Optimal Stackelberg Strategies for Financing A Supply Chain through. Online Peer-to-peer Lending

Transcription

1 Optial Stackelbeg Stategie fo Financing A Supply Chain though Online Pee-to-pee Lending Guang-Xin Gao a, Zhi-Ping Fan b, c, Ü, Xin Fang d, Yun Fong Li d a College of Econoic and Manageent, Nanjing Univeity of Aeonautic and Atonautic, Nanjing 11106, China b Depatent of Infoation Manageent and Deciion Science, School of Buine Adinitation, Notheaten Univeity, Shenyang , China c State Key Laboatoy of Synthetical Autoation fo Poce Indutie, Notheaten Univeity, Shenyang , China d Lee Kong Chian School of Buine, Singapoe Manageent Univeity, 50 Stafod Road, Singapoe , Singapoe Abtact In ecent yea, upply chain finance (SCF) though online pee-to-pee (PP) lending platfo ha gained it populaity. We tudy an SCF yte with a anufactue elling a poduct to a etaile that face uncetain deand ove a ingle peiod. We aue that eithe the etaile o the anufactue face a capital containt and ut boow capital though an online PP lending platfo. The platfo deteine a evice ate fo the loan, the anufactue et a wholeale pice fo the poduct, and the etaile chooe it ode quantity fo the poduct. We identify optial Stackelbeg tategie of the paticipant in the SCF yte. Fo an SCF yte with a capital-contained etaile, we find that the etaile optial ode quantity and the anufactue optial wholeale pice deceae with the platfo evice ate. Fo an SCF yte with a capital-contained anufactue, we find that a the platfo evice ate inceae, the anufactue optial wholeale pice inceae but the etaile optial ode quantity deceae. Ou analyi ugget that it i ipotant fo the etaile and the anufactue to take the online PP lending platfo financial deciion (uch a the evice ate) into account when aking thei opeational deciion. Keywod: Supply chain anageent; Supply chain finance; Online PP lending; Stackelbeg gae; Opeational and financial deciion Ü Coeponding autho at: Depatent of Infoation Manageent and Deciion Science. School of Buine Adinitation, Notheaten Univeity, Shenyang , China. Tel.: ; Fax: E-ail adde: zpfan@ail.neu.edu.cn (Z.-P. Fan), gxgao@nuaa.edu.cn (G.-X. Gao), xfang@u.edu.g (X. Fang), yfli@u.edu.g (Y.F. Li)

2 1. Intoduction Supply chain finance (SCF) i a new financial buine fo entepie in a upply chain. The ai of SCF ae to diveify the funding ouce of capital-contained entepie and ipove the entie upply chain financial efficiency (Buzacott and Zhang, 004; Yan et al., 016). A evidenced in the liteatue, SCF ha becoe a pevailing hot-te financing ouce fo thouand of all- and ediu-ize entepie (SME), which account fo oe than 90% of all fi (Yan et al., 016). Thee SME play an ipotant ole a they ay eve a upplie o etaile in a upply chain (Singh, 011). Fo exaple, 93.5% of the upplie in the Koean autoobile induty ae SME (Choi, 003; Lee, 008), and ot gocey etaile in China ae SME owned by local copanie (IGD, 013; Eng, 016). Regadle of thei ole in a upply chain, SME often face a hotage of fund to tun aound thei buine (Cai et al., 014; Yan et al., 016). Capital-contained SME in a upply chain taditionally boow loan fo two widely ued financing ouce: extenal bank cedit financing (BCF) and intenal tade cedit financing (TCF). BCF efe to one entepie in a upply chain accee financing fo bank, wheea TCF efe to one entepie in a upply chain extend it cedit to an uptea o downtea patne uing hot-te loan (that i, account payable o account eceivable) (Kouveli and Zhao, 01; Cai et al., 014; Yan et al., 016). It i geneally challenging fo SME to finance though BCF. Fo exaple, appoxiately 50% of SME facing financial containt in developing countie (uch a China) uffe fo liited acce to extenal financial eouce (Buzacott and Zhang, 004; Baa and Schooten, 006; Chen, Lai, and Lin, 014). Futheoe, it i a global phenoenon that bank ae geneally eluctant to povide cedit to SME (Baa and Schooten, 006; Ni et al., 017). On the othe hand, an uptea o downtea patne in a upply chain would egad an SME with low wealth (low woking capital o low collateal) a an entepie with low ceditwothine and high bankuptcy ik. To itigate potential ik, the uptea o downtea patne often chage the SME a high inteet ate, which eult in a high financing cot unde TCF. Thi caue the capital-contained SME to eek othe financing chee uch a online pee-to-pee (PP) lending. - -

3 With advance in infoation technologie and the Intenet, online PP lending ha becoe an ipotant uppleent to taditional financing ecently (Mild, Waitz, and Wöckl, 015). Online PP lending allow fi to diectly finance one anothe on an online platfo, which doe not involve taditional financial inteediaie (Guo et al., 016; Chen, Lai, and Lin, 014; Chen and Han, 01; Lin, Pabhala, and Viwanathan, 009; Bachann et al., 011). Copaed with the taditional financing ouce (uch a BCF and TCF), online PP lending function in a diffeent anne. Fo exaple, in an online PP lending platfo, catteed oney in the ociety i conolidated, financing activitie ae copleted though the Intenet, boowe would not be equied to ubit a high pledge o povide copehenive infoation, and boowe' ceditwothine i only aeed by a iple ik evaluation (Guo et al., 016). Specifically, the following chaacteitic diffeentiate online PP lending fo BCF and TCF: (1) The ouce of pofit fo online PP lending i coiion (deteined by it evice ate). In contat to BCF, which benefit fo the pead between depoit and loan, online PP lending benefit fo integating the infoation fo both boowe and inveto (Lin, 009). Note that online PP lending platfo ay alo bea the boowe default ik (Zhang, 017; Wang et al., 015; Wang et al., 014). They poie a pincipal efund to the inveto in cae of default o ovedue to attact the inveto that ae conevative about PP lending (Wang et al., 015). Fo exaple, the PP lending platfo in China, Hongling Capital ( Hydbet ( and Song ( attact any inveto a they guaantee thei inveto pincipal and inteet. () Boowe and lende can eaily pot and each fo infoation, and coplete tanaction on the online platfo with low tanaction cot (Lin, 009; Lin, Pabhala, and Viwanathan, 009, 013). (3) SME can acquie loan oe efficiently. (4) The lack of copehenive infoation and ik evaluation of the boowe lead to infoation ayety between the lende and the boowe (Lin, Pabhala, and Viwanathan, 009). A a new financing ode, SCF baed on online PP lending ha alteed the paticipant ole and elationhip in a upply chain. Thi ceate new challenge uch a identifying the paticipant optial opeational and financial deciion, a well a the inteaction between thee deciion

