Joint advertising and ordering decisions under the pre-sales crowdfunding

Our study considers a company launching new product on pre-sales crowdfunding platform and explores the joint advertising and ordering decisions in mass market thereafter. Under the “AON” mechanism, we construct profit-maximizing models to examine the impact of pledged amount on the company’s decision on advertising expense, order quantity and funding target. The results show that the optimal order quantity increases with the pledged amount in crowdfunding, and the company won’t promote new product in mass market until the pledged amount reaches a funding threshold, and the advertising expense increases with the pledged amount since then. In addition, we figure out the feasible scope for the funding target and suggest the company set the target within the scope, to ensure the project profitability. Furthermore, we demonstrate that the expected pledged amount won’t increase with the funding target, and the optimal funding target would be the lower bound in its feasible scope, to improve the campaign success. Finally, we conduct numerical analysis to verify our theoretical results.


Introduction
In recent years, the development of crowdfunding receives more attentions in China, and the State Council issued policies to promote "mass entrepreneurship and innovation" vigorously in 2019, clarifying specific measures to support the crowdfunding industry.Under the policy guidance, crowdfunding platforms develop with the direction shifted from initial scale to standardization gradually, and both campaign quantity and pledged amount are on the rise, in spite of the decreased crowdfunding platforms from 532 during the peak period to 145 at present 1 .According to the China Internet Crowdfunding Industry Development Report in 2022, at the end of June in 2022, there are 145 crowdfunding platforms still in operation, and the pre-sale crowdfunding platforms account for 46, including some influential crowdfunding platforms, such as JD.com, Tmall.com and Xiaomi.com.Pre-sales crowdfunding belongs to a common type in crowdfunding industry, referred to fundraisers initiating campaigns on crowdfunding platforms and soliciting funding for product launches, and delivering products to sponsors after the campaigns success (Bradford 2012).For simplicity, the crowdfunding mentioned below refers to pre-sales crowdfunding.Crowdfunding can pre-sale products and financing projects, but crowdfunding platforms in China highlight the pre-sales and adjust the crowdfunding process based on the actual situation in local market.On crowdfunding platforms abroad, e.g., Kickstarter and Indiegogo, an entrepreneur starts a funding campaign with the product underdeveloped Page 2 of 14 Zeng and Shen Management System Engineering (2023) 2:9 and continues the product completion after the campaign success, lowering the R&D and demand risk for the entrepreneur but increasing the delivery risk for sponsors.Instead, local crowdfunding platforms, such as JD crowdfunding and Tmall crowdfunding, necessitate the product completion and even the capabilities for mass production before the funding campaign launches, protecting the rights and interests of sponsors.For example, JD crowdfunding stipulates that fundraisers need to provide both product sample report and quality inspection report and ensure the products delivery in time after the campaign success.
Besides the product pre-sales and project financing, crowdfunding can assist in market research and product promotion.Compared with bank financing and venture capital, the pledged amount in crowdfunding is relatively low.For an example, in the first half of 2018, there were 7169 successful crowdfunding campaigns with a total pledging amount of $5.314 billion yuan, indicating an average pledged amount of only $750,000 yuan for each campaign2 .However, many companies still release products on crowdfunding platforms first, e.g., Haier launches the "magic mirror" crowdfunding campaign, Huawei releases portable photo printer through crowdfunding, while Lenovo launches a twistable micro-projector and a so-called "housekeeper" smart camera on crowdfunding platform.Obviously, with strong financing capabilities and well-established sales channels, these companies can release new products in mass market directly.However, they prefer to conduct pre-sales in crowdfunding market first, not for the pledged amounts and product pre-sales, but for the demand forecasting, expecting to optimize the advertising and ordering decisions in mass market.After the campaign success, some companies would launch advertisement in mass market (e.g., Haier smart mirror and Lenovo smart camera) and attract regular consumers to make more purchases, while some won't promote products in mass market even after the campaign success, such as Lenovo micro-projector.Actually, the advertising decision is a practical issue concerned by entrepreneurs in presales crowdfunding.Moreover, advertising is a daily-life element and stimulates demand for the product in mass market (Wang et al. 2015;Zhou et al. 2018), impacting the order quantity, so that the joint advertising and ordering decisions become a critical problem in crowdfunding.
Motivated by the practices in crowdfunding, we focus on the joint advertising and ordering decisions in presales crowdfunding and try to answer the questions as: "What is the optimal advertising expense and order quantity in mass market after the campaign success, and how to set the funding target at the crowdfunding stage".To address the questions, we divide the procedure into two stages, i.e., crowdfunding stage and sales stage, and consider the demands in crowdfunding market and mass market following the bivariate normal distribution and construct a generalized model to feature the impact of advertising investment on the demand in mass market.With the backward induction, we examine the optimal orderings with and without advertising expense respectively at the sales stage, and then compare profits to derive the entrepreneur advertising decision and the optimal advertising expense.In addition, we explore the pledged amount requirement for the advertising launch in mass market and figure out the feasible scope of the funding target and then obtain the optimal funding target at the crowdfunding stage.
Our study yields some interesting findings that the pledged amount in crowdfunding is decisive to the advertising expense and order quantity in mass market.Specifically, the advertising investment is preferred if the pledged amount exceeds a funding threshold, otherwise the sales revenue cannot compensate for the advertising expense, making the advertisement unnecessary.Conditional on the advertising investment, both advertising expense and order quantity increase with the pledged amount in crowdfunding.Furthermore, the advertising and ordering decisions at the sales stage also impact the funding target setting at the crowdfunding stage.Contingent on the advertising decision, there exist feasible scopes for the funding target, and setting the target within the scope ensures the project profitability.Moreover, we find that the optimal funding target would be the lower bound in the scope, maximizing the campaign success probability at the premise of project profitability.
Our study makes the following new contributes.First, the pledged amount in crowdfunding can serve as an important signal for the advertising decision in mass market, and we examine the optimal advertising expense in mass market based on the pledged amount in crowdfunding.Second, ordering decision is a critical issue in both crowdfunding market and mass market, and we investigate the optimal order quantity considering the pledged amount in crowdfunding and the advertisement in mass market.Third, the funding target is decisive to both the campaign success under the "AON" mechanism and the project profitability after the funding campaign success, and we derive the funding thresholds with and without subsequent advertisement in mass market respectively, regarded as the lower bounds of the target.Furthermore, we figure out the optimal funding target contingent on the funding thresholds and provide guidance for entrepreneurs in crowdfunding.
The remainder of our study is organized as follows.Section 2 reviews the relevant literature, and Sect. 3 is about the problem description and notations.In Sect.4, we construct models to explore the joint advertising and ordering decisions and examine the optimal funding target in crowdfunding.Section 5 conducts numerical experiments to verify our conclusions, and the last section provides managerial insights.

