

eSlowastTH

s3TH

ηT s2TH + ηTrσ21113872 1113873

eSlowastTL

(1 minus ρ)s3TL

s2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873ηT

(12)




L


H 1 andbSlowast












H le aSlowastL






EΠTH aSlowastH b


SlowastL b

SlowastL1113872 1113873 0

EΠTL aSlowastL b


SlowastH b

SlowastH1113872 1113873

bSlowastH minus b

SlowastL1113872 1113873 b

SlowastH + b

SlowastL1113872 1113873 sTH + sTL( 1113857

2ηT

gt 0

(13)











2TL1113872 1113873

2ηT

bSlowast 2L

(14)


L







H



L



TL


Oi




Oi


TH


TL




EΠO sT( 1113857 1113946sT

sT

1 minus bC

sT( 11138571113872 1113873 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873 minus a

CsT( 1113857 minus

ηOeC2O sT( 1113857

21113896 1113897f sT( 1113857dsT (15)


EΠT sT( 1113857 aC

sT( 1113857 + bC

sT( 1113857 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873

minusηTe

C2T sT( 1113857

2minus

rσ2bC2sT( 1113857

2





EΠT sT( 1113857 aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857

+12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857

(17)


maxaCbC

1113946sT

sT

1 minus bC

sT( 11138571113872 1113873 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873 minus a

CsT( 1113857 minus

ηOeC2O sT( 1113857

21113896 1113897f sT( 1113857dsT

st(IC)bCmiddot

sT( 1113857ge 0

(18)

(IC)aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857 +s

C2T

ηT


sT( 1113857 0 (19)


(IR)aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge ΠT (20)

where eCT(sT) (sC


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857 + 1113946sT

sT

ωηT

bC2

(ω)dω

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

eClowastO

sO

ηO


sCTb

ClowastsT( 1113857

ηT

(21)











i and aNlowasti in-



i and bNlowasti

















Oi |


5

45

aH

STH

4

35

305 06 07 08 09 1

r = 01r = 05r = 09

aHN lowast

aHN lowast

aHS lowast

aHSlowast

aHNlowast

aHS lowast

(a)

aLN lowast

aLN lowast

aLN lowast

aLS lowast

aLSlowast

aLS lowast

5

48

aL

STL

46

44

4202 03 04 05

r = 01r = 05r = 09

(b)


1

06

08

bH

STH

04

02

005 06 07 08 09 1

r = 01r = 05r = 09




(a)

bLNlowast

bLN lowast

bLNlowast

bLS lowast

bLS lowast

bLS lowast

08

06

bL

STL

04

02

002 03 04 05

r = 01r = 05r = 09

(b)













Appendix



01

0

005

∆EprodH

STH

ndash005

ndash0105 06 07 08 09 1

r = 01r = 05r = 09

∆EprodOHSN lowast


SN lowast

∆EprodTHSNlowast

∆EprodTHSN lowast

∆EprodOHSN lowast

(a)

0

times10ndash3

∆EprodL

STL

ndash5

ndash10

ndash1502 03 04 05

r = 01r = 05r = 09

∆EprodTHSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

(b)




EU(x) 1113946+infin

minus infinminus e

minus rx 12πn



(A1)







Ni ηT)




effort






Oi


L aNi b

Ni e

NOi1113872 1113873 1 minus b

Ni1113872 1113873 sOe

NOi + sTie

NTi + θ1113872 1113873 minus a

Ni minus

ηOeN2Oi

21113888 1113889

+ χ aNi + b

Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2minus ΠT

⎛⎝ ⎞⎠

(A2)




(aNlowasti bNlowast

i eNlowastOi )




i bSi )








Oi


maxaS

HbS

HeS

OHaS

LbS

LeS

OL


SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ) times1113896

1 minus bSL1113872 1113873 sOe

SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 11138891113897


SH sO( e

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2b2H2ge

aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTH1113872 1113873

2

2ηT

minusrσ2b2L2

(IR minus L)aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2b2L2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A3)




SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ)times


SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 1113889 + χ1 a

SL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2bS2

L

2minus ΠR

⎛⎝ ⎞⎠

+ χ2 aSH + b

SH sOe

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2bS2

H

2minus a

SL minus b

SL sOe

SOL + θ1113872 1113873 minus

bSLsTH1113872 1113873

2

2ηT

+rσ2bS2

L

2⎛⎝ ⎞⎠

(A4)


zL

zaSi

zL

zbSi

zL

zeSOi

0zL

zχ1ge 0

zL

zχ2ge 0 (A5)



H eSlowastOHa

SlowastL bSlowast


obtained


bSlowastH minus b

SlowastL

s2TH minus s

2TL1113872 1113873 ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

s2TH + ηTrσ21113872 1113873 s

2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

ge 0

aSlowastH minus a

SlowastL

s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 11138731113888 1113889 b

SlowastL minus b

SlowastH1113872 1113873

(A6)


L When rgt 0 let

G(r) s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 1113873

s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

bSlowastL + b


12

rσ2 bSlowastL + b

SlowastH1113872 1113873 (A7)



12

rσ2 bSlowastL + b

SlowastH1113872 1113873

12

rσ2s2TH

s2TH + ηTrσ2

+(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs


1113888 1113889

1

2ηT

s2TH + s

2TL minus

s4TH

s2TH + ηTrσ2

minus(1 minus ρ)s

2TL s

2TL minus 2ρs

2TL + ρs

2TH1113872 1113873

s2TL minus 2ρs

2TL + ρs


⎛⎝ ⎞⎠

(A8)





G(+infin) s2O

ηO

+ θ1113888 1113889

G(0) s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

1 +(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs

2TH

1113888 1113889

(A9)


L ge aSlowast


L aSlowastH


H minus aSlowast



L minus bNlowast



aSlowast

L minus aNlowast

L bNlowast

L minus bSlowast

L1113872 1113873s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TL

ηT

1113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889 (A10)


L minus aNlowast



Proof of eorem A4

(1) ΔEΠSNlowast

OH EΠSlowast

OH minus EΠNlowast


(bSlowast

L )2 le 0

ΔEΠSNlowast

OL EΠSlowast

OL minus EΠNlowast

OL bSlowast

L minus bNlowast

L1113872 1113873s2TL

ηT

minus12

s2TL

ηT

+ rσ21113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889

1ηT

bSlowast

L minus bNlowast

L1113872 1113873 s2TL minus

s2TL

2bNlowastL

bNlowast

L + bSlowast

L1113872 11138731113888 1113889 s2TL

2ηT

bSlowast

L minus bNlowast

L1113872 1113873 1 minusb

Slowast

L

bNlowast

L

⎛⎝ ⎞⎠

minuss2TL + ηTrσ21113872 1113873

2ηT

bSlowast

L minus bNlowast

L1113872 11138732le 0

(A11)

(2) ΔEΠSNlowast

TH EΠSlowast

TH minus EΠNlowast


(bSlowast

L )2 ge 0

ΔEΠSNlowast

TH EΠSlowast

TL minus EΠNlowast

TL 0 (A12)


harr) b(sTharr



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC

sTharr

1113872 1113873 + sOeCO sTharr

1113872 1113873 + θ1113872 1113873bC

sTharr

1113872 1113873 +12

s2T

ηT

minus rσ21113888 1113889bC2

sTharr

1113872 1113873

(A13)



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sTprime( 1113857

(A14)

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

sprime2T

ηT


sTprime( 1113857ge

aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

sprime2T

ηT


sT( 1113857

(A15)




CsTprime( 11138571113872 1113873ge 0 (A16)


bCmiddot


obtained aCmiddot

(sTharr

) + sOeCO

middot


middot



T isin H

there is

aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857

+s2T

ηT


sT( 1113857 0

(A17)



EΠT

middot

sT( 1113857 sT

ηT

bC2

sT( 1113857 (A18)

Bring EΠT

middot


maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A19)



EΠT sT( 1113857 1113946sT

sT

ωηT

bC2

(ω)dω + ΠT (A20)


sT[1113938

sT


1113938sT

sT[1113938

sT


sT

sT(1 minus F(sT)f



maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


f sT( 1113857

sT

ηT

bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A21)



eClowastO

sO

ηO

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

(A22)


2))



aClowast

sT1113872 1113873 ΠT minuss2O

ηO

+ θ1113888 1113889bClowast

sT1113872 1113873 minus12

sT

2

ηT


sT1113872 1113873

(A23)


obtained as

EΠT sT( 1113857 1113946sT

sT

ωηT

bClowast2

(ω)dω + ΠT (A24)


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857

+ 1113946sT

sT

ωηT

bC2

(ω)dω

(A25)

Data Availability




Acknowledgments


References









































2c(eT)de2T gt 0




q f sO eO( 1113857 + g sT eT( 1113857 + u(θ) + ξ (1)

where fsOgt 0 fsO


le 0 gsTgt 0 gsTsT

le 0geTgt 0 geTeT







minus a minusηOe

2O

2

(2)



2O

2 (3)



2T

2

(4)




2T

2minus

rσ2b2

2 (5)










problem P1 is

maxaN

ibN

ieN

Oi


NOi + sTie

NTi + θ1113872 1113873 minus a

Ni minus

ηOeN2Oi

21113888 1113889


Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(6)

where

eNTi argmax

eT

aNi + b

Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2⎛⎝ ⎞⎠ (7)









Ti are

bNlowasti

s2Ti

s2Ti + ηTrσ2

aNlowasti ΠT minus

s2O

ηO

+ θ1113888 1113889bNlowasti +

12

rσ2 minuss2Ti

ηT

1113888 1113889 bNlowasti1113872 1113873

2

eNlowastOi

sO

ηO

eNlowastTi

s3Ti

ηT s2Ti + ηTrσ21113872 1113873

sTi

ηT

bNlowasti

(8)


i








Ti increases








maxaS

HbS

HeS

OHaS

LbS

LeS

OL


SOH + sTHe

STH + θ1113872 1113873 minus a

SH minus

ηOeS2OH

21113888 1113889 +1113896


SOL + sTHe

STL + θ1113872 1113873 minus a

SL minus

ηOeS2OL

21113888 11138891113897


SH sOe

SOH + sTHe

STH + θ1113872 1113873 minus

ηTeS2TH

2minus

rσ2b2H2ge

aSL + b

SL sOe

SOS + sTHe

LTH + θ1113872 1113873 minus

ηTeL2TH

2minus

rσ2b2L2

(IC minus L)aSL + b

SL sOe

SOL + sTLe

STL + θ1113872 1113873 minus

ηTeS2TL

2minus

rσ2b2L2ge

aSH + b

SH sOe

SOH + sTLe

HTL + θ1113872 1113873 minus

ηTeH2TL

2minus

rσ2b2H2

(IR minus H)aSH + b

SH sOe

SOH + sTHe

STH + θ1113872 1113873 minus

ηTeS2TH

2minus

rσ2b2H2ge ΠT

(IR minus L)aSL + b

SL sOe

SOL + sTLe

STL + θ1113872 1113873 minus

ηTeS2TL

2minus

rσ2b2L2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)


where

eSTH argmax

e

aSH + b

SH sOe

SOH + sTHe

STH + θ1113872 1113873 minus

ηTeS2TH

2minus

rσ2b2H2

1113888 1113889 (10)

eSTL argmax

e

aSL + b

SL sOe

SOL + sTLe

STL + θ1113872 1113873 minus

ηTeS2TL

2minus

rσ2b2L2

1113888 1113889 (11)




L)((aS








aSlowastH ΠT minus

s2O

ηO

+ θ1113888 1113889bSlowastH +

12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastH1113872 1113873

2+

12ηT

s2TH minus s

2TL1113872 1113873 b

SlowastL1113872 1113873

2

bSlowastH

s2TH

s2TH + ηTrσ2

aSlowastL ΠT minus

s2O

ηO

+ θ1113888 1113889bSlowastL +

12

rσ2 minuss2TL

ηT

1113888 1113889 bSlowastL1113872 1113873

2

bSlowastL

(1 minus ρ)s2TL

s2TL minus 2ρs

2TL + ρs


eSlowastOH e

SlowastOL

sO

ηO

eSlowastTH

s3TH

ηT s2TH + ηTrσ21113872 1113873

eSlowastTL

(1 minus ρ)s3TL

s2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873ηT

(12)




L


H 1 andbSlowast












H le aSlowastL






EΠTH aSlowastH b


SlowastL b

SlowastL1113872 1113873 0

EΠTL aSlowastL b


SlowastH b

SlowastH1113872 1113873

bSlowastH minus b

SlowastL1113872 1113873 b

SlowastH + b

SlowastL1113872 1113873 sTH + sTL( 1113857

2ηT

gt 0

(13)











2TL1113872 1113873

2ηT

bSlowast 2L

(14)


L







H



L



TL


Oi




Oi


TH


TL




EΠO sT( 1113857 1113946sT

sT

1 minus bC

sT( 11138571113872 1113873 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873 minus a

CsT( 1113857 minus

ηOeC2O sT( 1113857

21113896 1113897f sT( 1113857dsT (15)


EΠT sT( 1113857 aC

sT( 1113857 + bC

sT( 1113857 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873

minusηTe

C2T sT( 1113857

2minus

rσ2bC2sT( 1113857

2





EΠT sT( 1113857 aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857

+12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857

(17)


maxaCbC

1113946sT

sT

1 minus bC

sT( 11138571113872 1113873 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873 minus a

