Analyzing the operation of cloud supply chain: adoption barriers and business model

Abstract

Although on-demand cloud services are considered to be cost-saving solution for small and medium-sized enterprises, data security and cloud interoperability remain the major barriers for the adoption of cloud services. In addition, the operation of a cloud supply chain consisting of an independent software vendor (ISV) and Software-as-a-service provider may be inefficient without an automatic centralized platform for bundling and delivering ISVs’ applications to subscribers. In fact, these concerns can be resolved if all applications follow the same procedures to work together. Therefore, this study examines the operation of a cloud supply chain in which an intermediary offers a packaging service. The service helps ISVs wrap their applications according to a specific industry standard. This can reduce cost and time needed for deploying and updating applications. By examining an ISV’s channel selection and an intermediary’s pricing offer, our approach helps identify the conditions under which the selling channel is more profitable for the ISV. In addition, the importance of dual channel is highlighted when the intermediary becomes the ISV’s cloud partner. However, the adoption of dual channel does not lead to the highest profit if the quality of the software product is inadequate. We also find that the intermediary has to lower the service fee when there are lots of barriers to cloud adoption, even if the importance of the packaging service increases with the adoption cost. Finally, we find that the R&D for reducing adoption barriers implemented by the ISV may increase or damage the profit of the intermediary, which depends upon the benefits gained from its packaging service.

Appendix

1.1 Proof of Lemma 1

Solving $U^{(B)} = 0$ yields $\hat{\theta }^{(B)} = {{(\kappa + p_{s} )} \mathord{\left/ {\vphantom {{(\kappa + p_{s} )} \upsilon }} \right. \kern-0pt} \upsilon }$. Solving ${{\partial \pi_{s}^{(B)} } \mathord{\left/ {\vphantom {{\partial \pi_{s}^{(B)} } {\partial p_{s} }}} \right. \kern-0pt} {\partial p_{s} }} = 0$ yields the optimal solution $p_{s}^{ \ast }$ because ${{\partial^{2} \pi_{s}^{(B)} } \mathord{\left/ {\vphantom {{\partial^{2} \pi_{s}^{(B)} } {\partial p_{s}^{2} }}} \right. \kern-0pt} {\partial p_{s}^{2} }} < 0$.

1.2 Proof of Lemma 2

By backward induction, we first solve (3.2) and then figure out (3.3) by incorporating the solution of (3.2) into (3.3). In stage two the SaaS provider can raise the usage price $p_{c}$ until $U^{(C)} = 0$. Thus, its optimal usage price is given by $p_{c}^{ \ast } = \upsilon - \phi$ and the aggregated usage is given by $\mu_{c} = \alpha \int_{0}^{1} {\theta d\theta } = {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}$. In stage one the maximal $\omega_{s}$ cannot be higher than $\upsilon - \phi - \beta \kappa$; otherwise, the SaaS provider breaks the deal due to $\pi_{s}^{(C)} < 0$. Thus, we have $\omega_{s}^{ \ast } = \upsilon - \phi - \beta \kappa$ so that $\pi_{s}^{(C)} = \alpha \left( {\frac{\upsilon - \phi - \beta \kappa }{2}} \right) - H$. Moreover, solving $\pi_{s}^{(C)} - \pi_{s}^{(B)} > 0$ yields the inequality $\phi < \frac{{(\upsilon + \kappa )^{2} - (1 - \alpha )2\upsilon^{2} - 2(\kappa^{2} + \alpha \beta \kappa \upsilon ) - 4\upsilon H}}{2\alpha \upsilon }$.

