Joint Production and Pricing Strategy in Robust Model of Crowdfunding Considering Uncertain Preference

Abstract

While our society began to recognize the importance to capture the uncertain preference of people, the existing literature has confined its research work only under a static preference framework in crowdfunding. This paper represents the attempt to incorporate people uncertain preference on product quality and attributes into crowdfunding for production and pricing. More specifically, with describe the uncertain preference of supporters on product quality with robust optimization method, the distribution of the uncertain preference in the worst cases is solved firstly and the corresponding optimal product quality is investigated. Then, according to the different preference of supporters for product attributes, the differentiated pricing strategy is considered in the second stage. Finally, the model is extended to the situation that the third-party platform is participation to illustrate the effectiveness of platform on sharing economy. The research provides the sponsor with the quality level of product production and the corresponding optimal prices for different product quality and attributes. And the results show that: (i) It is optimal for sponsor to increase the unit price of the taste product and not to decide the product quality at the early stage when the risk of investors uncertain preference on product quality is greater than a certain value; (ii) Compare with the low effort level of the platform, the price of the taste product is lower when the platform effort level is high.

Appendix

1.1 A. Proof of Theorem 1

From Eq. (4), the excepted profit of the sponsor is given as follows:

$$\begin{aligned} \Pi _s & = p_{1}\lambda ((\theta +\alpha d)\varepsilon )\nonumber \\&+p_{2}(1-\lambda )\phi \varepsilon e-m(\varepsilon -Q)^{+}+v(Q-\varepsilon )^{+}-kQ, \end{aligned}$$

(A.1)

For Fig. 1a, the intersection point is $\varepsilon _1=0$ and the cut-off point is $\varepsilon _2=u+\frac{\sigma ^2}{u-\varepsilon _1}$, and $\varepsilon _2>Q$, so $Q<\frac{u^2+\sigma ^2}{2u}$. Therefore, the followings equations can be written according the tangent properties as:

$$\begin{aligned} \left\{ \begin{array}{lll} m(\varepsilon _1)=\Pi _s(\varepsilon _1),&{}&{} m(\varepsilon _1)'=\Pi _s(\varepsilon _1)',\\ m(\varepsilon _2)=\Pi _s(\varepsilon _2),&{}&{} m(\varepsilon _2)'=\Pi _s(\varepsilon _2)'.\\ \end{array}\right. \end{aligned}$$

(A.2)

By solving the Eq. (A.2), the following equations can gotten as

$$\begin{aligned} \left\{ \begin{array}{lll} x_1+x_2\varepsilon _2+x_3\varepsilon _2^2=p_1\lambda (\theta +\alpha d)\varepsilon _2+p_2(1-\lambda )\phi e\varepsilon _2-m(\varepsilon _2-Q)-kQ,\\ x_2+2x_3\varepsilon _1=p_1\lambda (\theta +\alpha d)+p_2(1-\lambda )\phi e-m,\\ x_1=-(k-v)Q.\\ \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} \left\{ \begin{array}{lll} x_1=-(k-v)Q,\\ x_2=p_1\lambda (\theta +\alpha d)+p_2(1-\lambda )\phi e-m+\frac{(2m-k-v)Qu}{u^2+\sigma ^2},\\ x_3=\frac{-(m-k)Qu^2}{(u^2+\sigma ^2)^2}.\\ \end{array}\right. \end{aligned}$$

(A.3)

Taking the Eq. (A.3) into Eq. (A.1), the excepted profit of the sponsor is written as

$$\begin{aligned} \Pi _s= & {} \left[p_{1}\lambda ((\theta +\alpha d)\varepsilon )+p_{2}(1-\lambda )\phi \varepsilon e-m \right]u\\&-(k-v)Q+\frac{(m-v)u^2Q}{u^2+\sigma ^2}. \end{aligned}$$

Therefore, the Eq. (10) is proved.