CsT( 1113857 minus

ηOeC2O sT( 1113857

21113896 1113897f sT( 1113857dsT

st(IC)bCmiddot

sT( 1113857ge 0

(18)

(IC)aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857 +s

C2T

ηT


sT( 1113857 0 (19)


(IR)aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge ΠT (20)

where eCT(sT) (sC


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857 + 1113946sT

sT

ωηT

bC2

(ω)dω

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

eClowastO

sO

ηO


sCTb

ClowastsT( 1113857

ηT

(21)











i and aNlowasti in-



i and bNlowasti

















Oi |


5

45

aH

STH

4

35

305 06 07 08 09 1

r = 01r = 05r = 09

aHN lowast

aHN lowast

aHS lowast

aHSlowast

aHNlowast

aHS lowast

(a)

aLN lowast

aLN lowast

aLN lowast

aLS lowast

aLSlowast

aLS lowast

5

48

aL

STL

46

44

4202 03 04 05

r = 01r = 05r = 09

(b)


1

06

08

bH

STH

04

02

005 06 07 08 09 1

r = 01r = 05r = 09




(a)

bLNlowast

bLN lowast

bLNlowast

bLS lowast

bLS lowast

bLS lowast

08

06

bL

STL

04

02

002 03 04 05

r = 01r = 05r = 09

(b)













Appendix



01

0

005

∆EprodH

STH

ndash005

ndash0105 06 07 08 09 1

r = 01r = 05r = 09

∆EprodOHSN lowast


SN lowast

∆EprodTHSNlowast

∆EprodTHSN lowast

∆EprodOHSN lowast

(a)

0

times10ndash3

∆EprodL

STL

ndash5

ndash10

ndash1502 03 04 05

r = 01r = 05r = 09

∆EprodTHSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

(b)




EU(x) 1113946+infin

minus infinminus e

minus rx 12πn



(A1)







Ni ηT)




effort






Oi


L aNi b

Ni e

NOi1113872 1113873 1 minus b

Ni1113872 1113873 sOe

NOi + sTie

NTi + θ1113872 1113873 minus a

Ni minus

ηOeN2Oi

21113888 1113889

+ χ aNi + b

Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2minus ΠT

⎛⎝ ⎞⎠

(A2)




(aNlowasti bNlowast

i eNlowastOi )




i bSi )








Oi


maxaS

HbS

HeS

OHaS

LbS

LeS

OL


SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ) times1113896

1 minus bSL1113872 1113873 sOe

SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 11138891113897


SH sO( e

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2b2H2ge

aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTH1113872 1113873

2

2ηT

minusrσ2b2L2

(IR minus L)aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2b2L2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A3)




SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ)times


SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 1113889 + χ1 a

SL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2bS2

L

2minus ΠR

⎛⎝ ⎞⎠

+ χ2 aSH + b

SH sOe

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2bS2

H

2minus a

SL minus b

SL sOe

SOL + θ1113872 1113873 minus

bSLsTH1113872 1113873

2

2ηT

+rσ2bS2

L

2⎛⎝ ⎞⎠

(A4)


zL

zaSi

zL

zbSi

zL

zeSOi

0zL

zχ1ge 0

zL

zχ2ge 0 (A5)



H eSlowastOHa

SlowastL bSlowast


obtained


bSlowastH minus b

SlowastL

s2TH minus s

2TL1113872 1113873 ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

s2TH + ηTrσ21113872 1113873 s

2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

ge 0

aSlowastH minus a

SlowastL

s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 11138731113888 1113889 b

SlowastL minus b

SlowastH1113872 1113873

(A6)


L When rgt 0 let

G(r) s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 1113873

s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

bSlowastL + b


12

rσ2 bSlowastL + b

SlowastH1113872 1113873 (A7)



12

rσ2 bSlowastL + b

SlowastH1113872 1113873

12

rσ2s2TH

s2TH + ηTrσ2

+(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs


1113888 1113889

1

2ηT

s2TH + s

2TL minus

s4TH

s2TH + ηTrσ2

minus(1 minus ρ)s

2TL s

2TL minus 2ρs

2TL + ρs

2TH1113872 1113873

s2TL minus 2ρs

2TL + ρs


⎛⎝ ⎞⎠

(A8)





G(+infin) s2O

ηO

+ θ1113888 1113889

G(0) s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

1 +(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs

2TH

1113888 1113889

(A9)


L ge aSlowast


L aSlowastH


H minus aSlowast



L minus bNlowast



aSlowast

L minus aNlowast

L bNlowast

L minus bSlowast

L1113872 1113873s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TL

ηT

1113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889 (A10)


L minus aNlowast



Proof of eorem A4

(1) ΔEΠSNlowast

OH EΠSlowast

OH minus EΠNlowast


(bSlowast

L )2 le 0

ΔEΠSNlowast

OL EΠSlowast

OL minus EΠNlowast

OL bSlowast

L minus bNlowast

L1113872 1113873s2TL

ηT

minus12

s2TL

ηT

+ rσ21113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889

1ηT

bSlowast

L minus bNlowast

L1113872 1113873 s2TL minus

s2TL

2bNlowastL

bNlowast

L + bSlowast

L1113872 11138731113888 1113889 s2TL

2ηT

bSlowast

L minus bNlowast

L1113872 1113873 1 minusb

Slowast

L

bNlowast

L

⎛⎝ ⎞⎠

minuss2TL + ηTrσ21113872 1113873

2ηT

bSlowast

L minus bNlowast

L1113872 11138732le 0

(A11)

(2) ΔEΠSNlowast

TH EΠSlowast

TH minus EΠNlowast


(bSlowast

L )2 ge 0

ΔEΠSNlowast

TH EΠSlowast

TL minus EΠNlowast

TL 0 (A12)


harr) b(sTharr



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC

sTharr

1113872 1113873 + sOeCO sTharr

1113872 1113873 + θ1113872 1113873bC

sTharr

1113872 1113873 +12

s2T

ηT

minus rσ21113888 1113889bC2

sTharr

1113872 1113873

(A13)



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sTprime( 1113857

(A14)

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

sprime2T

ηT


sTprime( 1113857ge

aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

sprime2T

ηT


sT( 1113857

(A15)




CsTprime( 11138571113872 1113873ge 0 (A16)


bCmiddot


obtained aCmiddot

(sTharr

) + sOeCO

middot


middot



T isin H

there is

aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857

+s2T

ηT


sT( 1113857 0

(A17)



EΠT

middot

sT( 1113857 sT

ηT

bC2

sT( 1113857 (A18)

Bring EΠT

middot


maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A19)



EΠT sT( 1113857 1113946sT

sT

ωηT

bC2

(ω)dω + ΠT (A20)


sT[1113938

sT


1113938sT

sT[1113938

sT


sT

sT(1 minus F(sT)f



maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


f sT( 1113857

sT

ηT

bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A21)



eClowastO

sO

ηO

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

(A22)


2))



aClowast

sT1113872 1113873 ΠT minuss2O

ηO

+ θ1113888 1113889bClowast

sT1113872 1113873 minus12

sT

2

ηT


sT1113872 1113873

(A23)


obtained as

EΠT sT( 1113857 1113946sT

sT

ωηT

bClowast2

(ω)dω + ΠT (A24)


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857

+ 1113946sT

sT

ωηT

bC2

(ω)dω

(A25)

Data Availability




Acknowledgments


References




































2T

2minus

rσ2b2

2 (5)










problem P1 is

maxaN

ibN

ieN

Oi


NOi + sTie

NTi + θ1113872 1113873 minus a

Ni minus

ηOeN2Oi

21113888 1113889


Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(6)

where

eNTi argmax

eT

aNi + b

Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2⎛⎝ ⎞⎠ (7)









Ti are

bNlowasti

s2Ti

s2Ti + ηTrσ2

aNlowasti ΠT minus

s2O

ηO

+ θ1113888 1113889bNlowasti +

12

rσ2 minuss2Ti

ηT

1113888 1113889 bNlowasti1113872 1113873

2

eNlowastOi

sO

ηO

eNlowastTi

s3Ti

ηT s2Ti + ηTrσ21113872 1113873

sTi

ηT

bNlowasti

(8)


i








Ti increases








maxaS

HbS

HeS

OHaS

LbS

LeS

OL


SOH + sTHe

STH + θ1113872 1113873 minus a

SH minus

ηOeS2OH

21113888 1113889 +1113896


SOL + sTHe

STL + θ1113872 1113873 minus a

SL minus

ηOeS2OL

21113888 11138891113897


SH sOe

SOH + sTHe

STH + θ1113872 1113873 minus

ηTeS2TH

2minus

rσ2b2H2ge

aSL + b

SL sOe

SOS + sTHe

LTH + θ1113872 1113873 minus

ηTeL2TH

2minus

rσ2b2L2

(IC minus L)aSL + b

SL sOe

SOL + sTLe

STL + θ1113872 1113873 minus

ηTeS2TL

2minus

rσ2b2L2ge

aSH + b

SH sOe

SOH + sTLe

HTL + θ1113872 1113873 minus

ηTeH2TL

2minus

rσ2b2H2

(IR minus H)aSH + b

SH sOe

SOH + sTHe

STH + θ1113872 1113873 minus

ηTeS2TH

2minus

rσ2b2H2ge ΠT

(IR minus L)aSL + b

SL sOe

SOL + sTLe

STL + θ1113872 1113873 minus

ηTeS2TL

2minus

rσ2b2L2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)


where

eSTH argmax

e

aSH + b

SH sOe

SOH + sTHe

STH + θ1113872 1113873 minus

ηTeS2TH

2minus

rσ2b2H2

1113888 1113889 (10)

eSTL argmax

e

aSL + b

SL sOe

SOL + sTLe

STL + θ1113872 1113873 minus

ηTeS2TL

2minus

rσ2b2L2

1113888 1113889 (11)




L)((aS








aSlowastH ΠT minus

s2O

ηO

+ θ1113888 1113889bSlowastH +

12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastH1113872 1113873

2+

12ηT

s2TH minus s

2TL1113872 1113873 b

SlowastL1113872 1113873

2

bSlowastH

s2TH

s2TH + ηTrσ2

aSlowastL ΠT minus

s2O

ηO

+ θ1113888 1113889bSlowastL +

12

rσ2 minuss2TL

ηT

1113888 1113889 bSlowastL1113872 1113873

2

bSlowastL

(1 minus ρ)s2TL

s2TL minus 2ρs

2TL + ρs


eSlowastOH e

SlowastOL

sO

ηO

eSlowastTH

s3TH

ηT s2TH + ηTrσ21113872 1113873

eSlowastTL

(1 minus ρ)s3TL

s2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873ηT

(12)




L


H 1 andbSlowast












H le aSlowastL






EΠTH aSlowastH b


SlowastL b

SlowastL1113872 1113873 0

EΠTL aSlowastL b


SlowastH b

SlowastH1113872 1113873

bSlowastH minus b

SlowastL1113872 1113873 b

SlowastH + b

SlowastL1113872 1113873 sTH + sTL( 1113857

2ηT

gt 0

(13)











2TL1113872 1113873

2ηT

bSlowast 2L

(14)


L







H



L



TL


Oi




Oi


TH


TL




EΠO sT( 1113857 1113946sT

sT

1 minus bC

sT( 11138571113872 1113873 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873 minus a

CsT( 1113857 minus

ηOeC2O sT( 1113857

21113896 1113897f sT( 1113857dsT (15)


EΠT sT( 1113857 aC

sT( 1113857 + bC

sT( 1113857 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873

minusηTe

C2T sT( 1113857

2minus

rσ2bC2sT( 1113857

2





EΠT sT( 1113857 aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857

+12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857

(17)


maxaCbC

1113946sT

sT

1 minus bC

sT( 11138571113872 1113873 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873 minus a

CsT( 1113857 minus

ηOeC2O sT( 1113857

21113896 1113897f sT( 1113857dsT

st(IC)bCmiddot

sT( 1113857ge 0

(18)

(IC)aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857 +s

C2T

ηT


sT( 1113857 0 (19)


(IR)aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge ΠT (20)

where eCT(sT) (sC


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857 + 1113946sT

sT

ωηT

bC2

(ω)dω

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

eClowastO

sO

ηO


sCTb

ClowastsT( 1113857

ηT

(21)











i and aNlowasti in-



i and bNlowasti

















Oi |


5

45

aH

STH

4

35

305 06 07 08 09 1

r = 01r = 05r = 09

aHN lowast

aHN lowast

aHS lowast

aHSlowast

aHNlowast

aHS lowast

(a)

aLN lowast

aLN lowast

aLN lowast

aLS lowast

aLSlowast

aLS lowast

5

48

aL

STL

46

44

4202 03 04 05

r = 01r = 05r = 09

(b)


1

06

08

bH

STH

04

02

005 06 07 08 09 1

r = 01r = 05r = 09




(a)

bLNlowast

bLN lowast

bLNlowast

bLS lowast

bLS lowast

bLS lowast

08

06

bL

STL

04

02

002 03 04 05

r = 01r = 05r = 09

(b)













Appendix



01

0

005

∆EprodH

STH

ndash005

ndash0105 06 07 08 09 1

r = 01r = 05r = 09

∆EprodOHSN lowast


SN lowast

∆EprodTHSNlowast

∆EprodTHSN lowast

∆EprodOHSN lowast

(a)

