Research ArticleDynamic Strategies on Firm Production andPlatform Advertisement in Crowdfunding consideringInvestorrsquos Perception

Ying Ji JuWei ZhongWu Shaojian Qu and Baojun Zhang

Business School University of Shanghai for Science and Technology Shanghai China

Correspondence should be addressed to Ju Wei wj725496163com

Received 27 June 2018 Accepted 13 August 2018 Published 18 September 2018

Academic Editor Lu Zhen

Copyright copy 2018 Ying Ji et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Taking investorrsquos perception into account the optimal decisions about the product quality and platform advertisement areinvestigated in a dynamic model in the context of crowdfunding Researches in the literature however usually set investorrsquosperception as a fixed value and rarely consider the important phenomenon that the online information has some influences oninvestorrsquos perception Considering the effects of information about product quality and platform advertisement on the investorrsquosperception a dynamic decision model is proposed Firstly investment desire and reference price of the investor are introducedin two dynamic settings to describe investorrsquos perception Then the optimal decisions about the product quality and platformadvertisement are formulated under two circumstances the sponsor and the platform make decisions independently and theycooperate as a system Finally the influences of reference price and cost-sharing ratio on the optimal results are compared and thedata simulation experiment verifies the necessity of the study Some new insights can be drawn for the operations management ofthe firm in crowdfunding as follows (i) it is more profitable for the firm to cooperate with the platform when investors pay moreattention to their reference price (ii) it is optimal for the firm to share a larger proportion of platform cost when the profit-sharingratio is low

1 Introduction

As an effective way of financing crowdfunding has rapiddevelopment in the investment market Crowdfunding is ageneral term that refers to the activities of raising fundsfrom common investors through the Internet In one ofthe most popular crowdfunding websites Kickstartercommore than 35 million people have participated in 108437creative projects and over 247 billion dollars has beenpledged from the date Kickstartercom founded till nowMore than 115 numbers of crowdfunding platforms have beenbuilt and over 1 billion yuan has been raised at the end of2014 in China [1] Therefore how to produce and operateeffectively is becoming an increasingly prominent problemfor crowdfunding firms with the popularity of crowdfundingactivities

There are four types of crowdfunding projects dependingon what the investors can receive from the project including

donation-based reward-based lending and equity crowd-funding [2ndash4] In reward-based crowdfunding the sponsorpublishes the productrsquos material and function information onthe platform to introduce the products in detail The infor-mation provides accurate product quality and the cases thatthe sponsor provides faulty information to mislead readersare not considered in the present study After accessing tothe information investors will form a certain valuation aboutthe product quality and they will compare the price of theproduct with their reference price when they make purchasedecisions If the published online information can not attractinvestors effectively the investment will be failing

Investorrsquos perception frequently appears in the field offinancial investment When investors consider whether toparticipate in a crowdfunding project or not the process ofmaking decisions is similar to making purchase decisions infinancial marketTherefore it is reasonable to apply investorrsquosperception to analyze investorrsquos behavior in the context

HindawiDiscrete Dynamics in Nature and SocietyVolume 2018 Article ID 3726080 12 pageshttpsdoiorg10115520183726080

2 Discrete Dynamics in Nature and Society

Project NameThe sponsor

Project progress

Project Name

Project progress

Figure 1 One webpage on Kickstartercom

of crowdfunding Researches in the literature usually setinvestorrsquos perception as a fixed value and they rarely considerthe important phenomenon that the online information willcause some influences on investorrsquos perception

As themost important way of information disseminationpersonal networks and entrepreneurrsquos social network havesignificant effects on crowdfunding performance [5] Bi [1]illustrated that the signals of product quality and electronicword of mouth have caused great influences on funderrsquosinvestment decisions by solving ELM model Undoubtedlythe information about the product quality is the mostimportant factor that affects investors to buy the product ornot Figure 1 illustrates that the advertisements of the projecton platform can also cause some influences on the productvaluations of the investors

Figure 1 is onewebpage of art projects onKickstartercomThere are five art projects in the picture It is obvious thatthe project on the left side Upstate Photographs by TeamaStauffer can attract us mostly when we click on this webpagesince it takes more space than the other four projects

As shown in Figure 1 the location and space of theproject advertisement will cause some important influenceson investorrsquos perception Platforms of crowdfunding such asKickstartercom decide the location and space of the projectintroductions in a whole webpage Further platforms providethe services of matching fund raisers with investors and theyalso provide some investment style to fund raisersThereforeit is reasonable to suppose the advertisement on the platformas a risk factor that can cause some influences on investorrsquosperception

Considering the effects of online information oninvestorrsquos perception investorrsquos perception applied in thepresent study is that the investors will continuously changetheir own valuations about the product based on theinformation they got In detail the investment desire and

reference price of the investor are introduced in two dynamicsettings to describe investorrsquos perception

Therefore taking investorrsquos perception into account theproblems of product design are illustrated in the presentstudy Product design refers to firms which make suitabledesigns about product quality price marketing strategy etcProduct design has been proved to be an efficient tool forthe firms to produce their product and the advisable productdesign shown was an effective way for the actual firms tomaintain a long-term competitive advantages [6] Whendynamic investorrsquos perception is taken into consideration wewant to analyze the following four questions(1)How do the online information of the product qualityand platform advertisement affect investorrsquos investment deci-sions(2)What is the optimal decisions of product quality forthe sponsor in crowdfunding(3)What is the optimal decisions of platform advertise-ment in crowdfunding(4)Howdo the cost-sharing ratio profit-sharing ratio andreference price affect the optimal decisions

To solve the above questions we propose and analyzea dynamic model in which the sponsor designs the prod-uct quality to cater investorrsquos preference and the platformformulates the strategies about project advertisement tooptimize their own profit based on the changeable investorrsquosperception Firstly the information about product quality andplatform advertisement is identified as the two risk factorsthat affect investorrsquos perception Then investment desire andreference price are used to illustrate investorrsquos perception indetail which are described in two dynamic equations Finallythe optimal decisions about the product quality and platformadvertisement are formulated under two circumstances thesponsor and the platform make decisions independently andthey cooperate as a system

Discrete Dynamics in Nature and Society 3

The rest of the paper is organized as follows Literaturereviews are presented in Section 2 In Section 3 investorrsquosperception is introduced into the dynamic product qualitydesign model and the optimal results are given in two cir-cumstances respectively Based on the model the influencesof investorrsquos perception on the optimal results are analyzedand some profitable strategies for the sponsor are givenIn Section 4 some sensitive analyses for parameters aregiven and the simulation is described in Section 5 Section 6describes some concluding remarks of the present study

2 Literature Reviews

It is difficult for many innovative companies to get financialsupport by traditional financing methods since small inno-vative companies have no power to make public offeringsCrowdfunding provides a solution for the small innovativecompanies to raise the initial startup capital on the Internet[7 8] It is found that Internet financing was a good way tosolve the traditional financing difficulties of small companieseffectively [9]

The preliminary literatures of crowdfunding focus on thefollowing two aspects First some studies have discussedthe definition and the business model of the crowdfundingCrowdfunding was defined as a concept that invites individ-uals to invest in various section [10] The concept of crowd-funding originates from the idea of the crowdsourcing whichcan be seen as the wisdom of a crowd while the differencein the crowdfunding focuses on the financial perspective ofcrowdsourcing [11 12] For the businessmodel there are threeparticipators involved in crowdfunding Internet platformsfund raisers and investors Internet platforms provide theservices of matching fund raisers with investors and providesome investment style to fund raisers [13 14] As shownin the research of Lukkarinen [15] the driving force ofcrowdfunding can be summarized as two aspects One is thecharacteristics of the project such as the early fund raised bymedia networks and the value of minimum investment Theother includes the total target the projectrsquos duration and theprovision of financial information

Second some studies have investigated which factorsaffect the crowdfunding mostly The period of fund raisingand the amount of goal were identified as the directive factorsrelated to a project [3 16ndash19] Many researches revealedthat an appropriate fund raising goal and an appropriategeographical location of a crowdfunding project have causedsome valid influences on the success of crowdfunding [13 5 18 19] It was found that the amount of the target isnegative to the success of the project while the projects thatattract the interest of investors aremore likely to have a highersuccess rate of a crowdfunding [16 18] Some researchesclaimed frequent updates and a high level of communicationthrough updates and the information about the investors typealso caused great influence on the success of crowdfunding[20 21]

There are few researches focusing on the productionand operation problem of the firms in crowdfunding whenthe investorrsquos product valuation is changing over time Thepreliminary researches have focused on analyzing the effects

of the information on investorrsquos perception in financial mar-ket and accordingly some studies have explored the effectsof the investorrsquos perception on the investment decisions Itwas found that the low risk of fund and the liquidity offund scheme have a great impact on investorrsquos perceptionfor investing in the mutual fund [22] Joshi and Patel [23]found that the adverse investorrsquos perception has preventedthe information being reflected into the prices of stockInvestors may form their own valuations about the productaccording to their previous purchase experience and theinformation they mastered in the market How to producethe products with a reasonable product quality to attractinvestors to participate in a crowdfunding is our focus Hu[24] studied the optimal product pricing strategy in a reward-based crowdfunding and the product line was proved to bemore optimal than providing a single product Based on thestudy of Hu Chen [25] proposed the optimal pricing strategyfor the sponsor when facing strategic and myopic consumersin a crowdfunding of green product

Based on the above studies we give the dynamic modelto describe how to design the product quality when investorrsquosperception is taken into account in the next followingsections

3 The Basic Model

The decision system considered in the present study consistsof a sponsor and a platform The sponsor raises fundsthrough the crowdfunding on the platform and providessome rewards to the platform when the raised funds reachthe amount of target Different from the real sales investorsunderstand the product based on the online informationthey perceived from the platform In crowdfunding sponsordesigns the product quality to cater investorrsquos preferenceand platform makes decisions about the advertisement ofthe project Therefore investorrsquos perception is consideredas the function of the product quality and the platformadvertisement in the present study For understanding easilythe description of parameters is illustrated in Table 1

Since the information updates continuously investorrsquosperception changes over time In detail investment desireand reference price are introduced to describe investorrsquosperception First the changing of the investment desirefollows the Nerlove-Arrow framework [26]

119892 (119905) = 1205791119902 + 1205792119890 minus 120575119892119892 (0) = 1198920 (1)

where 119892(119905) is the accumulated investment desire over time 119905and 119892(0) gt 0 is the initial investment desire 1205791 and 1205792 arepositive constants that reflect both the product quality andplatform advertisement which have an positive effects on theinvestment desire and 120575 is the diminishing rate of investmentdesire

Then product quality and platform advertisement canalso affect investorrsquos reference price The reference priceis formed after investors get the information about theproduct attributes The changeable reference price in thepresent study is supposed to be affected by product quality

4 Discrete Dynamics in Nature and Society

Table 1 Parameters description

Symbol Explanation Variable type119902 The quality of product Decision variable119890 The advertisement of platform Decision variable119881119905 Investorrsquos product valuations (119881119905=H L) ndash119892 Investment desire ndash119903 Reference price ndash119889 The demand of product ndashΠ119904 The profit of the sponsor ndashΠ119901 The profit of the platform ndash119869119904 The present value of sponsorrsquos profit Decision goal119869119901 The present value of platformrsquos profit Decision goal120576 The ratio of the profit-sharing ndash120601 The ratio of the cost-sharing Decision variable119896 The probability of success ndash119898 1205791 1205792 120575 1205831 1205832 1205821 1205822 120572 120573 Parameters ndash

platform advertisement and product price The followingequation is introduced to illustrate the changeable processionof reference price

119903 (119905) = 120573 (119901 minus 119903) + 1205831119902 + 1205832119890119903 (0) = 1199030 (2)

where 119903(0) = 1199030 is the initial reference price and 120573 1205831 1205832 areall constants The item 120573(119901 minus 119903) means the price influence ahigher 120573 implies that the investors have short memory of theproduct The two items 1205831119902 and 1205832119890 represent the influencesof the product quality and the platform advertisement onthe reference price Higher product quality and platformadvertisement can enhance the consumerrsquos valuations on theproduct so 1205831 gt 0 and 1205832 gt 0 are supposed

Further investors are divided into short and long sighted-ness which means that investors have less or deep impressionon the product purchase experience in the past Based onthe research of119867119906 [24] combining the consumerrsquos type ieshort or long sightedness we give the following equation toillustrate the different investorrsquos product evaluations

119881119905 = 119867 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 120572 (1 minus 120573) 119871 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 120572120573 (3)

where 119867 gt 119871 gt 0 represents the product valuations of theinvestor

The price of product is determined by the sponsor atthe beginning of the crowdfunding and then the platformadvertises the product onlineThis paper focuses on choosingthe optimal product quality and platform advertisementwhen investorrsquos perception is taken into account The priceof product 119901 is assumed as a fixed value when the modelis solved and the different results caused by the changesof price will be discussed based on the research of 119867119906[24] 119867119906 provided several price strategies for the productcrowdfunding accordingly the optimal product quality andplatform advertisement are analyzed in the present studybased on his research

Generally the reference price the product quality andthe platform advertisement are assumed to have positiveeffects on the sale Thus the demand 119889(119905) of the product ina crowdfunding is supposed to satisfy the following equation

119889 (119905) = 119898 (119903 minus 119901) + 119892 + 1205821119902 + 1205822119890 (4)

where119898 1205821 and 1205822 are all positive constants The item119898(119903minus119901) represents the influence of reference price on demandThe influence on the demand is positive when 119903 gt 119901and the influence is negative otherwise A high 119898 impliesthat the investors are more sensitive to the gap between thereference price and the real priceTheother two items1205821119902 and1205822119890 indicate that both the product quality and the platformadvertisement have positive influences on the demand

Similar to the previous research of Joslashrgensen [27] thecost is in the form of quadratic function The product costof the sponsor is 119862119904 = (12)1199022 and the cost of the platformis 119862119901 = (12)1198902 Platforms can get some rewards whenthe raised funds reach the goal amount while there are norewards provided to the platform if the crowdfunding failsTherefore it is advisable for the sponsor to share a part of theplatformrsquos cost to encourage the platform tomake their effortsto improve the crowdfunding 120601 isin (0 1) is the cost-sharingratio that means the sponsor is willing to undertake theplatformrsquos advertisement cost then the profit of the sponsoris

Π119904 = 119896119901 (1 minus 120576) 119889 minus 121199022 minus 121206011198902 (5)

where 119896 isin (0 1) is the probability that the raised fundsreach the goal amount which is often called the successfulprobability of a crowdfunding 120576 isin (0 1) is called the ratioof profit sharing So the profit of the platform is

Π119901 = 119896119901120576119889 minus 12 (1 minus 120601) 1198902 (6)

where the item 119896120576119901119889 (120576 isin (0 1)) is the reward of the platformwhen the crowdfunding raised the goal amount of funds

Discrete Dynamics in Nature and Society 5

The goals of the sponsor and the platform are tomaximizethe present value of their profit respectively

max119889(119905)

119869119904= int+infin0

exp (minus120588119905) [119896119901 (1 minus 120576) 119889 (119905) minus 121199022 minus 121206011198902] 119889119905(7)

and

max119889(119905)

119869119901= int+infin0

exp (minus120588119905) [119896119901120576119889 (119905) minus 12 (1 minus 120601) 1198902] 119889119905 (8)

where 120588 is the discount rateIn the next sections the optimal product quality and

platformadvertisement are calculated in two different scenar-ios One scenario is that the sponsor and the platform makedecisions independently and the other is that the sponsor andthe platform are coordinate as a system to make decisions

31 The Optimal Decisions in Separation Separation meansthat the sponsor and the platform make decisions indepen-dently the sponsor offers a cost-sharing ratio 120601 firstly to theplatformThen the sponsor decides their product quality andthe platform decides their advertisement to maximize theirprofit It is reasonable to suppose that the sponsor and theplatform make decisions simultaneously since the demandchanges over time and the cost-sharing ratio 120601 is assumedas a fixed value when we calculate the values of decisionsabout the product quality and platform advertisement Theprofit of the two firms will be calculated after we obtain theoptimal equilibrium value of decisions and then the optimalcost-sharing ratio 120601 that maximizes the sponsorrsquos profit iscalculated

To maximize the profit of the sponsor and the platformthe function of Hamiltonian for the sponsor and the platformare given as the follows respectively

119867119904 = 119896119901 (1 minus 120576) [119898 (119903 minus 119901) + 119892 + 1205821119902 + 1205822119890] minus 121199022minus 121206011198902 + 1205741119904 (120573 (119901 minus 119903) + 1205831119902 + 1205832119890)+ 1205742119904 (1205791119902 + 1205792119890 minus 120575119892)

(9)

and

119867119901 = 119896119901120576 [119898 (119903 minus 119901) + 119892 + 1205821119902 + 1205822119890] minus 12 (1 minus 120601) 1198902+ 1205741119901 (120573 (119901 minus 119903) + 1205831119902 + 1205832119890)+ 1205742119901 (1205791119902 + 1205792119890 minus 120575119892)

(10)

where 1205741119904 1205742119904(1205741119901 1205742119901) represent the costate variablesCalculating the two Hamiltonian functions the equilib-

rium product quality and platform advertisement are givenin the following Theorem 1

Theorem 1 The equilibrium product quality is

119902 = 119896119901 (1 minus 120576) (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (11)

and the equilibrium platform advertisement is

119890 = 1198961199011205761 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (12)

Therefore the reference price and accumulated invest-ment desire can also be given as follows

119903 (119905) = 1198601119890minus120573119905 + 119903119899119892 (119905) = 1198602119890minus120575119905 + 119892119899

(13)

where1198601 = 1199030 minus 119903119899 119903119899 = 119901+ (1205831119902+1205832119890)1205731198602 = 1198920 minus119892119899 and119892119899 = (1205791119902 + 1205792119890)120575The proof of each theorem is illustrated in Appendixes

and some conclusions can be addressed from Theorem 1 asfollows

(i) There are three parts of equilibrium product qualitylevel in (11) the first part 1198961199011205821 represents the product qualityinfluences on the product demand which is the short-terminfluence on product sales The second part 1198961199011205791(120588 + 120575)expresses the long-term effect that means the product qualityhas the positive influence in accumulate investment desireThe third part 1198961199011198981205831(120588 + 120573) expresses the fact that theproduct quality has the effect on the reference price with1205831 = 0 The structure of (12) is the same as (11) so we donot repeat it here Since the product quality and platformadvertisement have positive effect on the reference price ie1205831 gt 0 and 1205832 gt 0 the sponsor will provide a higher qualityproduct and the platform will put more advertisement inadverting when they take the effect of reference price intoconsideration The higher 119898 means the consumers are moresensitive to the reference price therefore the decision makersshould pay more attention to this impact