The detail proof of Theorem 1 is given as following. According to Eq. (18), the expected profit function under the worst distribution is obtained as shown in the following:

(1) When $Q<\frac{u^2+\sigma ^2}{2u}$

$$\begin{aligned} \underline{\Pi _s^2}(Q,p_2)= & {} \left[p_1\lambda (\theta +\alpha d)+p_2(1-\lambda )\phi e-m \right]u\nonumber \\&-(k-v)Q+\frac{(m-v)u^2Q}{u^2+\sigma ^2}. \end{aligned}$$

(A.4)

$$\begin{aligned} \frac{\partial \underline{\Pi _s^2}(Q,p_2)}{\partial Q}= & {} -(k-v)+\frac{(m-v)u^2}{u^2+\sigma ^2}. \end{aligned}$$

(A.5)

From Eq. (A.5), when $\frac{u}{\sigma }=\sqrt{\frac{k-v}{m-k}},(m>k>v)$, $\frac{\partial \underline{\Pi _s^2}(Q,p_2)}{\partial Q}=0$. The profit function $\underline{\Pi _s^2}(Q,p_2)$ is a monotonically increasing function of Q when $\frac{u}{\sigma }<\sqrt{\frac{k-v}{m-k}}$, and $\underline{\Pi _s^2}(Q,p_2)$ is a monotonically decreasing function of Q when $\frac{u}{\sigma }\ge \sqrt{\frac{k-v}{m-k}}$. Therefore, the optimal product production quality level can be expressed by the following formula

$$\begin{aligned} Q_1^*=\left\{ \begin{array}{llll} 0,&{}&{} \text{if} \;\frac{u}{\sigma }<\sqrt{\frac{k-v}{m-k}}, \\ \frac{u^2+\sigma ^2}{2u},&{}&{} \text{if}\; \frac{u}{\sigma }\ge \sqrt{\frac{k-v}{m-k}}.\\ \end{array}\right. \end{aligned}$$

(A.6)

(2) When $Q\ge \frac{u^2+\sigma ^2}{2u}$

$$\begin{aligned} \underline{\Pi _s^1}(Q,p_2)= & {} \left[p_1\lambda (\theta +\alpha d)+p_2(1-\lambda )\phi e-\frac{v+m}{2} \right]u\nonumber \\&-(k-v)Q+\frac{m-v}{2} \left(\sqrt{(Q-u)^2+\sigma ^2}-Q \right). \end{aligned}$$

(A.7)

$$\begin{aligned} \frac{\partial \underline{\Pi _s^1}(Q,p_2)}{\partial Q}= & {} \frac{m+v-2k}{2}-\frac{(m-v)(Q-u)}{2\sqrt{(Q-u)^2+\sigma ^2}}, \end{aligned}$$

(A.8)

$$\begin{aligned} \frac{\partial ^2 \underline{\Pi _s^1}(Q,p_2)}{\partial Q^2}= & {} \frac{-(m-v\sigma ^2)}{\left[(Q-u)^2+\sigma ^2 \right]^{\frac{3}{2}}}<0. \end{aligned}$$

(A.9)

Since the second derivative of the expected profit function is less than zero, so $\underline{\Pi _s}(Q,p_2)$ is a concave function, and when $\frac{\partial \underline{\Pi _s^1}(Q,p_2)}{\partial Q}=0$, we can get $Q_2^*=u+\frac{\sigma }{2}(\sqrt{\frac{m-k}{k-v}}-\sqrt{\frac{k-v}{m-k}})$. Combining the condition $Q\ge \frac{u^2+\sigma ^2}{2u}$, we can get

$$\begin{aligned} Q_2^*=\left\{ \begin{array}{llll} \frac{u^2+\sigma ^2}{2u},&{}&{} \text{if} \; \frac{u}{\sigma }<\sqrt{\frac{k-v}{m-k}}, \\ u+\frac{\sigma }{2} \left(\sqrt{\frac{m-k}{k-v}}- \sqrt{\frac{k-v}{m-k}}\right),&{}&{} \text{if} \; \frac{u}{\sigma }\ge \sqrt{\frac{k-v}{m-k}}.\\ \end{array}\right. \end{aligned}$$

(A.10)

When $Q=\frac{u^2+\sigma ^2}{2u}$, the piecewise function is continuous, that is $\underline{\Pi _s^1}(Q,p_2)=\underline{\Pi _s^2}(Q,p_2)$. Therefore, combining Eqs. (A.6) and (A.10), the optimal product quality level can be written as

$$\begin{aligned} Q^*=\left\{ \begin{array}{llll} 0,&{}&{} \text{if} \; \frac{u}{\sigma }<\sqrt{\frac{k-v}{m-k}}, \\ u+\frac{\sigma }{2}(\sqrt{\frac{m-k}{k-v}}-\sqrt{\frac{k-v}{m-k}}),&{}&{} \text{if} \; \frac{u}{\sigma }\ge \sqrt{\frac{k-v}{m-k}}.\\ \end{array}\right. \end{aligned}$$

(A.11)

1.2 B. Proof of Theorem 2

(i) When $\frac{u}{\sigma }>e \sqrt{\frac{k-v}{m-k}}$, it is always hold $a>(1-\lambda )\phi e \sigma \sqrt{\frac{k-v}{m-k}}$, $\overline{p_2}=\frac{a}{b}-\frac{(1-\lambda )\phi e\sigma }{b}\sqrt{\frac{k-v}{m-k}}>0$, the pricing problem in the following two situations is discussed.