1.3 Addendum to double marginalization problem

To incorporate a heterogeneous adoption cost into our model, the payoff of a corporation using the cloud service is rewritten as $U(\theta ) = \theta (\upsilon - \delta \cdot \phi - p_{c} )$ where $\delta$ is an index uniformly distributed over [0,1]. As a result, the corporations with the parameters $\delta \in \left[ {0,\hat{\delta }} \right]$ will adopt the cloud service if $p_{c} = \upsilon - \hat{\delta } \cdot \phi$. In addition, their overall usage is given by $\mu_{c} = \alpha \cdot \hat{\delta }\int_{0}^{1} {\theta d\theta } = \left. {\alpha \cdot \hat{\delta }\frac{{\theta^{2} }}{2}} \right|_{0}^{1} = \frac{{\alpha \cdot \hat{\delta }}}{2}$. Accordingly, the SaaS provider’s profit maximization problem is $\mathop {Max}\limits_{{\hat{\delta }}} \;\pi_{c} = \alpha \cdot \left( {\upsilon - \hat{\delta }\phi - \beta \cdot \kappa - \omega_{s} } \right){{\hat{\delta }} \mathord{\left/ {\vphantom {{\hat{\delta }} 2}} \right. \kern-0pt} 2}$. Solving ${{\partial \pi_{c} } \mathord{\left/ {\vphantom {{\partial \pi_{c} } {\partial \hat{\delta }}}} \right. \kern-0pt} {\partial \hat{\delta }}} = 0$ yields $\hat{\delta }^{ \ast } = {{(\upsilon - \beta \cdot \kappa - \omega_{s} )} \mathord{\left/ {\vphantom {{(\upsilon - \beta \cdot \kappa - \omega_{s} )} {(2\phi )}}} \right. \kern-0pt} {(2\phi )}}$. As for the ISV, its profit maximization problem is $\mathop {Max}\limits_{{\omega_{s} }} \;\pi_{s} = \omega_{s} \cdot \mu_{c} - H$. Solving ${{\partial \pi_{s} } \mathord{\left/ {\vphantom {{\partial \pi_{s} } {\partial \omega_{s} }}} \right. \kern-0pt} {\partial \omega_{s} }} = 0$ yields $\omega_{s}^{ \ast } = {{(\upsilon - \beta \cdot \kappa )} \mathord{\left/ {\vphantom {{(\upsilon - \beta \cdot \kappa )} 2}} \right. \kern-0pt} 2}$; therefore, combining boundary conditions (namely, $\hat{\delta }^{ \ast } = 1$), we have

$$\omega_{s}^{ \ast } = \left\{ {\begin{array}{*{20}l} {\frac{\upsilon - \beta \cdot \kappa }{2},} & {\upsilon \le \hat{\upsilon }} \\ {\upsilon - \beta \cdot \kappa - 2\phi ,} & {\upsilon > \hat{\upsilon }} \\ \end{array} } \right.,\;\hat{\delta }^{ * } = \left\{ {\begin{array}{*{20}l} {\frac{\upsilon - \beta \cdot \kappa }{4\phi },} & {\upsilon \le \hat{\upsilon }} \\ {1,} & {\upsilon > \hat{\upsilon }} \\ \end{array} } \right.\;{\text{and}}\; p_{c}^{ * } = \left\{ {\begin{array}{*{20}c} {\frac{{3\upsilon + \beta \cdot \kappa }}{4},} & {\upsilon \le \hat{\upsilon }} \\ {\upsilon - \phi ,} & {\upsilon > \hat{\upsilon }}, \\ \end{array} } \right.$$

where $\hat{\upsilon } \equiv \beta \cdot \kappa + 4\phi$. Subsequently, we consider an integrated firm consisting of the ISV and SaaS provider. The integrated firm’s profit maximization problem is $\pi_{s}^{m} = (p_{c}^{m} - \beta \cdot \kappa )\mu_{c},$ where $\mu_{c} = \alpha \cdot \delta^{m} \int_{0}^{1} {\theta d\theta }$. By the same approach, we have $\delta^{m} = {{(\upsilon - \beta \cdot \kappa )} \mathord{\left/ {\vphantom {{(\upsilon - \beta \cdot \kappa )} {(2\phi )}}} \right. \kern-0pt} {(2\phi )}}$ and $p_{c}^{m} = {{(\upsilon + \beta \cdot \kappa )} \mathord{\left/ {\vphantom {{(\upsilon + \beta \cdot \kappa )} 2}} \right. \kern-0pt} 2}$. Combining boundary conditions, we have

$$p_{c}^{m} = \left\{ {\begin{array}{*{20}l} {{{(\upsilon + \beta \cdot \kappa )} \mathord{\left/ {\vphantom {{(\upsilon + \beta \cdot \kappa )} 2}} \right. \kern-0pt} 2}} & {\upsilon \le \beta \cdot \kappa + 2\phi } \\ {\upsilon - \phi ,} & {\upsilon > \beta \cdot \kappa + 2\phi } \\ \end{array} } \right.\;{\text{and}}\;\delta^{m} = \left\{ {\begin{array}{*{20}l} {{{(\upsilon - \beta \cdot \kappa )} \mathord{\left/ {\vphantom {{(\upsilon - \beta \cdot \kappa )} {(2\phi )}}} \right. \kern-0pt} {(2\phi )}},} & {\upsilon \le \beta \cdot \kappa + 2\phi } \\ {1,} & {\upsilon > \beta \cdot \kappa + 2\phi } \\ \end{array} } \right.$$