4 In thi pape, we conide a upply chain with a anufactue elling a poduct to a etaile that face uncetain deand ove a ingle peiod. We aue that eithe the etaile o the anufactue face a capital containt and ut boow capital though an online PP lending platfo. The platfo i eponible fo evaluating a boowe ceditwothine, and publihing the loan infoation online to invite individual lende to bid fo the loan. We aue that the boowe will be fully financed. The platfo deteine a evice ate fo the loan, the anufactue et a wholeale pice fo the poduct, and the etaile chooe it ode quantity fo the poduct. At the end of the elling peiod, the platfo will ean a pofit if the boowe epay the loan pincipal and inteet, and pay the evice coiion to the platfo. If the boowe evenue i not ufficient to cove it loan obligation and the evice coiion, then it declae bankuptcy and the platfo will ue the boowe liquidated aet and oe ik-eeve capital to epay the loan pincipal and inteet to the lende. Thi caue a lo to the platfo. We deteine the online PP platfo optial evice ate, the anufactue optial wholeale pice, and the etaile optial ode quantity unde two diffeent ituation decibed a follow. In the fit ituation, we aue a capital-contained etaile with bankuptcy ik. We foulate a ulti-level Stackelbeg gae odel with thee paticipant, whee the anufactue act a a leade, the online PP lending platfo act a a ubleade, and the capital-contained etaile act a a followe. We identify the anufactue optial wholeale pice, the platfo optial evice ate, and the etaile optial ode quantity. We alo pove oe popetie of thee optial deciion. In the econd ituation, we aue a capital-contained anufactue with bankuptcy ik. Siilaly, we contuct a ulti-level Stackelbeg gae odel with thee paticipant. In thi odel the platfo act a a leade, the capital-contained anufactue act a a ubleade, and the etaile act a a followe. We deteine the optial evice ate, wholeale pice, and ode quantity. We ake thee ain contibution in thi pape. Fit, we adde the gap in the liteatue on SCF with online PP lending by contucting a theoetical odel. Second, we obtain the optial Stackelbeg olution fo a upply chain with a capital-contained etaile o with a - 4 -

5 capital-contained anufactue. Thid, we povide inight by analyzing the inteaction between the opeational and the financial deciion. The et of thi pape i oganized a follow. Section eview the elevant liteatue. Section 3 define notation and pecifie auption. Section 4 and 5 analyze the optial Stackelbeg olution fo a upply chain with a capital-contained etaile and with a capital-contained anufactue epectively. Section 6 tudie the inteaction between the opeational and the financial deciion baed on ealitic paaete value fo all-ize anufactuing entepie in China. Section 7 extend the tudy to include two cae whee the online PP platfo i ik avee and whee the platfo doe not bea the boowe default ik. Section 8 give concluding eak.. Related liteatue Ou wok i elated to two tea of liteatue. The fit tea of liteatue tudie the equilibiu tategie fo SCF poble. The econd tea focue on online PP lending. We dicu the epaately below..1. Equilibiu tategie fo upply chain finance Many pape in thi tea of liteatue focu on the optial Stackelbeg tategie fo a capital-contained upply chain. Fo exaple, Buzacott and Zhang (004) ake the fit attept to incopoate aet-baed financing into poduction deciion by developing gae theoetical odel that captue the tade-off between financial and poduction deciion. They deontate that aet-baed financing can offe etaile bette etun than uing only thei own capital. Caldentey and Haugh (009) foulate a Stackelbeg gae odel coniting of a capital-contained etaile and a anufactue to analyze the equilibiu tategie fo two diffeent upply chain contact. They find that if the etaile i oe cah contained, the anufactue will chage a highe wholeale pice. Unde that cenaio, the anufactue alway pefe a flexible contact with hedging. Kouveli and Zhao (011) invetigate a capital-contained etaile optial ode quantity and a anufactue optial wholeale pice fo thee type of bankuptcy cot. They foulate a Stackelbeg gae odel in which the anufactue act a a leade and the etaile act a a followe

6 They find that in the peence of the etaile bankuptcy ik, the inceae in the etaile wealth lead to the inceae in the equilibiu wholeale pice. If thee i no bankuptcy ik, the inceae in the etaile wealth lead to the ae o a lowe equilibiu wholeale pice. To adde the poble of a thid-paty logitic (3PL) fi poviding both logitic and financial evice to a capital-contained etaile, Chen and Cai (011) conduct a Stackelbeg gae odel to obtain the etaile optial opeational deciion and the 3PL fi optial financial deciion. Copaed with upplie cedit financing, the 3PL financing can yield bette pefoance. Yan and Sun (013) foulate a ulti-level Stackelbeg gae odel in which a anufactue act a a leade, a bank act a a ubleade, and a capital-contained etaile act a a followe. Conideing the etaile bankuptcy ik, they analyze the anufactue optial wholeale pice, the bank optial cedit line, and the etaile optial ode quantity. Yan, Dai, and Sun (014) analyze the optial tategie fo a upply chain with a anufactue and a etaile, who ae both capital contained. They foulate the poble a a bi-level Stackelbeg gae in which the bank act a a leade and the anufactue act a a ubleade. They deive the bank optial inteet ate, the anufactue optial wholeale pice, and the etaile optial ode quantity. Yan and Sun (015) conduct a copaative analyi of the Stackelbeg equilibiu deciion fo SCF paticipant between two diffeent financing ode. Yan et al. (016) tudy the equilibiu tategie fo a upply chain with a capital-contained etaile, a anufactue, and a coecial bank. They deign a patial-cedit guaantee contact that cobine BCF and TCF, and peent the coodination condition. In contat, ou pape conide a capital-contained boowe that obtain loan though an online PP lending platfo. To the bet of ou knowledge, no pape povide a atheatical analyi of the equilibiu tategie fo SCF with online PP lending, and exploe the inteaction between the opeational and financial deciion... Online PP lending A a new platfo in the Intenet-baed financial cedit aket, online PP lending peent an altenative ouce of cedit fo individual boowe and a potential invetent oppotunity - 6 -

