RISK-SENSITIVE REVENUE-SHARING STRATEGY AND SENSITIVITY ANALYSIS IN E-COMMERCE. Masaru Unno 1 and Hua Xu 2 1 NTT Finance Corporation

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1 Inernaional Journal of Innovaive Compuing, Informaion and Conrol ICIC Inernaional c 213 ISSN Volume 9, Number 9, Sepember 213 pp RISK-SENSITIVE REVENUE-SHARING STRATEGY AND SENSITIVITY ANALYSIS IN E-COMMERCE Masaru Unno 1 and Hua Xu 2 1 NTT Finance Corporaion Seavans N 19h floor, Shibaura Minao-ku, Tokyo , Japan masaru.unno7@gmail.com 2 Graduae School of Business Sciences Universiy of Tsukuba , Osuka, Bunkyo-ku, Tokyo , Japan xu@mbaib.gsbs.sukuba.ac.jp Received July 212; revised February 213 Absrac. In his paper, we develop a new model o describe a dynamic revenue-sharing problem beween an online shopping mall and a sore in an E-commerce marke. We formulae he revenue-sharing problem as a dynamic principal-agen problem, which is hen ransformed o a risk-sensiive sochasic opimal conrol problem where he objecive of he risk-averse shopping mall is o find a risk-sensiive revenue-sharing sraegy and o advise an incenive-compaible effor o he sore. Sufficien condiions for he exisence of a risk-sensiive revenue-sharing sraegy and an incenive-compaible effor are obained. A numerical example is solved o show he exisence of such sraegy and is sensiiviy o he risk. Moreover, as a comparsion, we also discuss a myopic revene-sharing sraegy and find ha dynamic revenue-sharing sraegy is more effecive in expanding he expeced profi of he shopping mall. Keywords: E-commerce, Revenue-sharing sraegy, Dynamic principal-agen problem, Risk-sensiive sochasic conrol 1. Inroducion. E-commerce can be viewed as an online wo-sided marke. In wosided markes, plaforms play imporan roles. They provide infrasrucures and make business rules so ha differen user groups in he marke can conduc heir businesses smoohly. A ypical example of a plaform is Rakuen. Rakuen is he bigges online shopping mall operaor in Japan wih over 5 million regisered users. Rakuen brings sores and cusomers ogeher o form a wo-sided marke or a wo-sided nework. I is well known ha he so-called cross-side nework effecs exis in a wo-sided marke, and sellers and buyers in a plaform are araced o each oher. Because of he crossside nework effecs, increasing he number of users on one side will benefi he users on he plaform s oher side. In oher words, sellers will have more business chances as he number of buyers in a plaform increases, and buyers will have more choices and beer purchase condiions or beer services if more sellers join he plaform. In he early sudies on wo-sided markes, considerable aenion has been paid o how a plaform should charge wo differen user groups in a wo-sided marke. Theoreical frameworks have been esablished o explain how he srucure of prices is deermined. I is well known ha he pricing srucure in a wo-sided marke is asymmeric because of differen ypes of users on he wo sides [1-4. E-commerce is a special wo-sided marke where he online shopping mall only charges sores. The issues faced by a shopping mall in E-commerce are no only o induce paricipaions of sores and cusomers in he marke, bu also o make an incenive scheme o 3655