Literature review
Our study examines the funding target in pre-sales crowdfunding and investigates the advertising expense and order quantity in mass market after the campaign success, relating to two streams of literature including the pre-sales crowdfunding and joint advertising and ordering decisions.
Pre-sales crowdfunding belongs to a new financing mode, and many studies analyze the crowdfunding in various aspects, including crowdfunding type comparison (e.g., Belleflamme et al. 2014Belleflamme et al. , 2015;;Beaulieu et al. 2015;Bi et al. 2019;Nie et al. 2020;Guo et al. 2021), mechanism design (e.g., Chemla et al. 2020;Strausz 2017;Belavina et al. 2020), pricing decision (e.g., Guan et al. 2020a, b;Hu et al. 2015;Ellman and Hurkens 2019;Du et al. 2020;Peng et al. 2020;Zhang and Tian 2021;Bolandifar et al. 2023), product quality (e.g., Chakraborty et al. 2021;Liu et al. 2021;Gao et al. 2022), cooperation between entrepreneurs and VCs (e.g., Roma et al. 2018;Babich et al. 2021;Zeng 2023) and ordering and rationing decisions (Zeng et al. 2021).Similar with Guan et al. (2020a, b), we investigate the advertising investment in mass market after the campaign success in crowdfunding.However, Guan et al. (2020a, b) consider an entrepreneur investing in advertisement with the pledged amount from crowdfunding and discuss the pricing decision in this scenario.Instead, our study considers a company launching advertisement in mass market based on the pledged amount in crowdfunding and focuses on the joint advertising and ordering decisions after the campaign success.
Our research is also related to the joint advertising and ordering decisions, and many studies involve this issue (e.g., Lee and Hsu 2011;Chen 2011;Guler 2014;Wang et al. 2015;Cao et al. 2017;Zhou et al. 2017;Lu et al. 2018;Zhou et al. 2018;Xiao et al. 2019;Dye 2020).In a Newsvendor setting, Lee and Hsu (2011) examine the advertising and ordering decisions under the mean and variance of the demand, and Guler (2014) further figures out the necessary and sufficient conditions that advertising expense and order quantity increase with the advertising parameters.In a two-level supply chain, Chen (2011) derives the equilibrium on advertising expense and order quantity and designs a profit-sharing contract to achieve coordination, and Wang et al. (2015) further propose a two-period revenue-sharing contract to coordinate the supply chain.In addition, Zhou et al. (2018) consider both manufacturer and retailer risk-averse and investigate the cooperative advertising and ordering decisions, while Xiao et al. (2019) analyze the advertising and ordering decisions in a mixed manufacturer-retailers advertising cooperation.With a capital-constrained retailer, Cao et al. (2017) study the joint advertising, ordering and pricing decisions, while Zhou et al. (2017) examine the joint advertising, ordering and financing decisions.For perishable items, Lu et al. (2018) discuss the joint advertising, ordering and pricing decisions in an infinite horizon, while Dye (2020) further considers the psychic stock effect in this issue.
Different from the literature mentioned above, our study considers a company initiating a campaign in crowdfunding for the subsequent advertisement decision in mass market and mainly investigates the joint advertising and ordering decisions in this scenario.In presales crowdfunding, we examine the impact of pledged amount in crowdfunding on the advertising expense and order quantity in mass market, and then explore the funding target in crowdfunding based on the decisions.