7 fo individual lende (Chaffee and Rapp, 01; Chen, Zhou, and Wan, 016). Howeve, thee ae only liited tudie elated to online PP lending. Fo intance, Bege and Gleine (009) epiically exaine the ole of financial inteediaie in the electonic PP lending platfo. They find that the inteediaie in the electonic aket ipove the boowe cedit by educing infoation ayety. The eult ugget the attactivene of online PP lending. Puo et al. (010) eploy egeion odel and data-diven quey ethod to develop a deciion uppot yte fo boowe in online PP lending. The yte help the boowe to quantify thei tategic option. Chen and Han (01) ake a copaative analyi of online PP lending aket in the USA and China. They claify the cedit infoation into two type: had infoation and oft infoation. The autho find that thee two type of cedit infoation ay affect lending outcoe in both countie. Soe pape tudy cedit ik aeent in online PP lending. Fo exaple, Chen, Lai, and Lin (014) develop an integated tut odel to exaine tut in online PP lending. Baed on eal data collected fo a PP lending platfo in China, the tudy eveal that tut in the boowe can oe efficiently dive the lende intention to lend than tut in the inteediay. Eekte et al. (015) epiically evaluate the cedit ik of online PP loan. They find that a boowe cedit coe, debt-to-incoe atio, FICO coe, and utilization of evolving cedit line have ignificant ipact on the likelihood of thei loan default. Guo et al. (016) popoe a data-diven invetent deciion-aking faewok fo online PP lending to help individual inveto to effectively allocate thei oney aco diffeent loan. They popoe an intance-baed ik aeent odel and ue a potfolio optiization ethod to deteine invetent deciion in online PP lending. Othe pape on online PP lending tudy lende heding behavio and the ipact of uch behavio on lending outcoe. Hezentein, Dholakia, and Andew (011) conduct an epiical tudy of online PP lending and deontate that tategic heding behavio exit. They find that the heding behavio ha a poitive ipact on both the lende and boowe. Lee and Lee (01) eploy a ultinoial logit aket-hae odel to epiically invetigate the heding behavio in an online PP lending aket. They find that the lende in the online PP lending aket tend to exhibit heding behavio. Liu et al. (015) epiically exploe the ipact of fiendhip on econoic deciion aking in online PP lending. In paticula, - 7 -

8 fiend of the boowe have a tonge tendency to offe loan than tange. Endoeent by fiend of the boowe have a negative ipact on ubequent lende deciion. In contat, thi pape ainly focue on optial deciion in a upply chain with financing though online PP lending, which i not tudied in the liteatue. Futheoe, none of the peviou pape have explicitly dicued the integation of SCF and online PP lending. 3. Notation and auption 3.1. Notation The following notation i ued to epeent the paaete and deciion vaiable. Paaete: p : Retaile unit etail pice, which i noalized to 1. Thi i conitent with the liteatue (Bukat and Ellingen, 004; Mateut et al., 006; Cai et al., 014). c : D : Manufactue unit poduction cot. Maket deand, which i a nonnegative ando vaiable following a continuou cuulative ditibution function FD ( ) with denity function f( D ) > 0. B : B : i : i : q : q : h : Retaile initial capital. Manufactue initial capital. Intenet financing inteet ate fo the etaile. Intenet financing inteet ate fo the anufactue. Ceditwothine of the etaile. Ceditwothine of the anufactue. Online PP lending platfo inability to epond to the boowe bankuptcy, h Î [0,1] ( η =1 ean that the platfo ha no ik eeve capital and η = 0 coepond to the cae whee the platfo ha the axiu ik eeve capital). x : Online PP lending platfo axiu ecovey cot when the boowe i bankupt

9 It i woth noting that the Intenet financing inteet ate i and i depend only on the ceditwothine of the boowe, and ae independent of the deciion vaiable defined a follow. Deciion vaiable: Q : Retaile ode quantity fo the anufactue. : Online PP lending platfo evice ate povided to the etaile, Î [, ] and 0 < 1. : w : Online PP lending platfo evice ate povided to the anufactue, Î [, ] and 0, < 1. Manufactue wholeale pice fo each unit of the poduct. We aue that the aount of capital that the etaile need to boow i wq - B > 0. Siilaly, the aount of capital that the anufactue need to boow i cq - B > 0. To eliinate uninteeting cae, we aue that c < w and w(1 + i + ) < p= Auption We ake the following auption in the pape. Auption 1. The online PP lending platfo, the anufactue, and the etaile ae all ik-neutal, and they axiize thei expected pofit (Chen and Cai, 011; Cai et al., 014). Auption. All paaete ae coon knowledge to the thee paticipant (Kouveli and Zhao, 01). Auption 3. Both the etaile and the anufactue have difficulty in obtaining bank loan and theefoe the online PP lending platfo i the only financing option available (Chen and Han, 01). Auption 4. Let hd ( ) = f( D) FD ( ) and H( D) = Dh( D) denote the failue ate and the genealized failue ate, epectively, with FD ( ) = 1 - FD ( ). We aue the poduct deand D atifie the inceaing failue ate (IFR) condition: dh( D) dd > 0 (Chen and - 9 -

10 Cai, 011; Cai et al., 014; Yan et al., 016). Auption 5. The capital aket i pefect without taxe, opeation cot, and bankuptcy cot, and theefoe we oit thee cot in the pape (Kouveli and Zhao, 01). Accoding to Auption 3, the etaile and the anufactue with capital containt in the upply chain ae willing to be financed by the online PP lending platfo, even though the platfo ha a highe inteet ate copaed to the coecial bank (Chen and Han, 01). 4. Optial tategie fo financing a upply chain with a capital-contained etaile In thi ection, we conide a ituation whee the anufactue ha ufficient capital but the etaile i unde financial te and encounte tat-up difficultie. Thu, the etaile boow loan though online PP lending to axiize it pofit. A any loan ae not ecued by collateal, the platfo will ae the etaile ceditwothine to deteine the Intenet financing inteet ate i, which i a paaete of ou odel (Mild, Waitz, and Wöckl, 015). We deive the coeponding aket equilibiu fo an SCF yte hown in Fig. 1, which conit of thee paticipant (the anufactue, the online PP lending platfo, and the etaile). We ue Stackelbeg gae theoy to analyze the optial tategie and tudy the inteaction aong the thee paticipant. In the ulti-level Stackelbeg gae, the anufactue act a a leade, the online PP lending platfo act a a ubleade, and the etaile act a a followe. We have the following equence of event. Knowing the optial epone of the platfo and the etaile, the anufactue fit et a wholeale pice w fo the etaile. Afte obeving the wholeale pice w, the platfo chooe the evice ate by auing that the etaile will eact optially. Finally, the etaile deteine it optial ode quantity Q given w and. Fig. 1 illutate the detailed equence of the buine activitie of the SCF yte. We aue that the etaile boow the capital wq - B though the platfo. Fo notational convenience, let l = 1+ i +. The etaile pay the aount wq to the anufactue and ake the total epayent L= l ( wq- B ) to the platfo. Following