(ii) The reference price and investment desire in (13) willget their steady states 119903119899 and 119892119899 when 119905 997888rarr +infin The steadystates ie 119903119899 = 119901 + (1205831119902 + 1205832119890)120573 and 119892119899 = (1205791119902 + 1205792119890)120575are mainly influenced by the product price 119901 the productquality 119902 and the platform advertisement 119890 It is shown thatthe reference price and the investment desire are increasingwhen the product quality and platform advertisement aretaken into account In addition investors will have a largereference price when the value of 120573 is small Analyzing 119892119899similarly the investment desire is positively correlated withthe product quality and the platform advertisement

Based on the above results the profits of the sponsor andplatform are calculated as follows

119869119904 = 119896119901 (1 minus 120576) ( 1198631119898120588 + 120573 + 1198632120588 + 120575 + 119898 (119903119899 minus 119901)120588 + 119892119899120588

+ 1205821119902120588 + 1205822119890120588 ) minus 11990222120588 minus 12060111989022120588 (14)

6 Discrete Dynamics in Nature and Society

and

119869119901 = 119896119901120576( 1198631119898120588 + 120573 + 1198632120588 + 120575 + 119898 (119903119899 minus 119901)120588 + 119892119899120588 + 1205821119902119901

+ 1205822119890120588 ) minus (1 minus 120601) 11989022120588

(15)

Since sponsors make the decisions about the productprice strategy at the beginning of the crowdfunding they willdecided the optimal value of product quality and platformadvertisement for any specific price strategy The optimalresults in margin (H) and volume (L) price strategy areanalyzed respectively based on the research of 119867119906 [24]Therefore there will be two groups ie (119902 119890119867) (119902 119890 119871)to choose different situations In the following Theorem 2the optimal product quality and platform advertisement aregiven in detail in different price strategy

Theorem 2 (i) When

1198612 = (1 minus 120576)2 1198601 + 11989812058321198602120576 (1 minus 120576) + 12058221205731198602120576 (1 minus 120576)120588120573 (1 minus 120601)

+ 12057911198601 (1 minus 120576)2 (1 minus 120601) + 12057911198602120576120588120575 (1 minus 120601)+ 212058211198601 (1 minus 120576)2 minus 11986021 (1 minus 120576)2

2120588 minus 119860221205762(1 minus 120601)2 gt 0(16)

we obtain the following two conclusions(a) 0 lt 119867119897 lt 11205722(1 minus 120573)2 the optimal values of product

quality and platform advertisement are

119902 = (1 minus 120576) 119871 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (17)

and

119890 = 1205761198711 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (18)

(b) 119867119897 gt 11205722(1 minus 120573)2 the optimal values of productquality and platform advertisement are

119902 = (1 minus 120576)1198671205722 (1 minus 120573)2 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (19)

and

119890 = 1205761198671205722 (1 minus 120573)21 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (20)

(ii) When

1198612 = (1 minus 120576)2 1198601 + 11989812058321198602120576 (1 minus 120576) + 12058221205731198602120576 (1 minus 120576)120588120573 (1 minus 120601)+ 12057911198601 (1 minus 120576)2 (1 minus 120601) + 12057911198602120576120588120575 (1 minus 120601)+ 212058211198601 (1 minus 120576)2 minus 11986021 (1 minus 120576)2

2120588 minus 119860221205762(1 minus 120601)2 lt 0(21)

the optimal values of product quality and platform advertise-ment are

119902 = (120576 minus 1) 119861121198612 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (22)

and

119890 = 120576 (120576 minus 1) 119861121198612 (1 minus 120601) (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (23)

Note that the equilibrium value of product quality andplatform advertisement will be higher if the sponsor takes themargin price strategy compared to the volume price strategyIt is also clear that both two equilibriums are the increasingfunction of the consumerrsquos evaluation (119871 or 119867) When theevaluation of consumer is higher the sponsor should improvetheir product quality level and similarly the platform shouldpay more advertisement to advertise the product And thereare no differences of the sponsor profit in the two pricestrategies when119867119897 = 11205722(1 minus 120573)232 The Optimal Decisions in Coordination In this sectionthe sponsor and the platform cooperate as a system to makedecisions about product quality and platform advertisementWe calculate the optimal decision of the system to maximizethe present value of it Taking (5) and (6) in to account theHamiltonian function for the system is given as follows

119867 = 119901119896 [119898 (119903 minus 119901) + 119892 + 1205821119902 + 1205822119890] minus 121199022 minus 121198902+ 1205741 (120573 (119901 minus 119903) + 1205831119902 + 1205832119890)+ 1205742 (1205791119902 + 1205792119890 minus 120575119892)

(24)

where 1205741 1205742 represent the costate variables of the systemproblemassociating the reference price and investment desirewith the decisions Similar to the process of Theorem 1 theequilibrium of the system is shown as follows

119902 = 119896119901(1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (25)

and

119890 = 119896119901(1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (26)

It is clearly shown that the product quality is improved due tothe cooperation InTheorem 3 the optimal cost-sharing ratiois given and the advertisement of platform in two situationsis compared

Theorem 3 The optimal cost-sharing ratio 120601 is

120601 = 2 minus 31205762 minus 120576 119894119891 120576 le 23 0 119890119897119904119890 (27)

Taking (27) into (20) we get

119890 = 119896119901120576 (2 minus 120576)2120576 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (28)

Discrete Dynamics in Nature and Society 7

and

119890 minus 119890 = 1198961199011205762 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) gt 0 (29)

For the cooperation system the reference price andinvestment desire are also steady similar to the situation ofseparation as follows

119903 (119905) = 1198641119890minus120573119905 + 119903119888119892 (119905) = 1198642119890minus120575119905 + 119892119888

(30)

where 1198641 = 1199030 minus 119903119888 119903119888 = 119901 + (1205831 119902 + 1205832 119890)120573 1198642 = 1198920 minus 119892119888 and119892119888 = (1205791 119902+1205792 119890)120575The steady states of the reference price andinvestment desire are 119903119888 and 119892119888

From the above results we can find that both the steadyreference price and steady investment desire are highercompared to Theorem 2 since both the product quality andplatform advertisement are improved in cooperation systemSubstituting (30) into (7) we get the present value of the profitfor cooperation system as follows

119869 = 119896119901( 1198641119898120588 + 120573 + 1198642120588 + 120575 + 119898 (119903119888 minus 119901)120588 + 119892119888120588 + 1205821 119902119901

+ 1205822 119890120588 ) minus 11990222120588 minus 11989022120588 (31)

In the next section we will discuss the changes of theresults in ourmodel when relevant parameters are in differentvalues Analyzing these changes does not only build therobustness of our results but also deepen our understandingof this problem

4 Parameters Sensitivity Analysis

In this section we analyze the relations between the relevantparameters and the values of profit product quality andplatform advertisement Firstly the impact of parameter 119898on the present value of profit is illustrated in Figures 2 and 3The parameter119898 is seen as an item that reflects the degree ofthe reference price effects on the investorrsquos behaviorThere aretwo firms the sponsor and the platform in the research so itis worthwhile to analyze the influence of parameter119898 on theprofit of them because the effects of119898may be different

As shown in Figure 2 119869119904 119869119901 and 119869 represent the profitof the sponsor the profit of the platform and the totalprofit when the sponsor and the platform chose to cooperaterespectively The present value of the sponsorrsquos profit willdecrease if the investors pay more attention to referenceprice while the platformrsquos profit is an increasing function ofparameter 119898 when they make decisions independently It isclear that the profits of the two firms are increased whenthey cooperate as a system and the cooperation profit is anincreasing function of the parameter 119898 Note that parameter119898 denotes the degree of the reference price effects on theinvestorrsquos behavior the fact implied in Figure 2 shows that thelarger the effect of the reference price the more profit the twoitems can get when they cooperate as a system

ＭＪJ

05

101520253035404550

The p

rofit

01 02 03 04 05 06 07 08 09 10m

Figure 2 Profit changes

Δ ＭΔ Ｊ

01 02 03 04 05 06 07 08 09 10m

05

101520253035404550

The c

hang

e of p

rofit

Figure 3 Differences of profit in two situations

From Figure 3 we can find that cooperation is more prof-itable for the sponsor than the platform and the differences ofthe two profits are both the increasing function of parameter119898 Therefore the sponsor should choose to cooperate withthe platform when the investors pay more attention to theirreference price And the differences of platformrsquos profit are notobvious in twoways whichmeans that it has a little differencefor platform in two cases

Comparing Figure 2 with Figure 3 it is found that coop-eration with platform not only promotes the profit of sponsorbut also can help the sponsor to keep a growing trend of theprofit when investors pay more attention to their referenceprice

Then the impact of parameters 120576 120601 on the values ofproduct quality and platform advertisement are illustratedin Figures 4 and 5

As shown in Figures 4 and 5 it is clear that the platformwill increase their advertisement to improve crowdfundingproject with the profit-sharing ratio increasing And thistrend is more obvious when the value of cost-sharing ratio is

8 Discrete Dynamics in Nature and Society

qe

05

101520253035404550

qe

01 02 03 04 05 06 07 08 09 10

Figure 4The changes of product quality (platform advertisement)

=05=02

0

10

20

30

40

50

60

e

01 02 03 04 05 06 07 08 09 10

Figure 5 The changes of platform advertisement in two situations

Table 2 Impacts of 120576 120601 on the optimal results

parameters 119869119904 119869119901 e q(003 010) 8716 9063 1 485(120576 120601) (003 020) 29515 9036 1125 485(010 020) 39555 7811 375 45(020 030) 80175 5831 86 40

higher It is also clear that the advertisement of platform hasa complementary effect with product quality In Table 2 theimpact of 120576 120601 on the optimal results is illustrated in detail

As shown in Table 2 the present value of the sponsorrsquosprofit is an increasing function of 120576 and 120601 and the trend ismore obvious if both the two ratios increase simultaneouslyAnd the advertisement of platform will also increase if thesponsor shares more cost or profit with them The aboveresults illustrate that sponsor sharing the cost of the platformcan not only promote the development of crowdfunding butalso improve its own profit

5 Simulation

To demonstrate some external validity for our theoreticalmodel an empirical study about the impact of product qualityand platform advertisement on the success of a crowdfundingproject is conducted Data for our study is derived frompublicly available information on the Kickstartercomwebsite(httpswwwkickstartercom) Kickstarter has grown intoone of the best-known and oldest crowdfunding platformsin the world since its inception in April 2009 We writea computer program by using the language of Python toget the information on all projects posted on the platformOver six months the information for 1400 projects fromKickstartercom has been collected The projects for charityare removed because charity crowdfunding projects aredonation-based crowdfunding And some data for inaccura-cies and incomplete information are also removed After thatthe data for analysis purposes included 835 crowdfundingprojects in the categories art design and technology

Data about a given project includes themoneywhich havebeen raised the number of comments and the introductionof the project The introduction of the product includes thenumber of introduction pictures the number of videos andthe number of description words In the present study thedescription and introduction of the project are defined asthe factors that relate to the quality of the product And thenumber of comments is seen as an external influence

The correlation analysis of the variables is first performedand results of which are shown in Table 3 It is can beseen from the table that correlations between the dependentvariable and all independent variables are positive Thereforethe assumptions that the information about the productquality and platform advertisement will cause some influ-ences on the success of crowdfunding were supported InTable 4 some results of a regression analysis are illustratedin detail We choose the signal of introduction picture countto represent the signal of product quality and the signal ofcomments count is seen as the signal of external influencesAs can be seen from the table the introduction picture countexplains 446 of the raised money and the comment countexplains 435 of the raised money The F value is 42462and the values of VIF (lt10) for each variable all indicatethat there is no multicollinearity in the model The resultsof the numerical experiment support the theory and modelanalyzed in the present study and provide some externalvalidity for our theoretical model

6 Conclusion

Product design allows firms to improve their profit andmarket competitiveness in crowdfunding market Howeverthe successful crowdfunding is always hampered by lack ofguidance on how to design the product quality to attractinvestorsrsquo preference To solve this dilemma we describetwo items investment desire and reference price to representinvestorrsquos perception that can cause some effects on thesuccess of crowdfunding

A necessary consideration that investorrsquos investmentdesire and reference price have a significant effect on investor

Discrete Dynamics in Nature and Society 3

The rest of the paper is organized as follows Literaturereviews are presented in Section 2 In Section 3 investorrsquosperception is introduced into the dynamic product qualitydesign model and the optimal results are given in two cir-cumstances respectively Based on the model the influencesof investorrsquos perception on the optimal results are analyzedand some profitable strategies for the sponsor are givenIn Section 4 some sensitive analyses for parameters aregiven and the simulation is described in Section 5 Section 6describes some concluding remarks of the present study

2 Literature Reviews

It is difficult for many innovative companies to get financialsupport by traditional financing methods since small inno-vative companies have no power to make public offeringsCrowdfunding provides a solution for the small innovativecompanies to raise the initial startup capital on the Internet[7 8] It is found that Internet financing was a good way tosolve the traditional financing difficulties of small companieseffectively [9]

The preliminary literatures of crowdfunding focus on thefollowing two aspects First some studies have discussedthe definition and the business model of the crowdfundingCrowdfunding was defined as a concept that invites individ-uals to invest in various section [10] The concept of crowd-funding originates from the idea of the crowdsourcing whichcan be seen as the wisdom of a crowd while the differencein the crowdfunding focuses on the financial perspective ofcrowdsourcing [11 12] For the businessmodel there are threeparticipators involved in crowdfunding Internet platformsfund raisers and investors Internet platforms provide theservices of matching fund raisers with investors and providesome investment style to fund raisers [13 14] As shownin the research of Lukkarinen [15] the driving force ofcrowdfunding can be summarized as two aspects One is thecharacteristics of the project such as the early fund raised bymedia networks and the value of minimum investment Theother includes the total target the projectrsquos duration and theprovision of financial information

Second some studies have investigated which factorsaffect the crowdfunding mostly The period of fund raisingand the amount of goal were identified as the directive factorsrelated to a project [3 16ndash19] Many researches revealedthat an appropriate fund raising goal and an appropriategeographical location of a crowdfunding project have causedsome valid influences on the success of crowdfunding [13 5 18 19] It was found that the amount of the target isnegative to the success of the project while the projects thatattract the interest of investors aremore likely to have a highersuccess rate of a crowdfunding [16 18] Some researchesclaimed frequent updates and a high level of communicationthrough updates and the information about the investors typealso caused great influence on the success of crowdfunding[20 21]

There are few researches focusing on the productionand operation problem of the firms in crowdfunding whenthe investorrsquos product valuation is changing over time Thepreliminary researches have focused on analyzing the effects

of the information on investorrsquos perception in financial mar-ket and accordingly some studies have explored the effectsof the investorrsquos perception on the investment decisions Itwas found that the low risk of fund and the liquidity offund scheme have a great impact on investorrsquos perceptionfor investing in the mutual fund [22] Joshi and Patel [23]found that the adverse investorrsquos perception has preventedthe information being reflected into the prices of stockInvestors may form their own valuations about the productaccording to their previous purchase experience and theinformation they mastered in the market How to producethe products with a reasonable product quality to attractinvestors to participate in a crowdfunding is our focus Hu[24] studied the optimal product pricing strategy in a reward-based crowdfunding and the product line was proved to bemore optimal than providing a single product Based on thestudy of Hu Chen [25] proposed the optimal pricing strategyfor the sponsor when facing strategic and myopic consumersin a crowdfunding of green product

Based on the above studies we give the dynamic modelto describe how to design the product quality when investorrsquosperception is taken into account in the next followingsections

3 The Basic Model

The decision system considered in the present study consistsof a sponsor and a platform The sponsor raises fundsthrough the crowdfunding on the platform and providessome rewards to the platform when the raised funds reachthe amount of target Different from the real sales investorsunderstand the product based on the online informationthey perceived from the platform In crowdfunding sponsordesigns the product quality to cater investorrsquos preferenceand platform makes decisions about the advertisement ofthe project Therefore investorrsquos perception is consideredas the function of the product quality and the platformadvertisement in the present study For understanding easilythe description of parameters is illustrated in Table 1

Since the information updates continuously investorrsquosperception changes over time In detail investment desireand reference price are introduced to describe investorrsquosperception First the changing of the investment desirefollows the Nerlove-Arrow framework [26]

119892 (119905) = 1205791119902 + 1205792119890 minus 120575119892119892 (0) = 1198920 (1)

where 119892(119905) is the accumulated investment desire over time 119905and 119892(0) gt 0 is the initial investment desire 1205791 and 1205792 arepositive constants that reflect both the product quality andplatform advertisement which have an positive effects on theinvestment desire and 120575 is the diminishing rate of investmentdesire

Then product quality and platform advertisement canalso affect investorrsquos reference price The reference priceis formed after investors get the information about theproduct attributes The changeable reference price in thepresent study is supposed to be affected by product quality

4 Discrete Dynamics in Nature and Society

Table 1 Parameters description

Symbol Explanation Variable type119902 The quality of product Decision variable119890 The advertisement of platform Decision variable119881119905 Investorrsquos product valuations (119881119905=H L) ndash119892 Investment desire ndash119903 Reference price ndash119889 The demand of product ndashΠ119904 The profit of the sponsor ndashΠ119901 The profit of the platform ndash119869119904 The present value of sponsorrsquos profit Decision goal119869119901 The present value of platformrsquos profit Decision goal120576 The ratio of the profit-sharing ndash120601 The ratio of the cost-sharing Decision variable119896 The probability of success ndash119898 1205791 1205792 120575 1205831 1205832 1205821 1205822 120572 120573 Parameters ndash

platform advertisement and product price The followingequation is introduced to illustrate the changeable processionof reference price