0

times10ndash3

∆EprodL

STL

ndash5

ndash10

ndash1502 03 04 05

r = 01r = 05r = 09

∆EprodTHSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

(b)




EU(x) 1113946+infin

minus infinminus e

minus rx 12πn



(A1)







Ni ηT)




effort






Oi


L aNi b

Ni e

NOi1113872 1113873 1 minus b

Ni1113872 1113873 sOe

NOi + sTie

NTi + θ1113872 1113873 minus a

Ni minus

ηOeN2Oi

21113888 1113889

+ χ aNi + b

Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2minus ΠT

⎛⎝ ⎞⎠

(A2)




(aNlowasti bNlowast

i eNlowastOi )




i bSi )








Oi


maxaS

HbS

HeS

OHaS

LbS

LeS

OL


SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ) times1113896

1 minus bSL1113872 1113873 sOe

SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 11138891113897


SH sO( e

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2b2H2ge

aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTH1113872 1113873

2

2ηT

minusrσ2b2L2

(IR minus L)aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2b2L2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A3)




SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ)times


SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 1113889 + χ1 a

SL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2bS2

L

2minus ΠR

⎛⎝ ⎞⎠

+ χ2 aSH + b

SH sOe

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2bS2

H

2minus a

SL minus b

SL sOe

SOL + θ1113872 1113873 minus

bSLsTH1113872 1113873

2

2ηT

+rσ2bS2

L

2⎛⎝ ⎞⎠

(A4)


zL

zaSi

zL

zbSi

zL

zeSOi

0zL

zχ1ge 0

zL

zχ2ge 0 (A5)



H eSlowastOHa

SlowastL bSlowast


obtained


bSlowastH minus b

SlowastL

s2TH minus s

2TL1113872 1113873 ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

s2TH + ηTrσ21113872 1113873 s

2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

ge 0

aSlowastH minus a

SlowastL

s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 11138731113888 1113889 b

SlowastL minus b

SlowastH1113872 1113873

(A6)


L When rgt 0 let

G(r) s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 1113873

s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

bSlowastL + b


12

rσ2 bSlowastL + b

SlowastH1113872 1113873 (A7)



12

rσ2 bSlowastL + b

SlowastH1113872 1113873

12

rσ2s2TH

s2TH + ηTrσ2

+(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs


1113888 1113889

1

2ηT

s2TH + s

2TL minus

s4TH

s2TH + ηTrσ2

minus(1 minus ρ)s

2TL s

2TL minus 2ρs

2TL + ρs

2TH1113872 1113873

s2TL minus 2ρs

2TL + ρs


⎛⎝ ⎞⎠

(A8)





G(+infin) s2O

ηO

+ θ1113888 1113889

G(0) s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

1 +(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs

2TH

1113888 1113889

(A9)


L ge aSlowast


L aSlowastH


H minus aSlowast



L minus bNlowast



aSlowast

L minus aNlowast

L bNlowast

L minus bSlowast

L1113872 1113873s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TL

ηT

1113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889 (A10)


L minus aNlowast



Proof of eorem A4

(1) ΔEΠSNlowast

OH EΠSlowast

OH minus EΠNlowast


(bSlowast

L )2 le 0

ΔEΠSNlowast

OL EΠSlowast

OL minus EΠNlowast

OL bSlowast

L minus bNlowast

L1113872 1113873s2TL

ηT

minus12

s2TL

ηT

+ rσ21113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889

1ηT

bSlowast

L minus bNlowast

L1113872 1113873 s2TL minus

s2TL

2bNlowastL

bNlowast

L + bSlowast

L1113872 11138731113888 1113889 s2TL

2ηT

bSlowast

L minus bNlowast

L1113872 1113873 1 minusb

Slowast

L

bNlowast

L

⎛⎝ ⎞⎠

minuss2TL + ηTrσ21113872 1113873

2ηT

bSlowast

L minus bNlowast

L1113872 11138732le 0

(A11)

(2) ΔEΠSNlowast

TH EΠSlowast

TH minus EΠNlowast


(bSlowast

L )2 ge 0

ΔEΠSNlowast

TH EΠSlowast

TL minus EΠNlowast

TL 0 (A12)


harr) b(sTharr



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC

sTharr

1113872 1113873 + sOeCO sTharr

1113872 1113873 + θ1113872 1113873bC

sTharr

1113872 1113873 +12

s2T

ηT

minus rσ21113888 1113889bC2

sTharr

1113872 1113873

(A13)



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sTprime( 1113857

(A14)

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

sprime2T

ηT


sTprime( 1113857ge

aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

sprime2T

ηT


sT( 1113857

(A15)




CsTprime( 11138571113872 1113873ge 0 (A16)


bCmiddot


obtained aCmiddot

(sTharr

) + sOeCO

middot


middot



T isin H

there is

aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857

+s2T

ηT


sT( 1113857 0

(A17)



EΠT

middot

sT( 1113857 sT

ηT

bC2

sT( 1113857 (A18)

Bring EΠT

middot


maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A19)



EΠT sT( 1113857 1113946sT

sT

ωηT

bC2

(ω)dω + ΠT (A20)


sT[1113938

sT


1113938sT

sT[1113938

sT


sT

sT(1 minus F(sT)f



maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


f sT( 1113857

sT

ηT

bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A21)



eClowastO

sO

ηO

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

(A22)


2))



aClowast

sT1113872 1113873 ΠT minuss2O

ηO

+ θ1113888 1113889bClowast

sT1113872 1113873 minus12

sT

2

ηT


sT1113872 1113873

(A23)


obtained as

EΠT sT( 1113857 1113946sT

sT

ωηT

bClowast2

(ω)dω + ΠT (A24)


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857

+ 1113946sT

sT

ωηT

bC2

(ω)dω

(A25)

Data Availability




Acknowledgments


References






































Ti are

bNlowasti

s2Ti

s2Ti + ηTrσ2

aNlowasti ΠT minus

s2O

ηO

+ θ1113888 1113889bNlowasti +

12

rσ2 minuss2Ti

ηT

1113888 1113889 bNlowasti1113872 1113873

2

eNlowastOi

sO

ηO

eNlowastTi

s3Ti

ηT s2Ti + ηTrσ21113872 1113873

sTi

ηT

bNlowasti

(8)


i








Ti increases








maxaS

HbS

HeS

OHaS

LbS

LeS

OL


SOH + sTHe

STH + θ1113872 1113873 minus a

SH minus

ηOeS2OH

21113888 1113889 +1113896


SOL + sTHe

STL + θ1113872 1113873 minus a

SL minus

ηOeS2OL

21113888 11138891113897


SH sOe

SOH + sTHe

STH + θ1113872 1113873 minus

ηTeS2TH

2minus

rσ2b2H2ge

aSL + b

SL sOe

SOS + sTHe

LTH + θ1113872 1113873 minus

ηTeL2TH

2minus

rσ2b2L2

(IC minus L)aSL + b

SL sOe

SOL + sTLe

STL + θ1113872 1113873 minus

ηTeS2TL

2minus

rσ2b2L2ge

aSH + b

SH sOe

SOH + sTLe

HTL + θ1113872 1113873 minus

ηTeH2TL

2minus

rσ2b2H2

(IR minus H)aSH + b

SH sOe

SOH + sTHe

STH + θ1113872 1113873 minus

ηTeS2TH

2minus

rσ2b2H2ge ΠT

(IR minus L)aSL + b

SL sOe

SOL + sTLe

STL + θ1113872 1113873 minus

ηTeS2TL

2minus

rσ2b2L2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)


where

eSTH argmax

e

aSH + b

SH sOe

SOH + sTHe

STH + θ1113872 1113873 minus

ηTeS2TH

2minus

rσ2b2H2

1113888 1113889 (10)

eSTL argmax

e

aSL + b

SL sOe

SOL + sTLe

STL + θ1113872 1113873 minus

ηTeS2TL

2minus

rσ2b2L2

1113888 1113889 (11)




L)((aS








aSlowastH ΠT minus

s2O

ηO

+ θ1113888 1113889bSlowastH +

12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastH1113872 1113873

2+

12ηT

s2TH minus s

2TL1113872 1113873 b

SlowastL1113872 1113873

2

bSlowastH

s2TH

s2TH + ηTrσ2

aSlowastL ΠT minus

s2O

ηO

+ θ1113888 1113889bSlowastL +

12

rσ2 minuss2TL

ηT

1113888 1113889 bSlowastL1113872 1113873

2

bSlowastL

(1 minus ρ)s2TL

s2TL minus 2ρs

2TL + ρs


eSlowastOH e

SlowastOL

sO

ηO

eSlowastTH

s3TH

ηT s2TH + ηTrσ21113872 1113873

eSlowastTL

(1 minus ρ)s3TL

s2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873ηT

(12)




L


H 1 andbSlowast












H le aSlowastL






EΠTH aSlowastH b


SlowastL b

SlowastL1113872 1113873 0

EΠTL aSlowastL b


SlowastH b

SlowastH1113872 1113873

bSlowastH minus b

SlowastL1113872 1113873 b

SlowastH + b

SlowastL1113872 1113873 sTH + sTL( 1113857

2ηT

gt 0

(13)











2TL1113872 1113873

2ηT

bSlowast 2L

(14)


L







H



L



TL


Oi




Oi


TH


TL




EΠO sT( 1113857 1113946sT

sT

1 minus bC

sT( 11138571113872 1113873 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873 minus a

CsT( 1113857 minus

ηOeC2O sT( 1113857

21113896 1113897f sT( 1113857dsT (15)


EΠT sT( 1113857 aC

sT( 1113857 + bC

sT( 1113857 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873

minusηTe

C2T sT( 1113857

2minus

rσ2bC2sT( 1113857

2





EΠT sT( 1113857 aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857

+12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857

(17)


maxaCbC

1113946sT

sT

1 minus bC

sT( 11138571113872 1113873 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873 minus a

CsT( 1113857 minus

ηOeC2O sT( 1113857

21113896 1113897f sT( 1113857dsT

st(IC)bCmiddot

sT( 1113857ge 0

(18)

(IC)aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857 +s

C2T

ηT


sT( 1113857 0 (19)


(IR)aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge ΠT (20)

where eCT(sT) (sC


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857 + 1113946sT

sT

ωηT

bC2

(ω)dω

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

eClowastO

sO

ηO


sCTb

ClowastsT( 1113857

ηT

(21)











i and aNlowasti in-



i and bNlowasti

















Oi |


5

45

aH

STH

4

35

305 06 07 08 09 1

r = 01r = 05r = 09

aHN lowast

aHN lowast

aHS lowast

aHSlowast

aHNlowast

aHS lowast

(a)

aLN lowast

aLN lowast

aLN lowast

aLS lowast

aLSlowast

aLS lowast

5

48

aL

STL

46

44

4202 03 04 05

r = 01r = 05r = 09

(b)


1

06

08

bH

STH

04

02

005 06 07 08 09 1

r = 01r = 05r = 09




(a)

bLNlowast

bLN lowast

bLNlowast

bLS lowast

bLS lowast

bLS lowast

08

06

bL

STL

04

02

002 03 04 05

r = 01r = 05r = 09

(b)













Appendix



01

0

005

∆EprodH

STH

ndash005

ndash0105 06 07 08 09 1

r = 01r = 05r = 09

∆EprodOHSN lowast


SN lowast

∆EprodTHSNlowast

∆EprodTHSN lowast

∆EprodOHSN lowast

(a)

0

times10ndash3

∆EprodL

STL

ndash5

ndash10

ndash1502 03 04 05

r = 01r = 05r = 09

∆EprodTHSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

(b)




EU(x) 1113946+infin

minus infinminus e

minus rx 12πn



(A1)







Ni ηT)




effort






Oi


L aNi b

Ni e

NOi1113872 1113873 1 minus b

Ni1113872 1113873 sOe

NOi + sTie

NTi + θ1113872 1113873 minus a

Ni minus

ηOeN2Oi

21113888 1113889

+ χ aNi + b

Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2minus ΠT

⎛⎝ ⎞⎠

(A2)




(aNlowasti bNlowast

i eNlowastOi )




i bSi )








Oi


maxaS

HbS

HeS

OHaS

LbS

LeS

OL


SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ) times1113896

1 minus bSL1113872 1113873 sOe

SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 11138891113897


SH sO( e

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2b2H2ge

aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTH1113872 1113873

2

2ηT

minusrσ2b2L2

(IR minus L)aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2b2L2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A3)




SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ)times


SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 1113889 + χ1 a

SL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2bS2

L

2minus ΠR

⎛⎝ ⎞⎠

+ χ2 aSH + b

SH sOe

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2bS2

H

2minus a

SL minus b

SL sOe

SOL + θ1113872 1113873 minus

bSLsTH1113872 1113873

2

2ηT

+rσ2bS2

L

2⎛⎝ ⎞⎠

(A4)


zL

zaSi

zL

zbSi

zL

zeSOi

0zL

zχ1ge 0

zL

zχ2ge 0 (A5)



H eSlowastOHa

SlowastL bSlowast


obtained


bSlowastH minus b

SlowastL

s2TH minus s

2TL1113872 1113873 ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

s2TH + ηTrσ21113872 1113873 s

2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

ge 0

aSlowastH minus a

SlowastL

s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 11138731113888 1113889 b

SlowastL minus b

SlowastH1113872 1113873

(A6)


L When rgt 0 let

G(r) s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 1113873

s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

bSlowastL + b


12

rσ2 bSlowastL + b

SlowastH1113872 1113873 (A7)



