Abstract: In the development process of small and medium-sized creative enterprises, the difficulty of financing has not been overcome. With the accelerated penetration of the Internet and big data, crowdfunding, as a new business model combining crowdsourcing and micro finance, has the advantages of low cost, popularity and high efficiency. It complements traditional financing methods and gradually becomes one of the effective ways for small and medium-sized creative enterprises to carry out external financing. However, since the introduction of crowdfunding into the Chinese market in 2011, crowdfunding has gone through many stages, such as germination, explosion and reduction. After the industry reshuffle, it now presents the status quo of monopoly by Internet giants. The key to this transformation is the success of crowdfunding mechanism design optimization, which directly determines whether the income distribution of various participants in crowdfunding is reasonable, and determines the survival of the crowdfunding model.Product crowdfunding occupies the leading position in the crowdfunding market. However, this model is in a regulatory gap in the Chinese market. The lack of experience and ability of sponsors and the impact of imitator competition have become urgent problems to be solved. At present, there are many articles on the performance influencing factors of crowdfunding and the pricing strategies of sponsors, but most of them fail to show the internal correlation between the performance influencing factors, and they use static empirical methods, ignoring the continuity and dynamics of the crowdfunding process.
This paper applies differential equation and considers discriminatory pricing, network effect and initiator's effort level. By constructing a mathematical model with continuous variables, with the help of variation method, this paper explores the optimal dynamic path of public investors, and on this basis, studies how initiators design the dynamic mechanism of crowdfunding. This paper is committed to optimizing the product crowdfunding mechanism and will focus on the following points: What are the relevant influencing factors of crowdfunding deadline setting? Is there an appropriate interval? In the dynamic change process of crowdfunding stage, what is the change path of the number of investors and the psychological effect of its mapping? What factors affect the best effort level of the sponsor?
The research finds that: Not only there is an upper limit to the financing period of crowdfunding projects, but also the duration of financing is related to product pricing and manufacturing cost. With the increase of the amount of public investors' commitment, theproduction of crowdfunding products realizes scale economy. The popularity of crowdfunding platform, network effect and discriminatory pricing of products will all affect the initiators' optimal effort level. The proposition and inference of this paper provide some verifiable predictions for future empirical research. First of all, when considering the factors that affect the financing term, product pricing and production cost are one of the key factors. Secondly, the choice of the best effort level of the sponsors will be affected by the popularity of crowdfunding platforms, network effects and discriminatory pricing. Through the exploration of this article, it provides theoretical guidance for project sponsors to design the financing period of crowdfunding projects, preset the number of investors, and determine the level of effort, which will help promote the healthy development of crowdfunding industry.

Key words: variation method, optimal path, dynamic analysis, network effects, discriminatory pricing, effort level

摘要： 目前,对众筹成功影响因素的研究较为丰富,但对影响因素之间的关系以及在此基础上的动态机制设计的研究还略显不足。本文应用微分方程,考虑歧视性定价、网络效应和发起人努力水平,通过构建连续变量的数学模型,借助变分法求解,探究大众出资人到达的最优动态路径,并在此基础上研究发起人如何设计众筹动态机制。研究发现:众筹项目融资期限的设定,不仅存在上限,而且与产品定价、生产成本均有关联;随着大众出资人承诺出资数量的增加,众筹产品的生产存在规模化经济;项目发起人的最优努力水平受众筹平台知名度、网络效应和项目歧视性定价的影响。论文研究为众筹项目的动态机制设计提供了理论指导。

关键词: 变分法, 最优路径, 动态分析, 网络效应, 歧视性定价, 努力水平