119903 (119905) = 120573 (119901 minus 119903) + 1205831119902 + 1205832119890119903 (0) = 1199030 (2)

where 119903(0) = 1199030 is the initial reference price and 120573 1205831 1205832 areall constants The item 120573(119901 minus 119903) means the price influence ahigher 120573 implies that the investors have short memory of theproduct The two items 1205831119902 and 1205832119890 represent the influencesof the product quality and the platform advertisement onthe reference price Higher product quality and platformadvertisement can enhance the consumerrsquos valuations on theproduct so 1205831 gt 0 and 1205832 gt 0 are supposed

Further investors are divided into short and long sighted-ness which means that investors have less or deep impressionon the product purchase experience in the past Based onthe research of119867119906 [24] combining the consumerrsquos type ieshort or long sightedness we give the following equation toillustrate the different investorrsquos product evaluations

119881119905 = 119867 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 120572 (1 minus 120573) 119871 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 120572120573 (3)

where 119867 gt 119871 gt 0 represents the product valuations of theinvestor

The price of product is determined by the sponsor atthe beginning of the crowdfunding and then the platformadvertises the product onlineThis paper focuses on choosingthe optimal product quality and platform advertisementwhen investorrsquos perception is taken into account The priceof product 119901 is assumed as a fixed value when the modelis solved and the different results caused by the changesof price will be discussed based on the research of 119867119906[24] 119867119906 provided several price strategies for the productcrowdfunding accordingly the optimal product quality andplatform advertisement are analyzed in the present studybased on his research

Generally the reference price the product quality andthe platform advertisement are assumed to have positiveeffects on the sale Thus the demand 119889(119905) of the product ina crowdfunding is supposed to satisfy the following equation

119889 (119905) = 119898 (119903 minus 119901) + 119892 + 1205821119902 + 1205822119890 (4)

where119898 1205821 and 1205822 are all positive constants The item119898(119903minus119901) represents the influence of reference price on demandThe influence on the demand is positive when 119903 gt 119901and the influence is negative otherwise A high 119898 impliesthat the investors are more sensitive to the gap between thereference price and the real priceTheother two items1205821119902 and1205822119890 indicate that both the product quality and the platformadvertisement have positive influences on the demand

Similar to the previous research of Joslashrgensen [27] thecost is in the form of quadratic function The product costof the sponsor is 119862119904 = (12)1199022 and the cost of the platformis 119862119901 = (12)1198902 Platforms can get some rewards whenthe raised funds reach the goal amount while there are norewards provided to the platform if the crowdfunding failsTherefore it is advisable for the sponsor to share a part of theplatformrsquos cost to encourage the platform tomake their effortsto improve the crowdfunding 120601 isin (0 1) is the cost-sharingratio that means the sponsor is willing to undertake theplatformrsquos advertisement cost then the profit of the sponsoris

Π119904 = 119896119901 (1 minus 120576) 119889 minus 121199022 minus 121206011198902 (5)

where 119896 isin (0 1) is the probability that the raised fundsreach the goal amount which is often called the successfulprobability of a crowdfunding 120576 isin (0 1) is called the ratioof profit sharing So the profit of the platform is

Π119901 = 119896119901120576119889 minus 12 (1 minus 120601) 1198902 (6)

where the item 119896120576119901119889 (120576 isin (0 1)) is the reward of the platformwhen the crowdfunding raised the goal amount of funds

Discrete Dynamics in Nature and Society 5

The goals of the sponsor and the platform are tomaximizethe present value of their profit respectively

max119889(119905)

119869119904= int+infin0

exp (minus120588119905) [119896119901 (1 minus 120576) 119889 (119905) minus 121199022 minus 121206011198902] 119889119905(7)

and

max119889(119905)

119869119901= int+infin0

exp (minus120588119905) [119896119901120576119889 (119905) minus 12 (1 minus 120601) 1198902] 119889119905 (8)

where 120588 is the discount rateIn the next sections the optimal product quality and

platformadvertisement are calculated in two different scenar-ios One scenario is that the sponsor and the platform makedecisions independently and the other is that the sponsor andthe platform are coordinate as a system to make decisions

31 The Optimal Decisions in Separation Separation meansthat the sponsor and the platform make decisions indepen-dently the sponsor offers a cost-sharing ratio 120601 firstly to theplatformThen the sponsor decides their product quality andthe platform decides their advertisement to maximize theirprofit It is reasonable to suppose that the sponsor and theplatform make decisions simultaneously since the demandchanges over time and the cost-sharing ratio 120601 is assumedas a fixed value when we calculate the values of decisionsabout the product quality and platform advertisement Theprofit of the two firms will be calculated after we obtain theoptimal equilibrium value of decisions and then the optimalcost-sharing ratio 120601 that maximizes the sponsorrsquos profit iscalculated

To maximize the profit of the sponsor and the platformthe function of Hamiltonian for the sponsor and the platformare given as the follows respectively

119867119904 = 119896119901 (1 minus 120576) [119898 (119903 minus 119901) + 119892 + 1205821119902 + 1205822119890] minus 121199022minus 121206011198902 + 1205741119904 (120573 (119901 minus 119903) + 1205831119902 + 1205832119890)+ 1205742119904 (1205791119902 + 1205792119890 minus 120575119892)

(9)

and

119867119901 = 119896119901120576 [119898 (119903 minus 119901) + 119892 + 1205821119902 + 1205822119890] minus 12 (1 minus 120601) 1198902+ 1205741119901 (120573 (119901 minus 119903) + 1205831119902 + 1205832119890)+ 1205742119901 (1205791119902 + 1205792119890 minus 120575119892)

(10)

where 1205741119904 1205742119904(1205741119901 1205742119901) represent the costate variablesCalculating the two Hamiltonian functions the equilib-

rium product quality and platform advertisement are givenin the following Theorem 1

Theorem 1 The equilibrium product quality is

119902 = 119896119901 (1 minus 120576) (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (11)

and the equilibrium platform advertisement is

119890 = 1198961199011205761 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (12)

Therefore the reference price and accumulated invest-ment desire can also be given as follows

119903 (119905) = 1198601119890minus120573119905 + 119903119899119892 (119905) = 1198602119890minus120575119905 + 119892119899

(13)

where1198601 = 1199030 minus 119903119899 119903119899 = 119901+ (1205831119902+1205832119890)1205731198602 = 1198920 minus119892119899 and119892119899 = (1205791119902 + 1205792119890)120575The proof of each theorem is illustrated in Appendixes

and some conclusions can be addressed from Theorem 1 asfollows

(i) There are three parts of equilibrium product qualitylevel in (11) the first part 1198961199011205821 represents the product qualityinfluences on the product demand which is the short-terminfluence on product sales The second part 1198961199011205791(120588 + 120575)expresses the long-term effect that means the product qualityhas the positive influence in accumulate investment desireThe third part 1198961199011198981205831(120588 + 120573) expresses the fact that theproduct quality has the effect on the reference price with1205831 = 0 The structure of (12) is the same as (11) so we donot repeat it here Since the product quality and platformadvertisement have positive effect on the reference price ie1205831 gt 0 and 1205832 gt 0 the sponsor will provide a higher qualityproduct and the platform will put more advertisement inadverting when they take the effect of reference price intoconsideration The higher 119898 means the consumers are moresensitive to the reference price therefore the decision makersshould pay more attention to this impact

(ii) The reference price and investment desire in (13) willget their steady states 119903119899 and 119892119899 when 119905 997888rarr +infin The steadystates ie 119903119899 = 119901 + (1205831119902 + 1205832119890)120573 and 119892119899 = (1205791119902 + 1205792119890)120575are mainly influenced by the product price 119901 the productquality 119902 and the platform advertisement 119890 It is shown thatthe reference price and the investment desire are increasingwhen the product quality and platform advertisement aretaken into account In addition investors will have a largereference price when the value of 120573 is small Analyzing 119892119899similarly the investment desire is positively correlated withthe product quality and the platform advertisement

Based on the above results the profits of the sponsor andplatform are calculated as follows

119869119904 = 119896119901 (1 minus 120576) ( 1198631119898120588 + 120573 + 1198632120588 + 120575 + 119898 (119903119899 minus 119901)120588 + 119892119899120588

+ 1205821119902120588 + 1205822119890120588 ) minus 11990222120588 minus 12060111989022120588 (14)

6 Discrete Dynamics in Nature and Society

and

119869119901 = 119896119901120576( 1198631119898120588 + 120573 + 1198632120588 + 120575 + 119898 (119903119899 minus 119901)120588 + 119892119899120588 + 1205821119902119901

+ 1205822119890120588 ) minus (1 minus 120601) 11989022120588

(15)

Since sponsors make the decisions about the productprice strategy at the beginning of the crowdfunding they willdecided the optimal value of product quality and platformadvertisement for any specific price strategy The optimalresults in margin (H) and volume (L) price strategy areanalyzed respectively based on the research of 119867119906 [24]Therefore there will be two groups ie (119902 119890119867) (119902 119890 119871)to choose different situations In the following Theorem 2the optimal product quality and platform advertisement aregiven in detail in different price strategy

Theorem 2 (i) When

1198612 = (1 minus 120576)2 1198601 + 11989812058321198602120576 (1 minus 120576) + 12058221205731198602120576 (1 minus 120576)120588120573 (1 minus 120601)

+ 12057911198601 (1 minus 120576)2 (1 minus 120601) + 12057911198602120576120588120575 (1 minus 120601)+ 212058211198601 (1 minus 120576)2 minus 11986021 (1 minus 120576)2

2120588 minus 119860221205762(1 minus 120601)2 gt 0(16)

we obtain the following two conclusions(a) 0 lt 119867119897 lt 11205722(1 minus 120573)2 the optimal values of product

quality and platform advertisement are

119902 = (1 minus 120576) 119871 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (17)

and

119890 = 1205761198711 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (18)

(b) 119867119897 gt 11205722(1 minus 120573)2 the optimal values of productquality and platform advertisement are

119902 = (1 minus 120576)1198671205722 (1 minus 120573)2 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (19)

and

119890 = 1205761198671205722 (1 minus 120573)21 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (20)

(ii) When

1198612 = (1 minus 120576)2 1198601 + 11989812058321198602120576 (1 minus 120576) + 12058221205731198602120576 (1 minus 120576)120588120573 (1 minus 120601)+ 12057911198601 (1 minus 120576)2 (1 minus 120601) + 12057911198602120576120588120575 (1 minus 120601)+ 212058211198601 (1 minus 120576)2 minus 11986021 (1 minus 120576)2

2120588 minus 119860221205762(1 minus 120601)2 lt 0(21)

the optimal values of product quality and platform advertise-ment are

119902 = (120576 minus 1) 119861121198612 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (22)

and

119890 = 120576 (120576 minus 1) 119861121198612 (1 minus 120601) (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (23)

Note that the equilibrium value of product quality andplatform advertisement will be higher if the sponsor takes themargin price strategy compared to the volume price strategyIt is also clear that both two equilibriums are the increasingfunction of the consumerrsquos evaluation (119871 or 119867) When theevaluation of consumer is higher the sponsor should improvetheir product quality level and similarly the platform shouldpay more advertisement to advertise the product And thereare no differences of the sponsor profit in the two pricestrategies when119867119897 = 11205722(1 minus 120573)232 The Optimal Decisions in Coordination In this sectionthe sponsor and the platform cooperate as a system to makedecisions about product quality and platform advertisementWe calculate the optimal decision of the system to maximizethe present value of it Taking (5) and (6) in to account theHamiltonian function for the system is given as follows

119867 = 119901119896 [119898 (119903 minus 119901) + 119892 + 1205821119902 + 1205822119890] minus 121199022 minus 121198902+ 1205741 (120573 (119901 minus 119903) + 1205831119902 + 1205832119890)+ 1205742 (1205791119902 + 1205792119890 minus 120575119892)

(24)

where 1205741 1205742 represent the costate variables of the systemproblemassociating the reference price and investment desirewith the decisions Similar to the process of Theorem 1 theequilibrium of the system is shown as follows

119902 = 119896119901(1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (25)

and

119890 = 119896119901(1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (26)

It is clearly shown that the product quality is improved due tothe cooperation InTheorem 3 the optimal cost-sharing ratiois given and the advertisement of platform in two situationsis compared

Theorem 3 The optimal cost-sharing ratio 120601 is

120601 = 2 minus 31205762 minus 120576 119894119891 120576 le 23 0 119890119897119904119890 (27)

Taking (27) into (20) we get

119890 = 119896119901120576 (2 minus 120576)2120576 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (28)

Discrete Dynamics in Nature and Society 7

and

119890 minus 119890 = 1198961199011205762 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) gt 0 (29)

For the cooperation system the reference price andinvestment desire are also steady similar to the situation ofseparation as follows

119903 (119905) = 1198641119890minus120573119905 + 119903119888119892 (119905) = 1198642119890minus120575119905 + 119892119888

(30)

where 1198641 = 1199030 minus 119903119888 119903119888 = 119901 + (1205831 119902 + 1205832 119890)120573 1198642 = 1198920 minus 119892119888 and119892119888 = (1205791 119902+1205792 119890)120575The steady states of the reference price andinvestment desire are 119903119888 and 119892119888

From the above results we can find that both the steadyreference price and steady investment desire are highercompared to Theorem 2 since both the product quality andplatform advertisement are improved in cooperation systemSubstituting (30) into (7) we get the present value of the profitfor cooperation system as follows

119869 = 119896119901( 1198641119898120588 + 120573 + 1198642120588 + 120575 + 119898 (119903119888 minus 119901)120588 + 119892119888120588 + 1205821 119902119901

+ 1205822 119890120588 ) minus 11990222120588 minus 11989022120588 (31)

In the next section we will discuss the changes of theresults in ourmodel when relevant parameters are in differentvalues Analyzing these changes does not only build therobustness of our results but also deepen our understandingof this problem

4 Parameters Sensitivity Analysis

In this section we analyze the relations between the relevantparameters and the values of profit product quality andplatform advertisement Firstly the impact of parameter 119898on the present value of profit is illustrated in Figures 2 and 3The parameter119898 is seen as an item that reflects the degree ofthe reference price effects on the investorrsquos behaviorThere aretwo firms the sponsor and the platform in the research so itis worthwhile to analyze the influence of parameter119898 on theprofit of them because the effects of119898may be different

As shown in Figure 2 119869119904 119869119901 and 119869 represent the profitof the sponsor the profit of the platform and the totalprofit when the sponsor and the platform chose to cooperaterespectively The present value of the sponsorrsquos profit willdecrease if the investors pay more attention to referenceprice while the platformrsquos profit is an increasing function ofparameter 119898 when they make decisions independently It isclear that the profits of the two firms are increased whenthey cooperate as a system and the cooperation profit is anincreasing function of the parameter 119898 Note that parameter119898 denotes the degree of the reference price effects on theinvestorrsquos behavior the fact implied in Figure 2 shows that thelarger the effect of the reference price the more profit the twoitems can get when they cooperate as a system

ＭＪJ

05

101520253035404550

The p

rofit

01 02 03 04 05 06 07 08 09 10m

Figure 2 Profit changes

Δ ＭΔ Ｊ

01 02 03 04 05 06 07 08 09 10m

05

101520253035404550

The c

hang

e of p

rofit

Figure 3 Differences of profit in two situations

From Figure 3 we can find that cooperation is more prof-itable for the sponsor than the platform and the differences ofthe two profits are both the increasing function of parameter119898 Therefore the sponsor should choose to cooperate withthe platform when the investors pay more attention to theirreference price And the differences of platformrsquos profit are notobvious in twoways whichmeans that it has a little differencefor platform in two cases

Comparing Figure 2 with Figure 3 it is found that coop-eration with platform not only promotes the profit of sponsorbut also can help the sponsor to keep a growing trend of theprofit when investors pay more attention to their referenceprice

Then the impact of parameters 120576 120601 on the values ofproduct quality and platform advertisement are illustratedin Figures 4 and 5

As shown in Figures 4 and 5 it is clear that the platformwill increase their advertisement to improve crowdfundingproject with the profit-sharing ratio increasing And thistrend is more obvious when the value of cost-sharing ratio is

8 Discrete Dynamics in Nature and Society

qe

05

101520253035404550

qe

01 02 03 04 05 06 07 08 09 10

Figure 4The changes of product quality (platform advertisement)

=05=02

0

10

20

30

40

50

60

e

01 02 03 04 05 06 07 08 09 10

Figure 5 The changes of platform advertisement in two situations

Table 2 Impacts of 120576 120601 on the optimal results

parameters 119869119904 119869119901 e q(003 010) 8716 9063 1 485(120576 120601) (003 020) 29515 9036 1125 485(010 020) 39555 7811 375 45(020 030) 80175 5831 86 40

higher It is also clear that the advertisement of platform hasa complementary effect with product quality In Table 2 theimpact of 120576 120601 on the optimal results is illustrated in detail

As shown in Table 2 the present value of the sponsorrsquosprofit is an increasing function of 120576 and 120601 and the trend ismore obvious if both the two ratios increase simultaneouslyAnd the advertisement of platform will also increase if thesponsor shares more cost or profit with them The aboveresults illustrate that sponsor sharing the cost of the platformcan not only promote the development of crowdfunding butalso improve its own profit

5 Simulation

To demonstrate some external validity for our theoreticalmodel an empirical study about the impact of product qualityand platform advertisement on the success of a crowdfundingproject is conducted Data for our study is derived frompublicly available information on the Kickstartercomwebsite(httpswwwkickstartercom) Kickstarter has grown intoone of the best-known and oldest crowdfunding platformsin the world since its inception in April 2009 We writea computer program by using the language of Python toget the information on all projects posted on the platformOver six months the information for 1400 projects fromKickstartercom has been collected The projects for charityare removed because charity crowdfunding projects aredonation-based crowdfunding And some data for inaccura-cies and incomplete information are also removed After thatthe data for analysis purposes included 835 crowdfundingprojects in the categories art design and technology

Data about a given project includes themoneywhich havebeen raised the number of comments and the introductionof the project The introduction of the product includes thenumber of introduction pictures the number of videos andthe number of description words In the present study thedescription and introduction of the project are defined asthe factors that relate to the quality of the product And thenumber of comments is seen as an external influence