Problem description and notations
Consider a company developing a new product with the investment I for R&D, testing, samples and patents and then releasing the product on a crowdfunding platform.As shown in Fig. 1, there exist two stages for the project in pre-sales crowdfunding, including crowdfunding stage and sales stage.At the first stage (crowdfunding stage), the company initiates a campaign on crowdfunding platform and sets the funding target G, and then sponsors pre-order products at a unit price p 1 .If the funding target cannot be reached, the campaign fails and the pledged amount will be returned to sponsors, otherwise the company obtains the funding excluding the commissions and fulfills the product delivery on time, then the project proceeds to the second stage.At the second stage (sales stage), the company decides the advertising expense u and the order quantity Q in mass market.The order quantity for sponsors is confirmed, i.e., Q F = d F , so that the company decides the order quantity and fulfills the delivery for sponsors first, and then sells the rest products in mass market, ensuring that regular consumers could purchase products at a unit price p 2 .Meanwhile, the unit production cost is c, and leftovers own a unit salvage value s at the end of the period.
In pre-sales crowdfunding, sponsors and regular consumers come from different channels (i.e., crowdfunding market and mass market respectively), and the realized demand in crowdfunding market can be treated as a signal to reflect the demand potential in mass market.Referring to Prasad et al. (2011) and Zhao et al. (2016), we denote the demands in crowdfunding market and mass market as D F and D R respectively and con- sider , wherein ρ > 0 , indicating the positive correlation between the demands in different markets.Actually, a greater ρ presents a stronger correlation, so that the parameter ρ can be treated as the factors impacting the significant interaction in demands between the two markets, such as the word-of-mouth communication.At the second stage (sales stage), the company updates the regular consumer demand D R based on the realized demand D F = d F in crowdfunding market, wherein F R (•) and f R (•) present the probability distribution function and probability density function of the updated demand D R in mass market respectively.Meanwhile, the company could launch advertisement in mass market and stimulate the purchases from regular consumers, and we consider the demand in mass market ξ(u) = θ(u)D R after the adver- tising investment, wherein θ(u) reflects the degree the regular consumer demand affected by the advertisement in mass market and satisfies ∂θ(u)  ∂u ≥ 0 , ∂ 2 θ(u) ∂ 2 u ≤ 0 and θ(0) = 1 , indicating that the demand in mass mar- ket increases with the advertising expense decreasingly, and that the demand in mass market is D R without advertising investment.In addition, the advertising investment cannot expand the mass market without limit, so that there exists the upper bound θ m for the θ(u) , i.e., lim u→∞ θ(u) = θ m , referring to Agrawal and Seshadri (2000), Wang et al. (2010) and Zhou et al. (2017).Notations are summarized in Table 1.