12

rσ2 bSlowastL + b

SlowastH1113872 1113873

12

rσ2s2TH

s2TH + ηTrσ2

+(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs


1113888 1113889

1

2ηT

s2TH + s

2TL minus

s4TH

s2TH + ηTrσ2

minus(1 minus ρ)s

2TL s

2TL minus 2ρs

2TL + ρs

2TH1113872 1113873

s2TL minus 2ρs

2TL + ρs


⎛⎝ ⎞⎠

(A8)





G(+infin) s2O

ηO

+ θ1113888 1113889

G(0) s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

1 +(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs

2TH

1113888 1113889

(A9)


L ge aSlowast


L aSlowastH


H minus aSlowast



L minus bNlowast



aSlowast

L minus aNlowast

L bNlowast

L minus bSlowast

L1113872 1113873s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TL

ηT

1113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889 (A10)


L minus aNlowast



Proof of eorem A4

(1) ΔEΠSNlowast

OH EΠSlowast

OH minus EΠNlowast


(bSlowast

L )2 le 0

ΔEΠSNlowast

OL EΠSlowast

OL minus EΠNlowast

OL bSlowast

L minus bNlowast

L1113872 1113873s2TL

ηT

minus12

s2TL

ηT

+ rσ21113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889

1ηT

bSlowast

L minus bNlowast

L1113872 1113873 s2TL minus

s2TL

2bNlowastL

bNlowast

L + bSlowast

L1113872 11138731113888 1113889 s2TL

2ηT

bSlowast

L minus bNlowast

L1113872 1113873 1 minusb

Slowast

L

bNlowast

L

⎛⎝ ⎞⎠

minuss2TL + ηTrσ21113872 1113873

2ηT

bSlowast

L minus bNlowast

L1113872 11138732le 0

(A11)

(2) ΔEΠSNlowast

TH EΠSlowast

TH minus EΠNlowast


(bSlowast

L )2 ge 0

ΔEΠSNlowast

TH EΠSlowast

TL minus EΠNlowast

TL 0 (A12)


harr) b(sTharr



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC

sTharr

1113872 1113873 + sOeCO sTharr

1113872 1113873 + θ1113872 1113873bC

sTharr

1113872 1113873 +12

s2T

ηT

minus rσ21113888 1113889bC2

sTharr

1113872 1113873

(A13)



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sTprime( 1113857

(A14)

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

sprime2T

ηT


sTprime( 1113857ge

aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

sprime2T

ηT


sT( 1113857

(A15)




CsTprime( 11138571113872 1113873ge 0 (A16)


bCmiddot


obtained aCmiddot

(sTharr

) + sOeCO

middot


middot



T isin H

there is

aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857

+s2T

ηT


sT( 1113857 0

(A17)



EΠT

middot

sT( 1113857 sT

ηT

bC2

sT( 1113857 (A18)

Bring EΠT

middot


maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A19)



EΠT sT( 1113857 1113946sT

sT

ωηT

bC2

(ω)dω + ΠT (A20)


sT[1113938

sT


1113938sT

sT[1113938

sT


sT

sT(1 minus F(sT)f



maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


f sT( 1113857

sT

ηT

bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A21)



eClowastO

sO

ηO

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

(A22)


2))



aClowast

sT1113872 1113873 ΠT minuss2O

ηO

+ θ1113888 1113889bClowast

sT1113872 1113873 minus12

sT

2

ηT


sT1113872 1113873

(A23)


obtained as

EΠT sT( 1113857 1113946sT

sT

ωηT

bClowast2

(ω)dω + ΠT (A24)


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857

+ 1113946sT

sT

ωηT

bC2

(ω)dω

(A25)

Data Availability




Acknowledgments


References



































where

eSTH argmax

e

aSH + b

SH sOe

SOH + sTHe

STH + θ1113872 1113873 minus

ηTeS2TH

2minus

rσ2b2H2

1113888 1113889 (10)

eSTL argmax

e

aSL + b

SL sOe

SOL + sTLe

STL + θ1113872 1113873 minus

ηTeS2TL

2minus

rσ2b2L2

1113888 1113889 (11)




L)((aS








aSlowastH ΠT minus

s2O

ηO

+ θ1113888 1113889bSlowastH +

12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastH1113872 1113873

2+

12ηT

s2TH minus s

2TL1113872 1113873 b

SlowastL1113872 1113873

2

bSlowastH

s2TH

s2TH + ηTrσ2

aSlowastL ΠT minus

s2O

ηO

+ θ1113888 1113889bSlowastL +

12

rσ2 minuss2TL

ηT

1113888 1113889 bSlowastL1113872 1113873

2

bSlowastL

(1 minus ρ)s2TL

s2TL minus 2ρs

2TL + ρs


eSlowastOH e

SlowastOL

sO

ηO

eSlowastTH

s3TH

ηT s2TH + ηTrσ21113872 1113873

eSlowastTL

(1 minus ρ)s3TL

s2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873ηT

(12)




L


H 1 andbSlowast












H le aSlowastL






EΠTH aSlowastH b


SlowastL b

SlowastL1113872 1113873 0

EΠTL aSlowastL b


SlowastH b

SlowastH1113872 1113873

bSlowastH minus b

SlowastL1113872 1113873 b

SlowastH + b

SlowastL1113872 1113873 sTH + sTL( 1113857

2ηT

gt 0

(13)











2TL1113872 1113873

2ηT

bSlowast 2L

(14)


L







H



L



TL


Oi




Oi


TH


TL




EΠO sT( 1113857 1113946sT

sT

1 minus bC

sT( 11138571113872 1113873 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873 minus a

CsT( 1113857 minus

ηOeC2O sT( 1113857

21113896 1113897f sT( 1113857dsT (15)


EΠT sT( 1113857 aC

sT( 1113857 + bC

sT( 1113857 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873

minusηTe

C2T sT( 1113857

2minus

rσ2bC2sT( 1113857

2





EΠT sT( 1113857 aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857

+12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857

(17)


maxaCbC

1113946sT

sT

1 minus bC

sT( 11138571113872 1113873 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873 minus a

CsT( 1113857 minus

ηOeC2O sT( 1113857

21113896 1113897f sT( 1113857dsT

st(IC)bCmiddot

sT( 1113857ge 0

(18)

(IC)aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857 +s

C2T

ηT


sT( 1113857 0 (19)


(IR)aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge ΠT (20)

where eCT(sT) (sC


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857 + 1113946sT

sT

ωηT

bC2

(ω)dω

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

eClowastO

sO

ηO


sCTb

ClowastsT( 1113857

ηT

(21)











i and aNlowasti in-



i and bNlowasti

















Oi |


5

45

aH

STH

4

35

305 06 07 08 09 1

r = 01r = 05r = 09

aHN lowast

aHN lowast

aHS lowast

aHSlowast

aHNlowast

aHS lowast

(a)

aLN lowast

aLN lowast

aLN lowast

aLS lowast

aLSlowast

aLS lowast

5

48

aL

STL

46

44

4202 03 04 05

r = 01r = 05r = 09

(b)


1

06

08

bH

STH

04

02

005 06 07 08 09 1

r = 01r = 05r = 09




(a)

bLNlowast

bLN lowast

bLNlowast

bLS lowast

bLS lowast

bLS lowast

08

06

bL

STL

04

02

002 03 04 05

r = 01r = 05r = 09

(b)













Appendix



01

0

005

∆EprodH

STH

ndash005

ndash0105 06 07 08 09 1

r = 01r = 05r = 09

∆EprodOHSN lowast


SN lowast

∆EprodTHSNlowast

∆EprodTHSN lowast

∆EprodOHSN lowast

(a)

0

times10ndash3

∆EprodL

STL

ndash5

ndash10

ndash1502 03 04 05

r = 01r = 05r = 09

∆EprodTHSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

(b)




EU(x) 1113946+infin

minus infinminus e

minus rx 12πn



(A1)







Ni ηT)




effort






Oi


L aNi b

Ni e

NOi1113872 1113873 1 minus b

Ni1113872 1113873 sOe

NOi + sTie

NTi + θ1113872 1113873 minus a

Ni minus

ηOeN2Oi

21113888 1113889

+ χ aNi + b

Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2minus ΠT

⎛⎝ ⎞⎠

(A2)




(aNlowasti bNlowast

i eNlowastOi )




i bSi )








Oi


maxaS

HbS

HeS

OHaS

LbS

LeS

OL


SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ) times1113896

1 minus bSL1113872 1113873 sOe

SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 11138891113897


SH sO( e

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2b2H2ge

aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTH1113872 1113873

2

2ηT

minusrσ2b2L2

(IR minus L)aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2b2L2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A3)




SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ)times


SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 1113889 + χ1 a

SL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2bS2

L

2minus ΠR

⎛⎝ ⎞⎠

+ χ2 aSH + b

SH sOe

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2bS2

H

2minus a

SL minus b

SL sOe

SOL + θ1113872 1113873 minus

bSLsTH1113872 1113873

2

2ηT

+rσ2bS2

L

2⎛⎝ ⎞⎠

(A4)


zL

zaSi

zL

zbSi

zL

zeSOi

0zL

zχ1ge 0

zL

zχ2ge 0 (A5)



H eSlowastOHa

SlowastL bSlowast


obtained


bSlowastH minus b

SlowastL

s2TH minus s

2TL1113872 1113873 ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

s2TH + ηTrσ21113872 1113873 s

2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

ge 0

aSlowastH minus a

SlowastL

s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 11138731113888 1113889 b

SlowastL minus b

SlowastH1113872 1113873

(A6)


L When rgt 0 let

G(r) s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 1113873

s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

bSlowastL + b


12

rσ2 bSlowastL + b

SlowastH1113872 1113873 (A7)



12

rσ2 bSlowastL + b

SlowastH1113872 1113873

12

rσ2s2TH

s2TH + ηTrσ2

+(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs


1113888 1113889

1

2ηT

s2TH + s

2TL minus

s4TH

s2TH + ηTrσ2

minus(1 minus ρ)s

2TL s

2TL minus 2ρs

2TL + ρs

2TH1113872 1113873

s2TL minus 2ρs

2TL + ρs


⎛⎝ ⎞⎠

(A8)





G(+infin) s2O

ηO

+ θ1113888 1113889

G(0) s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

1 +(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs

2TH

1113888 1113889

(A9)


L ge aSlowast


L aSlowastH


H minus aSlowast



L minus bNlowast



aSlowast

L minus aNlowast

L bNlowast

L minus bSlowast

L1113872 1113873s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TL

ηT

1113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889 (A10)


L minus aNlowast



Proof of eorem A4

(1) ΔEΠSNlowast

OH EΠSlowast

OH minus EΠNlowast


(bSlowast

L )2 le 0

ΔEΠSNlowast

OL EΠSlowast

OL minus EΠNlowast

OL bSlowast

L minus bNlowast

L1113872 1113873s2TL

ηT

minus12

s2TL

ηT

+ rσ21113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889

1ηT

bSlowast

L minus bNlowast

L1113872 1113873 s2TL minus

s2TL

2bNlowastL

bNlowast

L + bSlowast

L1113872 11138731113888 1113889 s2TL

2ηT

bSlowast

L minus bNlowast

L1113872 1113873 1 minusb

Slowast

L

bNlowast

L

⎛⎝ ⎞⎠

minuss2TL + ηTrσ21113872 1113873

2ηT

bSlowast

L minus bNlowast

L1113872 11138732le 0

(A11)

(2) ΔEΠSNlowast

TH EΠSlowast

TH minus EΠNlowast


(bSlowast

L )2 ge 0

ΔEΠSNlowast

TH EΠSlowast

TL minus EΠNlowast

TL 0 (A12)


harr) b(sTharr



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC

sTharr

1113872 1113873 + sOeCO sTharr

1113872 1113873 + θ1113872 1113873bC

sTharr

1113872 1113873 +12

s2T

ηT

minus rσ21113888 1113889bC2

sTharr

1113872 1113873

(A13)



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sTprime( 1113857

(A14)

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

sprime2T

ηT


sTprime( 1113857ge

aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

sprime2T

ηT


sT( 1113857

(A15)




CsTprime( 11138571113872 1113873ge 0 (A16)


bCmiddot


obtained aCmiddot

(sTharr

) + sOeCO

middot


middot



T isin H

there is

aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857

+s2T

ηT


sT( 1113857 0

(A17)



EΠT

middot

sT( 1113857 sT

ηT

bC2

sT( 1113857 (A18)

Bring EΠT

middot


maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A19)



EΠT sT( 1113857 1113946sT

sT

ωηT

bC2

(ω)dω + ΠT (A20)


sT[1113938

sT


1113938sT

sT[1113938

sT


sT

sT(1 minus F(sT)f



maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


f sT( 1113857

sT

ηT

bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A21)



eClowastO

sO

ηO

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

(A22)


2))



aClowast

sT1113872 1113873 ΠT minuss2O

ηO

+ θ1113888 1113889bClowast

sT1113872 1113873 minus12

sT

2

ηT


sT1113872 1113873

(A23)


obtained as

EΠT sT( 1113857 1113946sT

sT

ωηT

bClowast2

(ω)dω + ΠT (A24)


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857

+ 1113946sT

sT

ωηT

bC2

(ω)dω

(A25)