The correlation analysis of the variables is first performedand results of which are shown in Table 3 It is can beseen from the table that correlations between the dependentvariable and all independent variables are positive Thereforethe assumptions that the information about the productquality and platform advertisement will cause some influ-ences on the success of crowdfunding were supported InTable 4 some results of a regression analysis are illustratedin detail We choose the signal of introduction picture countto represent the signal of product quality and the signal ofcomments count is seen as the signal of external influencesAs can be seen from the table the introduction picture countexplains 446 of the raised money and the comment countexplains 435 of the raised money The F value is 42462and the values of VIF (lt10) for each variable all indicatethat there is no multicollinearity in the model The resultsof the numerical experiment support the theory and modelanalyzed in the present study and provide some externalvalidity for our theoretical model

6 Conclusion

Product design allows firms to improve their profit andmarket competitiveness in crowdfunding market Howeverthe successful crowdfunding is always hampered by lack ofguidance on how to design the product quality to attractinvestorsrsquo preference To solve this dilemma we describetwo items investment desire and reference price to representinvestorrsquos perception that can cause some effects on thesuccess of crowdfunding

A necessary consideration that investorrsquos investmentdesire and reference price have a significant effect on investor

4 Discrete Dynamics in Nature and Society

Table 1 Parameters description

Symbol Explanation Variable type119902 The quality of product Decision variable119890 The advertisement of platform Decision variable119881119905 Investorrsquos product valuations (119881119905=H L) ndash119892 Investment desire ndash119903 Reference price ndash119889 The demand of product ndashΠ119904 The profit of the sponsor ndashΠ119901 The profit of the platform ndash119869119904 The present value of sponsorrsquos profit Decision goal119869119901 The present value of platformrsquos profit Decision goal120576 The ratio of the profit-sharing ndash120601 The ratio of the cost-sharing Decision variable119896 The probability of success ndash119898 1205791 1205792 120575 1205831 1205832 1205821 1205822 120572 120573 Parameters ndash

platform advertisement and product price The followingequation is introduced to illustrate the changeable processionof reference price

119903 (119905) = 120573 (119901 minus 119903) + 1205831119902 + 1205832119890119903 (0) = 1199030 (2)

where 119903(0) = 1199030 is the initial reference price and 120573 1205831 1205832 areall constants The item 120573(119901 minus 119903) means the price influence ahigher 120573 implies that the investors have short memory of theproduct The two items 1205831119902 and 1205832119890 represent the influencesof the product quality and the platform advertisement onthe reference price Higher product quality and platformadvertisement can enhance the consumerrsquos valuations on theproduct so 1205831 gt 0 and 1205832 gt 0 are supposed

Further investors are divided into short and long sighted-ness which means that investors have less or deep impressionon the product purchase experience in the past Based onthe research of119867119906 [24] combining the consumerrsquos type ieshort or long sightedness we give the following equation toillustrate the different investorrsquos product evaluations

119881119905 = 119867 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 120572 (1 minus 120573) 119871 119908119894119905ℎ 119901119903119900119887119886119887119894119897119894119905119910 120572120573 (3)

where 119867 gt 119871 gt 0 represents the product valuations of theinvestor

The price of product is determined by the sponsor atthe beginning of the crowdfunding and then the platformadvertises the product onlineThis paper focuses on choosingthe optimal product quality and platform advertisementwhen investorrsquos perception is taken into account The priceof product 119901 is assumed as a fixed value when the modelis solved and the different results caused by the changesof price will be discussed based on the research of 119867119906[24] 119867119906 provided several price strategies for the productcrowdfunding accordingly the optimal product quality andplatform advertisement are analyzed in the present studybased on his research

Generally the reference price the product quality andthe platform advertisement are assumed to have positiveeffects on the sale Thus the demand 119889(119905) of the product ina crowdfunding is supposed to satisfy the following equation

119889 (119905) = 119898 (119903 minus 119901) + 119892 + 1205821119902 + 1205822119890 (4)

where119898 1205821 and 1205822 are all positive constants The item119898(119903minus119901) represents the influence of reference price on demandThe influence on the demand is positive when 119903 gt 119901and the influence is negative otherwise A high 119898 impliesthat the investors are more sensitive to the gap between thereference price and the real priceTheother two items1205821119902 and1205822119890 indicate that both the product quality and the platformadvertisement have positive influences on the demand

Similar to the previous research of Joslashrgensen [27] thecost is in the form of quadratic function The product costof the sponsor is 119862119904 = (12)1199022 and the cost of the platformis 119862119901 = (12)1198902 Platforms can get some rewards whenthe raised funds reach the goal amount while there are norewards provided to the platform if the crowdfunding failsTherefore it is advisable for the sponsor to share a part of theplatformrsquos cost to encourage the platform tomake their effortsto improve the crowdfunding 120601 isin (0 1) is the cost-sharingratio that means the sponsor is willing to undertake theplatformrsquos advertisement cost then the profit of the sponsoris

Π119904 = 119896119901 (1 minus 120576) 119889 minus 121199022 minus 121206011198902 (5)

where 119896 isin (0 1) is the probability that the raised fundsreach the goal amount which is often called the successfulprobability of a crowdfunding 120576 isin (0 1) is called the ratioof profit sharing So the profit of the platform is

Π119901 = 119896119901120576119889 minus 12 (1 minus 120601) 1198902 (6)

where the item 119896120576119901119889 (120576 isin (0 1)) is the reward of the platformwhen the crowdfunding raised the goal amount of funds

Discrete Dynamics in Nature and Society 5

The goals of the sponsor and the platform are tomaximizethe present value of their profit respectively

max119889(119905)

119869119904= int+infin0

exp (minus120588119905) [119896119901 (1 minus 120576) 119889 (119905) minus 121199022 minus 121206011198902] 119889119905(7)

and

max119889(119905)

119869119901= int+infin0

exp (minus120588119905) [119896119901120576119889 (119905) minus 12 (1 minus 120601) 1198902] 119889119905 (8)

where 120588 is the discount rateIn the next sections the optimal product quality and

platformadvertisement are calculated in two different scenar-ios One scenario is that the sponsor and the platform makedecisions independently and the other is that the sponsor andthe platform are coordinate as a system to make decisions

31 The Optimal Decisions in Separation Separation meansthat the sponsor and the platform make decisions indepen-dently the sponsor offers a cost-sharing ratio 120601 firstly to theplatformThen the sponsor decides their product quality andthe platform decides their advertisement to maximize theirprofit It is reasonable to suppose that the sponsor and theplatform make decisions simultaneously since the demandchanges over time and the cost-sharing ratio 120601 is assumedas a fixed value when we calculate the values of decisionsabout the product quality and platform advertisement Theprofit of the two firms will be calculated after we obtain theoptimal equilibrium value of decisions and then the optimalcost-sharing ratio 120601 that maximizes the sponsorrsquos profit iscalculated

To maximize the profit of the sponsor and the platformthe function of Hamiltonian for the sponsor and the platformare given as the follows respectively

119867119904 = 119896119901 (1 minus 120576) [119898 (119903 minus 119901) + 119892 + 1205821119902 + 1205822119890] minus 121199022minus 121206011198902 + 1205741119904 (120573 (119901 minus 119903) + 1205831119902 + 1205832119890)+ 1205742119904 (1205791119902 + 1205792119890 minus 120575119892)

(9)

and

119867119901 = 119896119901120576 [119898 (119903 minus 119901) + 119892 + 1205821119902 + 1205822119890] minus 12 (1 minus 120601) 1198902+ 1205741119901 (120573 (119901 minus 119903) + 1205831119902 + 1205832119890)+ 1205742119901 (1205791119902 + 1205792119890 minus 120575119892)

(10)

where 1205741119904 1205742119904(1205741119901 1205742119901) represent the costate variablesCalculating the two Hamiltonian functions the equilib-

rium product quality and platform advertisement are givenin the following Theorem 1

Theorem 1 The equilibrium product quality is

119902 = 119896119901 (1 minus 120576) (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (11)

and the equilibrium platform advertisement is

119890 = 1198961199011205761 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (12)

Therefore the reference price and accumulated invest-ment desire can also be given as follows

119903 (119905) = 1198601119890minus120573119905 + 119903119899119892 (119905) = 1198602119890minus120575119905 + 119892119899

(13)

where1198601 = 1199030 minus 119903119899 119903119899 = 119901+ (1205831119902+1205832119890)1205731198602 = 1198920 minus119892119899 and119892119899 = (1205791119902 + 1205792119890)120575The proof of each theorem is illustrated in Appendixes

and some conclusions can be addressed from Theorem 1 asfollows

(i) There are three parts of equilibrium product qualitylevel in (11) the first part 1198961199011205821 represents the product qualityinfluences on the product demand which is the short-terminfluence on product sales The second part 1198961199011205791(120588 + 120575)expresses the long-term effect that means the product qualityhas the positive influence in accumulate investment desireThe third part 1198961199011198981205831(120588 + 120573) expresses the fact that theproduct quality has the effect on the reference price with1205831 = 0 The structure of (12) is the same as (11) so we donot repeat it here Since the product quality and platformadvertisement have positive effect on the reference price ie1205831 gt 0 and 1205832 gt 0 the sponsor will provide a higher qualityproduct and the platform will put more advertisement inadverting when they take the effect of reference price intoconsideration The higher 119898 means the consumers are moresensitive to the reference price therefore the decision makersshould pay more attention to this impact

(ii) The reference price and investment desire in (13) willget their steady states 119903119899 and 119892119899 when 119905 997888rarr +infin The steadystates ie 119903119899 = 119901 + (1205831119902 + 1205832119890)120573 and 119892119899 = (1205791119902 + 1205792119890)120575are mainly influenced by the product price 119901 the productquality 119902 and the platform advertisement 119890 It is shown thatthe reference price and the investment desire are increasingwhen the product quality and platform advertisement aretaken into account In addition investors will have a largereference price when the value of 120573 is small Analyzing 119892119899similarly the investment desire is positively correlated withthe product quality and the platform advertisement

Based on the above results the profits of the sponsor andplatform are calculated as follows

119869119904 = 119896119901 (1 minus 120576) ( 1198631119898120588 + 120573 + 1198632120588 + 120575 + 119898 (119903119899 minus 119901)120588 + 119892119899120588

+ 1205821119902120588 + 1205822119890120588 ) minus 11990222120588 minus 12060111989022120588 (14)

6 Discrete Dynamics in Nature and Society

and

119869119901 = 119896119901120576( 1198631119898120588 + 120573 + 1198632120588 + 120575 + 119898 (119903119899 minus 119901)120588 + 119892119899120588 + 1205821119902119901

+ 1205822119890120588 ) minus (1 minus 120601) 11989022120588

(15)

Since sponsors make the decisions about the productprice strategy at the beginning of the crowdfunding they willdecided the optimal value of product quality and platformadvertisement for any specific price strategy The optimalresults in margin (H) and volume (L) price strategy areanalyzed respectively based on the research of 119867119906 [24]Therefore there will be two groups ie (119902 119890119867) (119902 119890 119871)to choose different situations In the following Theorem 2the optimal product quality and platform advertisement aregiven in detail in different price strategy

Theorem 2 (i) When

1198612 = (1 minus 120576)2 1198601 + 11989812058321198602120576 (1 minus 120576) + 12058221205731198602120576 (1 minus 120576)120588120573 (1 minus 120601)

+ 12057911198601 (1 minus 120576)2 (1 minus 120601) + 12057911198602120576120588120575 (1 minus 120601)+ 212058211198601 (1 minus 120576)2 minus 11986021 (1 minus 120576)2

2120588 minus 119860221205762(1 minus 120601)2 gt 0(16)

we obtain the following two conclusions(a) 0 lt 119867119897 lt 11205722(1 minus 120573)2 the optimal values of product

quality and platform advertisement are

119902 = (1 minus 120576) 119871 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (17)

and

119890 = 1205761198711 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (18)

(b) 119867119897 gt 11205722(1 minus 120573)2 the optimal values of productquality and platform advertisement are

119902 = (1 minus 120576)1198671205722 (1 minus 120573)2 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (19)

and

119890 = 1205761198671205722 (1 minus 120573)21 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (20)

(ii) When

1198612 = (1 minus 120576)2 1198601 + 11989812058321198602120576 (1 minus 120576) + 12058221205731198602120576 (1 minus 120576)120588120573 (1 minus 120601)+ 12057911198601 (1 minus 120576)2 (1 minus 120601) + 12057911198602120576120588120575 (1 minus 120601)+ 212058211198601 (1 minus 120576)2 minus 11986021 (1 minus 120576)2

2120588 minus 119860221205762(1 minus 120601)2 lt 0(21)

the optimal values of product quality and platform advertise-ment are

119902 = (120576 minus 1) 119861121198612 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (22)

and

119890 = 120576 (120576 minus 1) 119861121198612 (1 minus 120601) (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (23)

Note that the equilibrium value of product quality andplatform advertisement will be higher if the sponsor takes themargin price strategy compared to the volume price strategyIt is also clear that both two equilibriums are the increasingfunction of the consumerrsquos evaluation (119871 or 119867) When theevaluation of consumer is higher the sponsor should improvetheir product quality level and similarly the platform shouldpay more advertisement to advertise the product And thereare no differences of the sponsor profit in the two pricestrategies when119867119897 = 11205722(1 minus 120573)232 The Optimal Decisions in Coordination In this sectionthe sponsor and the platform cooperate as a system to makedecisions about product quality and platform advertisementWe calculate the optimal decision of the system to maximizethe present value of it Taking (5) and (6) in to account theHamiltonian function for the system is given as follows

119867 = 119901119896 [119898 (119903 minus 119901) + 119892 + 1205821119902 + 1205822119890] minus 121199022 minus 121198902+ 1205741 (120573 (119901 minus 119903) + 1205831119902 + 1205832119890)+ 1205742 (1205791119902 + 1205792119890 minus 120575119892)

(24)

where 1205741 1205742 represent the costate variables of the systemproblemassociating the reference price and investment desirewith the decisions Similar to the process of Theorem 1 theequilibrium of the system is shown as follows

119902 = 119896119901(1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (25)

and

119890 = 119896119901(1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (26)

It is clearly shown that the product quality is improved due tothe cooperation InTheorem 3 the optimal cost-sharing ratiois given and the advertisement of platform in two situationsis compared

Theorem 3 The optimal cost-sharing ratio 120601 is

120601 = 2 minus 31205762 minus 120576 119894119891 120576 le 23 0 119890119897119904119890 (27)

Taking (27) into (20) we get

119890 = 119896119901120576 (2 minus 120576)2120576 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (28)

Discrete Dynamics in Nature and Society 7

and

119890 minus 119890 = 1198961199011205762 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) gt 0 (29)

For the cooperation system the reference price andinvestment desire are also steady similar to the situation ofseparation as follows

119903 (119905) = 1198641119890minus120573119905 + 119903119888119892 (119905) = 1198642119890minus120575119905 + 119892119888

(30)

where 1198641 = 1199030 minus 119903119888 119903119888 = 119901 + (1205831 119902 + 1205832 119890)120573 1198642 = 1198920 minus 119892119888 and119892119888 = (1205791 119902+1205792 119890)120575The steady states of the reference price andinvestment desire are 119903119888 and 119892119888

From the above results we can find that both the steadyreference price and steady investment desire are highercompared to Theorem 2 since both the product quality andplatform advertisement are improved in cooperation systemSubstituting (30) into (7) we get the present value of the profitfor cooperation system as follows

119869 = 119896119901( 1198641119898120588 + 120573 + 1198642120588 + 120575 + 119898 (119903119888 minus 119901)120588 + 119892119888120588 + 1205821 119902119901

+ 1205822 119890120588 ) minus 11990222120588 minus 11989022120588 (31)

In the next section we will discuss the changes of theresults in ourmodel when relevant parameters are in differentvalues Analyzing these changes does not only build therobustness of our results but also deepen our understandingof this problem

4 Parameters Sensitivity Analysis

In this section we analyze the relations between the relevantparameters and the values of profit product quality andplatform advertisement Firstly the impact of parameter 119898on the present value of profit is illustrated in Figures 2 and 3The parameter119898 is seen as an item that reflects the degree ofthe reference price effects on the investorrsquos behaviorThere aretwo firms the sponsor and the platform in the research so itis worthwhile to analyze the influence of parameter119898 on theprofit of them because the effects of119898may be different

As shown in Figure 2 119869119904 119869119901 and 119869 represent the profitof the sponsor the profit of the platform and the totalprofit when the sponsor and the platform chose to cooperaterespectively The present value of the sponsorrsquos profit willdecrease if the investors pay more attention to referenceprice while the platformrsquos profit is an increasing function ofparameter 119898 when they make decisions independently It isclear that the profits of the two firms are increased whenthey cooperate as a system and the cooperation profit is anincreasing function of the parameter 119898 Note that parameter119898 denotes the degree of the reference price effects on theinvestorrsquos behavior the fact implied in Figure 2 shows that thelarger the effect of the reference price the more profit the twoitems can get when they cooperate as a system

ＭＪJ

05

101520253035404550

The p

rofit

01 02 03 04 05 06 07 08 09 10m

Figure 2 Profit changes

Δ ＭΔ Ｊ

01 02 03 04 05 06 07 08 09 10m

05

101520253035404550

The c

hang

e of p

rofit

Figure 3 Differences of profit in two situations

From Figure 3 we can find that cooperation is more prof-itable for the sponsor than the platform and the differences ofthe two profits are both the increasing function of parameter119898 Therefore the sponsor should choose to cooperate withthe platform when the investors pay more attention to theirreference price And the differences of platformrsquos profit are notobvious in twoways whichmeans that it has a little differencefor platform in two cases

Comparing Figure 2 with Figure 3 it is found that coop-eration with platform not only promotes the profit of sponsorbut also can help the sponsor to keep a growing trend of theprofit when investors pay more attention to their referenceprice

Then the impact of parameters 120576 120601 on the values ofproduct quality and platform advertisement are illustratedin Figures 4 and 5

As shown in Figures 4 and 5 it is clear that the platformwill increase their advertisement to improve crowdfundingproject with the profit-sharing ratio increasing And thistrend is more obvious when the value of cost-sharing ratio is

8 Discrete Dynamics in Nature and Society

qe

05

101520253035404550

qe

01 02 03 04 05 06 07 08 09 10

Figure 4The changes of product quality (platform advertisement)