(1) When $p_2\le \overline{p_2}$, the expected profit function is $\underline{\Pi _s^1}$:

$$\begin{aligned} \frac{\partial \underline{\Pi _s^1}}{\partial p_2}= & {} -2b p_2+a-\frac{b(p_1\lambda (\theta +\alpha d)-k)}{(1-\lambda )\phi e}, \end{aligned}$$

(B.1)

$$\begin{aligned} \frac{\partial ^2\underline{\Pi _s^1}}{\partial p_2^2}= & {} -2b<0. \end{aligned}$$

(B.2)

From the Eq. (18), we can know the expected profit function is a concave function of $p_2$, and when $\frac{\partial \underline{\Pi _s^1}}{\partial p_2}=0$

$$\begin{aligned} p_{21}^*=\frac{a}{2b}-\frac{p_1\lambda (\theta +\alpha d)-k}{2(1-\lambda )\phi e}. \end{aligned}$$

(B.3)

When $p_{21}<p_{21}^*$, the expected profit function $\underline{\Pi _s^1}$ is a monotonically increase function of $p_2$, and the expected profit function $\underline{\Pi _s^1}$ is a monotonically decreasing function of $p_2$ when $p_{21}\ge p_{21}^*$. Therefore, when $\sigma <\sigma _1=\left(\frac{a}{2(1-\lambda ) \phi e}+\frac{[p_1\lambda (\theta +\alpha d-k)]b}{2[(1-\lambda )(\phi e)]^2}\right)\sqrt{\frac{m-k}{k-v}}$, we get $p_{21}^*< \overline{p_2}$, and the optimal price is $\overline{p_2}$. When $\sigma \ge \sigma _1=\left(\frac{a}{2(1-\lambda ) \phi e}+\frac{[p_1\lambda (\theta +\alpha d-k)]b}{2(1-\lambda )^3(\phi e)62}\right)\sqrt{\frac{m-k}{k-v}}$, we get $p_{21}^*\ge \overline{p_2}$, and the optimal price is $p_{21}^*$. So the optimal product price can be written as follows

$$\begin{aligned} \left\{ \begin{array}{llll} \overline{p_2},&{}&{} \text{if} \; \sigma <\sigma _1= \left(\frac{a}{2(1-\lambda ) \phi e}+\frac{[p_1\lambda (\theta +\alpha d)-k]b}{2[(1-\lambda )(\phi e)]^2} \right)\sqrt{\frac{m-k}{k-v}} , \\ p_{21}^*=\frac{a}{2b}-\frac{\left[p_1\lambda (\theta +\alpha d)-k\right]}{2(1-\lambda )\phi e} ,&{}&{} \text{if} \; \sigma \ge \sigma _1=\left(\frac{a}{2(1-\lambda ) \phi e}+\frac{[p_1\lambda (\theta +\alpha d-k)]b}{2[(1-\lambda )(\phi e)]^2}\right)\sqrt{\frac{m-k}{k-v}}.\\ \end{array}\right. \end{aligned}$$

(B.4)

(2) When $p_2 > \overline{p_2}$, the expected profit function is $\underline{\Pi _s^2}$:

$$\begin{aligned}&\frac{\partial \underline{\Pi _s^2}}{\partial p_2}=-2b p_2+a-\frac{b(p_1\lambda (\theta +\alpha d)-m)}{(1-\lambda )\phi e}, \end{aligned}$$

(B.5)

$$\begin{aligned}&\frac{\partial ^2\underline{\Pi _s^2}}{\partial p_2^2}=-2b<0. \end{aligned}$$

(B.6)

The expected profit function $\underline{\Pi _s^2}$ is a concave function of $p_2$, and when $\frac{\partial \underline{\Pi _s^1}}{\partial p_2}=0$

$$\begin{aligned} p_{22}^*=\frac{a}{2b}-\frac{p_1\lambda (\theta +\alpha d)-m}{2(1-\lambda )\phi e}. \end{aligned}$$