Data Availability




Acknowledgments


References











































H le aSlowastL






EΠTH aSlowastH b


SlowastL b

SlowastL1113872 1113873 0

EΠTL aSlowastL b


SlowastH b

SlowastH1113872 1113873

bSlowastH minus b

SlowastL1113872 1113873 b

SlowastH + b

SlowastL1113872 1113873 sTH + sTL( 1113857

2ηT

gt 0

(13)











2TL1113872 1113873

2ηT

bSlowast 2L

(14)


L







H



L



TL


Oi




Oi


TH


TL




EΠO sT( 1113857 1113946sT

sT

1 minus bC

sT( 11138571113872 1113873 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873 minus a

CsT( 1113857 minus

ηOeC2O sT( 1113857

21113896 1113897f sT( 1113857dsT (15)


EΠT sT( 1113857 aC

sT( 1113857 + bC

sT( 1113857 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873

minusηTe

C2T sT( 1113857

2minus

rσ2bC2sT( 1113857

2





EΠT sT( 1113857 aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857

+12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857

(17)


maxaCbC

1113946sT

sT

1 minus bC

sT( 11138571113872 1113873 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873 minus a

CsT( 1113857 minus

ηOeC2O sT( 1113857

21113896 1113897f sT( 1113857dsT

st(IC)bCmiddot

sT( 1113857ge 0

(18)

(IC)aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857 +s

C2T

ηT


sT( 1113857 0 (19)


(IR)aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge ΠT (20)

where eCT(sT) (sC


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857 + 1113946sT

sT

ωηT

bC2

(ω)dω

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

eClowastO

sO

ηO


sCTb

ClowastsT( 1113857

ηT

(21)











i and aNlowasti in-



i and bNlowasti

















Oi |


5

45

aH

STH

4

35

305 06 07 08 09 1

r = 01r = 05r = 09

aHN lowast

aHN lowast

aHS lowast

aHSlowast

aHNlowast

aHS lowast

(a)

aLN lowast

aLN lowast

aLN lowast

aLS lowast

aLSlowast

aLS lowast

5

48

aL

STL

46

44

4202 03 04 05

r = 01r = 05r = 09

(b)


1

06

08

bH

STH

04

02

005 06 07 08 09 1

r = 01r = 05r = 09




(a)

bLNlowast

bLN lowast

bLNlowast

bLS lowast

bLS lowast

bLS lowast

08

06

bL

STL

04

02

002 03 04 05

r = 01r = 05r = 09

(b)













Appendix



01

0

005

∆EprodH

STH

ndash005

ndash0105 06 07 08 09 1

r = 01r = 05r = 09

∆EprodOHSN lowast


SN lowast

∆EprodTHSNlowast

∆EprodTHSN lowast

∆EprodOHSN lowast

(a)

0

times10ndash3

∆EprodL

STL

ndash5

ndash10

ndash1502 03 04 05

r = 01r = 05r = 09

∆EprodTHSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

(b)




EU(x) 1113946+infin

minus infinminus e

minus rx 12πn



(A1)







Ni ηT)




effort






Oi


L aNi b

Ni e

NOi1113872 1113873 1 minus b

Ni1113872 1113873 sOe

NOi + sTie

NTi + θ1113872 1113873 minus a

Ni minus

ηOeN2Oi

21113888 1113889

+ χ aNi + b

Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2minus ΠT

⎛⎝ ⎞⎠

(A2)




(aNlowasti bNlowast

i eNlowastOi )




i bSi )








Oi


maxaS

HbS

HeS

OHaS

LbS

LeS

OL


SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ) times1113896

1 minus bSL1113872 1113873 sOe

SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 11138891113897


SH sO( e

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2b2H2ge

aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTH1113872 1113873

2

2ηT

minusrσ2b2L2

(IR minus L)aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2b2L2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A3)




SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ)times


SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 1113889 + χ1 a

SL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2bS2

L

2minus ΠR

⎛⎝ ⎞⎠

+ χ2 aSH + b

SH sOe

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2bS2

H

2minus a

SL minus b

SL sOe

SOL + θ1113872 1113873 minus

bSLsTH1113872 1113873

2

2ηT

+rσ2bS2

L

2⎛⎝ ⎞⎠

(A4)


zL

zaSi

zL

zbSi

zL

zeSOi

0zL

zχ1ge 0

zL

zχ2ge 0 (A5)



H eSlowastOHa

SlowastL bSlowast


obtained


bSlowastH minus b

SlowastL

s2TH minus s

2TL1113872 1113873 ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

s2TH + ηTrσ21113872 1113873 s

2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

ge 0

aSlowastH minus a

SlowastL

s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 11138731113888 1113889 b

SlowastL minus b

SlowastH1113872 1113873

(A6)


L When rgt 0 let

G(r) s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 1113873

s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

bSlowastL + b


12

rσ2 bSlowastL + b

SlowastH1113872 1113873 (A7)



12

rσ2 bSlowastL + b

SlowastH1113872 1113873

12

rσ2s2TH

s2TH + ηTrσ2

+(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs


1113888 1113889

1

2ηT

s2TH + s

2TL minus

s4TH

s2TH + ηTrσ2

minus(1 minus ρ)s

2TL s

2TL minus 2ρs

2TL + ρs

2TH1113872 1113873

s2TL minus 2ρs

2TL + ρs


⎛⎝ ⎞⎠

(A8)





G(+infin) s2O

ηO

+ θ1113888 1113889

G(0) s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

1 +(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs

2TH

1113888 1113889

(A9)


L ge aSlowast


L aSlowastH


H minus aSlowast



L minus bNlowast



aSlowast

L minus aNlowast

L bNlowast

L minus bSlowast

L1113872 1113873s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TL

ηT

1113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889 (A10)


L minus aNlowast



Proof of eorem A4

(1) ΔEΠSNlowast

OH EΠSlowast

OH minus EΠNlowast


(bSlowast

L )2 le 0

ΔEΠSNlowast

OL EΠSlowast

OL minus EΠNlowast

OL bSlowast

L minus bNlowast

L1113872 1113873s2TL

ηT

minus12

s2TL

ηT

+ rσ21113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889

1ηT

bSlowast

L minus bNlowast

L1113872 1113873 s2TL minus

s2TL

2bNlowastL

bNlowast

L + bSlowast

L1113872 11138731113888 1113889 s2TL

2ηT

bSlowast

L minus bNlowast

L1113872 1113873 1 minusb

Slowast

L

bNlowast

L

⎛⎝ ⎞⎠

minuss2TL + ηTrσ21113872 1113873

2ηT

bSlowast

L minus bNlowast

L1113872 11138732le 0

(A11)

(2) ΔEΠSNlowast

TH EΠSlowast

TH minus EΠNlowast


(bSlowast

L )2 ge 0

ΔEΠSNlowast

TH EΠSlowast

TL minus EΠNlowast

TL 0 (A12)


harr) b(sTharr



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC

sTharr

1113872 1113873 + sOeCO sTharr

1113872 1113873 + θ1113872 1113873bC

sTharr

1113872 1113873 +12

s2T

ηT

minus rσ21113888 1113889bC2

sTharr

1113872 1113873

(A13)



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sTprime( 1113857

(A14)

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

sprime2T

ηT


sTprime( 1113857ge

aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

sprime2T

ηT


sT( 1113857

(A15)




CsTprime( 11138571113872 1113873ge 0 (A16)


bCmiddot


obtained aCmiddot

(sTharr

) + sOeCO

middot


middot



T isin H

there is

aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857

+s2T

ηT


sT( 1113857 0

(A17)



EΠT

middot

sT( 1113857 sT

ηT

bC2

sT( 1113857 (A18)

Bring EΠT

middot


maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A19)



EΠT sT( 1113857 1113946sT

sT

ωηT

bC2

(ω)dω + ΠT (A20)


sT[1113938

sT


1113938sT

sT[1113938

sT


sT

sT(1 minus F(sT)f



maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


f sT( 1113857

sT

ηT

bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A21)



eClowastO

sO

ηO

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

(A22)


2))



aClowast

sT1113872 1113873 ΠT minuss2O

ηO

+ θ1113888 1113889bClowast

sT1113872 1113873 minus12

sT

2

ηT


sT1113872 1113873

(A23)


obtained as

EΠT sT( 1113857 1113946sT

sT

ωηT

bClowast2

(ω)dω + ΠT (A24)


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857

+ 1113946sT

sT

ωηT

bC2

(ω)dω

(A25)

Data Availability




Acknowledgments


References





































H



L



TL


Oi




Oi


TH


TL




EΠO sT( 1113857 1113946sT

sT

1 minus bC

sT( 11138571113872 1113873 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873 minus a

CsT( 1113857 minus

ηOeC2O sT( 1113857

21113896 1113897f sT( 1113857dsT (15)


EΠT sT( 1113857 aC

sT( 1113857 + bC

sT( 1113857 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873

minusηTe

C2T sT( 1113857

2minus

rσ2bC2sT( 1113857

2





EΠT sT( 1113857 aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857

+12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857

(17)


maxaCbC

1113946sT

sT

1 minus bC

sT( 11138571113872 1113873 sOeCO sT( 1113857 + s

CTe

CT sT( 1113857 + θ1113872 1113873 minus a

CsT( 1113857 minus

ηOeC2O sT( 1113857

21113896 1113897f sT( 1113857dsT

st(IC)bCmiddot

sT( 1113857ge 0

(18)

(IC)aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857 +s

C2T

ηT


sT( 1113857 0 (19)


(IR)aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge ΠT (20)

where eCT(sT) (sC


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857 + 1113946sT

sT

ωηT

bC2

(ω)dω

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

eClowastO

sO

ηO


sCTb

ClowastsT( 1113857

ηT

(21)











i and aNlowasti in-



i and bNlowasti

















Oi |


5

45

aH

STH

4

35

305 06 07 08 09 1

r = 01r = 05r = 09

aHN lowast

aHN lowast

aHS lowast

aHSlowast

aHNlowast

aHS lowast

(a)

aLN lowast

aLN lowast

aLN lowast

aLS lowast

aLSlowast

aLS lowast

5

48

aL

STL

46

44

4202 03 04 05

r = 01r = 05r = 09

(b)


1

06

08

bH

STH

04

02

005 06 07 08 09 1

r = 01r = 05r = 09




(a)

bLNlowast

bLN lowast

bLNlowast

bLS lowast

bLS lowast

bLS lowast

08

06

bL

STL

04

02

002 03 04 05

r = 01r = 05r = 09

(b)













Appendix



01

0

005

∆EprodH

STH

ndash005

ndash0105 06 07 08 09 1

r = 01r = 05r = 09

∆EprodOHSN lowast


SN lowast

∆EprodTHSNlowast

∆EprodTHSN lowast

∆EprodOHSN lowast

(a)

0

times10ndash3

∆EprodL

STL

ndash5

ndash10

ndash1502 03 04 05

r = 01r = 05r = 09

∆EprodTHSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

(b)




EU(x) 1113946+infin

minus infinminus e

minus rx 12πn



(A1)







Ni ηT)




effort






Oi


L aNi b

Ni e

NOi1113872 1113873 1 minus b

Ni1113872 1113873 sOe

NOi + sTie

NTi + θ1113872 1113873 minus a

Ni minus

ηOeN2Oi

21113888 1113889

+ χ aNi + b

Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2minus ΠT

⎛⎝ ⎞⎠

(A2)




(aNlowasti bNlowast

i eNlowastOi )




i bSi )








Oi


maxaS

HbS

HeS

OHaS

LbS

LeS

OL


SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ) times1113896

1 minus bSL1113872 1113873 sOe

SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 11138891113897


SH sO( e

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2b2H2ge

aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTH1113872 1113873

2

2ηT

minusrσ2b2L2

(IR minus L)aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2b2L2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A3)




SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ)times


SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 1113889 + χ1 a

SL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2bS2

L

2minus ΠR

⎛⎝ ⎞⎠

+ χ2 aSH + b

SH sOe

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2bS2

H

2minus a

SL minus b

SL sOe

SOL + θ1113872 1113873 minus

bSLsTH1113872 1113873

2

2ηT

+rσ2bS2

L

2⎛⎝ ⎞⎠

(A4)


zL

zaSi

zL

zbSi

zL

zeSOi

0zL

zχ1ge 0

zL

zχ2ge 0 (A5)



H eSlowastOHa

SlowastL bSlowast


obtained


bSlowastH minus b

SlowastL

s2TH minus s

2TL1113872 1113873 ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

s2TH + ηTrσ21113872 1113873 s

2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

ge 0

aSlowastH minus a

SlowastL

s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 11138731113888 1113889 b

SlowastL minus b

SlowastH1113872 1113873

(A6)


L When rgt 0 let

G(r) s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 1113873

s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

bSlowastL + b


12

rσ2 bSlowastL + b

SlowastH1113872 1113873 (A7)



12

rσ2 bSlowastL + b

SlowastH1113872 1113873

12

rσ2s2TH

s2TH + ηTrσ2

+(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs


1113888 1113889

1

2ηT

s2TH + s

2TL minus

s4TH

s2TH + ηTrσ2

minus(1 minus ρ)s

2TL s

2TL minus 2ρs

2TL + ρs

2TH1113872 1113873

s2TL minus 2ρs

2TL + ρs


⎛⎝ ⎞⎠

(A8)





G(+infin) s2O

ηO

+ θ1113888 1113889

G(0) s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

1 +(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs

2TH

1113888 1113889

(A9)