Modeling
In our study, the company sets the funding target and starts the campaign on crowdfunding platform first.Then, the company decides the advertising expense and order quantity in mass market if the funding target can be reached, otherwise the campaign fails and the advertising investment terminates, and Fig. 2 illustrates the sequence of the event.To solve the problem, we adopt backward induction and derive the optimal order quantity and advertising expense at the second stage (sales stage), and then figure out the optimal funding target at the first stage (crowdfunding stage).

The optimal advertising and ordering decisions
After the campaign success, the company obtains the pledged amount Ŵ = p 1 d F and updates the demand D R in mass market based on the real- ized demand . After the adver- tising investment in mass market, the company gains the profit as: is the production cost, u is the advertising expense in mass market, and I is the initial investment for the product project.After the reorganization, the profit function �(Q R , u) can be described as: Note that the order quantity for sponsors is confirmed, the company would decide the order quantity for regular consumers.Analyzing the profit function �(Q R , u) obtains the Lemma 1 as:

(see proof in Appendix)
Lemma 1 shows that, there exists an optimal solution (Q * R , u * ) for the profit function �(Q R , u) , wherein the optimal order quantity Q * R satisfies the Eq.(3).
And the optimal advertising expense u * satisfies the Eq. ( 4). (1) (2) (3) The Eq. ( 3) can be reorganized as ) , and inputting Q * R into the Eq. ( 4) derives the Eq. ( 5) as: If the marginal revenue from advertising exceeds the marginal advertising expense (i.e., ∂�(Q R ,u) ∂u | u=0 > 0 ), the company would launch advertisement in mass market after the campaign success, and ∂�(Q R ,u) ∂u | u=0 > 0 can be reorganized as: Solving the Eq. ( 6) derives the sufficient condition that companies won't launch advertisement in mass market, as shown in Theorem 1.
Theorem 1 After the campaign success, the company won't launch advertisement in mass market if the pledged In this case, the optimal advertising expense is u * = 0 , and the optimal order quantity is Theorem 1 shows that the company makes advertising decision contingent on the pledged amount in crowdfunding, and the advertisement launch should be abandoned unless the pledged amount reaches a funding threshold, and the campaign success cannot guarantee the subsequent advertising investment.Actually, the realized demand in crowdfunding market can reflect the demand potential of regular consumers, and a low pledged amount indicates a less-prospective mass market, then the ( 4) ( Fig. 2 Sequence of the event company won't launch advertisement further, since the revenue from advertisement launch cannot compensate for the advertising expense.In this case, the order quantity for regular consumers stays at is the sufficient condition to abandon advertising investment, but it doesn't mean that the company should launch advertisement in mass market when Next, we examine the sufficient and necessary conditions and denote (p 2 − s)θ ′ (0)

(See proof in Appendix)
The realized pledged amount in crowdfunding market reflects the demand potential in mass market, and Ŵ = Ŵ indicates that the mass market is large enough so that the increased revenue from the advertisement launch equals to the advertising expense, then the company won't launch advertisement in mass market until the pledged amount exceeds the funding threshold Ŵ .In addition, Theorem 2 shows that the advertising expense increases with the realized demand in crowdfunding market, since the increased funder order quantity implies a more prospective mass market, and the company would invest more in advertisement, resulting in more orders from regular consumers.
To conclude, the optimal order quantity and advertising expense at the second stage (sales stage) can be described as:

The optimal funding target
Based on the advertising and ordering decisions at the second stage (sales stage), this section explores the optimal funding target at the first stage (crowdfunding stage).The company should guarantee the project profitability in target setting, otherwise the project can be unprofitable even after the campaign success.Next, we examine the optimal funding target for the profitable projects screening and discuss the target setting with and without advertisement respectively.
Without advertisement launch in mass market, the optimal strategy is To analyze the impact of pledged amount on the profit �(Q R , 0) , we define a function as

(See proof in Appendix)
Based on Lemma 3, we examine the funding target without advertisement launch and obtain the Theorem 3 as: Theorem 3 Without advertisement launch in mass market (i.e., u * = 0), to guarantee the project profitability after the campaign success, the funding target G should be set in the scope [G 1 , +∞] , wherein G 1 = p 1 (d F 1 ) + .

