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doi: 10.3934/jimo.2022146
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 1 School of Finance, Anhui University of Finance and Economics, Bengbu 233030, China 2 IIF, School of Management, University of Science and Technology of China, Hefei 230026, China 3 College of Economics & Management, Anhui Agricultural University, Hefei 230036, China

*Corresponding author: Hongping Li

Received  September 2021 Revised  February 2022 Early access August 2022

As a new financing method, crowdfunding has alleviated the capital pressure for start-ups, and has attracted extensive attention of scholars. As an intermediary, crowdfunding platforms obtain benefits from providing services. However, the way of charging commission fees still remains a problem for platforms. This paper investigates the impacts of three service contracts, i.e., revenue sharing contract, per-unit fee contract, and fixed fee contract on the creator and platform. Using backward induction, we derive equilibrium results and get some conclusions. First, we find that under fixed fee contract, due to the trade-off between commission fee and crowdfunding success probability, the platform can only make decisions according to first-order conditions, and get partial profits from the creator. This contradicts the traditional view that the fixed fee contract allows leaders to get all the follower's profits in a stackelberg game. Further, we find that compared with revenue sharing contract and per-unit fee contract, the fixed fee contract is mostly although, not always worse (better) for the platform and system (the creator). Third, in the case of consumer homogeneity, we prove the equivalence of revenue sharing contract and per unit fee contract. However, we find that the efficiency of revenue sharing contract decreases when quality is endogenous, while when consumers are heterogeneous, the advantage of per unit fee contract for the platform decreases. In addition, we design a two-part tariff contract and prove that it can achieve coordination when the platform bears partial fixed cost. The two-part tariff contract can simultaneously improve the benefits of all parties with the appropriate parameters.