=05=02

0

10

20

30

40

50

60

e

01 02 03 04 05 06 07 08 09 10

Figure 5 The changes of platform advertisement in two situations

Table 2 Impacts of 120576 120601 on the optimal results

parameters 119869119904 119869119901 e q(003 010) 8716 9063 1 485(120576 120601) (003 020) 29515 9036 1125 485(010 020) 39555 7811 375 45(020 030) 80175 5831 86 40

higher It is also clear that the advertisement of platform hasa complementary effect with product quality In Table 2 theimpact of 120576 120601 on the optimal results is illustrated in detail

As shown in Table 2 the present value of the sponsorrsquosprofit is an increasing function of 120576 and 120601 and the trend ismore obvious if both the two ratios increase simultaneouslyAnd the advertisement of platform will also increase if thesponsor shares more cost or profit with them The aboveresults illustrate that sponsor sharing the cost of the platformcan not only promote the development of crowdfunding butalso improve its own profit

5 Simulation

To demonstrate some external validity for our theoreticalmodel an empirical study about the impact of product qualityand platform advertisement on the success of a crowdfundingproject is conducted Data for our study is derived frompublicly available information on the Kickstartercomwebsite(httpswwwkickstartercom) Kickstarter has grown intoone of the best-known and oldest crowdfunding platformsin the world since its inception in April 2009 We writea computer program by using the language of Python toget the information on all projects posted on the platformOver six months the information for 1400 projects fromKickstartercom has been collected The projects for charityare removed because charity crowdfunding projects aredonation-based crowdfunding And some data for inaccura-cies and incomplete information are also removed After thatthe data for analysis purposes included 835 crowdfundingprojects in the categories art design and technology

Data about a given project includes themoneywhich havebeen raised the number of comments and the introductionof the project The introduction of the product includes thenumber of introduction pictures the number of videos andthe number of description words In the present study thedescription and introduction of the project are defined asthe factors that relate to the quality of the product And thenumber of comments is seen as an external influence

The correlation analysis of the variables is first performedand results of which are shown in Table 3 It is can beseen from the table that correlations between the dependentvariable and all independent variables are positive Thereforethe assumptions that the information about the productquality and platform advertisement will cause some influ-ences on the success of crowdfunding were supported InTable 4 some results of a regression analysis are illustratedin detail We choose the signal of introduction picture countto represent the signal of product quality and the signal ofcomments count is seen as the signal of external influencesAs can be seen from the table the introduction picture countexplains 446 of the raised money and the comment countexplains 435 of the raised money The F value is 42462and the values of VIF (lt10) for each variable all indicatethat there is no multicollinearity in the model The resultsof the numerical experiment support the theory and modelanalyzed in the present study and provide some externalvalidity for our theoretical model

6 Conclusion

Product design allows firms to improve their profit andmarket competitiveness in crowdfunding market Howeverthe successful crowdfunding is always hampered by lack ofguidance on how to design the product quality to attractinvestorsrsquo preference To solve this dilemma we describetwo items investment desire and reference price to representinvestorrsquos perception that can cause some effects on thesuccess of crowdfunding

A necessary consideration that investorrsquos investmentdesire and reference price have a significant effect on investor

Discrete Dynamics in Nature and Society 5

The goals of the sponsor and the platform are tomaximizethe present value of their profit respectively

max119889(119905)

119869119904= int+infin0

exp (minus120588119905) [119896119901 (1 minus 120576) 119889 (119905) minus 121199022 minus 121206011198902] 119889119905(7)

and

max119889(119905)

119869119901= int+infin0

exp (minus120588119905) [119896119901120576119889 (119905) minus 12 (1 minus 120601) 1198902] 119889119905 (8)

where 120588 is the discount rateIn the next sections the optimal product quality and

platformadvertisement are calculated in two different scenar-ios One scenario is that the sponsor and the platform makedecisions independently and the other is that the sponsor andthe platform are coordinate as a system to make decisions

31 The Optimal Decisions in Separation Separation meansthat the sponsor and the platform make decisions indepen-dently the sponsor offers a cost-sharing ratio 120601 firstly to theplatformThen the sponsor decides their product quality andthe platform decides their advertisement to maximize theirprofit It is reasonable to suppose that the sponsor and theplatform make decisions simultaneously since the demandchanges over time and the cost-sharing ratio 120601 is assumedas a fixed value when we calculate the values of decisionsabout the product quality and platform advertisement Theprofit of the two firms will be calculated after we obtain theoptimal equilibrium value of decisions and then the optimalcost-sharing ratio 120601 that maximizes the sponsorrsquos profit iscalculated

To maximize the profit of the sponsor and the platformthe function of Hamiltonian for the sponsor and the platformare given as the follows respectively

119867119904 = 119896119901 (1 minus 120576) [119898 (119903 minus 119901) + 119892 + 1205821119902 + 1205822119890] minus 121199022minus 121206011198902 + 1205741119904 (120573 (119901 minus 119903) + 1205831119902 + 1205832119890)+ 1205742119904 (1205791119902 + 1205792119890 minus 120575119892)

(9)

and

119867119901 = 119896119901120576 [119898 (119903 minus 119901) + 119892 + 1205821119902 + 1205822119890] minus 12 (1 minus 120601) 1198902+ 1205741119901 (120573 (119901 minus 119903) + 1205831119902 + 1205832119890)+ 1205742119901 (1205791119902 + 1205792119890 minus 120575119892)

(10)

where 1205741119904 1205742119904(1205741119901 1205742119901) represent the costate variablesCalculating the two Hamiltonian functions the equilib-

rium product quality and platform advertisement are givenin the following Theorem 1

Theorem 1 The equilibrium product quality is

119902 = 119896119901 (1 minus 120576) (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (11)

and the equilibrium platform advertisement is

119890 = 1198961199011205761 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (12)

Therefore the reference price and accumulated invest-ment desire can also be given as follows

119903 (119905) = 1198601119890minus120573119905 + 119903119899119892 (119905) = 1198602119890minus120575119905 + 119892119899

(13)

where1198601 = 1199030 minus 119903119899 119903119899 = 119901+ (1205831119902+1205832119890)1205731198602 = 1198920 minus119892119899 and119892119899 = (1205791119902 + 1205792119890)120575The proof of each theorem is illustrated in Appendixes

and some conclusions can be addressed from Theorem 1 asfollows

(i) There are three parts of equilibrium product qualitylevel in (11) the first part 1198961199011205821 represents the product qualityinfluences on the product demand which is the short-terminfluence on product sales The second part 1198961199011205791(120588 + 120575)expresses the long-term effect that means the product qualityhas the positive influence in accumulate investment desireThe third part 1198961199011198981205831(120588 + 120573) expresses the fact that theproduct quality has the effect on the reference price with1205831 = 0 The structure of (12) is the same as (11) so we donot repeat it here Since the product quality and platformadvertisement have positive effect on the reference price ie1205831 gt 0 and 1205832 gt 0 the sponsor will provide a higher qualityproduct and the platform will put more advertisement inadverting when they take the effect of reference price intoconsideration The higher 119898 means the consumers are moresensitive to the reference price therefore the decision makersshould pay more attention to this impact

(ii) The reference price and investment desire in (13) willget their steady states 119903119899 and 119892119899 when 119905 997888rarr +infin The steadystates ie 119903119899 = 119901 + (1205831119902 + 1205832119890)120573 and 119892119899 = (1205791119902 + 1205792119890)120575are mainly influenced by the product price 119901 the productquality 119902 and the platform advertisement 119890 It is shown thatthe reference price and the investment desire are increasingwhen the product quality and platform advertisement aretaken into account In addition investors will have a largereference price when the value of 120573 is small Analyzing 119892119899similarly the investment desire is positively correlated withthe product quality and the platform advertisement

Based on the above results the profits of the sponsor andplatform are calculated as follows

119869119904 = 119896119901 (1 minus 120576) ( 1198631119898120588 + 120573 + 1198632120588 + 120575 + 119898 (119903119899 minus 119901)120588 + 119892119899120588

+ 1205821119902120588 + 1205822119890120588 ) minus 11990222120588 minus 12060111989022120588 (14)

6 Discrete Dynamics in Nature and Society

and

119869119901 = 119896119901120576( 1198631119898120588 + 120573 + 1198632120588 + 120575 + 119898 (119903119899 minus 119901)120588 + 119892119899120588 + 1205821119902119901

+ 1205822119890120588 ) minus (1 minus 120601) 11989022120588

(15)

Since sponsors make the decisions about the productprice strategy at the beginning of the crowdfunding they willdecided the optimal value of product quality and platformadvertisement for any specific price strategy The optimalresults in margin (H) and volume (L) price strategy areanalyzed respectively based on the research of 119867119906 [24]Therefore there will be two groups ie (119902 119890119867) (119902 119890 119871)to choose different situations In the following Theorem 2the optimal product quality and platform advertisement aregiven in detail in different price strategy

Theorem 2 (i) When

1198612 = (1 minus 120576)2 1198601 + 11989812058321198602120576 (1 minus 120576) + 12058221205731198602120576 (1 minus 120576)120588120573 (1 minus 120601)

+ 12057911198601 (1 minus 120576)2 (1 minus 120601) + 12057911198602120576120588120575 (1 minus 120601)+ 212058211198601 (1 minus 120576)2 minus 11986021 (1 minus 120576)2

2120588 minus 119860221205762(1 minus 120601)2 gt 0(16)

we obtain the following two conclusions(a) 0 lt 119867119897 lt 11205722(1 minus 120573)2 the optimal values of product

quality and platform advertisement are

119902 = (1 minus 120576) 119871 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (17)

and

119890 = 1205761198711 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (18)

(b) 119867119897 gt 11205722(1 minus 120573)2 the optimal values of productquality and platform advertisement are

119902 = (1 minus 120576)1198671205722 (1 minus 120573)2 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (19)

and

119890 = 1205761198671205722 (1 minus 120573)21 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (20)

(ii) When

1198612 = (1 minus 120576)2 1198601 + 11989812058321198602120576 (1 minus 120576) + 12058221205731198602120576 (1 minus 120576)120588120573 (1 minus 120601)+ 12057911198601 (1 minus 120576)2 (1 minus 120601) + 12057911198602120576120588120575 (1 minus 120601)+ 212058211198601 (1 minus 120576)2 minus 11986021 (1 minus 120576)2

2120588 minus 119860221205762(1 minus 120601)2 lt 0(21)

the optimal values of product quality and platform advertise-ment are

119902 = (120576 minus 1) 119861121198612 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (22)

and

119890 = 120576 (120576 minus 1) 119861121198612 (1 minus 120601) (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (23)

Note that the equilibrium value of product quality andplatform advertisement will be higher if the sponsor takes themargin price strategy compared to the volume price strategyIt is also clear that both two equilibriums are the increasingfunction of the consumerrsquos evaluation (119871 or 119867) When theevaluation of consumer is higher the sponsor should improvetheir product quality level and similarly the platform shouldpay more advertisement to advertise the product And thereare no differences of the sponsor profit in the two pricestrategies when119867119897 = 11205722(1 minus 120573)232 The Optimal Decisions in Coordination In this sectionthe sponsor and the platform cooperate as a system to makedecisions about product quality and platform advertisementWe calculate the optimal decision of the system to maximizethe present value of it Taking (5) and (6) in to account theHamiltonian function for the system is given as follows

119867 = 119901119896 [119898 (119903 minus 119901) + 119892 + 1205821119902 + 1205822119890] minus 121199022 minus 121198902+ 1205741 (120573 (119901 minus 119903) + 1205831119902 + 1205832119890)+ 1205742 (1205791119902 + 1205792119890 minus 120575119892)

(24)

where 1205741 1205742 represent the costate variables of the systemproblemassociating the reference price and investment desirewith the decisions Similar to the process of Theorem 1 theequilibrium of the system is shown as follows

119902 = 119896119901(1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (25)

and

119890 = 119896119901(1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (26)

It is clearly shown that the product quality is improved due tothe cooperation InTheorem 3 the optimal cost-sharing ratiois given and the advertisement of platform in two situationsis compared

Theorem 3 The optimal cost-sharing ratio 120601 is

120601 = 2 minus 31205762 minus 120576 119894119891 120576 le 23 0 119890119897119904119890 (27)

Taking (27) into (20) we get

119890 = 119896119901120576 (2 minus 120576)2120576 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (28)

Discrete Dynamics in Nature and Society 7

and

119890 minus 119890 = 1198961199011205762 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) gt 0 (29)

For the cooperation system the reference price andinvestment desire are also steady similar to the situation ofseparation as follows

119903 (119905) = 1198641119890minus120573119905 + 119903119888119892 (119905) = 1198642119890minus120575119905 + 119892119888

(30)

where 1198641 = 1199030 minus 119903119888 119903119888 = 119901 + (1205831 119902 + 1205832 119890)120573 1198642 = 1198920 minus 119892119888 and119892119888 = (1205791 119902+1205792 119890)120575The steady states of the reference price andinvestment desire are 119903119888 and 119892119888

From the above results we can find that both the steadyreference price and steady investment desire are highercompared to Theorem 2 since both the product quality andplatform advertisement are improved in cooperation systemSubstituting (30) into (7) we get the present value of the profitfor cooperation system as follows

119869 = 119896119901( 1198641119898120588 + 120573 + 1198642120588 + 120575 + 119898 (119903119888 minus 119901)120588 + 119892119888120588 + 1205821 119902119901

+ 1205822 119890120588 ) minus 11990222120588 minus 11989022120588 (31)

In the next section we will discuss the changes of theresults in ourmodel when relevant parameters are in differentvalues Analyzing these changes does not only build therobustness of our results but also deepen our understandingof this problem

4 Parameters Sensitivity Analysis

In this section we analyze the relations between the relevantparameters and the values of profit product quality andplatform advertisement Firstly the impact of parameter 119898on the present value of profit is illustrated in Figures 2 and 3The parameter119898 is seen as an item that reflects the degree ofthe reference price effects on the investorrsquos behaviorThere aretwo firms the sponsor and the platform in the research so itis worthwhile to analyze the influence of parameter119898 on theprofit of them because the effects of119898may be different

As shown in Figure 2 119869119904 119869119901 and 119869 represent the profitof the sponsor the profit of the platform and the totalprofit when the sponsor and the platform chose to cooperaterespectively The present value of the sponsorrsquos profit willdecrease if the investors pay more attention to referenceprice while the platformrsquos profit is an increasing function ofparameter 119898 when they make decisions independently It isclear that the profits of the two firms are increased whenthey cooperate as a system and the cooperation profit is anincreasing function of the parameter 119898 Note that parameter119898 denotes the degree of the reference price effects on theinvestorrsquos behavior the fact implied in Figure 2 shows that thelarger the effect of the reference price the more profit the twoitems can get when they cooperate as a system

ＭＪJ

05

101520253035404550

The p

rofit

01 02 03 04 05 06 07 08 09 10m

Figure 2 Profit changes

Δ ＭΔ Ｊ

01 02 03 04 05 06 07 08 09 10m

05

101520253035404550

The c

hang

e of p

rofit

Figure 3 Differences of profit in two situations

From Figure 3 we can find that cooperation is more prof-itable for the sponsor than the platform and the differences ofthe two profits are both the increasing function of parameter119898 Therefore the sponsor should choose to cooperate withthe platform when the investors pay more attention to theirreference price And the differences of platformrsquos profit are notobvious in twoways whichmeans that it has a little differencefor platform in two cases

Comparing Figure 2 with Figure 3 it is found that coop-eration with platform not only promotes the profit of sponsorbut also can help the sponsor to keep a growing trend of theprofit when investors pay more attention to their referenceprice

Then the impact of parameters 120576 120601 on the values ofproduct quality and platform advertisement are illustratedin Figures 4 and 5

As shown in Figures 4 and 5 it is clear that the platformwill increase their advertisement to improve crowdfundingproject with the profit-sharing ratio increasing And thistrend is more obvious when the value of cost-sharing ratio is

8 Discrete Dynamics in Nature and Society

qe

05

101520253035404550

qe

01 02 03 04 05 06 07 08 09 10

Figure 4The changes of product quality (platform advertisement)

=05=02

0

10

20

30

40

50

60

e

01 02 03 04 05 06 07 08 09 10

Figure 5 The changes of platform advertisement in two situations

Table 2 Impacts of 120576 120601 on the optimal results

parameters 119869119904 119869119901 e q(003 010) 8716 9063 1 485(120576 120601) (003 020) 29515 9036 1125 485(010 020) 39555 7811 375 45(020 030) 80175 5831 86 40

higher It is also clear that the advertisement of platform hasa complementary effect with product quality In Table 2 theimpact of 120576 120601 on the optimal results is illustrated in detail

As shown in Table 2 the present value of the sponsorrsquosprofit is an increasing function of 120576 and 120601 and the trend ismore obvious if both the two ratios increase simultaneouslyAnd the advertisement of platform will also increase if thesponsor shares more cost or profit with them The aboveresults illustrate that sponsor sharing the cost of the platformcan not only promote the development of crowdfunding butalso improve its own profit

5 Simulation

To demonstrate some external validity for our theoreticalmodel an empirical study about the impact of product qualityand platform advertisement on the success of a crowdfundingproject is conducted Data for our study is derived frompublicly available information on the Kickstartercomwebsite(httpswwwkickstartercom) Kickstarter has grown intoone of the best-known and oldest crowdfunding platformsin the world since its inception in April 2009 We writea computer program by using the language of Python toget the information on all projects posted on the platformOver six months the information for 1400 projects fromKickstartercom has been collected The projects for charityare removed because charity crowdfunding projects aredonation-based crowdfunding And some data for inaccura-cies and incomplete information are also removed After thatthe data for analysis purposes included 835 crowdfundingprojects in the categories art design and technology

Data about a given project includes themoneywhich havebeen raised the number of comments and the introductionof the project The introduction of the product includes thenumber of introduction pictures the number of videos andthe number of description words In the present study thedescription and introduction of the project are defined asthe factors that relate to the quality of the product And thenumber of comments is seen as an external influence