Obviously,$p_{c}^{m} < p_{c}^{ * }$ when $\upsilon < \hat{\upsilon }$. Though $p_{c}^{m} = p_{c}^{ * } = \upsilon - \phi$ when $\upsilon > \hat{\upsilon }$, the ISV intending to sell its service to all l-type corporations has to charge a moderate $\omega_{s}^{ * }$ as an incentive to coordinate the SaaS provider. Consequently, the profit of the cloud supply chains is always less than that of the integrated firm when a heterogeneous adoption cost is considered.

1.4 Proof of Lemma 3

Solving $U^{(B)} = U^{(C)}$ yields $\hat{\theta }^{(M)} = {{(\kappa + p_{s} )} \mathord{\left/ {\vphantom {{(\kappa + p_{s} )} {(p_{c} + \phi )}}} \right. \kern-0pt} {(p_{c} + \phi )}}$. In stage two, solving ${{\partial \pi_{s}^{(M)} } \mathord{\left/ {\vphantom {{\partial \pi_{s}^{(M)} } {\partial p_{s} }}} \right. \kern-0pt} {\partial p_{s} }} = 0$ and ${{\partial \pi_{c}^{(M)} } \mathord{\left/ {\vphantom {{\partial \pi_{c}^{(M)} } {\partial p_{c} }}} \right. \kern-0pt} {\partial p_{c} }} = 0$ simultaneously yields $p_{c}^{ * } = \phi + 2\beta \kappa + 2\omega_{s}$ and $p_{s}^{ * } = \frac{{2\Delta_{1} \left( {2\Delta_{1} \Delta_{2} - \alpha \upsilon \kappa } \right) + \omega_{s} \left( {(4\omega_{s} + 8\Delta_{1} )\Delta_{2} - \alpha \upsilon \kappa } \right)}}{{4\Delta_{1} \left( {2(1 - \alpha )\Delta_{1} + \alpha \upsilon } \right) + \omega_{s} \left( {8\omega_{s} (1 - \alpha ) + 3\alpha \upsilon + 16(1 - \alpha )\Delta_{1} } \right),}}$ where $\Delta_{1} \equiv \phi + \beta \kappa$ and $\Delta_{2} \equiv \upsilon - (1 - \alpha )\kappa$. Note that

$$\begin{aligned} \frac{{\partial \pi_{s}^{(M)} (p_{s}^{ * } ,p_{c}^{ * } )}}{{\partial \omega_{s} }} = \hfill \\ \frac{\alpha }{2} \cdot \frac{{\left( {2(\Delta_{1} + \omega_{s} )(\upsilon + (1 - \alpha )\kappa ) + \alpha \upsilon \kappa } \right) \cdot \left( {2(5\upsilon - (1 - \alpha )3\kappa } \right)\Delta_{1} + \omega_{s} \left( {6\upsilon - (1 - \alpha )10\kappa ) - 3\alpha \upsilon \kappa } \right)}}{{\left( {4\Delta_{1} (2(1 - \alpha )\Delta_{1} + \alpha \upsilon } \right) + \omega_{s} \left( {8(\omega_{s} + 2\Delta_{1} )(1 - \alpha ) + 3\alpha \upsilon )} \right)^{2} }} \hfill \\ \end{aligned}$$