Citation: Yang Xu, Qiang Zhou, Xu Wang, Hongping Li. Platform contract selection and coordination contract design in reward-based crowdfunding. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022146
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##### References:
The sequence of the crowdfunding campaign
The profits of the platform and creator
The effects of product costs and product value on total profits of each contract
The effects of unit production cost, fixed production cost, and product value on efficiency of each contract
The effects of technology complexity and fixed production cost on the platform and creator's optimal profit under endogenous product value
The effects of technology complexity and fixed production cost on the total profits of each contract under endogenous product value
The effects of technology complexity and fixed production cost on contract efficiency under endogenous product value
The effects of heterogeneity on the platform and creator's optimal profits of each contract
The effects of heterogeneity on the total profits of each contract
Literature review
 Field Research Orientation Literature Crowdfunding Pricing Hu et al., 2015; Yang et al., 2020; Peng et al., 2020; Chakraborty and Swinney, 2021; Li and Cao, 2021; Strausz, 2014; Tang et al., 2021; Wu et al., 2021 Mode Selection Belleflamme and Lambert, 2014; Bi et al., 2019; Cumming et al., 2015; Du, 2020; Gao, 2021; Zhang and Tian, 2021 Advantages Da, 2018; Xu et al., 2018; Chemla and Tinn, 2020; Kumar et al., 2020; Xu et al., 2020 Others Burtch et al., 2021; Zhang and Tian, 2021; Zhou et al., 2021 Contract Design Revenue Sharing Contract & Per-unit Fee Contract Dana and Spier, 1999; Cachon and Lariviere, 2005; Yao et al., 2008; Linh and Hong, 2009; Xu et al., 2014; Zhang et al., 2012; Zhao et al., 2020; Zhang et al., 2012; Zhao et al., 2020; Panda, 2014; Pasternack, 1999; Abhishek et al., 2016; Jiang et al., 2011; Hagiu and Wright, 2015; Mantin et al., 2014; Ryan et al., 2012; Zhang et al., 2019 Fixed Fee Contract Sullivan, 1989; Chu, 1992; Sullivan, 1992; Lariviere and Padmanabhan, 1997; Marx and Shaffer, 2010; Shaffer, 1991; Dhar, 2013; Kuksov and Pazgal, 2007
 Field Research Orientation Literature Crowdfunding Pricing Hu et al., 2015; Yang et al., 2020; Peng et al., 2020; Chakraborty and Swinney, 2021; Li and Cao, 2021; Strausz, 2014; Tang et al., 2021; Wu et al., 2021 Mode Selection Belleflamme and Lambert, 2014; Bi et al., 2019; Cumming et al., 2015; Du, 2020; Gao, 2021; Zhang and Tian, 2021 Advantages Da, 2018; Xu et al., 2018; Chemla and Tinn, 2020; Kumar et al., 2020; Xu et al., 2020 Others Burtch et al., 2021; Zhang and Tian, 2021; Zhou et al., 2021 Contract Design Revenue Sharing Contract & Per-unit Fee Contract Dana and Spier, 1999; Cachon and Lariviere, 2005; Yao et al., 2008; Linh and Hong, 2009; Xu et al., 2014; Zhang et al., 2012; Zhao et al., 2020; Zhang et al., 2012; Zhao et al., 2020; Panda, 2014; Pasternack, 1999; Abhishek et al., 2016; Jiang et al., 2011; Hagiu and Wright, 2015; Mantin et al., 2014; Ryan et al., 2012; Zhang et al., 2019 Fixed Fee Contract Sullivan, 1989; Chu, 1992; Sullivan, 1992; Lariviere and Padmanabhan, 1997; Marx and Shaffer, 2010; Shaffer, 1991; Dhar, 2013; Kuksov and Pazgal, 2007
Descriptions of notation
 $v$ Product value $c$ Unit production cost $K$ Fixed production cost $a$ Technology complexity of production $x$ Demand for the product, which is uniformly distributed on $[0,1]$ $f(x), F(x)$ Probability density function and probability distribution function of $x$ $G$ Crowdfunding threshold $\lambda$ Commission rate under revenue sharing contract $r$ Commission fee under per-unit fee contact $T$ Commission fee under fixed fee contact $(r_t,T_t)$ Two-part tariff contract, where $r_t$ is the commission fee and $T_t$ is the transfer payment $e$ Contract efficiency $\Pi_{c},\Pi_{r},\Pi_{p},\Pi_{f}$ The total profits under centralized decision, revenue sharing contract, per-unit fee contract, and fixed fee contract $\Pi^{ex}_{pt},\Pi^{ex}_{ct}$ Profits of the platform and creator under two-part tariff contract $\Pi_{pr},\Pi_{pp},\Pi_{pf}$ Platform's profits under revenue sharing contract, per-unit fee contract, and fixed fee contract $\Pi_{cr},\Pi_{cp},\Pi_{cf}$ Creator's profits under revenue sharing contract, per-unit fee contract, and fixed fee contract