2 3656 M. UNNO AND H. XU sores such ha sores can make more effors o improve he qualiy of heir producs or services. In his paper, we develop a new model o describe a revenue-sharing problem beween an online shopping mall and a sore. We formulae he problem as a dynamic principal-agen problem where he shopping mall is he principal and he sore is he agen. Differen from [6, we also assume ha he shopping mall is a risk-averse decision maker. Shopping mall s problem is o find a risk-sensiive revenue-sharing sraegy and o advise an incenive compaible effor o he sore [7. Sufficien condiions for he exisence of a risk-sensiive revenue-sharing sraegy and an incenive-compaible effor o he sore are obained. I is believed ha he resuls obained in his paper can be used as a kind of benchmark for he plaform and he seller o deermine heir conracing condiion in pracice. A numerical example is solved o show he exisence of such sraegy and is sensiiviy o he risk. 2. Problem Formulaion. Consider an E-commerce marke which consiss of one plaform online shopping mall, many sellers sores and numerous buyers cusomers. Sellers who wish o join he elecronic commerce marke are required o sign a conrac wih he plaform on he raio of revenue-sharing. Wihou loss of generaliy, we assume ha he plaform will sign he conrac wih one seller because homogeneous sellers will be considered in his paper. Suppose ha he iniial number of plaform s regisered members a ime = is N, which is known by boh he plaform and he seller. The number of buyers who purchase he seller s producs or services, simply he buyers, varies depending on he iniial number N, he seller s coninuous effors o improve producs or services and some oher uncerain facors in he marke. If he seller makes more effors o improve he qualiy of producs or services, he number of he buyers will increase. I is assumed in his paper ha he number of he buyers can be observed by boh he plaform and he seller. However, he seller s real effor level is no observable o he plaform. The cumulaive number of he buyers a ime is denoed by X which evolves according o dx = qa N d + σn dz, 1 where Z = {Z, F ; < } is a sandard Brownian moion, σ is a consan, and {F ; < } is he filraion deermined by {X ; < }. a is he seller s choice of effor level and qa is he qualiy funcion of he seller s producs or services which is affeced by he seller s effor level a. The seller s effor range is denoed by a [, ā where ā is he upper bound of he effor level. qa [, 1 is a coninuous, sricly increasing and concave funcion of a which is known by he plaform. Since qa can be used o describe he degree of aracion of producs or services, i is assumed ha all conrac members will purchase he seller s producs or services if q = 1. For simpliciy, he price of he seller s producs or services is normalized o one. Hence, X is equal o he seller s cumulaive sales a ime which is observable o he plaform. The seller s sales X will be allocaed o he seller and o he plaform under he condiions of he revenue-sharing conrac. Le γ [, denoe he revenue-sharing sraegy a ime made by he plaform and he seller. The revenue-sharing sraegy γ specifies he revenue allocaed o he seller. Since he seller s expeced sales depend on qa and he conrac number N, here is an upper bound o he revenue-sharing sraegy γ, ha is, γ γ = N. Suppose ha he seller obains he uiliy uγ from he revenue-sharing sraegy γ, where uγ is a normalized increasing, concave and C 2 funcion saisfying u =. On he oher hand, he seller incurs he cos of effor ha, measured in he same uni as

3 RISK-SENSITIVE REVENUE-SHARING STRATEGY 3657 ha of he uiliy of revenue-sharing sraegy, where ha is a coninuous, increasing and convex funcion of a. I is assumed ha he plaform knows he seller s uiliy funcion and he cos funcion. The risk-averse plaform incurs he operaional cos βdx which depends on he number of he buyers, where β > is a consan. For simpliciy, i is assumed ha boh he plaform and he seller discoun he flow of profi and uiliy a a common rae r. If he seller chooses an effor level a, <, he seller s oal expeced uiliy is given by [ E e uγ r ha d, and he plaform s oal expeced profi is [ E e r dx e r γ d e r βdx [ = E e r 1 βqa N γ d. Since he plaform is he risk-averse decision maker, we define he following uiliy funcion as he plaform s objecive funcion: { [ } E exp ρ e r 1 βqa N γ d + 1, where ρ is a posiive parameer o denoe he risk sensiiviy of he plaform The plaform s problem. Under he condiion of he revenue-sharing conrac, he seller will choose an effor a o maximize is expeced uiliy. Knowing he behavior of he seller, he plaform s problem is o offer a revenue-sharing conrac o he seller, which includes an incenive-compaible advice of effor {a, < } o he seller and a revenue-sharing sraegy {γ, < } such ha he plaform s uiliy funcion E { exp [ ρ e r 1 βqa N γ d } + 1 is maximized. The effor {a, } is referred o as incenive-compaible wih respec o he revenue-sharing sraegy {γ, } if i saisfies [ a arg maxã E e uγ r hã d, 3 and [ E e uγ r ha d. 4 I is obvious ha an incenive-compaible effor is relevan o he revenue-sharing sraegy γ from The seller s coninuaion value. In order o make he seller choose a recommended incenive-compaible effor, he plaform is required o design a revenue-sharing sraegy γ which can change he allocaion of revenue o he seller according o is effor. Insead of designing a sraegy ha depends on he sales of he seller, we consider a sraegy which depends on he seller s coninuaion value W. The coninuaion value W is he oal expeced uiliy received by he seller from ime onwards. Suppose ha a revenue-sharing sraegy γ = {γ } and an effor a = {a } are given. The seller s coninuaion value is [ W γ, a = E a e uγ rs s ha s ds F, 5 2