(See proof in Appendix)
Theorem 3 figures out the feasible scope for the funding target without advertisement launch, and the company could ensure the project profitability by setting the target within the scope.The increased pledged amount indicates a larger demand in both crowdfunding market (7) and mass market, so that the company gains more profits as the pledged amount increases but cannot profit from the project until the pledged amount Ŵ reaches G 1 .Therefore, to avoid the case that the pledged amount Ŵ is less than G 1 , the company cannot set the target below the funding threshold G 1 , otherwise projects with low profit- ability would succeed in crowdfunding campaign.
When the company prefers the advertisement launch in mass market, the optimal strategy would be , and the company obtains the profit in this case.To examine the impact of pledged amount on the profit �(Q * R , u * ) , we define a function as

(See proof in Appendix)
Based on Lemma 4, we examine the funding target with advertisement launch and obtain Theorem 4 as: Theorem 4 With advertisement launch in mass market, to guarantee the project profitability after the campaign success, the funding target G should be set in the scope [G 2 , +∞] , wherein G 2 = p 1 (d F ) + .

(See proof in Appendix)
Theorem 4 shows that the funding target should be set within the scope [G 2 , +∞] when launching adver- tisement, ensuring the project profitability.Note that the company earns more profits as the pledged amount increases but cannot profit from the project until the amount reaches the funding threshold G 2 , i.e., the target Ŵ should be set above the threshold G 2 , otherwise the project remains unprofitable even after the campaign success, which should be avoided by companies.
Theorems 3 and 4 give the feasible scopes for the funding target with/without advertisement launch respectively and show that the company invests in advertisement contingent on the pledged amount.However, the pledged amount is unknown until the end of the funding campaign, indicating that the company cannot decide the advertising expense when setting target.Based on the Theorems 3 and 4, we analyze the funding target further and obtain the Lemma 5.
Lemma 5 To guarantee the project profitability after the campaign success, the company would set the funding target G within the feasible scope [min(G 1 , G 2 ), +∞).

(See proof in Appendix)
Lemma 5 shows that the company should set the target in the scope [min(G 1 , G 2 ), +∞) , guaranteeing the project profitability after the campaign success, regardless of the advertisement launch in mass market.Next, we discuss the optimal funding target under the "AON" mechanism.
Most crowdfunding platforms adopt the "All or Nothing (AON)" mechanism, wherein a fundraiser cannot obtain the pledges unless the funding target can be reached.Under the "AON" mechanism, the pledged amount Ŵ depends on the funding target G and can be described as: .
Note that the sponsor demand follows the Normal distribution as D F ∼ N (µ F , σ 2 F ) , the expected pledged amount can be described as: Analyzing the expected pledged amount E(Ŵ) derives the conclusion in Lemma 6.
Lemma 6 Under the "AON" mechanism, the expected pledged amount Ŵ won't increase with the target G.

(See proof in Appendix)
Under the "AON" mechanism, funding target necessitates sufficient pre-orders from sponsors, and the campaign is more likely to succeed as the target stays low, resulting in a higher success probability for the company to obtain the pledges.However, as the target increases, the demand in crowdfunding market is less likely to achieve the funding threshold, and the campaign success probability declines since then, i.e., the company is unlikely to obtain the pledges.Lemma 5 gives the feasible scope for the funding target, and companies can ensure the project profitability by setting the target in the feasible scope, regardless of the advertisement launch, while Lemma 6 suggests that companies should lower the target as much as it is allowed, increasing the campaign success probability.Based on Lemmas 5 and 6, we derive the optimal funding target, concluded in Theorem 5. (8 Theorem 5 The optimal funding target should be the lower bound of its feasible scope, i.e., G * = min(G 1 , G 2 ).

(See proof in Appendix)
To guarantee the project profitability after the campaign success, the company cannot set the funding target below the pledged amount min(G 1 , G 2 ) .In addition, the company should lower the target as much as it is allowed, since the expected pledged amount won't increase with the funding threshold.Therefore, the company should set the target as the lower bound in its feasible scope, i.e., G * = min(G 1 , G 2 ) , helping to guarantee the project prof- itability and improve the crowdfunding success.