Because of $\upsilon > \kappa$, ${{\partial \pi_{s} } \mathord{\left/ {\vphantom {{\partial \pi_{s} } {\partial \omega_{s} }}} \right. \kern-0pt} {\partial \omega_{s} }} > 0$ when $\omega_{s} = 0$. Therefore, ${{\partial \pi_{s} } \mathord{\left/ {\vphantom {{\partial \pi_{s} } {\partial \omega_{s} }}} \right. \kern-0pt} {\partial \omega_{s} }} > 0$ for each $\omega_{s}$ as long as $\upsilon > 2\kappa$ (namely, $6\upsilon - (1 - {{\upalpha }})10\kappa > 0$). In fact, the restriction $\upsilon > 2\kappa$ can be replaced with $\upsilon > \kappa$ when $\alpha > {2 \mathord{\left/ {\vphantom {2 5}} \right. \kern-0pt} 5}$. In stage one the ISV can enhance its profit by raising $\omega_{s}$ due to ${{\partial \pi_{s} } \mathord{\left/ {\vphantom {{\partial \pi_{s} } {\partial \omega_{s} }}} \right. \kern-0pt} {\partial \omega_{s} }} > 0$. $p_{c}^{ * } \le \upsilon - \phi$ because $U^{(C)}$ cannot be lower than zero. In case ${{\omega_{s} \ge \upsilon } \mathord{\left/ {\vphantom {{\omega_{s} \ge \upsilon } 2}} \right. \kern-0pt} 2} - \phi - \beta \kappa$, the SaaS provider cannot raise its price anymore so that $p_{c}^{ * } = \upsilon - \phi$. Notice that the maximal $\omega_{s}^{ * }$ cannot be higher than $\upsilon - \phi - \beta \kappa$; otherwise, we have $\pi_{c}^{(M)} < 0$. As a result, the ISV will extract all of the SaaS provider’s surplus by demanding $\omega_{s}^{ * } = \upsilon - \phi - \beta \kappa$ and then the SaaS provider has to respond by charging $p_{c}^{ * } = \upsilon - \phi$. Thus, incorporating $\omega_{s}^{ * }$ and $p_{c}^{ * }$ into (3.4), we have $p_{s}^{ * } = \frac{{\upsilon^{2} - ( 1- \alpha )\upsilon \kappa - \alpha (\phi + \beta \kappa )\kappa }}{(2 - \alpha )\upsilon + \alpha (\phi + \beta \kappa )}$ by solving ${{\partial \pi_{s}^{(M)} } \mathord{\left/ {\vphantom {{\partial \pi_{s}^{(M)} } {\partial p_{s} }}} \right. \kern-0pt} {\partial p_{s} }} = 0$. Based on $p_{s}^{ * }$ and $p_{c}^{ * }$, we have $\hat{\theta }^{(M)} = {{(\kappa + p_{s} )} \mathord{\left/ {\vphantom {{(\kappa + p_{s} )} {(p_{c} + \phi )}}} \right. \kern-0pt} {(p_{c} + \phi )}} = \frac{\kappa + \upsilon }{(2 - \alpha )\upsilon + \alpha (\phi + \beta \kappa )}$. Note that $\hat{\theta }^{(B)} = {{(\kappa + p_{s} )} \mathord{\left/ {\vphantom {{(\kappa + p_{s} )} \upsilon }} \right. \kern-0pt} \upsilon }$ can be found in Lemma 1. Thus, we have the following equations:

$$\eta_{s}^{ * } = \alpha (1 - \hat{\theta }^{(M)} ) + (1 - \alpha )(1 - \hat{\theta }^{(B)} ) = \frac{(1 - \alpha )\upsilon + \alpha (\phi + \beta \kappa ) - \kappa }{(2 - \alpha )\upsilon + \alpha (\phi + \beta \kappa )},{\text{ and}}$$

$$\mu_{c} = \alpha \int_{0}^{{\hat{\theta }^{(M)} }} {\theta d\theta } = {{\alpha (\hat{\theta }^{(M)} )^{2} } \mathord{\left/ {\vphantom {{\alpha (\hat{\theta }^{(M)} )^{2} } 2}} \right. \kern-0pt} 2} = {{\alpha \left( {\frac{\kappa + \upsilon }{(2 - \alpha )\upsilon + \alpha (\phi + \beta \kappa )}} \right)^{2} } \mathord{\left/ {\vphantom {{\alpha \left( {\frac{\kappa + \upsilon }{(2 - \alpha )\upsilon + \alpha (\phi + \beta \kappa )}} \right)^{2} } 2}} \right. \kern-0pt} 2}$$

$${\text{As a result}},\pi_{s}^{(M)} = p_{s} \eta_{s} + \omega_{s} \mu_{c} - H = \frac{{(\kappa + \upsilon )^{2} - 2\kappa \left( {(2 - \alpha )\upsilon + \alpha (\phi + \beta \kappa )} \right)}}{2((2 - \alpha )\upsilon + \alpha (\phi + \beta \kappa ))} - H$$