L ge aSlowast


L aSlowastH


H minus aSlowast



L minus bNlowast



aSlowast

L minus aNlowast

L bNlowast

L minus bSlowast

L1113872 1113873s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TL

ηT

1113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889 (A10)


L minus aNlowast



Proof of eorem A4

(1) ΔEΠSNlowast

OH EΠSlowast

OH minus EΠNlowast


(bSlowast

L )2 le 0

ΔEΠSNlowast

OL EΠSlowast

OL minus EΠNlowast

OL bSlowast

L minus bNlowast

L1113872 1113873s2TL

ηT

minus12

s2TL

ηT

+ rσ21113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889

1ηT

bSlowast

L minus bNlowast

L1113872 1113873 s2TL minus

s2TL

2bNlowastL

bNlowast

L + bSlowast

L1113872 11138731113888 1113889 s2TL

2ηT

bSlowast

L minus bNlowast

L1113872 1113873 1 minusb

Slowast

L

bNlowast

L

⎛⎝ ⎞⎠

minuss2TL + ηTrσ21113872 1113873

2ηT

bSlowast

L minus bNlowast

L1113872 11138732le 0

(A11)

(2) ΔEΠSNlowast

TH EΠSlowast

TH minus EΠNlowast


(bSlowast

L )2 ge 0

ΔEΠSNlowast

TH EΠSlowast

TL minus EΠNlowast

TL 0 (A12)


harr) b(sTharr



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC

sTharr

1113872 1113873 + sOeCO sTharr

1113872 1113873 + θ1113872 1113873bC

sTharr

1113872 1113873 +12

s2T

ηT

minus rσ21113888 1113889bC2

sTharr

1113872 1113873

(A13)



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sTprime( 1113857

(A14)

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

sprime2T

ηT


sTprime( 1113857ge

aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

sprime2T

ηT


sT( 1113857

(A15)




CsTprime( 11138571113872 1113873ge 0 (A16)


bCmiddot


obtained aCmiddot

(sTharr

) + sOeCO

middot


middot



T isin H

there is

aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857

+s2T

ηT


sT( 1113857 0

(A17)



EΠT

middot

sT( 1113857 sT

ηT

bC2

sT( 1113857 (A18)

Bring EΠT

middot


maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A19)



EΠT sT( 1113857 1113946sT

sT

ωηT

bC2

(ω)dω + ΠT (A20)


sT[1113938

sT


1113938sT

sT[1113938

sT


sT

sT(1 minus F(sT)f



maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


f sT( 1113857

sT

ηT

bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A21)



eClowastO

sO

ηO

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

(A22)


2))



aClowast

sT1113872 1113873 ΠT minuss2O

ηO

+ θ1113888 1113889bClowast

sT1113872 1113873 minus12

sT

2

ηT


sT1113872 1113873

(A23)


obtained as

EΠT sT( 1113857 1113946sT

sT

ωηT

bClowast2

(ω)dω + ΠT (A24)


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857

+ 1113946sT

sT

ωηT

bC2

(ω)dω

(A25)

Data Availability




Acknowledgments


References



































(IR)aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge ΠT (20)

where eCT(sT) (sC


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857 + 1113946sT

sT

ωηT

bC2

(ω)dω

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

eClowastO

sO

ηO


sCTb

ClowastsT( 1113857

ηT

(21)











i and aNlowasti in-



i and bNlowasti

















Oi |


5

45

aH

STH

4

35

305 06 07 08 09 1

r = 01r = 05r = 09

aHN lowast

aHN lowast

aHS lowast

aHSlowast

aHNlowast

aHS lowast

(a)

aLN lowast

aLN lowast

aLN lowast

aLS lowast

aLSlowast

aLS lowast

5

48

aL

STL

46

44

4202 03 04 05

r = 01r = 05r = 09

(b)


1

06

08

bH

STH

04

02

005 06 07 08 09 1

r = 01r = 05r = 09




(a)

bLNlowast

bLN lowast

bLNlowast

bLS lowast

bLS lowast

bLS lowast

08

06

bL

STL

04

02

002 03 04 05

r = 01r = 05r = 09

(b)













Appendix



01

0

005

∆EprodH

STH

ndash005

ndash0105 06 07 08 09 1

r = 01r = 05r = 09

∆EprodOHSN lowast


SN lowast

∆EprodTHSNlowast

∆EprodTHSN lowast

∆EprodOHSN lowast

(a)

0

times10ndash3

∆EprodL

STL

ndash5

ndash10

ndash1502 03 04 05

r = 01r = 05r = 09

∆EprodTHSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

(b)




EU(x) 1113946+infin

minus infinminus e

minus rx 12πn



(A1)







Ni ηT)




effort






Oi


L aNi b

Ni e

NOi1113872 1113873 1 minus b

Ni1113872 1113873 sOe

NOi + sTie

NTi + θ1113872 1113873 minus a

Ni minus

ηOeN2Oi

21113888 1113889

+ χ aNi + b

Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2minus ΠT

⎛⎝ ⎞⎠

(A2)




(aNlowasti bNlowast

i eNlowastOi )




i bSi )








Oi


maxaS

HbS

HeS

OHaS

LbS

LeS

OL


SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ) times1113896

1 minus bSL1113872 1113873 sOe

SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 11138891113897


SH sO( e

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2b2H2ge

aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTH1113872 1113873

2

2ηT

minusrσ2b2L2

(IR minus L)aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2b2L2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A3)




SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ)times


SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 1113889 + χ1 a

SL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2bS2

L

2minus ΠR

⎛⎝ ⎞⎠

+ χ2 aSH + b

SH sOe

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2bS2

H

2minus a

SL minus b

SL sOe

SOL + θ1113872 1113873 minus

bSLsTH1113872 1113873

2

2ηT

+rσ2bS2

L

2⎛⎝ ⎞⎠

(A4)


zL

zaSi

zL

zbSi

zL

zeSOi

0zL

zχ1ge 0

zL

zχ2ge 0 (A5)



H eSlowastOHa

SlowastL bSlowast


obtained


bSlowastH minus b

SlowastL

s2TH minus s

2TL1113872 1113873 ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

s2TH + ηTrσ21113872 1113873 s

2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

ge 0

aSlowastH minus a

SlowastL

s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 11138731113888 1113889 b

SlowastL minus b

SlowastH1113872 1113873

(A6)


L When rgt 0 let

G(r) s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 1113873

s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

bSlowastL + b


12

rσ2 bSlowastL + b

SlowastH1113872 1113873 (A7)



12

rσ2 bSlowastL + b

SlowastH1113872 1113873

12

rσ2s2TH

s2TH + ηTrσ2

+(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs


1113888 1113889

1

2ηT

s2TH + s

2TL minus

s4TH

s2TH + ηTrσ2

minus(1 minus ρ)s

2TL s

2TL minus 2ρs

2TL + ρs

2TH1113872 1113873

s2TL minus 2ρs

2TL + ρs


⎛⎝ ⎞⎠

(A8)





G(+infin) s2O

ηO

+ θ1113888 1113889

G(0) s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

1 +(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs

2TH

1113888 1113889

(A9)


L ge aSlowast


L aSlowastH


H minus aSlowast



L minus bNlowast



aSlowast

L minus aNlowast

L bNlowast

L minus bSlowast

L1113872 1113873s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TL

ηT

1113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889 (A10)


L minus aNlowast



Proof of eorem A4

(1) ΔEΠSNlowast

OH EΠSlowast

OH minus EΠNlowast


(bSlowast

L )2 le 0

ΔEΠSNlowast

OL EΠSlowast

OL minus EΠNlowast

OL bSlowast

L minus bNlowast

L1113872 1113873s2TL

ηT

minus12

s2TL

ηT

+ rσ21113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889

1ηT

bSlowast

L minus bNlowast

L1113872 1113873 s2TL minus

s2TL

2bNlowastL

bNlowast

L + bSlowast

L1113872 11138731113888 1113889 s2TL

2ηT

bSlowast

L minus bNlowast

L1113872 1113873 1 minusb

Slowast

L

bNlowast

L

⎛⎝ ⎞⎠

minuss2TL + ηTrσ21113872 1113873

2ηT

bSlowast

L minus bNlowast

L1113872 11138732le 0

(A11)

(2) ΔEΠSNlowast

TH EΠSlowast

TH minus EΠNlowast


(bSlowast

L )2 ge 0

ΔEΠSNlowast

TH EΠSlowast

TL minus EΠNlowast

TL 0 (A12)


harr) b(sTharr



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC

sTharr

1113872 1113873 + sOeCO sTharr

1113872 1113873 + θ1113872 1113873bC

sTharr

1113872 1113873 +12

s2T

ηT

minus rσ21113888 1113889bC2

sTharr

1113872 1113873

(A13)



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sTprime( 1113857

(A14)

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

sprime2T

ηT


sTprime( 1113857ge

aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

sprime2T

ηT


sT( 1113857

(A15)




CsTprime( 11138571113872 1113873ge 0 (A16)


bCmiddot


obtained aCmiddot

(sTharr

) + sOeCO

middot


middot



T isin H

there is

aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857

+s2T

ηT


sT( 1113857 0

(A17)



EΠT

middot

sT( 1113857 sT

ηT

bC2

sT( 1113857 (A18)

Bring EΠT

middot


maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A19)



EΠT sT( 1113857 1113946sT

sT

ωηT

bC2

(ω)dω + ΠT (A20)


sT[1113938

sT


1113938sT

sT[1113938

sT


sT

sT(1 minus F(sT)f



maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


f sT( 1113857

sT

ηT

bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A21)



eClowastO

sO

ηO

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

(A22)


2))



aClowast

sT1113872 1113873 ΠT minuss2O

ηO

+ θ1113888 1113889bClowast

sT1113872 1113873 minus12

sT

2

ηT


sT1113872 1113873

(A23)


obtained as

EΠT sT( 1113857 1113946sT

sT

ωηT

bClowast2

(ω)dω + ΠT (A24)


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857

+ 1113946sT

sT

ωηT

bC2

(ω)dω

(A25)

Data Availability




Acknowledgments


References











































Oi |


5

45

aH

STH

4

35

305 06 07 08 09 1

r = 01r = 05r = 09

aHN lowast

aHN lowast

aHS lowast

aHSlowast

aHNlowast

aHS lowast

(a)

aLN lowast

aLN lowast

aLN lowast

aLS lowast

aLSlowast

aLS lowast

5

48

aL

STL

46

44

4202 03 04 05

r = 01r = 05r = 09

(b)


1

06

08

bH

STH

04

02

005 06 07 08 09 1

r = 01r = 05r = 09




(a)

bLNlowast

bLN lowast

bLNlowast

bLS lowast

bLS lowast

bLS lowast

08

06

bL

STL

04

02

002 03 04 05

r = 01r = 05r = 09

(b)













Appendix



01

0

005

∆EprodH

STH

ndash005

ndash0105 06 07 08 09 1

r = 01r = 05r = 09

∆EprodOHSN lowast


SN lowast

∆EprodTHSNlowast

∆EprodTHSN lowast

∆EprodOHSN lowast

(a)

0

times10ndash3

∆EprodL

STL

ndash5

ndash10

ndash1502 03 04 05

r = 01r = 05r = 09

∆EprodTHSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

(b)




EU(x) 1113946+infin

minus infinminus e

minus rx 12πn



(A1)







Ni ηT)




effort






Oi


L aNi b

Ni e

NOi1113872 1113873 1 minus b

Ni1113872 1113873 sOe

NOi + sTie

NTi + θ1113872 1113873 minus a

Ni minus

ηOeN2Oi

21113888 1113889

+ χ aNi + b

Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2minus ΠT

⎛⎝ ⎞⎠

(A2)




(aNlowasti bNlowast

i eNlowastOi )




i bSi )








Oi


maxaS

HbS

HeS

OHaS

LbS

LeS

OL


SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ) times1113896

1 minus bSL1113872 1113873 sOe

SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 11138891113897


SH sO( e

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2b2H2ge

aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTH1113872 1113873

2

2ηT

minusrσ2b2L2

(IR minus L)aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2b2L2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A3)




SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ)times


SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 1113889 + χ1 a

SL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2bS2

L

2minus ΠR

⎛⎝ ⎞⎠

+ χ2 aSH + b

SH sOe

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2bS2

H

2minus a

SL minus b

SL sOe

SOL + θ1113872 1113873 minus

bSLsTH1113872 1113873

2

2ηT

+rσ2bS2

L

2⎛⎝ ⎞⎠

(A4)


zL

zaSi

zL

zbSi

zL

zeSOi

0zL

zχ1ge 0

zL

zχ2ge 0 (A5)



H eSlowastOHa

SlowastL bSlowast


obtained


bSlowastH minus b

SlowastL

s2TH minus s

2TL1113872 1113873 ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

s2TH + ηTrσ21113872 1113873 s

2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

ge 0

aSlowastH minus a

SlowastL

s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 11138731113888 1113889 b

SlowastL minus b

SlowastH1113872 1113873

(A6)


L When rgt 0 let

G(r) s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 1113873

s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

bSlowastL + b


12

rσ2 bSlowastL + b

SlowastH1113872 1113873 (A7)



12

rσ2 bSlowastL + b

SlowastH1113872 1113873

12

rσ2s2TH

s2TH + ηTrσ2

+(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs


1113888 1113889

1

2ηT

s2TH + s

2TL minus

s4TH

s2TH + ηTrσ2

minus(1 minus ρ)s

2TL s

2TL minus 2ρs

2TL + ρs

2TH1113872 1113873

s2TL minus 2ρs

2TL + ρs


⎛⎝ ⎞⎠

(A8)