(B.7)

When $p_{22}<p_{22}^*$, the expected profit function $\underline{\Pi _s^2}$ is a monotonically increase function of $p_2$, and the expected profit function $\underline{\Pi _s^2}$ is a monotonically decreasing function of $p_2$ when $p_{22}\ge p_{22}^*$. Therefore, when $\sigma <\sigma _2= \left(\frac{a}{2(1-\lambda ) \phi e}+\frac{[p_1\lambda (\theta +\alpha d-m)]b}{2[(1-\lambda )(\phi e)]^2}\right)\sqrt{\frac{m-k}{k-v}}$, we get $p_{22}^*< \overline{p_2}$ and the optimal price is $p_{22}^*$. When $\sigma \ge \sigma _2= \left(\frac{a}{2(1-\lambda ) \phi e}+\frac{[p_1\lambda (\theta +\alpha d-m)]b}{2[(1-\lambda )(\phi e)]^2}\right)\sqrt{\frac{m-k}{k-v}}$, we get $p_{22}^*\ge \overline{p_2}$

and the optimal product price is $\overline{p_2}$. So the optimal price can be written as follows

$$\begin{aligned} \left\{ \begin{array}{llll} p_{22}^*=\frac{a}{2b}-\frac{[p_1\lambda (\theta +\alpha d)-m]}{2(1-\lambda )\phi e}, &{}&{} \text{if} \; \sigma <\sigma _2= \left(\frac{a}{2(1-\lambda ) \phi e}+\frac{[p_1\lambda (\theta +\alpha d-m)]b}{2[(1-\lambda )(\phi e)]^2}\right)\sqrt{\frac{m-k}{k-v}} , \\ \overline{p_2} ,&{}&{} \text{if} \; \sigma \ge \sigma _2= \left(\frac{a}{2(1-\lambda ) \phi e}+\frac{[p_1\lambda (\theta +\alpha d)-m]b}{2[(1-\lambda )(\phi e)]^2}\right)\sqrt{\frac{m-k}{k-v}}.\\ \end{array}\right. \end{aligned}$$

(B.8)

Combing Eqs. (B.4) and (B.8), the intersection of two profit functions $\underline{\Pi _s^1}$ and $\underline{\Pi _s^2}$ has three different cases due to the difference values of $\sigma$. In the following Fig. 4, the graphic of the intersection is given to illustrate the optimal product price.

Fig. 4

The expected profit of the sponsor and optimal price for taste product $p_2$

In Fig. 4a, $\sigma >\sigma _1$, and the maximum value of the $\underline{\Pi _s^1}$ is get at $p=p_{21}^*$ and the maximum value of the $\underline{\Pi _s^2}$ is get at $p=\overline{p_2}$. The two profit functions are continuous at $p=\overline{p_2}$ and $\underline{\Pi _s^1}(p_{21}^*)>\underline{\Pi _s^2}(\overline{p_2})$, therefore the optimal price is $p=p_{21}^*$. In Fig. 4b, $\sigma \le \sigma _2$, and the maximum value of the $\underline{\Pi _s^1}$ is get at $p=\overline{p_2}$ and the maximum value of the $\underline{\Pi _s^2}$ is get at $p=p_{22}^*$. The two profit functions are continuous at $p=\overline{p_2}$ and $\underline{\Pi _s^2}(p_{22}^*)>\underline{\Pi _s^1}(\overline{p_2})$, therefore the optimal price is $p=p_{22}^*$. In the situation of Fig. 2c, we need to compare the values of $\underline{\Pi _s^1}(p_{21}^*)$, $\underline{\Pi _s^2}(p_{22}^*)$. When $\underline{\Pi _s^1}(p_{21}^*)= \underline{\Pi _s^2}(p_{22}^*)$, we get the followings

$$\begin{aligned}\underline{\Pi _s^1}(p_{21}^*)-\underline{\Pi _s^2}(p_{22}^*)&= \frac{b(m-k)(2p_1\lambda (\theta +\alpha d)-k-m)}{4((1-\lambda )\phi e)^2}\nonumber \\&\quad +\frac{a(m-k)}{2(1-\lambda )\phi e}-\sigma \sqrt{(m-k)(k-v)}=0. \end{aligned}$$

(B.9)