1.5 Proof of Proposition 1

Comparing $\pi_{s}^{(M)}$ with $\pi_{s}^{(B)}$, we have $\pi_{s}^{(M)} > \pi_{s}^{(B)}$ when $\phi < Q_{2}$, where $Q_{2} \equiv \frac{{\alpha (\kappa + \upsilon )^{2} (\upsilon - \beta \kappa ) - 4H\upsilon \left( {(2 - {{\alpha }})\upsilon + \alpha \beta \kappa } \right)}}{{\alpha \left( {(\kappa + \upsilon )^{2} + 4H\upsilon } \right)}}$. Likewise, $\phi > Q_{3}$ may imply $\pi_{s}^{(M)} > \pi_{s}^{(C)}$, where $Q_{3} \equiv {{\left( {(1 - \alpha \beta )\kappa - (1 - \alpha )\upsilon } \right)} \mathord{\left/ {\vphantom {{\left( {(1 - \alpha \beta )\kappa - (1 - \alpha )\upsilon } \right)} \alpha }} \right. \kern-0pt} \alpha }$. Solving $Q_{2} - Q_{3} < 0$ yields the result.

1.6 Proof of Corollary 1

$$\frac{{\partial \eta_{c}^{ * } }}{\partial \kappa } = \frac{{\alpha \left( {(2 - \alpha )\upsilon + \alpha (\phi - \beta \upsilon )} \right)}}{{\left( {(2 - \alpha )\upsilon + \alpha (\phi + \beta \kappa )} \right)^{2} }} > 0,\;\frac{{\partial \eta_{c}^{ * } }}{\partial \upsilon } = \alpha \frac{{\left( {\alpha (\phi + \beta \kappa ) - (2 - \alpha )\kappa } \right)}}{{\left( {(2 - \alpha )\upsilon + \alpha (\phi + \beta \kappa )} \right)^{2} }},\;{{\partial \eta_{c}^{ * } } \mathord{\left/ {\vphantom {{\partial \eta_{c}^{ * } } {\partial \phi }}} \right. \kern-0pt} {\partial \phi }} < 0\;,{\text{ and}}\;{{\partial \eta_{c}^{ * } } \mathord{\left/ {\vphantom {{\partial \eta_{c}^{ * } } {\partial \beta }}} \right. \kern-0pt} {\partial \beta }} < 0,$$

where $\eta_{c}^{ * } = \alpha \hat{\theta }^{(M)}$ is specified in the proof of Lemma 3.

1.7 Proof of Lemma 4

Note that $\mu_{c} = {\alpha \mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-0pt} 2}$ and $\eta_{c} = \alpha$ if the ISV intends to adopt the pure cloud model. In addition, $\phi$ is replaced with $\sigma \phi$ when the packaging service is introduced into the cloud supply chain. Consequently, the maximal $\omega_{s}^{ * }$ cannot be higher than $\upsilon - \sigma \phi - \beta \kappa$; otherwise, we have $\pi_{c}^{ * } < 0$. Thus, given $\omega_{s}^{ * } = \upsilon - \sigma \phi - \beta \kappa$, we have $p_{c}^{ * } = \upsilon - \sigma \phi$ and $\pi_{s}^{(CI)} = {{\alpha (\upsilon - \sigma \phi - \beta \kappa - f)} \mathord{\left/ {\vphantom {{\alpha (\upsilon - \sigma \phi - \beta \kappa - f)} 2}} \right. \kern-0pt} 2}$. Subsequently, the demands in the mixed cloud model can be specified by $\eta_{c}^{(MI)} = \alpha \hat{\theta }^{(MI)}$ and $\eta_{s}^{(MI)} = \alpha (1 - \hat{\theta }^{(MI)} ) + (1 - \alpha )(1 - \hat{\theta }^{(BI)} )$, where $\hat{\theta }^{(MI)} = \frac{{\kappa + p_{s} }}{{p_{c} + \sigma \phi }}$ is derived by solving $U^{(B)} = U^{(C)}$ and $\hat{\theta }^{(BI)}$ is derived by solving $U^{(B)} = 0$. The next step is to solve ${{\partial \pi_{s}^{(MI)} } \mathord{\left/ {\vphantom {{\partial \pi_{s}^{(MI)} } {\partial p_{s} }}} \right. \kern-0pt} {\partial p_{s} }} = 0$ and ${{\partial \pi_{c}^{(MI)} } \mathord{\left/ {\vphantom {{\partial \pi_{c}^{(MI)} } {\partial p_{c} }}} \right. \kern-0pt} {\partial p_{c} }} = 0$ simultaneously to yield $p_{s}^{ * }$ as well as $p_{c}^{ * }$ and then consider ${{\partial \pi_{s}^{(MI)} (p_{s}^{ * } ,p_{c}^{ * } )} \mathord{\left/ {\vphantom {{\partial \pi_{s}^{(MI)} (p_{s}^{ * } ,p_{c}^{ * } )} {\partial \omega_{s} }}} \right. \kern-0pt} {\partial \omega_{s} }}$. Just like Lemma 3, ${{\partial \pi_{s}^{(MI)} } \mathord{\left/ {\vphantom {{\partial \pi_{s}^{(MI)} } {\partial \omega_{s} }}} \right. \kern-0pt} {\partial \omega_{s} }} > 0$ for each $\omega_{s}$, which can be examined by the same approach. Therefore, we incorporate $\omega_{s}^{ * } = \upsilon - \sigma \phi - \beta \kappa$ and $p_{c}^{ * } = \upsilon - \sigma \phi$ into $\pi_{s}^{(MI)}$, and then solve $\partial {{\pi_{s}^{(MI)} } \mathord{\left/ {\vphantom {{\pi_{s}^{(MI)} } {\partial p_{s} = 0}}} \right. \kern-0pt} {\partial p_{s} = 0}}$ to yield $p_{s}^{ * } = \frac{{\upsilon^{2} - ( 1- \alpha )\upsilon \kappa - \alpha (\sigma \phi + \beta \kappa )\kappa - \alpha \kappa f}}{(2 - \alpha )\upsilon + \alpha (\sigma \phi + \beta \kappa + f)}$. Subsequently, we can find the f satisfying $\pi_{s}^{(MI)} - \pi_{s}^{(CI)} = 0$, where $\pi_{s}^{(MI)} = \frac{1}{2}\left( {\frac{{(\upsilon + \kappa )^{2} - 2\left( {\alpha f + (2 - \alpha )\upsilon + \alpha (\beta \kappa + \sigma \phi )} \right)\kappa }}{{(2 - \alpha )\upsilon + \alpha \left( {\sigma \phi + \beta \kappa + f} \right)}}} \right)$.