Numerical analysis
Founded in 2014, JD Crowdfunding becomes a wellknown pre-sale crowdfunding platform in China and supports projects ranging from technology to household appliance products.In November 2014, Lenovo released a smart camera, so-called "Kanbao Mini", on the JD crowdfunding platform and set the funding target at 10 thousand yuan.Meanwhile, Lenovo provides a price menu, including the unit price 199 yuan for the "Kanbao Mini (Pearl White)" version with a limit of 2999 units and the unit 229 yuan for normal version without limit.Lasted for 36 days, at the end of December, the campaign has attracted more than 10 thousand sponsors pledging and raised 760 thousand yuan, far exceeding the funding target.After the campaign success, Lenovo initiates the mass production for the camera and starts delivery in January 2015.In mass market, regular consumers can purchase "Kanbao Mini" cameras in retail channels, e.g., JD.com and Tmall.com, at a unit retail price 699 yuan.In Tmall.com, for example, consumers receive a coupon worth 200 yuan by pre-ordering and enjoy a discounted price 499 yuan.
Regarding the numerical analysis on advertising expense and order quantity in pre-sales crowdfunding, we refer to the real data in the "Kanbao Mini" campaign initiated by Lenovo and configure the parameters as: p 1 = 199 yuan/unit, p 2 = 499 yuan/unit, d F = 10.303thousand, G = 10 thousand, Ŵ = 763.861thousand.The other parameters are configured as: In addition, referring to Zhou et al. (2017), we consider , which satisfy the prop- erty of the function θ(u) , and reflects the degree the advertising investment impacting on the demand in mass market.A greater value of indicates a more effective advertising performance, and we consider = 0.01 in this experiment.
Figure 3 shows the change of the advertising expense with the pledged amount.When the pledged amount is less than the funding threshold Ŵ , the regular con- sumer demand stays low, and the company won't invest in advertisement in this case, since the increased revenue from advertising is less than the expense.Until the pledged amount exceeds the threshold Ŵ , the advertise- ment launch is preferred, and the company would invest more in advertising as the pledged amount increases, since more pledges in crowdfunding indicate greater Fig. 3 The optimal advertising expense under various pledged amounts potential in mass market.However, as the advertising expense increases, the marginal revenue from advertising declines, i.e., the growth in mass market increases, then the optimal advertising expense increases with the pledged amount decreasingly.
In Fig. 4, customers can be categorized into sponsors and regular consumers, and the company prefers larger order quantity in both markets as the pledged amount increases.Obviously, the growth in pledged amount indicates more orders by sponsors, resulting in the increased order quantity in crowdfunding market.Moreover, a larger pledged amount also implies a greater demand potential for regular consumers, so that the order quantity increases in mass market.Therefore, the company prefers to increase the order quantity as the pledged amount rises.
Figure 5 shows the impact of order quantity and advertising expense on the profit.When the advertising expense is given, we find that the company gains more profits as the order quantity increases at first.In this case, the company cannot satisfy regular consumers with a low inventory, resulting in the decreased revenue in mass market.However, the profit declines as the order quantity increases further, since overstock results in more leftovers, higher inventory cost and lower profit.When the order quantity is given, the company obtains Fig. 4 The optimal order quantity under various pledged amounts Fig. 5 The company's profit under different order quantity and advertising expense more profits as the advertising expense grows at first.In this case, the increase in advertising expense stimulates more purchases from regular consumers and consumes the inventory, resulting in higher revenue in mass market and lower inventory cost.Nevertheless, further increasing the expense in advertisement cannot generate more revenue in mass market, due to the limited inventory and the increased advertising expense, and the profit starts declining since then.In our experiment, the revenue gained from the advertisement launch exceeds the expense, then the advertising investment is more preferable.However, the profit decreases with the advertising expense as the investment reaches a certain level, since the marginal revenue is less than the marginal expense since then.It should be noted that the optimal advertising investment stays in a medium range, and the optimal decisions on order quantity and advertising expense (Q * R , u * ) is remarked in Fig. 5.As shown in Fig. 6, the company gains more profits as the pledged amount increases.Actually, the revenue comes from two channels, including crowdfunding market and mass market, and the increased pledged amount means that the company obtains more profits in crowdfunding market and also indicates a greater demand potential in mass market.Therefore, the profit increases with the pledged amount.In addition, Fig. 6 also shows Fig. 6 The company's profit under various pledged amount 7 The pledged amount under various funding targets that, the company benefits from the advertisement launch in this case, resulted from a large mass market, and the increased revenue from advertisement launch compensates for the advertising expense.
Figure 7 shows that the pledged amount declines with the funding target.When the funding target stays low, the demand in crowdfunding market can meet the requirement for the target, and the company is more likely to obtain the funding.In this case, the funding target won't be the constraint for the campaign success, resulted in a stable funding amount.However, as the funding target grows, the expected pledged amount declines gradually, since a higher funding target indicates a lower success probability for the crowdfunding campaign.As the funding target grows to a certain level, it will be more difficult for the demand in crowdfunding market to reach the funding target, and the pledged amount declines since then.