11 peviou eeach (Kouveli and Zhao, 011; Yan et al., 016), we ue backwad induction to obtain the equilibiu olution of the SCF yte. Fig. 1. An SCF yte with a capital-contained etaile baed on online PP lending 4.1. Retaile optial deciion Auing yetic infoation, the capital-contained etaile boow the aount wq - B though the online PP lending platfo. The etaile upload it authentication infoation equied by the platfo. Afte evaluating the etaile ceditwothine q, the platfo chooe an Intenet financing inteet ate i ( q ). Futheoe, auing the etaile will eact optially, the platfo alo et a evice ate fo the loan. Given the wholeale pice w and the evice ate, the etaile deteine the ode quantity Q to axiize it expected pofit: Õ ( Q) = E{ pin[ D, Q] + ( wq-b )(1- )-wq-( wq- B )(1 + i ( q ))}. (1) Eq. (1) can be ewitten a + Õ ( Q) = E{[in[ D, Q] -L] - B }. () Fo Eq. (), we know that if in[ DQ, ] ³ L the etaile will not go bankupt (that i, at the end of the elling eaon, the etaile evenue fo the aket can cove it loan

12 obligation and the evice chage). Othewie, the etaile announce bankuptcy and pay the aount in[ DQ, ] to the platfo. In the latte cae, the platfo will epay lende uing it ik eeve capital and the etaile liquidated aet. The following popoition chaacteize the etaile optial ode quantity and it baic popetie. All poof can be found in the Appendix. Popoition 1. Given the wholeale pice w, the Intenet financing inteet ate i ( q ), and the evice ate, we have the following eult: (1) The etaile optial ode quantity i Q (w, ) = F 1 [wλ F(L)], whee l = 1 + i ( q ) +. () Given the initial capital B, the etaile optial ode quantity Q (w, ) deceae with the wholeale pice w. (3) The etaile optial ode quantity Q (w, ) deceae with the Intenet financing inteet ate i and the evice ate, but inceae with it ceditwothine (4) The etaile optial ode quantity Q (w, ) deceae with it initial capital B. Popoition 1 how that the etaile optial ode quantity depend not only on the opeational deciion (the wholeale pice), but alo on the Intenet financing inteet ate and the evice chage. Thi i one of the key diffeence between online PP lending and offline taditional financing (e.g., BCF). Pat of Popoition 1 how that if the wholeale pice inceae, then the etaile will ode le. Since the anufactue pofit i popotional to both the wholeale pice w and the ode quantity Q, thi iplie that the anufactue face a tade-off when chooing the wholeale pice w. Thi tade-off i addeed in Section 4.3. Unde online PP lending, both the Intenet financing inteet ate and the evice ate deteine the etaile financing cot and bankuptcy ik. Pat 3 of Popoition 1 how that the etaile will ode le (hence boow le) when the Intenet financing inteet and evice ate ae highe. Fotunately, Pat 3 of Popoition 1 enue that the etaile can inceae it q

13 ode quantity by ipoving it ceditwothine. Pat 4 of Popoition 1 how that the etaile ode quantity deceae with it initial capital, which iplie the etaile will boow le when it initial capital i highe. Fo exaple, if the etaile initial capital i ufficiently lage uch that B wf 1 (w), then it doe not need to boow though online PP lending, and the etaile poble educe to the claic newvendo poble without capital containt (Chen and Cai, 011). 4.. Online PP lending platfo optial deciion A one of the paticipant in the SCF yte, the online PP lending platfo act a the ubleade of the ulti-level Stackelbeg gae. At the end of the ale eaon, the etaile hould pay the loan pincipal and inteet ( (wq B )(1+ i ) ) and the evice chage ( (wq B ) ) to the platfo fo it evenue. The platfo then ean the evice chage, and epay the pincipal and inteet to the inveto. If the etaile evenue i not ufficient to cove it loan obligation and the evice chage, the platfo will ue the etaile liquidated aet p in[d,q] a a epayent. If the liquidated aet i till inufficient to epay the pincipal, the inteet, and the evice chage, then the platfo will ue the ik eeve capital to fill the gap. We aue the platfo ha liited ik eeve capital and ha a ignificant ecovey cot fo the etaile bankuptcy. Let h Î [0,1] denote the platfo inability to epond to the etaile bankuptcy, and let ξ denote the cot pe ecovey fo the platfo. We define the expected ecovey cot a ξ F(ηL / p). Note that if h = 1, it ean that the platfo ha no ik eeve capital. Thu, the platfo ha to eek othe capital to epay the inveto, which incu an expected ecovey cot ξ F(L / p). If h = 0, it ean that the platfo ha the axiu ability to epond to the etaile bankuptcy. Thi coepond to the cae whee the platfo ha ufficient ik eeve capital uch that it doe not need any exta capital to fill the gap, which incu an expected ecovey cot ξ F(0). It i woth noting that in pactice the ik eeve capital i anaged by an independent tut

14 intitution. Thi i to enue the afety and benefit of the Intenet inveto. Thu, the ik eeve capital cannot be viewed a the platfo pofit. Given the etaile optial epone Q (w, ) and it initial capital, the online PP lending platfo expected pofit can be expeed a follow: Õ q x h o( ) = E{in[in[ D, Q ], L] -( wq - B)(1 + i( ))} - F( L) L = F(D)dD (wq B )(1+ i (θ )) ξ F(ηL). (3) 0 The fit te of Eq. (3) epeent the expected evenue of the platfo. Specifically, if in[ DQ, ] ³ L, then the platfo ean a evenue of ( wq B) -. Othewie, the etaile i bankupt, and the platfo liquidate the etaile and obtain an aount of in[d,q ] (wq B )(1+ i (θ )). The econd te of Eq. (3) epeent the expected ecovey cot. The following popoition chaacteize the PP lending platfo optial evice ate and it baic popetie. Popoition. Given the etaile bet epone, the initial capital B, the anufactue wholeale pice w, and the Intenet financing inteet ate i ( q ), the online PP lending platfo optial evice ate i ì if µ ( ) ³ 1 + i( q) ï = í if µ ( ) 1 + i( q). ï ïî ˆ if µ ( ) < 1 + i( q ) and µ ( ) > 1 + i( q) In the iddle cae, whee µ ( ) < 1 + i ( q ) and µ ( ) > 1 + i ( q ), a unique ˆ i iplied by µ ( ) = 1 + i ( q ), whee inceae with Î [, ]. wq - B FQ ( )[1-xh( hl)][1 - hq ( )] µ ( ) = w w[1 - Lh( L)] and µ ( ) Popoition ugget that the PP lending platfo optial evice ate heavily elie on µ ( ) which can be egaded a the coefficient that indicate the enitivity of the ode