The correlation analysis of the variables is first performedand results of which are shown in Table 3 It is can beseen from the table that correlations between the dependentvariable and all independent variables are positive Thereforethe assumptions that the information about the productquality and platform advertisement will cause some influ-ences on the success of crowdfunding were supported InTable 4 some results of a regression analysis are illustratedin detail We choose the signal of introduction picture countto represent the signal of product quality and the signal ofcomments count is seen as the signal of external influencesAs can be seen from the table the introduction picture countexplains 446 of the raised money and the comment countexplains 435 of the raised money The F value is 42462and the values of VIF (lt10) for each variable all indicatethat there is no multicollinearity in the model The resultsof the numerical experiment support the theory and modelanalyzed in the present study and provide some externalvalidity for our theoretical model

6 Conclusion

Product design allows firms to improve their profit andmarket competitiveness in crowdfunding market Howeverthe successful crowdfunding is always hampered by lack ofguidance on how to design the product quality to attractinvestorsrsquo preference To solve this dilemma we describetwo items investment desire and reference price to representinvestorrsquos perception that can cause some effects on thesuccess of crowdfunding

A necessary consideration that investorrsquos investmentdesire and reference price have a significant effect on investor

6 Discrete Dynamics in Nature and Society

and

119869119901 = 119896119901120576( 1198631119898120588 + 120573 + 1198632120588 + 120575 + 119898 (119903119899 minus 119901)120588 + 119892119899120588 + 1205821119902119901

+ 1205822119890120588 ) minus (1 minus 120601) 11989022120588

(15)

Since sponsors make the decisions about the productprice strategy at the beginning of the crowdfunding they willdecided the optimal value of product quality and platformadvertisement for any specific price strategy The optimalresults in margin (H) and volume (L) price strategy areanalyzed respectively based on the research of 119867119906 [24]Therefore there will be two groups ie (119902 119890119867) (119902 119890 119871)to choose different situations In the following Theorem 2the optimal product quality and platform advertisement aregiven in detail in different price strategy

Theorem 2 (i) When

1198612 = (1 minus 120576)2 1198601 + 11989812058321198602120576 (1 minus 120576) + 12058221205731198602120576 (1 minus 120576)120588120573 (1 minus 120601)

+ 12057911198601 (1 minus 120576)2 (1 minus 120601) + 12057911198602120576120588120575 (1 minus 120601)+ 212058211198601 (1 minus 120576)2 minus 11986021 (1 minus 120576)2

2120588 minus 119860221205762(1 minus 120601)2 gt 0(16)

we obtain the following two conclusions(a) 0 lt 119867119897 lt 11205722(1 minus 120573)2 the optimal values of product

quality and platform advertisement are

119902 = (1 minus 120576) 119871 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (17)

and

119890 = 1205761198711 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (18)

(b) 119867119897 gt 11205722(1 minus 120573)2 the optimal values of productquality and platform advertisement are

119902 = (1 minus 120576)1198671205722 (1 minus 120573)2 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (19)

and

119890 = 1205761198671205722 (1 minus 120573)21 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (20)

(ii) When

1198612 = (1 minus 120576)2 1198601 + 11989812058321198602120576 (1 minus 120576) + 12058221205731198602120576 (1 minus 120576)120588120573 (1 minus 120601)+ 12057911198601 (1 minus 120576)2 (1 minus 120601) + 12057911198602120576120588120575 (1 minus 120601)+ 212058211198601 (1 minus 120576)2 minus 11986021 (1 minus 120576)2

2120588 minus 119860221205762(1 minus 120601)2 lt 0(21)

the optimal values of product quality and platform advertise-ment are

119902 = (120576 minus 1) 119861121198612 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (22)

and

119890 = 120576 (120576 minus 1) 119861121198612 (1 minus 120601) (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (23)

Note that the equilibrium value of product quality andplatform advertisement will be higher if the sponsor takes themargin price strategy compared to the volume price strategyIt is also clear that both two equilibriums are the increasingfunction of the consumerrsquos evaluation (119871 or 119867) When theevaluation of consumer is higher the sponsor should improvetheir product quality level and similarly the platform shouldpay more advertisement to advertise the product And thereare no differences of the sponsor profit in the two pricestrategies when119867119897 = 11205722(1 minus 120573)232 The Optimal Decisions in Coordination In this sectionthe sponsor and the platform cooperate as a system to makedecisions about product quality and platform advertisementWe calculate the optimal decision of the system to maximizethe present value of it Taking (5) and (6) in to account theHamiltonian function for the system is given as follows

119867 = 119901119896 [119898 (119903 minus 119901) + 119892 + 1205821119902 + 1205822119890] minus 121199022 minus 121198902+ 1205741 (120573 (119901 minus 119903) + 1205831119902 + 1205832119890)+ 1205742 (1205791119902 + 1205792119890 minus 120575119892)

(24)

where 1205741 1205742 represent the costate variables of the systemproblemassociating the reference price and investment desirewith the decisions Similar to the process of Theorem 1 theequilibrium of the system is shown as follows

119902 = 119896119901(1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (25)

and

119890 = 119896119901(1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (26)

It is clearly shown that the product quality is improved due tothe cooperation InTheorem 3 the optimal cost-sharing ratiois given and the advertisement of platform in two situationsis compared

Theorem 3 The optimal cost-sharing ratio 120601 is

120601 = 2 minus 31205762 minus 120576 119894119891 120576 le 23 0 119890119897119904119890 (27)

Taking (27) into (20) we get

119890 = 119896119901120576 (2 minus 120576)2120576 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (28)

Discrete Dynamics in Nature and Society 7

and

119890 minus 119890 = 1198961199011205762 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) gt 0 (29)

For the cooperation system the reference price andinvestment desire are also steady similar to the situation ofseparation as follows

119903 (119905) = 1198641119890minus120573119905 + 119903119888119892 (119905) = 1198642119890minus120575119905 + 119892119888

(30)

where 1198641 = 1199030 minus 119903119888 119903119888 = 119901 + (1205831 119902 + 1205832 119890)120573 1198642 = 1198920 minus 119892119888 and119892119888 = (1205791 119902+1205792 119890)120575The steady states of the reference price andinvestment desire are 119903119888 and 119892119888

From the above results we can find that both the steadyreference price and steady investment desire are highercompared to Theorem 2 since both the product quality andplatform advertisement are improved in cooperation systemSubstituting (30) into (7) we get the present value of the profitfor cooperation system as follows

119869 = 119896119901( 1198641119898120588 + 120573 + 1198642120588 + 120575 + 119898 (119903119888 minus 119901)120588 + 119892119888120588 + 1205821 119902119901

+ 1205822 119890120588 ) minus 11990222120588 minus 11989022120588 (31)

In the next section we will discuss the changes of theresults in ourmodel when relevant parameters are in differentvalues Analyzing these changes does not only build therobustness of our results but also deepen our understandingof this problem

4 Parameters Sensitivity Analysis

In this section we analyze the relations between the relevantparameters and the values of profit product quality andplatform advertisement Firstly the impact of parameter 119898on the present value of profit is illustrated in Figures 2 and 3The parameter119898 is seen as an item that reflects the degree ofthe reference price effects on the investorrsquos behaviorThere aretwo firms the sponsor and the platform in the research so itis worthwhile to analyze the influence of parameter119898 on theprofit of them because the effects of119898may be different

As shown in Figure 2 119869119904 119869119901 and 119869 represent the profitof the sponsor the profit of the platform and the totalprofit when the sponsor and the platform chose to cooperaterespectively The present value of the sponsorrsquos profit willdecrease if the investors pay more attention to referenceprice while the platformrsquos profit is an increasing function ofparameter 119898 when they make decisions independently It isclear that the profits of the two firms are increased whenthey cooperate as a system and the cooperation profit is anincreasing function of the parameter 119898 Note that parameter119898 denotes the degree of the reference price effects on theinvestorrsquos behavior the fact implied in Figure 2 shows that thelarger the effect of the reference price the more profit the twoitems can get when they cooperate as a system

ＭＪJ

05

101520253035404550

The p

rofit

01 02 03 04 05 06 07 08 09 10m

Figure 2 Profit changes

Δ ＭΔ Ｊ

01 02 03 04 05 06 07 08 09 10m

05

101520253035404550

The c

hang

e of p

rofit

Figure 3 Differences of profit in two situations

From Figure 3 we can find that cooperation is more prof-itable for the sponsor than the platform and the differences ofthe two profits are both the increasing function of parameter119898 Therefore the sponsor should choose to cooperate withthe platform when the investors pay more attention to theirreference price And the differences of platformrsquos profit are notobvious in twoways whichmeans that it has a little differencefor platform in two cases

Comparing Figure 2 with Figure 3 it is found that coop-eration with platform not only promotes the profit of sponsorbut also can help the sponsor to keep a growing trend of theprofit when investors pay more attention to their referenceprice

Then the impact of parameters 120576 120601 on the values ofproduct quality and platform advertisement are illustratedin Figures 4 and 5

As shown in Figures 4 and 5 it is clear that the platformwill increase their advertisement to improve crowdfundingproject with the profit-sharing ratio increasing And thistrend is more obvious when the value of cost-sharing ratio is

8 Discrete Dynamics in Nature and Society

qe

05

101520253035404550

qe

01 02 03 04 05 06 07 08 09 10

Figure 4The changes of product quality (platform advertisement)

=05=02

0

10

20

30

40

50

60

e

01 02 03 04 05 06 07 08 09 10

Figure 5 The changes of platform advertisement in two situations

Table 2 Impacts of 120576 120601 on the optimal results

parameters 119869119904 119869119901 e q(003 010) 8716 9063 1 485(120576 120601) (003 020) 29515 9036 1125 485(010 020) 39555 7811 375 45(020 030) 80175 5831 86 40

higher It is also clear that the advertisement of platform hasa complementary effect with product quality In Table 2 theimpact of 120576 120601 on the optimal results is illustrated in detail

As shown in Table 2 the present value of the sponsorrsquosprofit is an increasing function of 120576 and 120601 and the trend ismore obvious if both the two ratios increase simultaneouslyAnd the advertisement of platform will also increase if thesponsor shares more cost or profit with them The aboveresults illustrate that sponsor sharing the cost of the platformcan not only promote the development of crowdfunding butalso improve its own profit

5 Simulation

To demonstrate some external validity for our theoreticalmodel an empirical study about the impact of product qualityand platform advertisement on the success of a crowdfundingproject is conducted Data for our study is derived frompublicly available information on the Kickstartercomwebsite(httpswwwkickstartercom) Kickstarter has grown intoone of the best-known and oldest crowdfunding platformsin the world since its inception in April 2009 We writea computer program by using the language of Python toget the information on all projects posted on the platformOver six months the information for 1400 projects fromKickstartercom has been collected The projects for charityare removed because charity crowdfunding projects aredonation-based crowdfunding And some data for inaccura-cies and incomplete information are also removed After thatthe data for analysis purposes included 835 crowdfundingprojects in the categories art design and technology

Data about a given project includes themoneywhich havebeen raised the number of comments and the introductionof the project The introduction of the product includes thenumber of introduction pictures the number of videos andthe number of description words In the present study thedescription and introduction of the project are defined asthe factors that relate to the quality of the product And thenumber of comments is seen as an external influence

The correlation analysis of the variables is first performedand results of which are shown in Table 3 It is can beseen from the table that correlations between the dependentvariable and all independent variables are positive Thereforethe assumptions that the information about the productquality and platform advertisement will cause some influ-ences on the success of crowdfunding were supported InTable 4 some results of a regression analysis are illustratedin detail We choose the signal of introduction picture countto represent the signal of product quality and the signal ofcomments count is seen as the signal of external influencesAs can be seen from the table the introduction picture countexplains 446 of the raised money and the comment countexplains 435 of the raised money The F value is 42462and the values of VIF (lt10) for each variable all indicatethat there is no multicollinearity in the model The resultsof the numerical experiment support the theory and modelanalyzed in the present study and provide some externalvalidity for our theoretical model

6 Conclusion

Product design allows firms to improve their profit andmarket competitiveness in crowdfunding market Howeverthe successful crowdfunding is always hampered by lack ofguidance on how to design the product quality to attractinvestorsrsquo preference To solve this dilemma we describetwo items investment desire and reference price to representinvestorrsquos perception that can cause some effects on thesuccess of crowdfunding

A necessary consideration that investorrsquos investmentdesire and reference price have a significant effect on investor

Discrete Dynamics in Nature and Society 7

and

119890 minus 119890 = 1198961199011205762 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) gt 0 (29)

For the cooperation system the reference price andinvestment desire are also steady similar to the situation ofseparation as follows

119903 (119905) = 1198641119890minus120573119905 + 119903119888119892 (119905) = 1198642119890minus120575119905 + 119892119888

(30)

where 1198641 = 1199030 minus 119903119888 119903119888 = 119901 + (1205831 119902 + 1205832 119890)120573 1198642 = 1198920 minus 119892119888 and119892119888 = (1205791 119902+1205792 119890)120575The steady states of the reference price andinvestment desire are 119903119888 and 119892119888

From the above results we can find that both the steadyreference price and steady investment desire are highercompared to Theorem 2 since both the product quality andplatform advertisement are improved in cooperation systemSubstituting (30) into (7) we get the present value of the profitfor cooperation system as follows

119869 = 119896119901( 1198641119898120588 + 120573 + 1198642120588 + 120575 + 119898 (119903119888 minus 119901)120588 + 119892119888120588 + 1205821 119902119901

+ 1205822 119890120588 ) minus 11990222120588 minus 11989022120588 (31)

In the next section we will discuss the changes of theresults in ourmodel when relevant parameters are in differentvalues Analyzing these changes does not only build therobustness of our results but also deepen our understandingof this problem

4 Parameters Sensitivity Analysis

In this section we analyze the relations between the relevantparameters and the values of profit product quality andplatform advertisement Firstly the impact of parameter 119898on the present value of profit is illustrated in Figures 2 and 3The parameter119898 is seen as an item that reflects the degree ofthe reference price effects on the investorrsquos behaviorThere aretwo firms the sponsor and the platform in the research so itis worthwhile to analyze the influence of parameter119898 on theprofit of them because the effects of119898may be different

As shown in Figure 2 119869119904 119869119901 and 119869 represent the profitof the sponsor the profit of the platform and the totalprofit when the sponsor and the platform chose to cooperaterespectively The present value of the sponsorrsquos profit willdecrease if the investors pay more attention to referenceprice while the platformrsquos profit is an increasing function ofparameter 119898 when they make decisions independently It isclear that the profits of the two firms are increased whenthey cooperate as a system and the cooperation profit is anincreasing function of the parameter 119898 Note that parameter119898 denotes the degree of the reference price effects on theinvestorrsquos behavior the fact implied in Figure 2 shows that thelarger the effect of the reference price the more profit the twoitems can get when they cooperate as a system

ＭＪJ

05

101520253035404550

The p

rofit

01 02 03 04 05 06 07 08 09 10m

Figure 2 Profit changes

Δ ＭΔ Ｊ

01 02 03 04 05 06 07 08 09 10m

05

101520253035404550

The c

hang

e of p

rofit

Figure 3 Differences of profit in two situations

From Figure 3 we can find that cooperation is more prof-itable for the sponsor than the platform and the differences ofthe two profits are both the increasing function of parameter119898 Therefore the sponsor should choose to cooperate withthe platform when the investors pay more attention to theirreference price And the differences of platformrsquos profit are notobvious in twoways whichmeans that it has a little differencefor platform in two cases

Comparing Figure 2 with Figure 3 it is found that coop-eration with platform not only promotes the profit of sponsorbut also can help the sponsor to keep a growing trend of theprofit when investors pay more attention to their referenceprice

Then the impact of parameters 120576 120601 on the values ofproduct quality and platform advertisement are illustratedin Figures 4 and 5

As shown in Figures 4 and 5 it is clear that the platformwill increase their advertisement to improve crowdfundingproject with the profit-sharing ratio increasing And thistrend is more obvious when the value of cost-sharing ratio is

8 Discrete Dynamics in Nature and Society

qe

05

101520253035404550

qe

01 02 03 04 05 06 07 08 09 10

Figure 4The changes of product quality (platform advertisement)

=05=02

0

10

20

30

40

50

60

e

01 02 03 04 05 06 07 08 09 10

Figure 5 The changes of platform advertisement in two situations

Table 2 Impacts of 120576 120601 on the optimal results

parameters 119869119904 119869119901 e q(003 010) 8716 9063 1 485(120576 120601) (003 020) 29515 9036 1125 485(010 020) 39555 7811 375 45(020 030) 80175 5831 86 40

higher It is also clear that the advertisement of platform hasa complementary effect with product quality In Table 2 theimpact of 120576 120601 on the optimal results is illustrated in detail

As shown in Table 2 the present value of the sponsorrsquosprofit is an increasing function of 120576 and 120601 and the trend ismore obvious if both the two ratios increase simultaneouslyAnd the advertisement of platform will also increase if thesponsor shares more cost or profit with them The aboveresults illustrate that sponsor sharing the cost of the platformcan not only promote the development of crowdfunding butalso improve its own profit

5 Simulation

To demonstrate some external validity for our theoreticalmodel an empirical study about the impact of product qualityand platform advertisement on the success of a crowdfundingproject is conducted Data for our study is derived frompublicly available information on the Kickstartercomwebsite(httpswwwkickstartercom) Kickstarter has grown intoone of the best-known and oldest crowdfunding platformsin the world since its inception in April 2009 We writea computer program by using the language of Python toget the information on all projects posted on the platformOver six months the information for 1400 projects fromKickstartercom has been collected The projects for charityare removed because charity crowdfunding projects aredonation-based crowdfunding And some data for inaccura-cies and incomplete information are also removed After thatthe data for analysis purposes included 835 crowdfundingprojects in the categories art design and technology

Data about a given project includes themoneywhich havebeen raised the number of comments and the introductionof the project The introduction of the product includes thenumber of introduction pictures the number of videos andthe number of description words In the present study thedescription and introduction of the project are defined asthe factors that relate to the quality of the product And thenumber of comments is seen as an external influence