Therefore we can get

$$\begin{aligned} \sigma _3= & {} \left( \frac{a}{2(1-\lambda )\phi e}+\frac{bp_1\lambda (\theta +\alpha d)}{2((1-\lambda )\phi e)^2}\right. \nonumber \\&\left. -\frac{b(k+m)}{(2(1-\lambda )\phi e)^2}\right) \sqrt{\frac{m-k}{k-v}}. \end{aligned}$$

(B.10)

When $\sigma \le \sigma _3$, we get $\underline{\Pi _s^1}(p_{21}^*)-\underline{\Pi _s^2}(p_{22}^*)>0$, the optimal price is $p_{21}^*$. And when $\sigma >\sigma _3$, $\underline{\Pi _s^1}(p_{21}^*)-\underline{\Pi _s^2}(p_{22}^*)<0$, the optimal price is $p_{22}^*$. By comparing the values of $\sigma _1$, $\sigma _2$ and $\sigma _3$, we can find that $\sigma _1>\sigma _3>\sigma _2$. Therefore the optimal price is given as

$$\begin{aligned} p_2^*=\left\{ \begin{array}{llll} p_{21}^*,&{}&{} \text{if} \;\sigma <\sigma _3\\ p_{22}^*,&{}&{} \text{if} \;\sigma \ge \sigma _3.\\ \end{array}\right. \end{aligned}$$

(B.11)

Combining Eqs. (B.4), (B.8) and (B.11), we analysis the price in the following two as

• (1) When $\sigma <\sigma _3$, the optimal price of the product is $p_{21}^*$. If $\frac{u}{\sigma }\ge \sqrt{\frac{k-v}{m-k}}$, we get $\sigma \le \sigma _1$, $Q^*=\frac{a-bp_2}{(1-\lambda )\phi e}+\frac{\sigma }{2} \left(\sqrt{\frac{m-k}{k-v}}-\sqrt{\frac{k-v}{m-k}}\right)$. If $\sigma >\sigma _1$, $Q^*=0$, this result does not match the condition $\sigma <\sigma _3$. Therefore, the optimal price and initial quality level of the product is $p=p_{21}^*$, $Q^*=\frac{a-bp_2}{(1-\lambda )\phi e}+\frac{\sigma }{2}(\sqrt{\frac{m-k}{k-v}}-\sqrt{\frac{k-v}{m-k}})$.

• (2) When $\sigma \ge \sigma _3$, the optimal price of the product is $p_{21}^*$. If $\frac{u}{\sigma }\ge \sqrt{\frac{k-v}{m-k}}$, we get $\sigma \le \sigma _2$, which is not match the condition $\sigma \ge \sigma _3(\sigma _3>\sigma _2)$. Therefore, when $\frac{u}{\sigma }< \sqrt{\frac{k-v}{m-k}}$, we get the optimal price and initial quality of the product is $p=p_{22}^*$,$Q^*=0$.

Combining the above analysis, the price and initial quality of the product is given as follows

$$\begin{aligned} \left\{ \begin{array}{cccc} p_{21}^*=\frac{a}{2b}-\frac{[p_1\lambda (\theta +\alpha d)-k]}{2(1-\lambda )\phi e},&{} Q^*=\frac{a-bp_2}{(1-\lambda )\phi e}+\frac{\sigma }{2} \left(\sqrt{\frac{m-k}{k-v}}-\sqrt{\frac{k-v}{m-k}}\right),&{} \text{if} \;\sigma <\sigma _3,\\ p_{22}^*=\frac{a}{2b}-\frac{p_1\lambda (\theta +\alpha d)-m}{2(1-\lambda )\phi e},&{}Q^*=0,&{} \text{if} \;\sigma \ge \sigma _3.\\ \end{array}\right. \end{aligned}$$

(B.12)

From the Eq. (B.12), when the supporter product quality preference uncertainty risk is greater than $\sigma _3$, the company will chooses to increase the product price and not determine the production quality of the product in advance to avoid the risk.

(ii) When $\frac{u}{\sigma }<e \sqrt{\frac{k-v}{m-k}}$, it hold $\overline{p_2}=\frac{a}{b}-\frac{(1-\lambda )\phi e\sigma }{b}\sqrt{\frac{k-v}{m-k}}\le 0$, the sponsor will not provide the product with taste attributes, the product quality level is $Q^*$ in Theorem 1 and the price is $p_1$.