15 quantity to the evice ate change. When µ ( ) ³ 1 + i ( q ), the ode quantity i conideed vey enitive to the evice ate change, which iplie that a ino inceae in the evice ate will eult in a ajo deceae in ode quantity and the aount of the etaile loan. In thi cae, the online PP lending platfo pofit deceae with the evice ate, uch that d Õ o( ) 0. Thu, the online PP lending platfo will et the evice ate at the lowet d level to encouage the etaile to place oe ode. When µ ( ) 1 + i ( q ), the ode quantity i inenitive to the evice ate change, and the etaile ode deciion i thu only lightly affected by the evice ate. In thi cae d Õ o( ) ³ 0, and the online PP lending d platfo will et the evice ate at the highet level to axiize it pofit. Except fo the above two extee cae, when µ ( ) < 1 + i ( q ) and µ ( ) > 1 + i ( q ), a unique evice ate ˆ, defined in Popoition, can balance the tade-off between the online PP lending platfo pofit and the etaile ode quantity Manufactue optial deciion A the uptea fi in the upply chain, the anufactue without capital containt ai to chooe an optial wholeale pice to axiize it expected pofit. Hence, the anufactue deciion poble can be foulated a follow. Õ ( ) = ( - ). (4) w w c Q Popoition 3. Given the etaile optial ode quantity Q, the optial evice ate and the Intenet financing inteet ate i ( q ), the anufactue would et the optial wholeale pice that uniquely atifie w = c w Q [w λ h(l) h(q )] 1 w λ Q h(l) Popoition 3 ugget that the expeion of the optial wholeale pice unde online PP lending i oe coplex than that unde taditional offline financing. A the leade of the ulti-level Stackelbeg gae, the anufactue need to decide how to et a uitable

16 wholeale pice in epone to both the etaile and the online PP lending platfo. Theefoe, the wholeale pice depend not only on the etaile ode quantity but alo on the online PP lending platfo evice ate and the Intenet financing inteet ate (i.e., the etaile ceditwothine). In addition, the optial wholeale pice can effectively guaantee that the anufactue pofit i ove zeo. 5. Optial tategie fo financing a upply chain with a capital-contained anufactue In thi ection, we analyze anothe ituation whee the etaile ha ufficient capital to pocue poduct, but the anufactue ha capital containt in the poduction poce. To ovecoe financing difficultie, the anufactue obtain capital fo online PP lending. Fig. decibe the inteaction aong the online PP lending platfo, the capital-contained anufactue, and the etaile in the SCF yte. We foulate the poble a a ulti-level Stackelbeg gae in which the online PP lending platfo act a a leade in etting the evice ate and the Intenet financing inteet ate i. In epone to the platfo deciion, the anufactue, acting a a ubleade, decide the wholeale pice w. Finally, the etaile, acting a a followe, decide the ode quantity Q baed on the deciion of the platfo and the anufactue. Fo notational convenience, the epayent of the full loan pincipal, the inteet, and the evice chage i expeed a l ( cq - B ), whee l = 1+ i +. Siila to Section 4, we ue backwad induction to obtain the equilibiu olution of the SCF yte. We fit deteine the etaile and anufactue optial deciion. Then, anticipating the optial epone of the etaile and the anufactue, we find the optial deciion of the platfo

17 Fig.. An SCF yte with a capital-contained anufactue baed on online PP lending 5.1. Retaile and anufactue optial deciion We fit olve the etaile poble, which i a claic newvendo poble whee the etaile pocue the poduct fo the anufactue and then ell it to the aket without knowing the actual deand. At the end of the ale eaon, the etaile yield a evenue of in[ DQ, ] and pay the pocueent cot wq to the anufactue. The etaile chooe it ode quantity Q to axiize it expected pofit expeed a follow: Q Õ ( Q) = E{in[ D, Q] - wq} = F( D) dd-wq ò. (5) 0 Since the etaile ha no capital containt, it optial deciion i identical to the claic newvendo, which i given in the following popoition. Popoition 4. Given the wholeale pice w, the etaile optial ode quantity i - Q = F 1 ( w), and Q deceae with the wholeale pice w. Given the ode quantity Q by the etaile, the anufactue incu a poduction cot cq and boow cq - B fo the platfo. Afte eceiving the payent wq fo the etaile, the anufactue pay the platfo the loan pincipal, the inteet, and the evice chage: l ( cq - B ). The anufactue optiize it wholeale pice w to axiize it expected pofit expeed a follow:

18 Õ ( w) = wq-cq-( cq- B )( i ( q ) + ) = [ wq-l ( cq-b )]- B. (6) The following popoition povide an equilibiu olution to the anufactue and the etaile poble. Popoition 5. Given the wholeale pice w, the etaile optial ode quantity atifie Q FQ c =, which deceae with i and. The capital-contained anufactue ( )- l f( Q ) unique optial wholeale pice i w Q f Q l c = ( ) +. Popoition 5 ugget that the etaile optial ode quantity depend on the inteet ate i and the evice ate. Thi i quite diffeent fo the taditional upply chain without capital containt. A the inteet ate i and the evice ate inceae, the capital-contained anufactue tanfe the financing cot to the etaile by etting a highe wholeale pice, which in tun caue the etaile to place a alle ode. Thi exhibit an inexticable elationhip between financing and opeational deciion. 5.. Online PP lending platfo optial deciion Fo the online PP lending platfo deciion poble, we conide a ituation whee the platfo collect a full epayent l ( cq - B ) fo the anufactue at the end of the ale eaon. Thi occu only if wq > l ( cq - B ), which lead to condition on B tated in the following lea. Lea 1. Given the wholeale pice w, the online PP lending platfo collect a full epayent if the anufactue initial capital atifie c( F( Q )- w) c( F( Q ) + w) < B <. f( Q ) f( Q ) It i woth noting that if the anufactue initial capital i ufficient to cove the poduction cot ( B c( F( Q ) + w) ³ ), thee i no need to boow fo the platfo. If the f( Q ) anufactue initial capital i elatively all ( B c( F( Q )- w) ), the anufactue f( Q )