The correlation analysis of the variables is first performedand results of which are shown in Table 3 It is can beseen from the table that correlations between the dependentvariable and all independent variables are positive Thereforethe assumptions that the information about the productquality and platform advertisement will cause some influ-ences on the success of crowdfunding were supported InTable 4 some results of a regression analysis are illustratedin detail We choose the signal of introduction picture countto represent the signal of product quality and the signal ofcomments count is seen as the signal of external influencesAs can be seen from the table the introduction picture countexplains 446 of the raised money and the comment countexplains 435 of the raised money The F value is 42462and the values of VIF (lt10) for each variable all indicatethat there is no multicollinearity in the model The resultsof the numerical experiment support the theory and modelanalyzed in the present study and provide some externalvalidity for our theoretical model

6 Conclusion

Product design allows firms to improve their profit andmarket competitiveness in crowdfunding market Howeverthe successful crowdfunding is always hampered by lack ofguidance on how to design the product quality to attractinvestorsrsquo preference To solve this dilemma we describetwo items investment desire and reference price to representinvestorrsquos perception that can cause some effects on thesuccess of crowdfunding

A necessary consideration that investorrsquos investmentdesire and reference price have a significant effect on investor

8 Discrete Dynamics in Nature and Society

qe

05

101520253035404550

qe

01 02 03 04 05 06 07 08 09 10

Figure 4The changes of product quality (platform advertisement)

=05=02

0

10

20

30

40

50

60

e

01 02 03 04 05 06 07 08 09 10

Figure 5 The changes of platform advertisement in two situations

Table 2 Impacts of 120576 120601 on the optimal results

parameters 119869119904 119869119901 e q(003 010) 8716 9063 1 485(120576 120601) (003 020) 29515 9036 1125 485(010 020) 39555 7811 375 45(020 030) 80175 5831 86 40

higher It is also clear that the advertisement of platform hasa complementary effect with product quality In Table 2 theimpact of 120576 120601 on the optimal results is illustrated in detail

As shown in Table 2 the present value of the sponsorrsquosprofit is an increasing function of 120576 and 120601 and the trend ismore obvious if both the two ratios increase simultaneouslyAnd the advertisement of platform will also increase if thesponsor shares more cost or profit with them The aboveresults illustrate that sponsor sharing the cost of the platformcan not only promote the development of crowdfunding butalso improve its own profit

5 Simulation

To demonstrate some external validity for our theoreticalmodel an empirical study about the impact of product qualityand platform advertisement on the success of a crowdfundingproject is conducted Data for our study is derived frompublicly available information on the Kickstartercomwebsite(httpswwwkickstartercom) Kickstarter has grown intoone of the best-known and oldest crowdfunding platformsin the world since its inception in April 2009 We writea computer program by using the language of Python toget the information on all projects posted on the platformOver six months the information for 1400 projects fromKickstartercom has been collected The projects for charityare removed because charity crowdfunding projects aredonation-based crowdfunding And some data for inaccura-cies and incomplete information are also removed After thatthe data for analysis purposes included 835 crowdfundingprojects in the categories art design and technology

Data about a given project includes themoneywhich havebeen raised the number of comments and the introductionof the project The introduction of the product includes thenumber of introduction pictures the number of videos andthe number of description words In the present study thedescription and introduction of the project are defined asthe factors that relate to the quality of the product And thenumber of comments is seen as an external influence

The correlation analysis of the variables is first performedand results of which are shown in Table 3 It is can beseen from the table that correlations between the dependentvariable and all independent variables are positive Thereforethe assumptions that the information about the productquality and platform advertisement will cause some influ-ences on the success of crowdfunding were supported InTable 4 some results of a regression analysis are illustratedin detail We choose the signal of introduction picture countto represent the signal of product quality and the signal ofcomments count is seen as the signal of external influencesAs can be seen from the table the introduction picture countexplains 446 of the raised money and the comment countexplains 435 of the raised money The F value is 42462and the values of VIF (lt10) for each variable all indicatethat there is no multicollinearity in the model The resultsof the numerical experiment support the theory and modelanalyzed in the present study and provide some externalvalidity for our theoretical model

6 Conclusion

Product design allows firms to improve their profit andmarket competitiveness in crowdfunding market Howeverthe successful crowdfunding is always hampered by lack ofguidance on how to design the product quality to attractinvestorsrsquo preference To solve this dilemma we describetwo items investment desire and reference price to representinvestorrsquos perception that can cause some effects on thesuccess of crowdfunding

A necessary consideration that investorrsquos investmentdesire and reference price have a significant effect on investor

Discrete Dynamics in Nature and Society 9

Table 3 Correlations of variables

1 2 3 4 5(1) Raised money ($) 1(2) No of comment 06624 1(3) Introduction picture count 03943 04637 1(4) Introduction video count 03621 03889 03556 1(5) Introduction word count 06571 04960 05173 03741 1

Table 4 Results of regression analysis

Variable Total sample120573 VIF

Dependent variable Raised money ($)Signals of platforms advertisement No of comment 0435 132Signals of product quality Introduction picture count 0446 1319

F 42462Summary of the model specified 1198772 069Δ1198772 0687

behavior is shown in our paper Further the product qualityand platform advertisement which are supposed have someeffects on investorrsquos investment desire and reference priceThis supposition is different from the previous literature thatthey assumed that there are only two types of investorshigh evaluation and low evaluation investors Accordingly adynamic model about the product quality is investigated bytaking the investorrsquos perception into account

In this paper the optimal product quality and platformadvertisement are calculated in two situations separation andcooperation respectively The results offer some new insightsthat will be useful to firms to research the product qualitydecisions in crowdfunding considering investorrsquos perceptionFirst firms should pay more attention to the potentialbehavior of investors influenced by online information whenthey make decisions about the product quality Then it is anoptimal strategy for firms to cooperate with platform wheninvestors pay more attention to their reference price Finallysharing the cost of the platform can not only promote thedevelopment of crowdfunding but also improve the profit offirms

Appendix

A Proof of Theorem 1

In model the necessary conditions for equilibrium of thesponsor profit are given by

120597119867119904120597119902 = 0 (A1)

1205971198671199041205971205741119904 = 119903 (119905) (A2)

1205971198671199041205971205742119904 = 119892 (119905) (A3)

1205741119904 = 1205881205741119904 minus 120597119867119904120597119903 (A4)

1205742119904 = 1205881205742119904 minus 120597119867119904120597119892 (A5)

From (A1) we get

119902 = 1198961199011205821 + 12058311205741119904 + 12057911205742119904 (A6)

Combining (A4) (A5) and (A6) we can get

119902 = (120588 + 120573) 12058311205741119904 + (120588 + 120575) 12057911205741119904 minus 1198961199011198981205831 minus 1198961199011205791 (A7)

Substituting 12058311205741119904 into (A7) by (A6) we get119902 = (120588 + 120573) 119902 + (120575 minus 120573) 12057911205742119904 minus (120588 + 120573) 1198961199011205821

minus 1198961199011198981205831 minus 1198961199011205791 (A8)

Similarly combining (A6) (A8) we get the differential of(A8) as follows

119902 = (2120588 + 120573 + 120575) 119902 minus (120588 + 120575) (120588 + 120573) 119902+ (120588 + 120575) (120588 + 120573) 1198961199011205821 + (120588 + 120575) 1198961199011198981205831+ (120588 + 120573) 1198961199011205791

(A9)

Calculating (A9) we get

119902 (119905) = 1198631119890(120588+120575)119905 + 1198632119890(120588+120573)119905 + 119902 (A10)

where 119902 = 119896119901(1205821 + 1205791(120588 + 120575) + 1198981205831(120588 + 120573)) and 1198631 1198632 areparameters to be determined The value of 119902 will be infinitewhen 119905 997888rarr infin which will obey our rules so we have 1198631 =1198632 = 0Therefore the equilibrium value of 119902(119905) can expressedas follows

119902 = 119896119901(1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (A11)

10 Discrete Dynamics in Nature and Society

Similarly the equilibrium platform advertisement can be gotas follows

119890 = 1205761198961199011 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (A12)

Taking (A11) (A12) into we get

120597119892 (119905)120597119905 = 1205791119902 + 1205792119890 minus 120575119892 (A13)

and120597119903 (119905)120597119905 = 120573 (119901 minus 119903) + 1205831119902 + 1205832119890 (A14)

Solving the differential equations (A13) (A14) the generalsolutions can be got

119903 (119905) = 1198641119890minus120573119905 + 119903119899119892 (119905) = 1198642119890minus120575119905 + 119892119899

(A15)

where 1198641 = 1199030 minus 119903119899 119903119899 = 119901 + (1205831119902 + 1205832119890)120573 1198642 = 1198920 minus 119892119899 and119892119899 = (1205791119902 + 1205792119890)120575B Proof of Theorem 2

Based on the research of 119867119906 with a volume strategy thesponsor sets the price 119901 = 119871 which is the lower limit ofconsumer valuation Since the price is the lower limits ofconsumer valuation we suppose that all consumers will takepart in the crowdfunding and the success rete 119896 = 1 Withthe margin strategy the sponsor sets the price 119901 = 119867 thehigh-type consumers will buy the product and the low-typeconsumer will decline The success rate in margin strategy is119896 = 1205722(1minus120573)2 Sponsor will make decisions about the productprice so we compare the profit of the sponsor in differentprice strategy to maximize sponsorrsquos profit as follows From(7) we can get the fact that the profit of the sponsor is

119869119904 = 119896119901 (1 minus 120576)( 1198631119898120588 + 120573 + 1198632120588 + 120575 + 119898 (119903119899 minus 119901)120588 + 119892119899120588

+ 1205821119902120588 + 1205822119890120588 ) minus 11990222120588 minus 12060111989022120588 (B1)

Taking (25) and (26) in to (B1) we can get

119869119904 = 119896119901 (1 minus 120576) 1198611 + (119896119901)2((1 minus 120576)2 1198601+ 11989812058321198602120576 (1 minus 120576) + 12058221205731198602120576 (1 minus 120576)

120588120573 (1 minus 120601)+ 12057911198601 (1 minus 120576)2 (1 minus 120601) + 12057911198602120576120588120575 (1 minus 120601)+ 212058211198601 (1 minus 120576)2 minus 11986021 (1 minus 120576)2

2120588 minus 119860221205762(1 minus 120601)2)

(B2)

where1198601 = 1205821+1205791(120588+120575)+1198981205831(120588+120573) gt 01198602 = 1205822+1205792(120588+120575)+1198981205832(120588+120573) gt 0 and 1198611 = 1198631119898(120588+120573)+1198632(120588+120575) gt 0

(i) When the coefficient of the quadratic term is positive

(1 minus 120576)2 1198601 + 11989812058321198602120576 (1 minus 120576) + 12058221205731198602120576 (1 minus 120576)120588120573 (1 minus 120601)

+ 12057911198601 (1 minus 120576)2 (1 minus 120601) + 12057911198602120576120588120575 (1 minus 120601)+ 212058211198601 (1 minus 120576)2 minus 11986021 (1 minus 120576)2

2120588 minus 119860221205762(1 minus 120601)2gt 0

(B3)

The value of inequity (B3) is positive which means that thefunction of (B2) is the monotonically increasing function ofthe item 119896119901 when 119896119901 gt 0 so the following can be got easily

(a)When 0 lt 119867119897 lt 11205722(1minus120573)2 we get 119871 gt 1198671205722(1minus120573)2so the volume strategy is better than margin strategy for thesponsor and the price is 119901 = 119871 Taking 119901 = 119871 119896 = 1 into (19)and (20) we can get

119902 = (1 minus 120576) 119871 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (B4)

and

119890 = 1205761198711 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (B5)

(b) When 119867119897 gt 11205722(1 minus 120573)2 we get 119871 lt 1198671205722(1 minus 120573)2and the margin strategy is better than the volume strategySimilarly the optimal values of product quality and platformadvertisement are

119902 = (1 minus 120576)1198671205722 (1 minus 120573)2 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (B6)

and

119890 = 1205761198671205722 (1 minus 120573)21 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (B7)

(ii)When the coefficient of the quadratic term is negativethat is

1198612 = (1 minus 120576)2 1198601+ 11989812058321198602120576 (1 minus 120576) + 12058221205731198602120576 (1 minus 120576)

120588120573 (1 minus 120601)+ 12057911198601 (1 minus 120576)2 (1 minus 120601) + 12057911198602120576120588120575 (1 minus 120601)+ 212058211198601 (1 minus 120576)2 minus 11986021 (1 minus 120576)2

2120588 minus 119860221205762(1 minus 120601)2lt 0

(B8)

the optimal price is constant

119896119901 = (120576 minus 1) 119861121198612 (B9)

Discrete Dynamics in Nature and Society 11

where 1198611 = 1198631119898(120588 + 120573) + 1198632(120588 + 120575) gt 0 Taking (B4) into(11) and (12) we get

119902 = (120576 minus 1) 119861121198612 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (B10)

and

119890 = 120576 (120576 minus 1) 119861121198612 (1 minus 120601) (1205822 +1205792120588 + 120575 + 1198981205832120588 + 120573) (B11)

C Proof of Theorem 3

Since the sponsor decides the optimal 120601 to get the presentvalue of his profit by the first-order optimal condition

119889119869119904119889120601 = 0 (C1)

we get

120601 = 2 minus 31205762 minus 120576 (C2)

and combining 120601 isin [0 1] we get120601 =

2 minus 31205762 minus 120576 119894119891 120576 le 23 0 119890119897119904119890 (C3)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no conflicts of interest

Acknowledgments

The authors gratefully acknowledge the support from theNational Social Science Foundation of China (17BGL083)

References

[1] S Bi Z Liu and K Usman ldquoThe influence of online infor-mation on investing decisions of reward-based crowdfundingrdquoJournal of Business Research vol 71 pp 10ndash18 2017

[2] A Agrawal C Catalini and A Goldfarb ldquoSome simple eco-nomics of crowdfundingrdquo Innovation Policy and the Economyvol 14 no 1 pp 63ndash97 2014

[3] P Belleflamme T Lambert and A Schwienbacher ldquoCrowd-funding Tapping the right crowdrdquo Journal of Business Ventur-ing vol 29 no 5 pp 585ndash609 2014

[4] M Cholakova and B Clarysse ldquoDoes the Possibility toMake Equity Investments in Crowdfunding Projects CrowdOut Reward-Based Investmentsrdquo Entrepreneurship Theory andPractice vol 39 no 1 pp 145ndash172 2015

[5] E Mollick ldquoThe dynamics of crowdfunding An exploratorystudyrdquo Journal of Business Venturing vol 29 no 1 pp 1ndash16 2014

[6] S P Anderson and L Celik ldquoProduct line designrdquo Journal ofEconomic Theory vol 157 pp 517ndash526 2015

[7] L Bottazzi and M Da Rin ldquoVenture capital in Europe and thefinancing of innovative companiesrdquo Economic Policy no 34 pp229ndash269 2002

[8] D Irwin and JM Scott ldquoBarriers faced by SMEs in raising bankfinancerdquo International Journal of Entrepreneurial Behavior andResearch vol 16 no 3 pp 245ndash259 2010

[9] A Schwienbacher and B Larralde ldquoCrowdfunding of SmallEntrepreneurial Venturesrdquo SSRN Electronic Journal 2010

[10] F Kleeman G G Voss and K Rieder ldquoUnder paid innovatorsThe commercial utilization of consumer work through crowd-sourcing Sciencerdquo Science Technology and Innovation Studiesvol 4 no 1 pp 5ndash26 2011

[11] B Armendariz and J MorduchThe Economics of Microfinancevol 1 Mit Press Books 2nd edition 2010

[12] JM Leimeister ldquoCrowdsourcing Crowdfunding crowdvotingcrowdcreationrdquo Zeitschrift fur Controlling und Managementvol 56 pp 388ndash392 2012

[13] A Tomczak and A Brem ldquoA conceptualized investment modelof crowdfundingrdquo Venture Capital vol 15 no 4 pp 335ndash3592013

[14] M Cho and G Kim ldquoA cross-cultural comparative analysis ofcrowdfunding projects in the United States and South KoreardquoComputers in Human Behavior vol 72 pp 312ndash320 2017

[15] A Lukkarinen J E Teich H Wallenius and J WalleniusldquoSuccess drivers of online equity crowdfunding campaignsrdquoDecision Support Systems vol 87 pp 26ndash38 2016

[16] T H Allison B C Davis J C Short and J W Webb ldquoCrowd-funding in a prosocial microlending environment Examiningthe role of intrinsic versus extrinsic cuesrdquo EntrepreneurshipTheory and Practice vol 39 no 1 pp 53ndash73 2015

[17] D Frydrych A J Bock T Kinder and B Koeck ldquoExploringentrepreneurial legitimacy in reward-based crowdfundingrdquoVenture Capital vol 16 no 3 pp 247ndash269 2014

[18] K Hyeog-In S Lee andY Na ldquoSuccess and failure case analysisof crowdfunding Focused on community art projectrdquo TheJournal of the Korea Contents Association vol 14 no 7 pp 125ndash136 2014

[19] V Alfredo ldquoCrowdfunding in Arts and Culture from fundingto engagementrdquoEconomia dellaCultura vol 1 pp 121ndash130 2017

[20] M Beier and K Wagner ldquoCrowdfunding Success of TourismProjects - Evidence from Switzerlandrdquo SSRN Electronic Journal2014

[21] S Bernstein A Korteweg and K Laws ldquoAttracting Early-StageInvestors Evidence from a Randomized Field ExperimentrdquoJournal of Finance vol 72 no 2 pp 509ndash538 2017

[22] K Sharma ldquoMutual fund purchases by high net worth individ-uals in Indiardquo Journal of Management Accounting Research vol6 no 2 pp 59ndash71 2006

[23] J Joshi and N N Patel ldquoInvestors perception towards thefinancial information disclosed in annual reports of listedcompanies of national stock exchange with special referencevadodara cityrdquo Optics Expresss vol 36 no 1 p 43 2013

[24] M Hu X Li and M Shi ldquoProduct and pricing decisions incrowdfundingrdquo Marketing Science vol 34 no 3 pp 331ndash3452015

[25] Y Chen R Zhang and B Liu ldquoJoint Decisions on Productionand Pricing with Strategic Consumers for Green CrowdfundingProductsrdquo International Journal of Environmental Research andPublic Health vol 14 no 9 2017

12 Discrete Dynamics in Nature and Society

[26] Z Li and D-q Tan ldquoTwo-stage dynamic pricing and advertis-ing strategies for online video servicesrdquo Discrete Dynamics inNature and Society vol 2017 Article ID 1349315 8 pages 2017