4 3658 M. UNNO AND H. XU where E a denoes he expecaion under he probabiliy measure P a induced by he seller s effor a = {a }. In he plaform s revenue-sharing sraegy, W is he unique sae variable ha deermines how much he seller s allocaion of he sales is, wha effor he seller is advised o choose, and how W iself evolves wih he realizaion of he sales. The plaform is required o use W as he sae feedback o design a revenue-sharing sraegy γ and a recommended effor a o achieve wo objecives. Firs, he seller mus have sufficien incenive o choose he recommended effor. Second, he plaform s profi is maximized. I is worh noing ha, no maer how much he coninuaion value W is, he plaform has he opion o sop he revenue-sharing conrac wih he seller. Suppose ha he plaform is willing o pay he cancellaion cos o he seller. The cancellaion cos is deermined by he coninuaion value W a he ime of cancellaion. The plaform s profi funcion a he ime of cancellaion is ΩW = δγ, where Ω = and δ is a consan. Since he seller can choose zero effor afer conrac cancellaion, he seller s coninuaion value a ime is W = uδγ. If he seller s coninuaion value W is exremely high, he plaform will cancel he conrac wih he seller. The reason is ha he coninuaion value W will increase as he allocaion of he sales o he seller increases. However, if he allocaion o he seller is oo high, he allocaion o he plaform will be below he operaional cos incurred by he plaform. Therefore, here mus exis a coninuaion value W > such ha he plaform is willing o pay he cancellaion cos ΩW o sop he conrac. 3. Risk-Sensiive Revenue-Sharing Sraegy. In his secion, we will derive he opimal soluion o he problem formulaed in he above secion. Firs, as a preliminary resul, we give he following proposiion, which is proved formally in [6 o describe he evoluion of he seller s coninuaion value W. Proposiion 3.1. Suppose ha a revenue-sharing sraegy γ = {γ } and an effor a = {a } afer ime > are given. Then, here exiss a F -progressively measurable process Y such ha he seller s coninuaion value W γ, a defined by 5 can be described by he sochasic differenial equaion dw γ, a = [ rw γ, a uγ + ha d + σn Y dz. 6 Second, we give he following proposiion, which is proved formally in [6 oo, o describe he incenive-compaibiliy condiion on he seller s effor. Proposiion 3.2. Suppose ha Y is a progressively measurable process defined by Proposiion 3.1. Then, he seller s effor a is opimal if and only if almos everywhere. a arg max Y qã N hã, < 7 ã [,ā From Proposiion 3.2, i is shown ha Y is he funcion of he seller s incenive compaible effor a, ha is, Y = h a q a N = ya >. 8 ya is an increasing funcion of a. Since Y of 6 represens he volailiy of he seller s coninuaion value W γ, a, he seller s risk will increase as effor increases.

5 RISK-SENSITIVE REVENUE-SHARING STRATEGY The risk-sensiive sochasic conrol problem. Making use of he monooniciy of a logarihmic funcion, we know ha maximizing he uiliy funcion 2 is equivalen o he problem of maximizing he following objecive funcion { [ JW = ρ 1 ln E exp ρ e rs 1 βqa s N γ s ds } Suppose ha he evoluion of he seller s coninuaion value W is known. The plaform s conrol problem o find he opimal revenue-sharing sraegy γ and he recommended effor a, which saisfies he incenive compaibiliy condiion, can be formulaed as a risk-sensiive sochasic conrol problem: subjec o where ΠW = max JW = a,γ ρ 1 ln ψw dw = [ rw uγ + ha d + σn ya dz, 11 { [ ψw = max exp ρ } e rs 1 βqa s N γ s ds Furhermore, defining ΨW = ψw 1, we have { [ } ΨW = max E exp ρ e rs 1 βqa s N γ s ds. 13 Therefore, he problem formulaed by 1, 11 is equivalen o he problem of maximizing 13 subjec o 11. This problem is solved by using dynamic programming, and he Hamilon-Jacobi-Bellman HJB equaion is obained below, max a,γ rw uγ + ha Ψ W ρ 1 βqan γ ΨW σ2 N 2 ya 2 Ψ W =. 14 Using he ransformaions Ψ W = ρπ W ΨW and Ψ W = ρπ W ΨW ρ 2 Π W 2 ΨW, we arrive a he HJB equaion rw uγ + ha Π W + 1 βqan γ ρσ2 N 2 ya 2 Π W 2 max a,γ from 14. The HJB Equaion 15 is solved under he iniial condiion and he final condiions σ2 N 2 ya 2 Π W = 15 Π =, 16 ΠW = ΩW, Π W = Ω W, 17 a a ime τ, where = τ is he ime poin when he plaform cancels he conrac wih he seller and Ω is he plaform s value funcion when he seller chooses zero effor. 1 ΠW = ΨW is called he value-maching condiion and Π W = Ψ W is called he smooh-pasing condiion.

6 366 M. UNNO AND H. XU 3.2. Soluions of risk-sensiive sochasic conrol problem. Suppose ha he soluion ΠW o 15 exiss. We have he following proposiion, which is proved formally in Appendix A. Proposiion 3.3. Suppose ha ΠW saisfies he HJB Equaion 15 wih respec o W [, W in [, τ, he iniial condiion 16 and he final condiions 17 a = τ. If a and γ are he seller s recommended effor and he plaform s revenue-sharing sraegy which maximize he lef-hand side of 15, hen a and γ are he soluions of he risk-sensiive sochasic conrol problem formulaed in Secion 3.1. From Proposiion 3.3, he opimal recommended effor aw is obained as he funcion of W by maximizing max haπ W + 1 a 2 ρσ2 N 2 ya 2 Π W σ2 N 2 ya 2 Π W + 1 βqan 18 where 1 βqan is he revenue flow, haπ W is he effor compensaion o he seller, and 1 2 ρσ2 N 2 ya 2 Π W σ2 N 2 ya 2 Π W is he risk premium paid o he seller in an uncerain business environmen. Similarly, he opimal revenue-sharing sraegy is obained by maximizing max γ uγπ W. 19 γ From he firs-order condiion Π W = 1/u γ, γw is obained as he funcion of he coninuaion value W. Π W represens he plaform s marginal decrease in he value funcion wih respec o he coninuaion value. 1/u γ = dγ/duγ represens he plaform s marginal revenue share wih respec o he seller s uiliy. Moreover, when W W where W is a poin such ha Π W =, since uγ and Π W >, γ = from 19. The soluion ΠW of he HJB Equaion 15 can be obained hrough numerical compuaion. In order o show he exisence of he soluion of 15 and he exisence of a corresponding risk-sensiive revenue-sharing sraegy and a recommended effor, an illusraive example is solved. The funcions and he parameers appeared in he problem formulaion are defined as follows: and qa = a, uγ = γ, ha =.5a 2 +.5a N = 1, r =.1, ρ =.1, σ = 1, β =.1, δ = 235. The plaform s opimal revenue-sharing sraegy can be obained from γ = Π W 2 /4, and he seller s opimal recommended effor is a =.9/Π W + ρπ W 2 + Π W.5. The numerical resuls of he plaform s value funcion, he opimal revenue-sharing sraegy and he opimal recommended effor are shown in Figure 1. Moreover, W = 1.256, ΠW = Discussion Sensiiviy analysis of plaform s sraegies. Since ρ denoes he risk sensiiviy of he plaform, he higher he value of ρ is, he more averse he plaform is o he risk. In his secion, we will analyze how he value of ρ affecs he various sraegies obained above by numerical simulaion.

7 RISK-SENSITIVE REVENUE-SHARING STRATEGY 3661 Figure 1. a Plaform s value funcion, b effor, c risk-sensiive revenue-sharing sraegy From 18, i is shown ha, when W = W, 1 max a 2 σ2 N 2 ya 2 Π W + 1 βqan is independen of ρ, where W is he seller s coninuaion value when Π W =. Therefore, he plaform s aiude o he risk will no influence he seller s effor level a he poin of W = W. Obviously, W will be differen as he change of ρ. I is sill no clear how he differen ρ will affec he value funcion ΠW. In general, if a decision maker is more risk-averse, he/she migh no pursue a bigger profi because he uiliy will change only a lile even if he profi increases a lo. In oher words, a relaively smaller profi will lead o he same saisfacion as ha of a bigger profi if a decision maker is more risk-averse. Hence, i is esimaed ha a more risk-averse plaform wih less profi may have he same saisfacion as ha of a less risk-averse plaform wih bigger profi. However, if he risk is ransferable, he esimaion above may no be correc. In he following, we will discuss he issues using he numerical example above. Figure 2 shows curves of ΠW, aw and γw when ρ =.5,.1 and.2, respecively. I is found from Figure 2 ha he resuls are differen from he esimaion above. The higher he value of ρ is, he bigger he value of ΠW, W W is. Moreover, he higher he value of ρ is, he higher he value of Π and he value of W are. These resuls can be explained as follows. Firs, he higher Π means a higher iniial effor aw = a. In fac, since γ = a =, Π depends on a from he definiion of Π. The simulaion Figure 2 shows he same resuls. The higher he value of ρ is, he values of a and W become higher. Now, he risk-averse plaform may expec a higher W o he seller, which are of grea advanages in excluding weak sellers and posponing he allocaion of revenue o he seller. I is required ha he seller makes much more effor a he beginning if a higher W is expeced. Tha also means he increase of he volailiy of he seller s coninuaion value, ha is, he seller s risk see 6.

8 3662 M. UNNO AND H. XU Figure 2. Resuls of sensiiviy analysis As he resul, he risk-averse plaform will expec a higher W o he seller, which will ransfer he risk from he plaform o he seller. If W is high, he seller is required o make much more effor a he beginning which leads o a high Π and a high ΠW Myopic revenue-sharing sraegy. Wha is he effec on revenue of he plaform and effor of he seller when revenue-sharing sraegy is deermined myopically, i.e., allocaion γ is deermined wih respec o sales X only a ime? There are a leas wo mehods of allocaion ha he plaform can selec. One mehod is o deermine revenue-sharing sraegy γ wih respec o he coninuaion value, aking ino accoun he fuure expeced revenue and seller s risk, as we have demonsraed before. Anoher mehod is o myopically deermine revenue-sharing sraegy γ only hrough he revenue a ime, X. There are pros and cons o boh mehods. The former allows endowmen of incenive for he seller o pu in greaer effor such as long-erm expansion as well as paymen of risk premium for sochasic demand on fuure revenue, bu compuaion of expeced revenue is raher complicaed. On he oher hand, he laer is simple o calculae bu promoes he seller o focus on shor-erm sales increase and also does no ake ino accoun he randomness exogenous o he seller s effor. In his secion, we will analyze he implicaion of choosing such myopic revenue-sharing scheme over he expeced revenue scheme as described by his paper. Similar o he problem formulaion, we assume ha one plaform in an E-commerce marke will sign he revenue-sharing conrac wih one seller myopically under he condiion ha γ. Firsly, consider he case where uncerainy does no exis in he sales X, ha is, dx = qa N d. 2 Since he allocaion o he seller is γqa N, he uiliy of he seller is uγqa N. Moreover, he seller incurs he cos ha. Thus, he seller s myopic opimal effor level wih respec o he given γ is [ a m = arg max uγqã N hã. 21 ã The opimal effor level a m γ is obained as he funcion of γ from he firs-order condiion duγqa N /da h a =, where, a m γ = a m γ is consan. Applying his o he

9 RISK-SENSITIVE REVENUE-SHARING STRATEGY 3663 numerical example in Secion 3.2, we find ha a m γ saisfies γ = 4a m γ a m γ +.5 2, a m γ a. Thus, he allocaion o he seller a any ime is 4 a m γ 2 a m γ Since he plaform s myopic expeced profi for an arbirary γ is [ Π m = E e r 1 βq a m γ N γq a m γ N d = 1 [1 βq a m γ N γq a m γ N, r 22 Π m = 1 [.9a m γ 4 a m γ 2 a m γ when i is applied o he numerical example. Therefore, a m γ =.1872, γ =.3536, Π m = Compared wih he resuls using he dynamic revenue-sharing sraegy in Secion 3.2, we find ha boh he effor level and he plaform s expeced profi are decreasing. Furhermore, since he seller s sales is uncerain, he risk-premium should also be added when considering he revenue-sharing sraegy. Suppose ha he seller s risk-averse level is λ = u. The risk premium is u γ λ qan = 1 2 λσ2 qan. 24 Including he risk-premium o he seller, he plaform s myopic expeced profi becomes Π mλ = 1 [1 βq a m γ N γq a M γ N 12 r λσ2 qan. 25 Applying his ino he numerical example in Secion 3.2, we find ha Π mλ = 1 [.9a m γ 4 a m γ 2 a m γ , 26.1 and Π mλ = 1.477, which means ha he plaform s profi is decreasing if he risk-premium is paid. However, if he plaform does no pay he risk-premium o he seller, he seller will no have he incenive o improve is effor, which will also affec sales in a long-erm perspecive. 5. Conclusion. In his paper, we have considered he risk-sensiive revenue-sharing problem beween a risk-averse plaform and a seller in E-commerce. We have formulaed he problem as a dynamic principal-agen problem, and hen ransformed i o a risk-sensiive sochasic conrol problem where he objecive of he plaform is o find a risk-sensiive revenue-sharing sraegy and o advise an incenive-compaible effor o he seller. Sufficien condiions for he exisence of a risk-sensiive revenue-sharing sraegy and an incenive-compaible effor are obained. A numerical example is solved o show he exisence of he sraegy and he effor, and heir sensiiviies o he risk. I is believed ha he resuls obained in his paper can be used as a kind of benchmark for he plaform and he seller o deermine heir conracing condiion in pracice. In paricular, in markes such as elecronic music marke or elecronic book marke where buyers purchase producs repeaedly, plaforms s dynamic incenive sraegy seems necessary which can enforce seller s effors o improve he produc qualiy. Resuls in Secion 4.2 show ha he myopic revenue-sharing sraegy will deeriorae he expeced profi of he plaform compared wih he dynamic revenue-sharing sraegy. The plaform s dynamic revenue-sharing sraegy is more effecive in giving incenive o he seller for is