19 cannot ake the full epayent to the platfo. Anticipating the anufactue and the etaile bet epone, the platfo, acting a a leade, optiize the evice ate to axiize it expected pofit: Õ = -. (7) o( ) ( cq B) Unde the condition in Lea 1, the following popoition deteine the platfo optial deciion. Popoition 6. Given the anufactue and the etaile bet epone, the online PP lending platfo optial evice ate i f ( Q )( cq - B ) =. c Popoition 6 ugget that the optial evice ate inceae with both the ode quantity and the loan ize. The foe i deteined by the etaile, while the latte i deteined by the capital-contained anufactue. Thi explain the diffeent evice ate chaged by online PP lending platfo, uch a Renendai ( and CeditEae ( fo diffeent boowe in pactice. 6. Nueical tudy We pefo a nueical tudy to exploe the elationhip between opeational and financial deciion fo the two diffeent financing cenaio coeponding to Section 4 and 5. In the Stackelbeg gae odel, we et the value of the paaete baed on eal data fo China' all-ize indutial entepie in the anufactuing ecto Paaete We focu on China' all-ize indutial entepie in the anufactuing ecto with data in the peiod of We lit the paaete ued in the nueical tudy a follow. Sevice ate: Following "The guidance on the PP platfo' fee chage tandad" publihed by the Intenet Financial Aociation of China, we et the online PP lending platfo evice ate povided to the boowe ( o ) in the ange of [6%,1%]. Intenet financing inteet ate: Accoding to the data publihed by the Minity of Induty and Infoation Technology of China ( in , we

20 et the Intenet financing inteet ate ( i o i ) in the ange of [8%,1%], whee 8% indicate a low level fo inteet ate, 10% tand fo a odeate level, and 1% ean a high level. Initial capital: Following the liteatue (ee, fo exaple, Cai et al., 014), we ue the aveage liquid aet to appoxiate a fi' initial capital. Table 1 how the detailed data on the total liquid aet, the nube of entepie, and the aveage liquid aet of China' all-ize indutial entepie in the anufactuing ecto in We ue the data in 015 and et the boowe initial capital equal to the aveage liquid aet ( B = B = 0.45). We obeve iila eult in ou nueical tudy when we ue the data in othe yea. Supplie unit poduction cot: Following Buzacott and Zhang (004), we et c = 0.4. Baed on thi etting, we have the wholeale pice wî [0.4,1]. Maket deand: We aue the deand D follow an exponential ditibution with a ean of 10 unit, which i conitent with the liteatue (ee Buzacott and Zhang, 004; Yan et al., 016). Table 1. Liquid aet of China' all-ize indutial entepie in Yea Total liquid aet 87, , , , ,74.87 Total nube of entepie 64,6 80, ,99 31, ,445 Aveage liquid aet Data ouce: National Bueau of Statitic of China ( The unit of total and aveage liquid aet i 100 illion Yuan. 6.. Nueical eult Baed on the above paaete etting, we tudy the ipact of financial deciion (the platfo' Intenet inteet ate and evice ate) on the opeational deciion (the etaile ode quantity and the anufactue' wholeale pice) Inteaction between opeational and financial deciion fo the SCF yte with a capital-contained etaile Ou fit et of expeient illutate the inteaction between the opeational and financial - 0 -

21 deciion fo the SCF yte with a capital-contained etaile. Fig. 3 how how the etaile optial ode quantity vaie with the platfo evice ate fo diffeent Intenet financing inteet ate (i = 0.08,0.1,0.1). We et the wholeale pice equal to the id-point value w = 0.7 of it ange. Fig 3 ugget that fo a fixed i, the etaile optial ode quantity deceae with the platfo evice ate. Futheoe, fo a given, the optial ode quantity deceae with i. We have thee obevation becaue a highe evice ate o a highe Intenet financing inteet ate iplie a highe financing cot, which eult in a alle loan aount and ode quantity. Thi i conitent with Popoition 1. Fig 4 illutate the ipact of the platfo evice ate and Intenet financing inteet ate on the anufactue optial wholeale pice. We et the etaile ode quantity equal to it ean value Q = 10. Fo a fixed i, the optial wholeale pice deceae with. Futheoe, by copaing thee cenaio of i = 0.08, i = 0.1, and i = 0.1, we find that the optial wholeale pice deceae with i fo a given. Thee obevation can be explained a follow. A the platfo evice ate o Intenet financing inteet ate inceae, the capital-contained etaile' financing cot inceae. Thi foce the anufactue to educe it wholeale pice in ode to aintain it ale. Thi i conitent with Popoition Q ( i = 0.08) Q ( i 0.1) = Q ( i = 0.1) Q Fig. 3. The capital-contained etaile optial ode quantity unde diffeent evice ate and Intenet financing inteet ate - 1 -

22 w w ( i = 0.08) w ( i = 0.1) w ( i 0.1) = Fig. 4. The anufactue optial wholeale pice unde diffeent evice ate and Intenet financing inteet ate 6... Inteaction between opeational and financial deciion fo the SCF yte with a capital-contained anufactue We alo conduct nueical expeient to analyze the inteaction between the opeational and financial deciion fo the SCF yte with a capital-contained anufactue. Fig. 5 illutate the ipact of the platfo evice ate and Intenet financing inteet ate on the anufactue' optial wholeale pice. Fo a given i, the optial wholeale pice inceae with. Siilaly, fo a fixed, the optial wholeale pice inceae with i. A the evice ate o the Intenet financing inteet ate inceae, the financing cot of the anufactue becoe highe. A a eult, the anufactue will tanfe the financing cot to the etaile by etting a highe wholeale pice. Thee obevation ae conitent with Popoition 5. w w ( i = 0.08) w ( i = 0.1) w ( i = 0.1) Fig. 5. The capital-contained anufactue optial wholeale pice unde diffeent evice ate and Intenet financing inteet ate

23 A the wholeale pice inceae with both the evice ate and the Intenet financing inteet ate, the etaile epond to the highe wholeale pice by loweing it ode quantity. Thi i evidenced in Fig. 6, which how that the etaile optial ode quantity deceae with both and i. Thi obevation i alo confied in Popoition 5. Q Q ( i = 0.08) Q ( i = 0.1) Q ( i = 0.1) Fig. 6. The etaile optial ode quantity unde diffeent evice ate and Intenet financing inteet ate The above eult deontate that a capital-contained boowe opeational deciion depend not only on thei uptea o downtea patne deciion in a upply chain, but alo on the online PP lending platfo' financial deciion. 7. Extenion We futhe tudy two extenion of the baic odel of the SCF yte. The fit extenion conide a ik-avee platfo. The econd extenion conide an online PP lending platfo that doe not bea the boowe' default ik A ik-avee online PP lending platfo Ou baic odel aue that the online PP lending platfo i ik neutal. Howeve, to potect individual inveto benefit, the platfo can be ik avee in pactice. In thi ection, we conide a ik-avee platfo that ha a deceaing aginal utility (Fibich, Gabiou, and Sela, 006; Gii, 011; Kouveli and Zhao, 01). We odel the ik aveion with a geneal utility function u () that i inceaing and concave. That i, u () ³ 0 and u () 0. The following coollaie illutate the eult of the SCF yte with a ik-avee platfo