 $v$ Product value $c$ Unit production cost $K$ Fixed production cost $a$ Technology complexity of production $x$ Demand for the product, which is uniformly distributed on $[0,1]$ $f(x), F(x)$ Probability density function and probability distribution function of $x$ $G$ Crowdfunding threshold $\lambda$ Commission rate under revenue sharing contract $r$ Commission fee under per-unit fee contact $T$ Commission fee under fixed fee contact $(r_t,T_t)$ Two-part tariff contract, where $r_t$ is the commission fee and $T_t$ is the transfer payment $e$ Contract efficiency $\Pi_{c},\Pi_{r},\Pi_{p},\Pi_{f}$ The total profits under centralized decision, revenue sharing contract, per-unit fee contract, and fixed fee contract $\Pi^{ex}_{pt},\Pi^{ex}_{ct}$ Profits of the platform and creator under two-part tariff contract $\Pi_{pr},\Pi_{pp},\Pi_{pf}$ Platform's profits under revenue sharing contract, per-unit fee contract, and fixed fee contract $\Pi_{cr},\Pi_{cp},\Pi_{cf}$ Creator's profits under revenue sharing contract, per-unit fee contract, and fixed fee contract
Algorithm to solve equilibrium results under heterogenous valuations (revenue sharing contract)
 Input: $v_h, v_l, c, s$ Output: $\lambda^*, \Pi^{dv*}_{pr}, \Pi^{dv*}_{cr}$ Begin else $\lambda_1=argmax \; \int_{\frac{K}{s((1-\lambda)v_h-c)}}^{1} (s\lambda v_hx)f(x)dx$ $\quad\Pi^{dv2}_{pr}=\int_{\frac{K}{((1-\lambda_2)v_l-c)}}^{1} (\lambda_2 v_lx)f(x)dx$ $\lambda_2=argmax \; \int_{\frac{K}{((1-\lambda)v_l-c)}}^{1} (\lambda v_lx)f(x)dx$ $\quad\lambda_l=\lambda_2$ $\lambda_3=solve(\Pi^{dv}_{cr}(p=v_h)=\Pi^{dv}_{cr}(p=v_l))$ end if $\lambda_1>\lambda_3$ if $\Pi^{dv1}_{pr}>\Pi^{dv2}_{pr}$ $\quad\Pi^{dv1}_{pr}= \int_{\frac{K}{s((1-\lambda_1)v_h-c)}}^{1} (s\lambda_1 v_hx)f(x)dx$ $\quad\lambda^*=\lambda_h$ $\quad\lambda_h=\lambda_1$ $\quad\Pi^{dv*}_{pr}=\Pi^{dv1}_{pr}$ else $\quad\Pi^{dv*}_{cr}=\Pi^{dv}_{cr}(p=v_h, \lambda=\lambda_h)$ $\quad\Pi^{dv1}_{pr}= \int_{\frac{K}{s((1-\lambda_3)v_h-c)}}^{1} (s\lambda_3 v_hx)f(x)dx$ else $\quad \lambda_h=\lambda_3$ $\quad\lambda^*=\lambda_l$ end $\quad\Pi^{dv*}_{pr}=\Pi^{dv2}_{pr}$ if $\lambda_2>\lambda_3$ $\quad\Pi^{dv*}_{cr}=\Pi^{dv}_{cr}(p=v_l, \lambda=\lambda_l)$ $\quad \Pi^{dv2}_{pr}=\int_{\frac{K}{((1-\lambda_3)v_l-c)}}^{1} (\lambda_3 v_lx)f(x)dx$ end $\quad\lambda_l=\lambda_3$ End
 Input: $v_h, v_l, c, s$ Output: $\lambda^*, \Pi^{dv*}_{pr}, \Pi^{dv*}_{cr}$ Begin else $\lambda_1=argmax \; \int_{\frac{K}{s((1-\lambda)v_h-c)}}^{1} (s\lambda v_hx)f(x)dx$ $\quad\Pi^{dv2}_{pr}=\int_{\frac{K}{((1-\lambda_2)v_l-c)}}^{1} (\lambda_2 v_lx)f(x)dx$ $\lambda_2=argmax \; \int_{\frac{K}{((1-\lambda)v_l-c)}}^{1} (\lambda v_lx)f(x)dx$ $\quad\lambda_l=\lambda_2$ $\lambda_3=solve(\Pi^{dv}_{cr}(p=v_h)=\Pi^{dv}_{cr}(p=v_l))$ end if $\lambda_1>\lambda_3$ if $\Pi^{dv1}_{pr}>\Pi^{dv2}_{pr}$ $\quad\Pi^{dv1}_{pr}= \int_{\frac{K}{s((1-\lambda_1)v_h-c)}}^{1} (s\lambda_1 v_hx)f(x)dx$ $\quad\lambda^*=\lambda_h$ $\quad\lambda_h=\lambda_1$ $\quad\Pi^{dv*}_{pr}=\Pi^{dv1}_{pr}$ else $\quad\Pi^{dv*}_{cr}=\Pi^{dv}_{cr}(p=v_h, \lambda=\lambda_h)$ $\quad\Pi^{dv1}_{pr}= \int_{\frac{K}{s((1-\lambda_3)v_h-c)}}^{1} (s\lambda_3 v_hx)f(x)dx$ else $\quad \lambda_h=\lambda_3$ $\quad\lambda^*=\lambda_l$ end $\quad\Pi^{dv*}_{pr}=\Pi^{dv2}_{pr}$ if $\lambda_2>\lambda_3$ $\quad\Pi^{dv*}_{cr}=\Pi^{dv}_{cr}(p=v_l, \lambda=\lambda_l)$ $\quad \Pi^{dv2}_{pr}=\int_{\frac{K}{((1-\lambda_3)v_l-c)}}^{1} (\lambda_3 v_lx)f(x)dx$ end $\quad\lambda_l=\lambda_3$ End

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