G(+infin) s2O

ηO

+ θ1113888 1113889

G(0) s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

1 +(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs

2TH

1113888 1113889

(A9)


L ge aSlowast


L aSlowastH


H minus aSlowast



L minus bNlowast



aSlowast

L minus aNlowast

L bNlowast

L minus bSlowast

L1113872 1113873s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TL

ηT

1113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889 (A10)


L minus aNlowast



Proof of eorem A4

(1) ΔEΠSNlowast

OH EΠSlowast

OH minus EΠNlowast


(bSlowast

L )2 le 0

ΔEΠSNlowast

OL EΠSlowast

OL minus EΠNlowast

OL bSlowast

L minus bNlowast

L1113872 1113873s2TL

ηT

minus12

s2TL

ηT

+ rσ21113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889

1ηT

bSlowast

L minus bNlowast

L1113872 1113873 s2TL minus

s2TL

2bNlowastL

bNlowast

L + bSlowast

L1113872 11138731113888 1113889 s2TL

2ηT

bSlowast

L minus bNlowast

L1113872 1113873 1 minusb

Slowast

L

bNlowast

L

⎛⎝ ⎞⎠

minuss2TL + ηTrσ21113872 1113873

2ηT

bSlowast

L minus bNlowast

L1113872 11138732le 0

(A11)

(2) ΔEΠSNlowast

TH EΠSlowast

TH minus EΠNlowast


(bSlowast

L )2 ge 0

ΔEΠSNlowast

TH EΠSlowast

TL minus EΠNlowast

TL 0 (A12)


harr) b(sTharr



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC

sTharr

1113872 1113873 + sOeCO sTharr

1113872 1113873 + θ1113872 1113873bC

sTharr

1113872 1113873 +12

s2T

ηT

minus rσ21113888 1113889bC2

sTharr

1113872 1113873

(A13)



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sTprime( 1113857

(A14)

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

sprime2T

ηT


sTprime( 1113857ge

aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

sprime2T

ηT


sT( 1113857

(A15)




CsTprime( 11138571113872 1113873ge 0 (A16)


bCmiddot


obtained aCmiddot

(sTharr

) + sOeCO

middot


middot



T isin H

there is

aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857

+s2T

ηT


sT( 1113857 0

(A17)



EΠT

middot

sT( 1113857 sT

ηT

bC2

sT( 1113857 (A18)

Bring EΠT

middot


maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A19)



EΠT sT( 1113857 1113946sT

sT

ωηT

bC2

(ω)dω + ΠT (A20)


sT[1113938

sT


1113938sT

sT[1113938

sT


sT

sT(1 minus F(sT)f



maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


f sT( 1113857

sT

ηT

bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A21)



eClowastO

sO

ηO

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

(A22)


2))



aClowast

sT1113872 1113873 ΠT minuss2O

ηO

+ θ1113888 1113889bClowast

sT1113872 1113873 minus12

sT

2

ηT


sT1113872 1113873

(A23)


obtained as

EΠT sT( 1113857 1113946sT

sT

ωηT

bClowast2

(ω)dω + ΠT (A24)


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857

+ 1113946sT

sT

ωηT

bC2

(ω)dω

(A25)

Data Availability




Acknowledgments


References













































Appendix



01

0

005

∆EprodH

STH

ndash005

ndash0105 06 07 08 09 1

r = 01r = 05r = 09

∆EprodOHSN lowast


SN lowast

∆EprodTHSNlowast

∆EprodTHSN lowast

∆EprodOHSN lowast

(a)

0

times10ndash3

∆EprodL

STL

ndash5

ndash10

ndash1502 03 04 05

r = 01r = 05r = 09

∆EprodTHSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

∆EprodOLSN lowast

(b)




EU(x) 1113946+infin

minus infinminus e

minus rx 12πn



(A1)







Ni ηT)




effort






Oi


L aNi b

Ni e

NOi1113872 1113873 1 minus b

Ni1113872 1113873 sOe

NOi + sTie

NTi + θ1113872 1113873 minus a

Ni minus

ηOeN2Oi

21113888 1113889

+ χ aNi + b

Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2minus ΠT

⎛⎝ ⎞⎠

(A2)




(aNlowasti bNlowast

i eNlowastOi )




i bSi )








Oi


maxaS

HbS

HeS

OHaS

LbS

LeS

OL


SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ) times1113896

1 minus bSL1113872 1113873 sOe

SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 11138891113897


SH sO( e

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2b2H2ge

aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTH1113872 1113873

2

2ηT

minusrσ2b2L2

(IR minus L)aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2b2L2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A3)




SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ)times


SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 1113889 + χ1 a

SL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2bS2

L

2minus ΠR

⎛⎝ ⎞⎠

+ χ2 aSH + b

SH sOe

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2bS2

H

2minus a

SL minus b

SL sOe

SOL + θ1113872 1113873 minus

bSLsTH1113872 1113873

2

2ηT

+rσ2bS2

L

2⎛⎝ ⎞⎠

(A4)


zL

zaSi

zL

zbSi

zL

zeSOi

0zL

zχ1ge 0

zL

zχ2ge 0 (A5)



H eSlowastOHa

SlowastL bSlowast


obtained


bSlowastH minus b

SlowastL

s2TH minus s

2TL1113872 1113873 ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

s2TH + ηTrσ21113872 1113873 s

2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

ge 0

aSlowastH minus a

SlowastL

s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 11138731113888 1113889 b

SlowastL minus b

SlowastH1113872 1113873

(A6)


L When rgt 0 let

G(r) s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 1113873

s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

bSlowastL + b


12

rσ2 bSlowastL + b

SlowastH1113872 1113873 (A7)



12

rσ2 bSlowastL + b

SlowastH1113872 1113873

12

rσ2s2TH

s2TH + ηTrσ2

+(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs


1113888 1113889

1

2ηT

s2TH + s

2TL minus

s4TH

s2TH + ηTrσ2

minus(1 minus ρ)s

2TL s

2TL minus 2ρs

2TL + ρs

2TH1113872 1113873

s2TL minus 2ρs

2TL + ρs


⎛⎝ ⎞⎠

(A8)





G(+infin) s2O

ηO

+ θ1113888 1113889

G(0) s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

1 +(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs

2TH

1113888 1113889

(A9)


L ge aSlowast


L aSlowastH


H minus aSlowast



L minus bNlowast



aSlowast

L minus aNlowast

L bNlowast

L minus bSlowast

L1113872 1113873s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TL

ηT

1113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889 (A10)


L minus aNlowast



Proof of eorem A4

(1) ΔEΠSNlowast

OH EΠSlowast

OH minus EΠNlowast


(bSlowast

L )2 le 0

ΔEΠSNlowast

OL EΠSlowast

OL minus EΠNlowast

OL bSlowast

L minus bNlowast

L1113872 1113873s2TL

ηT

minus12

s2TL

ηT

+ rσ21113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889

1ηT

bSlowast

L minus bNlowast

L1113872 1113873 s2TL minus

s2TL

2bNlowastL

bNlowast

L + bSlowast

L1113872 11138731113888 1113889 s2TL

2ηT

bSlowast

L minus bNlowast

L1113872 1113873 1 minusb

Slowast

L

bNlowast

L

⎛⎝ ⎞⎠

minuss2TL + ηTrσ21113872 1113873

2ηT

bSlowast

L minus bNlowast

L1113872 11138732le 0

(A11)

(2) ΔEΠSNlowast

TH EΠSlowast

TH minus EΠNlowast


(bSlowast

L )2 ge 0

ΔEΠSNlowast

TH EΠSlowast

TL minus EΠNlowast

TL 0 (A12)


harr) b(sTharr



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC

sTharr

1113872 1113873 + sOeCO sTharr

1113872 1113873 + θ1113872 1113873bC

sTharr

1113872 1113873 +12

s2T

ηT

minus rσ21113888 1113889bC2

sTharr

1113872 1113873

(A13)



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sTprime( 1113857

(A14)

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

sprime2T

ηT


sTprime( 1113857ge

aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

sprime2T

ηT


sT( 1113857

(A15)




CsTprime( 11138571113872 1113873ge 0 (A16)


bCmiddot


obtained aCmiddot

(sTharr

) + sOeCO

middot


middot



T isin H

there is

aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857

+s2T

ηT


sT( 1113857 0

(A17)



EΠT

middot

sT( 1113857 sT

ηT

bC2

sT( 1113857 (A18)

Bring EΠT

middot


maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A19)



EΠT sT( 1113857 1113946sT

sT

ωηT

bC2

(ω)dω + ΠT (A20)


sT[1113938

sT


1113938sT

sT[1113938

sT


sT

sT(1 minus F(sT)f



maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


f sT( 1113857

sT

ηT

bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A21)



eClowastO

sO

ηO

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

(A22)


2))



aClowast

sT1113872 1113873 ΠT minuss2O

ηO

+ θ1113888 1113889bClowast

sT1113872 1113873 minus12

sT

2

ηT


sT1113872 1113873

(A23)


obtained as

EΠT sT( 1113857 1113946sT

sT

ωηT

bClowast2

(ω)dω + ΠT (A24)


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857

+ 1113946sT

sT

ωηT

bC2

(ω)dω

(A25)

Data Availability




Acknowledgments


References



































EU(x) 1113946+infin

minus infinminus e

minus rx 12πn



(A1)







Ni ηT)




effort






Oi


L aNi b

Ni e

NOi1113872 1113873 1 minus b

Ni1113872 1113873 sOe

NOi + sTie

NTi + θ1113872 1113873 minus a

Ni minus

ηOeN2Oi

21113888 1113889

+ χ aNi + b

Ni sOe

NOi + sTie

NTi + θ1113872 1113873 minus

ηTeN2Ti

2minus

rσ2 bNi1113872 1113873

2

2minus ΠT

⎛⎝ ⎞⎠

(A2)




(aNlowasti bNlowast

i eNlowastOi )




i bSi )








Oi


maxaS

HbS

HeS

OHaS

LbS

LeS

OL


SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ) times1113896

1 minus bSL1113872 1113873 sOe

SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 11138891113897


SH sO( e

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2b2H2ge

aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTH1113872 1113873

2

2ηT

minusrσ2b2L2

(IR minus L)aSL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2b2L2ge ΠT

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(A3)




SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ)times


SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 1113889 + χ1 a

SL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2bS2

L

2minus ΠR

⎛⎝ ⎞⎠

+ χ2 aSH + b

SH sOe

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2bS2

H

2minus a

SL minus b

SL sOe

SOL + θ1113872 1113873 minus

bSLsTH1113872 1113873

2

2ηT

+rσ2bS2

L

2⎛⎝ ⎞⎠

(A4)


zL

zaSi

zL

zbSi

zL

zeSOi

0zL

zχ1ge 0

zL

zχ2ge 0 (A5)



H eSlowastOHa

SlowastL bSlowast


obtained


bSlowastH minus b

SlowastL

s2TH minus s

2TL1113872 1113873 ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

s2TH + ηTrσ21113872 1113873 s

2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

ge 0

aSlowastH minus a

SlowastL

s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 11138731113888 1113889 b

SlowastL minus b

SlowastH1113872 1113873

(A6)


L When rgt 0 let

G(r) s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 1113873

s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

bSlowastL + b


12

rσ2 bSlowastL + b

SlowastH1113872 1113873 (A7)



12

rσ2 bSlowastL + b

SlowastH1113872 1113873

12

rσ2s2TH

s2TH + ηTrσ2

+(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs


1113888 1113889

1

2ηT

s2TH + s

2TL minus

s4TH

s2TH + ηTrσ2

minus(1 minus ρ)s

2TL s

2TL minus 2ρs

2TL + ρs

2TH1113872 1113873

s2TL minus 2ρs

2TL + ρs


⎛⎝ ⎞⎠

(A8)





G(+infin) s2O

ηO

+ θ1113888 1113889

G(0) s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

1 +(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs

2TH

1113888 1113889

(A9)


L ge aSlowast


L aSlowastH


H minus aSlowast



L minus bNlowast



aSlowast

L minus aNlowast

L bNlowast

L minus bSlowast

L1113872 1113873s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TL

ηT

1113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889 (A10)


L minus aNlowast



Proof of eorem A4

(1) ΔEΠSNlowast

OH EΠSlowast

OH minus EΠNlowast


(bSlowast

L )2 le 0

ΔEΠSNlowast

OL EΠSlowast

OL minus EΠNlowast

OL bSlowast

L minus bNlowast

L1113872 1113873s2TL

ηT

minus12

s2TL

ηT

+ rσ21113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889

1ηT

bSlowast

L minus bNlowast

L1113872 1113873 s2TL minus

s2TL

2bNlowastL

bNlowast

L + bSlowast

L1113872 11138731113888 1113889 s2TL

2ηT

bSlowast

L minus bNlowast

L1113872 1113873 1 minusb

Slowast

L

bNlowast

L

⎛⎝ ⎞⎠

minuss2TL + ηTrσ21113872 1113873

2ηT

bSlowast

L minus bNlowast

L1113872 11138732le 0

(A11)