24 Coollay 1. Fo the SCF yte with a ik-avee PP lending platfo and a capital-contained etaile, the online PP lending platfo optial evice ate i identical to that found in Popoition. Coollay. Fo the SCF yte with a ik-avee PP lending platfo and a capital-contained anufactue, the online PP lending platfo optial evice ate i identical to that found in Popoition An online PP lending platfo that doe not bea the boowe' default ik The online PP lending platfo in ou odel i eponible fo inveto lo in cae of boowe default. Howeve, oe platfo eve only a infoation inteediay without beaing the boowe default ik (Zhang, 017). It i woth noting that the cae whee the platfo doe not bea the boowe default ik coepond to the cae with h = 0 in ou odel. The following coollay deteine the optial evice ate fo thi cae. Coollay 3. Fo the SCF yte with a capital-contained etaile, if in[ DQ, ] ³ L and h = 0, then the platfo optial evice ate i 1 = -(1 + i ( q )). ( wq - B ) h( L) Coollay 3 how that the optial evice ate depend on the loan ize and the Intenet financing inteet ate. In pactice, a platfo fit chooe an Intenet financing inteet ate by evaluating a boowe ceditwothine, and then chage the boowe a evice fee accoding to the loan ize. Coollay 4. Fo the SCF yte with a capital-contained anufactue, if h = 0, then the platfo optial evice ate i f ( Q )( cq - B ) =. c Siila to Popoition 6, Coollay 4 how that the platfo optial evice ate depend not only on the etaile ode quantity, but alo on the anufactue loan ize. 8. Concluion We tudy an SCF yte that conit of a anufactue, an online PP lending platfo, and a etaile. We analyze the optial Stackelbeg deciion of the paticipant in the SCF - 4 -

25 yte. Specifically, we conide two diffeent cenaio: (i) an SCF yte with a capital-contained etaile and (ii) an SCF yte with a capital-contained anufactue. Ou analyi how that the optial tategie ae quite diffeent unde thee two cenaio. Fo the SCF yte with a capital-contained etaile, we find that the etaile optial ode quantity deceae with the platfo evice ate and Intenet financing inteet ate. Thi i becaue a highe evice ate o a highe Intenet financing inteet ate iplie a highe financing cot, which eult in a alle loan aount and ode quantity. We alo find that the anufactue optial wholeale pice deceae with the platfo evice ate and Intenet financing inteet ate. Thi can be explained a follow. A the platfo evice ate o Intenet financing inteet ate inceae, the capital-contained etaile financing cot inceae. Thi foce the anufactue to educe it wholeale pice in ode to aintain it ale. The above obevation ae poved in Popoition 1 and 3, and confied in ou nueical tudy. Fo the SCF yte with a capital-contained anufactue, the anufactue optial wholeale pice inceae with the platfo evice ate and Intenet financing inteet ate. A the evice ate o the Intenet financing inteet ate inceae, the financing cot of the anufactue becoe highe. A a eult, the anufactue will tanfe the financing cot to the etaile by etting a highe wholeale pice. In epone to the highe wholeale pice, the etaile lowe it ode quantity. Thee obevation ae poved in Popoition 5 and confied in ou nueical tudy. We extend ou odel to conide a ik-avee online PP lending platfo. We find that even if the platfo i ik avee, it till et the ae evice ate a that of the ik-neutal platfo. Futheoe, we tudy the cae whee the online PP lending platfo doe not bea the boowe' default ik. We identify the platfo optial evice ate fo thi cae. Ou odel captue the inteaction between the opeational and the financial deciion. Though ou analytical and nueical tudie, we conclude that it i ipotant fo the etaile and the anufactue to take the platfo financial deciion (the evice ate and the Intenet financing inteet ate) into conideation when aking thei opeational deciion. Thi pape ha oe liitation, which ay eve a avenue fo futue eeach. Fit, iila to the ajoity of the liteatue, we aue that infoation i yetical aong the - 5 -

26 paticipant of the SCF yte. In eality, only the boowe know exactly it capital need and the othe paticipant in the upply chain ay not have the acce to thi infoation. Thi iplie that the infoation i ayetical in the SCF yte. To tudy an SCF yte with ayetic infoation, we need an entiely diffeent odel that ay povide additional inight. Fo exaple, the online PP lending platfo ay chage a highe evice ate to the boowe to itigate the effect of infoation ayety in the lending poce. Second, we aue a pefect capital aket in ou odel. Howeve, the capital aket in pactice contain taxe, opeation cot, and bankuptcy cot. The analyi of deciion aking fo an SCF yte with an ipefect capital aket including thee coplication would be eaningful and intiguing. Thid, thi pape tudie the optial Stackelbeg tategie fo an SCF yte eithe with a capital-contained etaile o with a capital-contained anufactue. In pactice, both the etaile and the anufactue ay be capital-contained at the ae tie. An analyi of thi oe coplex ituation would be inteeting. Acknowledgeent We thank the edito and the two anonyou eviewe fo thei contuctive coent, which have ignificantly ipoved the quality of the pape. Thi wok wa patially uppoted by the National Natual Science Foundation of China [gant nube , , ] and the 111 Poject (B16009). Yun Fong Li i gateful fo the geneou uppot fo the Lee Kong Chian School of Buine at the Singapoe Manageent Univeity and the Minity of Education, Singapoe unde the MOE Tie 1 Acadeic Reeach Fund. Refeence Baa, T., & Schooten, M. (006). Relationhip banking and SME: A theoetical analyi. Sall Buine Econoic, 7(), Bachann, A., Becke, A., Bueckne, D., Hille, M., Kock, F., Lehann, M., Tibutiu, P., & Funk, B. (011). Online pee-to-pee lending? A liteatue eview. Jounal of Intenet Banking and Coece, 16, Bege, S. C., & Gleine, F. (009). Eegence of financial inteediaie in electonic - 6 -