Concluding remarks
Pre-sales crowdfunding provides services including project financing, product pre-sales and demand forecasting.Many companies prefer to release new products in crowdfunding market and optimize the subsequent advertising and ordering decisions in mass market, given crowdfunding platforms can help to assess the demand potential of new products.Our study examines the joint advertising and ordering decisions under pre-sales crowdfunding and figures out the optimal order quantity, advertising expense and funding target.The results show that, as the pledged amount in crowdfunding rises, the advertising expense in mass market also increases, resulting in a higher order quantity.However, advertising investment brings no profit until the pledged amount reaches a funding threshold, and the company can guarantee the project profitability by setting the funding target within its feasible scope.Moreover, we find that the pledged amount won't increase with the funding target, and the optimal target would be the lower bound in its feasible scope.
Based on the conclusions above, we provide some managerial insights for companies in practice.First, releasing new product on crowdfunding platforms helps to examine the demand potential of regular consumers, so as to optimize the advertising and ordering decisions in mass market.Specifically, companies can evaluate the size of the mass market based on the pledged amount in crowdfunding and are suggested to abandon the mass market until the pledged amount reaches a certain level, since a small amount indicates a low demand of regular consumers.Conditional on a sufficient pledged amount in crowdfunding, companies are encouraged to enter the mass market and decide the advertising expense and order quantity contingent on the pledged amount.In this case, it is recommended to invest more in advertising and prepare more orders, as the pledged amount increases.In addition, regarding the funding target, companies should set the target above a funding threshold to ensure the project profitability after the campaign success.However, a higher target lowers the campaign success probability under the "AON" mechanism, so we suggest companies lower the target as possible as it is allowed and set the funding threshold as the target, to maximize the success probability of the funding campaign.
Our study focuses on the joint advertising and ordering decisions in crowdfunding and would explore the joint pricing, advertising and ordering problem in crowdfunding in future research.

Appendix
Proof of Lemma 1: For tractability, we define the function as To illustrate the concavity of the function �(Q R , u) , we need to prove that �(q, u, d F ) is a bivariate concave function about (q, u) for any d F .
Due to ∀(q 1 , u 1 ) , (q 2 , u 2 ) and ∈ (0, 1) , we have Proof of Theorem 1: After the funding campaign success, the company updates the demand distribution in the mass market as and prefers to launch advertisement when (p 2 − s)θ ′ (0) ≤ 1 , and the company won't launch adver- tisement in this case.Since (p 2 − s)θ After the derivation on d F , we have > 1 , we have As shown in Lemma 3, the profit increases with the funders' order quantity d F in this case, and the company can profit from the project when the order quantity d F ≥ (d F 1 ) + , i.e., �(Q R , 0) ≥ 0 .Therefore, only when the funding amount Ŵ ≥ p 1 (d F 1 ) + , the project can be profitable after the funding campaign success.Next, we will show that the feasible scope of the funding amount is [G 1 , +∞) , wherein G 1 = p 1 (d F 1 ) + .
If the funding target G < G 1 , then the company may face the case G ≤ Ŵ < G 1 .In this case, even if the fund- ing campaign succeeds, the company cannot profit from the project.Therefore, to guarantee the project profitability, the company should set the funding target G in the feasible scope [G 1 , +∞).

Fig. 1
Fig. 1 Procedure in pre-sales crowdfunding dx as �(d F ) for tractability.Analyzing the function �(d F ) derives Lemma 2. Lemma 2 �(d F ) increases with the realized demand d F in crowdfunding market, and ∃ d F ∈ (−∞, +∞) satisfies the equation �( d F ) = 1.(See proof in Appendix)Based on Lemma 2, we analyze the sufficient and necessary conditions for advertising investment and obtain the Theorem 2 as: Theorem 2 Until the pledged amount Ŵ ≥ Ŵ , wherein Ŵ = p 1 ( d F ) + , the company would launch advertisement in mass market, and both advertising expense u * and order quantity Q * R in mass market would increase with the realized demand d F in crowdfunding market.
and analyzing the function derives its property in Lemma 3.
Lemma 4: We first prove that H 2 (d F ) increases with d F .H 2 (d F ) can be reorganized as θ(u * ) (p 2 − c)F −1 R