(2) ΔEΠSNlowast

TH EΠSlowast

TH minus EΠNlowast


(bSlowast

L )2 ge 0

ΔEΠSNlowast

TH EΠSlowast

TL minus EΠNlowast

TL 0 (A12)


harr) b(sTharr



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC

sTharr

1113872 1113873 + sOeCO sTharr

1113872 1113873 + θ1113872 1113873bC

sTharr

1113872 1113873 +12

s2T

ηT

minus rσ21113888 1113889bC2

sTharr

1113872 1113873

(A13)



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sTprime( 1113857

(A14)

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

sprime2T

ηT


sTprime( 1113857ge

aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

sprime2T

ηT


sT( 1113857

(A15)




CsTprime( 11138571113872 1113873ge 0 (A16)


bCmiddot


obtained aCmiddot

(sTharr

) + sOeCO

middot


middot



T isin H

there is

aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857

+s2T

ηT


sT( 1113857 0

(A17)



EΠT

middot

sT( 1113857 sT

ηT

bC2

sT( 1113857 (A18)

Bring EΠT

middot


maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A19)



EΠT sT( 1113857 1113946sT

sT

ωηT

bC2

(ω)dω + ΠT (A20)


sT[1113938

sT


1113938sT

sT[1113938

sT


sT

sT(1 minus F(sT)f



maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


f sT( 1113857

sT

ηT

bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A21)



eClowastO

sO

ηO

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

(A22)


2))



aClowast

sT1113872 1113873 ΠT minuss2O

ηO

+ θ1113888 1113889bClowast

sT1113872 1113873 minus12

sT

2

ηT


sT1113872 1113873

(A23)


obtained as

EΠT sT( 1113857 1113946sT

sT

ωηT

bClowast2

(ω)dω + ΠT (A24)


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857

+ 1113946sT

sT

ωηT

bC2

(ω)dω

(A25)

Data Availability




Acknowledgments


References





































SOH +

bSHs

2TH

ηT


ηOeS2OH

21113888 1113889 +(1 minus ρ)times


SOL +

bSLs

2TL

ηT


ηOeS2OL

21113888 1113889 + χ1 a

SL + b

SL sOe

SOL + θ1113872 1113873 +

bSLsTL1113872 1113873

2

2ηT

minusrσ2bS2

L

2minus ΠR

⎛⎝ ⎞⎠

+ χ2 aSH + b

SH sOe

SOH + θ1113872 1113873 +

bSHsTH1113872 1113873

2

2ηT

minusrσ2bS2

H

2minus a

SL minus b

SL sOe

SOL + θ1113872 1113873 minus

bSLsTH1113872 1113873

2

2ηT

+rσ2bS2

L

2⎛⎝ ⎞⎠

(A4)


zL

zaSi

zL

zbSi

zL

zeSOi

0zL

zχ1ge 0

zL

zχ2ge 0 (A5)



H eSlowastOHa

SlowastL bSlowast


obtained


bSlowastH minus b

SlowastL

s2TH minus s

2TL1113872 1113873 ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

s2TH + ηTrσ21113872 1113873 s

2TL minus 2ρs

2TL + ρs

2TH +(1 minus ρ)ηTrσ21113872 1113873

ge 0

aSlowastH minus a

SlowastL

s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 11138731113888 1113889 b

SlowastL minus b

SlowastH1113872 1113873

(A6)


L When rgt 0 let

G(r) s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TH

ηT

1113888 1113889 bSlowastL + b

SlowastH1113872 1113873

s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

bSlowastL + b


12

rσ2 bSlowastL + b

SlowastH1113872 1113873 (A7)



12

rσ2 bSlowastL + b

SlowastH1113872 1113873

12

rσ2s2TH

s2TH + ηTrσ2

+(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs


1113888 1113889

1

2ηT

s2TH + s

2TL minus

s4TH

s2TH + ηTrσ2

minus(1 minus ρ)s

2TL s

2TL minus 2ρs

2TL + ρs

2TH1113872 1113873

s2TL minus 2ρs

2TL + ρs


⎛⎝ ⎞⎠

(A8)





G(+infin) s2O

ηO

+ θ1113888 1113889

G(0) s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

1 +(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs

2TH

1113888 1113889

(A9)


L ge aSlowast


L aSlowastH


H minus aSlowast



L minus bNlowast



aSlowast

L minus aNlowast

L bNlowast

L minus bSlowast

L1113872 1113873s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TL

ηT

1113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889 (A10)


L minus aNlowast



Proof of eorem A4

(1) ΔEΠSNlowast

OH EΠSlowast

OH minus EΠNlowast


(bSlowast

L )2 le 0

ΔEΠSNlowast

OL EΠSlowast

OL minus EΠNlowast

OL bSlowast

L minus bNlowast

L1113872 1113873s2TL

ηT

minus12

s2TL

ηT

+ rσ21113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889

1ηT

bSlowast

L minus bNlowast

L1113872 1113873 s2TL minus

s2TL

2bNlowastL

bNlowast

L + bSlowast

L1113872 11138731113888 1113889 s2TL

2ηT

bSlowast

L minus bNlowast

L1113872 1113873 1 minusb

Slowast

L

bNlowast

L

⎛⎝ ⎞⎠

minuss2TL + ηTrσ21113872 1113873

2ηT

bSlowast

L minus bNlowast

L1113872 11138732le 0

(A11)

(2) ΔEΠSNlowast

TH EΠSlowast

TH minus EΠNlowast


(bSlowast

L )2 ge 0

ΔEΠSNlowast

TH EΠSlowast

TL minus EΠNlowast

TL 0 (A12)


harr) b(sTharr



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC

sTharr

1113872 1113873 + sOeCO sTharr

1113872 1113873 + θ1113872 1113873bC

sTharr

1113872 1113873 +12

s2T

ηT

minus rσ21113888 1113889bC2

sTharr

1113872 1113873

(A13)



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sTprime( 1113857

(A14)

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

sprime2T

ηT


sTprime( 1113857ge

aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

sprime2T

ηT


sT( 1113857

(A15)




CsTprime( 11138571113872 1113873ge 0 (A16)


bCmiddot


obtained aCmiddot

(sTharr

) + sOeCO

middot


middot



T isin H

there is

aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857

+s2T

ηT


sT( 1113857 0

(A17)



EΠT

middot

sT( 1113857 sT

ηT

bC2

sT( 1113857 (A18)

Bring EΠT

middot


maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A19)



EΠT sT( 1113857 1113946sT

sT

ωηT

bC2

(ω)dω + ΠT (A20)


sT[1113938

sT


1113938sT

sT[1113938

sT


sT

sT(1 minus F(sT)f



maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


f sT( 1113857

sT

ηT

bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A21)



eClowastO

sO

ηO

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

(A22)


2))



aClowast

sT1113872 1113873 ΠT minuss2O

ηO

+ θ1113888 1113889bClowast

sT1113872 1113873 minus12

sT

2

ηT


sT1113872 1113873

(A23)


obtained as

EΠT sT( 1113857 1113946sT

sT

ωηT

bClowast2

(ω)dω + ΠT (A24)


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857

+ 1113946sT

sT

ωηT

bC2

(ω)dω

(A25)

Data Availability




Acknowledgments


References






































G(+infin) s2O

ηO

+ θ1113888 1113889

G(0) s2O

ηO

+ θ1113888 1113889 +s2TH

2ηT

1 +(1 minus ρ)s

2TL

s2TL minus 2ρs

2TL + ρs

2TH

1113888 1113889

(A9)


L ge aSlowast


L aSlowastH


H minus aSlowast



L minus bNlowast



aSlowast

L minus aNlowast

L bNlowast

L minus bSlowast

L1113872 1113873s2O

ηO

+ θ1113888 1113889 minus12

rσ2 minuss2TL

ηT

1113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889 (A10)


L minus aNlowast



Proof of eorem A4

(1) ΔEΠSNlowast

OH EΠSlowast

OH minus EΠNlowast


(bSlowast

L )2 le 0

ΔEΠSNlowast

OL EΠSlowast

OL minus EΠNlowast

OL bSlowast

L minus bNlowast

L1113872 1113873s2TL

ηT

minus12

s2TL

ηT

+ rσ21113888 1113889 bNlowast

L + bSlowast

L1113872 11138731113888 1113889

1ηT

bSlowast

L minus bNlowast

L1113872 1113873 s2TL minus

s2TL

2bNlowastL

bNlowast

L + bSlowast

L1113872 11138731113888 1113889 s2TL

2ηT

bSlowast

L minus bNlowast

L1113872 1113873 1 minusb

Slowast

L

bNlowast

L

⎛⎝ ⎞⎠

minuss2TL + ηTrσ21113872 1113873

2ηT

bSlowast

L minus bNlowast

L1113872 11138732le 0

(A11)

(2) ΔEΠSNlowast

TH EΠSlowast

TH minus EΠNlowast


(bSlowast

L )2 ge 0

ΔEΠSNlowast

TH EΠSlowast

TL minus EΠNlowast

TL 0 (A12)


harr) b(sTharr



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC

sTharr

1113872 1113873 + sOeCO sTharr

1113872 1113873 + θ1113872 1113873bC

sTharr

1113872 1113873 +12

s2T

ηT

minus rσ21113888 1113889bC2

sTharr

1113872 1113873

(A13)



aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sTprime( 1113857

(A14)

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

sprime2T

ηT


sTprime( 1113857ge

aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

sprime2T

ηT


sT( 1113857

(A15)




CsTprime( 11138571113872 1113873ge 0 (A16)


bCmiddot


obtained aCmiddot

(sTharr

) + sOeCO

middot


middot



T isin H

there is

aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857

+s2T

ηT


sT( 1113857 0

(A17)



EΠT

middot

sT( 1113857 sT

ηT

bC2

sT( 1113857 (A18)

Bring EΠT

middot


maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A19)



EΠT sT( 1113857 1113946sT

sT

ωηT

bC2

(ω)dω + ΠT (A20)


sT[1113938

sT


1113938sT

sT[1113938

sT


sT

sT(1 minus F(sT)f



maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


f sT( 1113857

sT

ηT

bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A21)



eClowastO

sO

ηO

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

(A22)


2))



aClowast

sT1113872 1113873 ΠT minuss2O

ηO

+ θ1113888 1113889bClowast

sT1113872 1113873 minus12

sT

2

ηT


sT1113872 1113873

(A23)


obtained as

EΠT sT( 1113857 1113946sT

sT

ωηT

bClowast2

(ω)dω + ΠT (A24)


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857

+ 1113946sT

sT

ωηT

bC2

(ω)dω

(A25)

Data Availability




Acknowledgments


References




































aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sT( 1113857ge

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

s2T

ηT

minus rσ21113888 1113889bC2

sTprime( 1113857

(A14)

aC


CO sTprime( 1113857 + θ1113872 1113873b

CsTprime( 1113857 +

12

sprime2T

ηT


sTprime( 1113857ge

aC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

CsT( 1113857 +

12

sprime2T

ηT


sT( 1113857

(A15)




CsTprime( 11138571113872 1113873ge 0 (A16)


bCmiddot


obtained aCmiddot

(sTharr

) + sOeCO

middot


middot



T isin H

there is

aCmiddot

sT( 1113857 + sOeCO

middot

sT( 1113857bC

sT( 1113857 + sOeCO sT( 1113857 + θ1113872 1113873b

Cmiddot

sT( 1113857

+s2T

ηT


sT( 1113857 0

(A17)



EΠT

middot

sT( 1113857 sT

ηT

bC2

sT( 1113857 (A18)

Bring EΠT

middot


maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A19)



EΠT sT( 1113857 1113946sT

sT

ωηT

bC2

(ω)dω + ΠT (A20)


sT[1113938

sT


1113938sT

sT[1113938

sT


sT

sT(1 minus F(sT)f



maxaCbC

1113946sT

sT

sOeCO sT( 1113857 +

s2Tb

CsT( 1113857

ηT

+ θ minus12

s2T

ηT

+ rσ21113888 1113889bC2


f sT( 1113857

sT

ηT

bC2


C2O sT( 1113857

21113896 1113897f sT( 1113857dsT (A21)



eClowastO

sO

ηO

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

(A22)


2))



aClowast

sT1113872 1113873 ΠT minuss2O

ηO

+ θ1113888 1113889bClowast

sT1113872 1113873 minus12

sT

2

ηT


sT1113872 1113873

(A23)


obtained as

EΠT sT( 1113857 1113946sT

sT

ωηT

bClowast2

(ω)dω + ΠT (A24)


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857

+ 1113946sT

sT

ωηT

bC2

(ω)dω

(A25)

Data Availability




Acknowledgments


References




































eClowastO

sO

ηO

bClowast

sCT1113872 1113873

11 + ηTrσ2sC2

T1113872 1113873 + 2 1 minus F sCT1113872 11138731113872 1113873sC

Tf sCT1113872 11138731113872 1113873

(A22)


2))



aClowast

sT1113872 1113873 ΠT minuss2O

ηO

+ θ1113888 1113889bClowast

sT1113872 1113873 minus12

sT

2

ηT


sT1113872 1113873

(A23)


obtained as

EΠT sT( 1113857 1113946sT

sT

ωηT

bClowast2

(ω)dω + ΠT (A24)


aClowast


ηO

+ θ1113888 1113889bClowast

sT( 1113857 minus12

s2T

ηT


sT( 1113857

+ 1113946sT

sT

ωηT

bC2

(ω)dω

(A25)

Data Availability




Acknowledgments


References



































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