27 aket: The cae of online PP lending. BuR-Buine Reeach, (1), Bown, C. M. (008). I pee-to-pee lending ight fo you. Black Entepie, 39, 146. Buzacott, J. A., & Zhang, R. Q. (004). Inventoy anageent with aet-baed financing. Manageent Science, 50(9), Cai G. G., Chen X., & Xiao Z. (014). The ole of bank and tade cedit: Theoetical analyi and epiical evidence. Poduction and Opeation Manageent, 3(4), Caldentey, R., & Haugh, M. B. (009). Supply contact with financial hedging. Opeation Reeach, 57(1), Chaffee, E. C., & Rapp, G. C. (01). Regulating online pee-to-pee lending in the afteath of Dodd-Fank: In each of an evolving egulatoy egie fo an evolving induty. Wahington and Lee Law Review, 69, 485. Chen, X., & Cai, G. G. (011). Joint logitic and financial evice by a 3PL fi. Euopean Jounal of Opeational Reeach, 14(3), Chen, D., & Han, C. (01). A Copaative tudy of online PP lending in the USA and China. Jounal of Intenet Banking and Coece, 17(), Chen, D., Lai, F., & Lin, Z. (014). A tut odel fo online pee-to-pee lending: a lende pepective. Infoation Technology and Manageent, 15(4), Chen, X., Zhou, L., & Wan, D. (016). Goup ocial capital and lending outcoe in the financial cedit aket: An epiical tudy of online pee-to-pee lending. Electonic Coece Reeach and Application, 15, Eekte, R., Tu, Y., Jiaakuldech, B., & Lu, M. (015). Evaluating cedit ik and loan pefoance in online Pee-to-Pee (PP) lending. Applied Econoic, 47(1), Eng, T. Y. (016). An epiical tudy of Chinee SME gocey etaile ditibution capabilitie. Supply Chain Manageent: An Intenational Jounal, 1(1), Fibich, G., Gaviou, A., & Sela, A. (006). All-pay auction with ik-avee playe. Intenational Jounal of Gae Theoy, 34(4), Gii, B. C. (011). Managing inventoy with two upplie unde yield uncetainty and ik aveion. Intenational Jounal of Poduction Econoic, 133(1), Guo, Y., Zhou, W., Luo, C., Liu, C., & Xiong, H. (016). Intance-baed cedit ik aeent fo invetent deciion in PP lending. Euopean Jounal of Opeational Reeach, 49(), Hezentein, M., Dholakia, U. M., & Andew, R. L. (011). Stategic heding behavio in pee-to-pee loan auction. Jounal of Inteactive Maketing, 5(1), Kouveli, P., & Zhao, W. (011). The newvendo poble and pice-only contact when - 7 -

28 bankuptcy cot exit. Poduction and Opeation Manageent, 0(6), Kouveli, P., & Zhao, W. (01). Financing the newvendo: upplie v. bank, and the tuctue of optial tade cedit contact. Opeation Reeach, 60(3), Lee, E., & Lee, B. (01). Heding behavio in online PP lending: An epiical invetigation. Electonic Coece Reeach and Application, 11(5), Lee, S. Y. (008). Dive fo the paticipation of all and ediu-ized upplie in geen upply chain initiative. Supply Chain Manageent: An Intenational Jounal, 13(3), Lin, M. (009). Pee-to-pee lending: An epiical tudy. AMCIS 009 Doctoal Conotiu, San Fancicon, Califonia, 1-7. Lin, M., Pabhala, N. R., & Viwanathan, S. (009). Social netwok a ignaling echani: Evidence fo online pee-to-pee lending. WISE 009. Lin, M., Pabhala, N. R., & Viwanathan, S. (013). Judging boowe by the copany they keep: fiendhip netwok and infoation ayety in online pee-to-pee lending. Manageent Science, 59(1), Liu, D., Ba, D., Lu, Y., & Chen, D. (015). Fiendhip in online pee-to-pee lending: Pipe, pi, and elational heding. MIS Quately, 39(3), Mild, A., Waitz, M., & Wöckl, J. (015). How low can you go? - Ovecoing the inability of lende to et pope inteet ate on unecued pee-to-pee lending aket. Jounal of Buine Reeach, 68(6), Ni, J., Chu, L. K., & Li, Q. (017). Capacity deciion with debt financing: The effect of agency poble. Euopean Jounal of Opeational Reeach, 61(3), Puo, L., Teich, J. E., Walleniu, H., & Walleniu. J. (010). Boowe deciion aid fo people-to-people lending. Deciion Suppot Syte, 49(1), Singh, R. K. (011). Developing the faewok fo coodination in upply chain of SME. Buine Poce Manageent Jounal, 17(4), Wang, Y., & Hua, R. (014). Guiding the healthy developent of the PP induty and pooting SME financing. 014 Intenational Confeence on Manageent of e-coece and e-govenent (ICMeCG), Beijing, Wang, J. G., Xu, H., & Ma, J. (015). Financing the undefinanced: Online lending in China. Spinge. Yan, N., & Sun, B. (013). Coodinating loan tategie fo upply chain financing with liited cedit. OR Spectu, 35(4),

29 Yan, N., Dai, H., & Sun, B. (014). Optial bi-level Stackelbeg tategie fo upply chain financing with both capital-contained buye and elle. Applied Stochatic Model in Buine and Induty, 30(6), Yan, N., & Sun, B. (015). Copaative analyi of upply chain financing tategie between diffeent financing ode. Jounal of Indutial and Manageent Optiization, 11(4), Yan, N., Sun, B., Zhang, H., & Liu, C. (016). A patial cedit guaantee contact in a capital-contained upply chain: Financing equilibiu and coodinating tategy. Intenational Jounal of Poduction Econoic, 173, Zhang, N. (017). Rationality of inveto in PP online lending platfo with guaantee echani: Evidence in China. Jounal of Applied Finance & Banking, 7(3), Appendix Poof of Popoition 1 Pat (1): Fo Eq. (), the etaile expected pofit function can be ewitten a Q ò ò ò. (A.1) Õ ( Q) = [ ( D- L) df( D) + ( Q-L) df( D)] - B = F( D) dd-b L Q L Taking the fit-ode and the econd-ode deivative of Õ ( Q ) with epect to Q in Eq. Q (A.1), we have d Õ ( Q) = FQ ( )- wl FL ( ) and d Õ ( Q) =- f( Q) + ( wl ) ( ) f L. Applying the iplicit function theoe of FQ ( ) = wl FL ( ), we obtain d Õ ( Q) - f( Q) ( wl) f( L) = FQ ( )[ + ] = FQ ( )[- hq ( ) + wl hl ( )]. (A.) F( Q) F( Q) d A IFR ditibution of deand and wl < 1, we have hq ( ) > wl hl ( ). Hence, Õ ( Q) < 0 hold, which ean that the etaile expected pofit function i concave and that thee exit a unique optial olution Q. Fo the fit-ode condition, i.e., d Õ ( Q) = 0,we can obtain the etaile optial ode quantity, i.e., - Q = F 1 [ wl F( L)]