10 3664 M. UNNO AND H. XU effor and in expanding he expeced profi. In his paper, we have assumed ha here exiss a monopolisic plaform and homogeneous sellers in he marke. Furher researches are under way o expand he model o he cases of compeiive plaforms and sellers wih differen cos srucures. REFERENCES [1 J. Roche and J. Tirol, Plaform compeiion in wo-sided markes, Journal of European Economic Associaion, vol.1, no.4, pp , 23. [2 J. Roche and J. Tirol, Two-sided markes: A progress repor, Rand Journal of Economics, vol.37, no.3, pp , 26. [3 M. Armsrong, Compeiion in wo-sided markes, Rand Journal of Economics, vol.37, no.3, pp , 26. [4 A. Hagiu, Two-sided plaform: Produc variey and pricing srucures, Journal of Economics & Managemen Sraegy, vol.18, no.4, pp , 29. [5 Y. Sannikov, A coninuous ime version of he principal-agen problem, Review of Economic Journal Sudies, vol.75, no.3, pp , 28. [6 M. Unno and H. Xu, Dynamic opimal revenue-sharing sraegy in E-commerce, in Knowledge- Based and Inelligen Informaion and Engineering Sysems, Par III, Lecure Noes in Arificial Inelligence, A. König e al. eds., Berlin, Springer-Verlag, 211. [7 T. Başar and P. Bernhard, H -Opimal Conrol and Relaed Minimax Design Problems: A Dynamic Game Approach, 2nd Ediion, Birkhäuser, [8 L. Karazas and S. Shreve, Brownian Moion and Sochasic Calculus, Springer-Verlag, Appendix A. Proof of Proposiion 3.3. Suppose ha γ, a are he soluion of 12. We have { [ + ψw = max E exp ρ e rs 1 βqa s N γ s ds [ exp ρ { [ = max E exp ρ = max E = max E + + ψw } e rs 1 βqa s N γ s ds + 1 e rs 1 βqa s N γ s ds } + 1 { [ 1 + ρe r 1 βqa N γ } ψw ψw { ψw + ψw ρ 1 βqa N γ ψw + ρr 1 βqa N γ ψw 2 ρ 1 βqa N γ ψw + ρr 1 βqa N γ ψw 2

11 RISK-SENSITIVE REVENUE-SHARING STRATEGY 3665 ρ 1 βqa N γ + ρr 1 βqa N γ }. 2 Le, hen 2 and ψw converge o as goes o. Thus, { } max E dψw ρ 1 βqa N γ ψw 1d =. Furhermore, using Io lemma, we have { [rw dψw = uγ + ha ψ W + 1 } 2 σ2 N 2 ya 2 ψ W d + σn ya ψ W dz. Therefore, we arrive a he HJB equaion [ rw uγ + ha ψ W σ2 N 2 ya 2 ψ W max ρ 1 βqa N γ ψw 1 =. Leing ΨW = ψw 1, Ψ W = ψ W and Ψ W = ψ W gives [ max rw uγ + ha Ψ W σ2 N 2 ya 2 Ψ W ρ 1 βqa N γ ΨW =. Therefore, γ, a can be obained hrough he soluions of 14, which consiue he risk-sensiive revenue-sharing sraegy and he incenive-compaible effor. Q.E.D