[27] S Joslashrgensen S Taboubi and G Zaccour ldquoCooperative adver-tising in a marketing channelrdquo Journal of Optimization Theoryand Applications vol 110 no 1 pp 145ndash158 2001

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10 Discrete Dynamics in Nature and Society

Similarly the equilibrium platform advertisement can be gotas follows

119890 = 1205761198961199011 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (A12)

Taking (A11) (A12) into we get

120597119892 (119905)120597119905 = 1205791119902 + 1205792119890 minus 120575119892 (A13)

and120597119903 (119905)120597119905 = 120573 (119901 minus 119903) + 1205831119902 + 1205832119890 (A14)

Solving the differential equations (A13) (A14) the generalsolutions can be got

119903 (119905) = 1198641119890minus120573119905 + 119903119899119892 (119905) = 1198642119890minus120575119905 + 119892119899

(A15)

where 1198641 = 1199030 minus 119903119899 119903119899 = 119901 + (1205831119902 + 1205832119890)120573 1198642 = 1198920 minus 119892119899 and119892119899 = (1205791119902 + 1205792119890)120575B Proof of Theorem 2

Based on the research of 119867119906 with a volume strategy thesponsor sets the price 119901 = 119871 which is the lower limit ofconsumer valuation Since the price is the lower limits ofconsumer valuation we suppose that all consumers will takepart in the crowdfunding and the success rete 119896 = 1 Withthe margin strategy the sponsor sets the price 119901 = 119867 thehigh-type consumers will buy the product and the low-typeconsumer will decline The success rate in margin strategy is119896 = 1205722(1minus120573)2 Sponsor will make decisions about the productprice so we compare the profit of the sponsor in differentprice strategy to maximize sponsorrsquos profit as follows From(7) we can get the fact that the profit of the sponsor is

119869119904 = 119896119901 (1 minus 120576)( 1198631119898120588 + 120573 + 1198632120588 + 120575 + 119898 (119903119899 minus 119901)120588 + 119892119899120588

+ 1205821119902120588 + 1205822119890120588 ) minus 11990222120588 minus 12060111989022120588 (B1)

Taking (25) and (26) in to (B1) we can get

119869119904 = 119896119901 (1 minus 120576) 1198611 + (119896119901)2((1 minus 120576)2 1198601+ 11989812058321198602120576 (1 minus 120576) + 12058221205731198602120576 (1 minus 120576)

120588120573 (1 minus 120601)+ 12057911198601 (1 minus 120576)2 (1 minus 120601) + 12057911198602120576120588120575 (1 minus 120601)+ 212058211198601 (1 minus 120576)2 minus 11986021 (1 minus 120576)2

2120588 minus 119860221205762(1 minus 120601)2)

(B2)

where1198601 = 1205821+1205791(120588+120575)+1198981205831(120588+120573) gt 01198602 = 1205822+1205792(120588+120575)+1198981205832(120588+120573) gt 0 and 1198611 = 1198631119898(120588+120573)+1198632(120588+120575) gt 0

(i) When the coefficient of the quadratic term is positive

(1 minus 120576)2 1198601 + 11989812058321198602120576 (1 minus 120576) + 12058221205731198602120576 (1 minus 120576)120588120573 (1 minus 120601)

+ 12057911198601 (1 minus 120576)2 (1 minus 120601) + 12057911198602120576120588120575 (1 minus 120601)+ 212058211198601 (1 minus 120576)2 minus 11986021 (1 minus 120576)2

2120588 minus 119860221205762(1 minus 120601)2gt 0

(B3)

The value of inequity (B3) is positive which means that thefunction of (B2) is the monotonically increasing function ofthe item 119896119901 when 119896119901 gt 0 so the following can be got easily

(a)When 0 lt 119867119897 lt 11205722(1minus120573)2 we get 119871 gt 1198671205722(1minus120573)2so the volume strategy is better than margin strategy for thesponsor and the price is 119901 = 119871 Taking 119901 = 119871 119896 = 1 into (19)and (20) we can get

119902 = (1 minus 120576) 119871 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (B4)

and

119890 = 1205761198711 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (B5)

(b) When 119867119897 gt 11205722(1 minus 120573)2 we get 119871 lt 1198671205722(1 minus 120573)2and the margin strategy is better than the volume strategySimilarly the optimal values of product quality and platformadvertisement are

119902 = (1 minus 120576)1198671205722 (1 minus 120573)2 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (B6)

and

119890 = 1205761198671205722 (1 minus 120573)21 minus 120601 (1205822 + 1205792120588 + 120575 + 1198981205832120588 + 120573) (B7)

(ii)When the coefficient of the quadratic term is negativethat is

1198612 = (1 minus 120576)2 1198601+ 11989812058321198602120576 (1 minus 120576) + 12058221205731198602120576 (1 minus 120576)

120588120573 (1 minus 120601)+ 12057911198601 (1 minus 120576)2 (1 minus 120601) + 12057911198602120576120588120575 (1 minus 120601)+ 212058211198601 (1 minus 120576)2 minus 11986021 (1 minus 120576)2

2120588 minus 119860221205762(1 minus 120601)2lt 0

(B8)

the optimal price is constant

119896119901 = (120576 minus 1) 119861121198612 (B9)

Discrete Dynamics in Nature and Society 11

where 1198611 = 1198631119898(120588 + 120573) + 1198632(120588 + 120575) gt 0 Taking (B4) into(11) and (12) we get

119902 = (120576 minus 1) 119861121198612 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (B10)

and

119890 = 120576 (120576 minus 1) 119861121198612 (1 minus 120601) (1205822 +1205792120588 + 120575 + 1198981205832120588 + 120573) (B11)

C Proof of Theorem 3

Since the sponsor decides the optimal 120601 to get the presentvalue of his profit by the first-order optimal condition

119889119869119904119889120601 = 0 (C1)

we get

120601 = 2 minus 31205762 minus 120576 (C2)

and combining 120601 isin [0 1] we get120601 =

2 minus 31205762 minus 120576 119894119891 120576 le 23 0 119890119897119904119890 (C3)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no conflicts of interest

Acknowledgments

The authors gratefully acknowledge the support from theNational Social Science Foundation of China (17BGL083)

References

[1] S Bi Z Liu and K Usman ldquoThe influence of online infor-mation on investing decisions of reward-based crowdfundingrdquoJournal of Business Research vol 71 pp 10ndash18 2017

[2] A Agrawal C Catalini and A Goldfarb ldquoSome simple eco-nomics of crowdfundingrdquo Innovation Policy and the Economyvol 14 no 1 pp 63ndash97 2014

[3] P Belleflamme T Lambert and A Schwienbacher ldquoCrowd-funding Tapping the right crowdrdquo Journal of Business Ventur-ing vol 29 no 5 pp 585ndash609 2014

[4] M Cholakova and B Clarysse ldquoDoes the Possibility toMake Equity Investments in Crowdfunding Projects CrowdOut Reward-Based Investmentsrdquo Entrepreneurship Theory andPractice vol 39 no 1 pp 145ndash172 2015

[5] E Mollick ldquoThe dynamics of crowdfunding An exploratorystudyrdquo Journal of Business Venturing vol 29 no 1 pp 1ndash16 2014

[6] S P Anderson and L Celik ldquoProduct line designrdquo Journal ofEconomic Theory vol 157 pp 517ndash526 2015

[7] L Bottazzi and M Da Rin ldquoVenture capital in Europe and thefinancing of innovative companiesrdquo Economic Policy no 34 pp229ndash269 2002

[8] D Irwin and JM Scott ldquoBarriers faced by SMEs in raising bankfinancerdquo International Journal of Entrepreneurial Behavior andResearch vol 16 no 3 pp 245ndash259 2010

[9] A Schwienbacher and B Larralde ldquoCrowdfunding of SmallEntrepreneurial Venturesrdquo SSRN Electronic Journal 2010

[10] F Kleeman G G Voss and K Rieder ldquoUnder paid innovatorsThe commercial utilization of consumer work through crowd-sourcing Sciencerdquo Science Technology and Innovation Studiesvol 4 no 1 pp 5ndash26 2011

[11] B Armendariz and J MorduchThe Economics of Microfinancevol 1 Mit Press Books 2nd edition 2010

[12] JM Leimeister ldquoCrowdsourcing Crowdfunding crowdvotingcrowdcreationrdquo Zeitschrift fur Controlling und Managementvol 56 pp 388ndash392 2012

[13] A Tomczak and A Brem ldquoA conceptualized investment modelof crowdfundingrdquo Venture Capital vol 15 no 4 pp 335ndash3592013

[14] M Cho and G Kim ldquoA cross-cultural comparative analysis ofcrowdfunding projects in the United States and South KoreardquoComputers in Human Behavior vol 72 pp 312ndash320 2017

[15] A Lukkarinen J E Teich H Wallenius and J WalleniusldquoSuccess drivers of online equity crowdfunding campaignsrdquoDecision Support Systems vol 87 pp 26ndash38 2016

[16] T H Allison B C Davis J C Short and J W Webb ldquoCrowd-funding in a prosocial microlending environment Examiningthe role of intrinsic versus extrinsic cuesrdquo EntrepreneurshipTheory and Practice vol 39 no 1 pp 53ndash73 2015

[17] D Frydrych A J Bock T Kinder and B Koeck ldquoExploringentrepreneurial legitimacy in reward-based crowdfundingrdquoVenture Capital vol 16 no 3 pp 247ndash269 2014

[18] K Hyeog-In S Lee andY Na ldquoSuccess and failure case analysisof crowdfunding Focused on community art projectrdquo TheJournal of the Korea Contents Association vol 14 no 7 pp 125ndash136 2014

[19] V Alfredo ldquoCrowdfunding in Arts and Culture from fundingto engagementrdquoEconomia dellaCultura vol 1 pp 121ndash130 2017

[20] M Beier and K Wagner ldquoCrowdfunding Success of TourismProjects - Evidence from Switzerlandrdquo SSRN Electronic Journal2014

[21] S Bernstein A Korteweg and K Laws ldquoAttracting Early-StageInvestors Evidence from a Randomized Field ExperimentrdquoJournal of Finance vol 72 no 2 pp 509ndash538 2017

[22] K Sharma ldquoMutual fund purchases by high net worth individ-uals in Indiardquo Journal of Management Accounting Research vol6 no 2 pp 59ndash71 2006

[23] J Joshi and N N Patel ldquoInvestors perception towards thefinancial information disclosed in annual reports of listedcompanies of national stock exchange with special referencevadodara cityrdquo Optics Expresss vol 36 no 1 p 43 2013

[24] M Hu X Li and M Shi ldquoProduct and pricing decisions incrowdfundingrdquo Marketing Science vol 34 no 3 pp 331ndash3452015

[25] Y Chen R Zhang and B Liu ldquoJoint Decisions on Productionand Pricing with Strategic Consumers for Green CrowdfundingProductsrdquo International Journal of Environmental Research andPublic Health vol 14 no 9 2017

12 Discrete Dynamics in Nature and Society

[26] Z Li and D-q Tan ldquoTwo-stage dynamic pricing and advertis-ing strategies for online video servicesrdquo Discrete Dynamics inNature and Society vol 2017 Article ID 1349315 8 pages 2017

[27] S Joslashrgensen S Taboubi and G Zaccour ldquoCooperative adver-tising in a marketing channelrdquo Journal of Optimization Theoryand Applications vol 110 no 1 pp 145ndash158 2001

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Discrete Dynamics in Nature and Society 11

where 1198611 = 1198631119898(120588 + 120573) + 1198632(120588 + 120575) gt 0 Taking (B4) into(11) and (12) we get

119902 = (120576 minus 1) 119861121198612 (1205821 + 1205791120588 + 120575 + 1198981205831120588 + 120573) (B10)

and

119890 = 120576 (120576 minus 1) 119861121198612 (1 minus 120601) (1205822 +1205792120588 + 120575 + 1198981205832120588 + 120573) (B11)

C Proof of Theorem 3

Since the sponsor decides the optimal 120601 to get the presentvalue of his profit by the first-order optimal condition

119889119869119904119889120601 = 0 (C1)

we get

120601 = 2 minus 31205762 minus 120576 (C2)

and combining 120601 isin [0 1] we get120601 =

2 minus 31205762 minus 120576 119894119891 120576 le 23 0 119890119897119904119890 (C3)

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare no conflicts of interest

Acknowledgments

The authors gratefully acknowledge the support from theNational Social Science Foundation of China (17BGL083)

References

[1] S Bi Z Liu and K Usman ldquoThe influence of online infor-mation on investing decisions of reward-based crowdfundingrdquoJournal of Business Research vol 71 pp 10ndash18 2017

[2] A Agrawal C Catalini and A Goldfarb ldquoSome simple eco-nomics of crowdfundingrdquo Innovation Policy and the Economyvol 14 no 1 pp 63ndash97 2014

[3] P Belleflamme T Lambert and A Schwienbacher ldquoCrowd-funding Tapping the right crowdrdquo Journal of Business Ventur-ing vol 29 no 5 pp 585ndash609 2014

[4] M Cholakova and B Clarysse ldquoDoes the Possibility toMake Equity Investments in Crowdfunding Projects CrowdOut Reward-Based Investmentsrdquo Entrepreneurship Theory andPractice vol 39 no 1 pp 145ndash172 2015

[5] E Mollick ldquoThe dynamics of crowdfunding An exploratorystudyrdquo Journal of Business Venturing vol 29 no 1 pp 1ndash16 2014

[6] S P Anderson and L Celik ldquoProduct line designrdquo Journal ofEconomic Theory vol 157 pp 517ndash526 2015

[7] L Bottazzi and M Da Rin ldquoVenture capital in Europe and thefinancing of innovative companiesrdquo Economic Policy no 34 pp229ndash269 2002

[8] D Irwin and JM Scott ldquoBarriers faced by SMEs in raising bankfinancerdquo International Journal of Entrepreneurial Behavior andResearch vol 16 no 3 pp 245ndash259 2010

[9] A Schwienbacher and B Larralde ldquoCrowdfunding of SmallEntrepreneurial Venturesrdquo SSRN Electronic Journal 2010

[10] F Kleeman G G Voss and K Rieder ldquoUnder paid innovatorsThe commercial utilization of consumer work through crowd-sourcing Sciencerdquo Science Technology and Innovation Studiesvol 4 no 1 pp 5ndash26 2011

[11] B Armendariz and J MorduchThe Economics of Microfinancevol 1 Mit Press Books 2nd edition 2010

[12] JM Leimeister ldquoCrowdsourcing Crowdfunding crowdvotingcrowdcreationrdquo Zeitschrift fur Controlling und Managementvol 56 pp 388ndash392 2012

[13] A Tomczak and A Brem ldquoA conceptualized investment modelof crowdfundingrdquo Venture Capital vol 15 no 4 pp 335ndash3592013

[14] M Cho and G Kim ldquoA cross-cultural comparative analysis ofcrowdfunding projects in the United States and South KoreardquoComputers in Human Behavior vol 72 pp 312ndash320 2017

[15] A Lukkarinen J E Teich H Wallenius and J WalleniusldquoSuccess drivers of online equity crowdfunding campaignsrdquoDecision Support Systems vol 87 pp 26ndash38 2016

[16] T H Allison B C Davis J C Short and J W Webb ldquoCrowd-funding in a prosocial microlending environment Examiningthe role of intrinsic versus extrinsic cuesrdquo EntrepreneurshipTheory and Practice vol 39 no 1 pp 53ndash73 2015

[17] D Frydrych A J Bock T Kinder and B Koeck ldquoExploringentrepreneurial legitimacy in reward-based crowdfundingrdquoVenture Capital vol 16 no 3 pp 247ndash269 2014

[18] K Hyeog-In S Lee andY Na ldquoSuccess and failure case analysisof crowdfunding Focused on community art projectrdquo TheJournal of the Korea Contents Association vol 14 no 7 pp 125ndash136 2014

[19] V Alfredo ldquoCrowdfunding in Arts and Culture from fundingto engagementrdquoEconomia dellaCultura vol 1 pp 121ndash130 2017

[20] M Beier and K Wagner ldquoCrowdfunding Success of TourismProjects - Evidence from Switzerlandrdquo SSRN Electronic Journal2014

[21] S Bernstein A Korteweg and K Laws ldquoAttracting Early-StageInvestors Evidence from a Randomized Field ExperimentrdquoJournal of Finance vol 72 no 2 pp 509ndash538 2017

[22] K Sharma ldquoMutual fund purchases by high net worth individ-uals in Indiardquo Journal of Management Accounting Research vol6 no 2 pp 59ndash71 2006

[23] J Joshi and N N Patel ldquoInvestors perception towards thefinancial information disclosed in annual reports of listedcompanies of national stock exchange with special referencevadodara cityrdquo Optics Expresss vol 36 no 1 p 43 2013

[24] M Hu X Li and M Shi ldquoProduct and pricing decisions incrowdfundingrdquo Marketing Science vol 34 no 3 pp 331ndash3452015

[25] Y Chen R Zhang and B Liu ldquoJoint Decisions on Productionand Pricing with Strategic Consumers for Green CrowdfundingProductsrdquo International Journal of Environmental Research andPublic Health vol 14 no 9 2017

12 Discrete Dynamics in Nature and Society

[26] Z Li and D-q Tan ldquoTwo-stage dynamic pricing and advertis-ing strategies for online video servicesrdquo Discrete Dynamics inNature and Society vol 2017 Article ID 1349315 8 pages 2017

[27] S Joslashrgensen S Taboubi and G Zaccour ldquoCooperative adver-tising in a marketing channelrdquo Journal of Optimization Theoryand Applications vol 110 no 1 pp 145ndash158 2001

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

12 Discrete Dynamics in Nature and Society

[26] Z Li and D-q Tan ldquoTwo-stage dynamic pricing and advertis-ing strategies for online video servicesrdquo Discrete Dynamics inNature and Society vol 2017 Article ID 1349315 8 pages 2017

[27] S Joslashrgensen S Taboubi and G Zaccour ldquoCooperative adver-tising in a marketing channelrdquo Journal of Optimization Theoryand Applications vol 110 no 1 pp 145ndash158 2001

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom