Pricing of a Three-Stage Supply Chain with a Big Data Company

Abstract

As Big Data is crucial to the competition and growth of companies, this paper considers a multi-stage supply chain with one Big Data company, multiple manufacturers, and one retailer. Demands of different products are dependent on the price and volume of each other, and such information can be obtained by the Big Data company at a cost. Manufacturers will purchase the demand information from the Big Data company and produce and sell products to the retailer. We intend to examine interactions among these firms. Specifically, various power structures are considered, and the equilibrium pricing decisions are analyzed and expressed explicitly through theoretical study. Effects of parameters under different power structures are explored and compared with each other through both special case studies and numerical experiments. This study enables the derivation of managerial insights related to Big Data that are useful for practical applications.

Appendix

1.1 Proof of Theorem 1

The profit of the retailer is

$$\begin{aligned} \Pi _r= & \; \sum _m(p_m-\omega _m)\cdot q_m\nonumber \\= & \; \sum _m(p_m-\omega _m)\cdot (Q_m+\sum _{m'}\theta _{m'm}p_{m'})\nonumber \\= & \; (p-\omega )'\cdot (Q+\theta 'p). \end{aligned}$$

(9)

The first-order derivatives of Eq. (9) are

$$\begin{aligned} \frac{\partial \Pi _r}{\partial p}=Q+(\theta '+\theta )p-\theta \omega . \end{aligned}$$

(10)

Similarly, the second-order derivatives are

$$\begin{aligned} \frac{\partial ^2\Pi _r}{\partial ^2p}=\theta '+\theta . \end{aligned}$$

Assume that $\theta '+\theta$ and $\mathbf {B}\theta '(\theta '+\theta )^{-1}\theta [\theta '(\theta '+\theta )^{-1}\theta +D]^{-1}D\mathbf {B}$ are negative definite and that the diagonal elements of $\theta '(\theta '+\theta )^{-1}\theta$ are negative. It can be derived from [36] that the optimal values of $p^*$ can be obtained by making Eq. (10) equal 0 since the second-order derivative matrix is negative semi-definite under the adopted assumption. The result is shown as follows:

$$\begin{aligned} p^*=(\theta '+\theta )^{-1}(\theta \omega -Q). \end{aligned}$$

(11)

The profit of manufacturer m is

$$\begin{aligned} \pi _m= & \; (\omega _m-c_m)q_m-p_m^b\beta _mq_m\nonumber \\= & \; (\omega _m-c_m-p_m^b\beta _m)q_m\nonumber \\= & \; (\omega _m-c_m-p_m^b\beta _m)(Q_m+\sum _{m'}\theta _{m'm}p_{m'}). \end{aligned}$$

(12)

Let A be the matrix whose principal diagonal elements are $\omega _m-c_m-p_m^b\beta _m$ and the other elements are 0. Then Eq. (12) can be expressed in vector space as follows:

$$\begin{aligned} \Pi _m= & \; A(Q+\theta 'p), \end{aligned}$$

(13)

where $\Pi _m=(\pi _1,\pi _2,\ldots ,\pi _M)'$.

Based on Eq. (11), it can be concluded that

$$\begin{aligned} \frac{\partial p^*}{\partial \omega '}= & \; \left( \begin{array}{cccc} \frac{\partial \rho _1}{\partial \omega _1} &{} \frac{\partial \rho _1}{\partial \omega _2} &{} \ldots &{} \frac{\partial \rho _1}{\partial \omega _M} \\ \frac{\partial \rho _2}{\partial \omega _1} &{} \frac{\partial \rho _2}{\partial \omega _2} &{} \ldots &{} \frac{\partial \rho _2}{\partial \omega _M} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \frac{\partial \rho _M}{\partial \omega _1} &{} \frac{\partial \rho _M}{\partial \omega _2} &{} \ldots &{} \frac{\partial \rho _M}{\partial \omega _M} \\ \end{array} \right) \nonumber \\= & \; (\theta '+\theta )^{-1}\theta . \end{aligned}$$

(14)

Let B and C be the vector whose elements are the principal diagonal of $A\theta '\frac{\partial \rho ^*}{\partial \omega }$ and $2\theta '\frac{\partial \rho ^*}{\partial \omega }$, respectively. Then, the first- and the second-order derivatives of Eq. (13) regarding $\omega$ are

$$\begin{aligned} \frac{\partial \Pi _m}{\partial \omega '}\overset{\bigtriangleup }{=}&\; \left[\frac{\partial \pi _1}{\partial \omega _1},\frac{\partial \pi _2}{\partial \omega _2},\ldots ,\frac{\partial \pi _M}{\partial \omega _M}\right]'\nonumber \\= & \; Q+\theta 'p+B, \end{aligned}$$

(15)

$$\begin{aligned} \frac{\partial ^2\Pi _m}{\partial ^2\omega '}\overset{\bigtriangleup }{=}&\;\left[\frac{\partial ^2\pi _1}{\partial ^2\omega _1},\frac{\partial ^2\pi _2}{\partial ^2\omega _2},\ldots ,\frac{\partial ^2\pi _M}{\partial ^2\omega _M}\right]'\nonumber \\= & \; C. \end{aligned}$$

(16)

Therefore, given the adopted assumption, the optimal wholesale price $\omega ^*$ can be obtained by making Eq. (15) equal 0, and there exists

$$\begin{aligned} \omega ^*=[\theta '(\theta '+\theta )^{-1}\theta +D]^{-1}[D(c+\mathbf {B} p^b)-\theta (\theta '+\theta )^{-1}Q], \end{aligned}$$

(17)

where D is a diagonal matrix whose principal diagonal elements are the same as $\theta '(\theta '+\theta )^{-1}\theta$, and $\mathbf {B}$ is a diagonal matrix whose principal diagonal elements are $\beta$.

The profit of the Big Data company is

$$\begin{aligned} \Pi _b= & \; \sum _m(p_m^b-c_m^b)\beta _mq_m\nonumber \\= & \; \sum _m(p_m^b-c_m^b)\beta _m(Q_m+\sum _{m'}\theta _{m'm}p_{m'})\nonumber \\= & \; (p^b-c^b)'\mathbf {B}(Q+\theta 'p). \end{aligned}$$

(18)

Since

$$\begin{aligned} \frac{\partial p}{\partial p^{b'}}\overset{\bigtriangleup }{=}&\;\left( \begin{array}{cccc} \frac{\partial p_1}{\partial p_1^b} &{} \frac{\partial p_1}{\partial p_2^b} &{} \ldots &{} \frac{\partial p_1}{\partial p_M^b} \\ \frac{\partial p_2}{\partial p_1^b} &{} \frac{\partial p_2}{\partial p_2^b} &{} \ldots &{} \frac{\partial p_2}{\partial p_M^b} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \frac{\partial p_M}{\partial p_1^b} &{} \frac{\partial p_M}{\partial p_2^b} &{} \ldots &{} \frac{\partial p_M}{\partial p_M^b} \\ \end{array} \right) \nonumber \\= & \; \frac{\partial p}{\partial \omega '}\cdot \frac{\partial \omega }{\partial p^{b'}}\nonumber \\= & \; (\theta '+\theta )^{-1}\theta \cdot [\theta '(\theta '+\theta )^{-1}\theta +D]^{-1}D\mathbf {B}, \end{aligned}$$

(19)

there exist

$$\begin{aligned} \frac{\partial \Pi _b}{\partial p^b}= & \; \mathbf {B}(Q+\theta 'p)+[(p^b-c^b)'\mathbf {B}\theta '\frac{\partial p}{\partial p^b}]'\nonumber \\= & \; \mathbf {B}(Q+\theta 'p)+\mathbf {B}D[\theta '(\theta '+\theta )^{-1}\theta \\& +D]^{-1}\theta '(\theta '+\theta )^{-1}\theta \mathbf {B}(p^b-c^b),\\ \frac{\partial ^2\Pi _b}{\partial ^2p^b}= & \; \mathbf {B}\theta '\frac{\partial p}{\partial p^b}+(\frac{\partial p}{\partial p^b})'\theta \mathbf {B}.\nonumber \end{aligned}$$

(20)

Let Eq. (20) equal 0, and there exists

$$\begin{aligned} p^{b*}= & {} -\mathbf {B}^{-1}\theta ^{-1}(\theta '+\theta )(\theta ')^{-1}[\theta '(\theta '+\theta )^{-1}\theta +D]D^{-1}(Q+\theta 'p)+c^b\nonumber \\= & {} -\mathbf {B}^{-1}[D^{-1}+\theta ^{-1}(\theta '+\theta )(\theta ')^{-1}](Q+\theta 'p)+c^b\nonumber \\= & {} -\mathbf {B}^{-1}[D^{-1}+\theta ^{-1}(\theta '+\theta )(\theta ')^{-1}]D(c+\mathbf {B}p^b-\omega )+c^b. \end{aligned}$$

(21)

Based on Eqs. (11), (17) and (21), the explicit expressions of the equilibrium solution can be obtained.