1.8 Proof of Lemma 5

In this lemma and subsequent lemmas, we need to ensure that the ISV won’t reject this offer and the chosen f is the largest one and best for the intermediary. In addition, we need to examine both the mixed cloud model and the pure cloud model because the ISV accepting the contract may choose one of them as its business model.

1.8.1 Case 1: mixed cloud model

Note that $J$ is the intermediary’s optimal service fee in the mixed cloud model without any constraints. Solving $\pi_{s}^{(MI)} = \pi_{s}^{(C)}$ yields $f^{(MI)} = \frac{{(\upsilon + \kappa )^{2} }}{{\alpha \left( {\alpha (\upsilon - \phi - \beta \kappa ) + 2(\kappa - H)} \right)}} - J$, which is less than $J$ due to $\phi < Q_{1}$.

1.8.2 Case 2: pure cloud model

Solving $\pi_{s}^{(CI)} = \pi_{s}^{(C)}$ yields $f^{(CI)} = ( 1- \sigma )\phi { + }{{ 2H} \mathord{\left/ {\vphantom {{ 2H} \alpha }} \right. \kern-0pt} \alpha }$, which is the maximal service fee the intermediary may charge in the pure cloud model.

Results

Once $f^{ * } = \frac{(1 - \alpha \beta )\kappa - \alpha \sigma \phi - (1 - \alpha )\upsilon }{\alpha }$ holds, we have $\pi_{s}^{(C)} = \pi_{s}^{(MI)} = \pi_{s}^{(CI)}$, which is a degenerated case of the mixed cloud model. Thus, solving $f^{(MI)} = \frac{(1 - \alpha \beta )\kappa - \alpha \sigma \phi - (1 - \alpha )\upsilon }{\alpha }$ yields $\hat{\phi } = \frac{(1 - \alpha \beta )\kappa - 2H - ( 1- \alpha )\upsilon }{\alpha }$. Note that $f^{ * }$ is a continuous function when $\phi < Min\{ Q_{1} ,Q_{3} \}$ because $f^{(MI)} = f^{(CI)}$ when $\phi = \hat{\phi }$.

1.9 Proof of Lemma 6

1.9.1 Case 1: mixed cloud model

Note that $J$ is the intermediary’s optimal service fee in the mixed cloud model without any constraints. Solving $\pi_{s}^{(MI)} = \pi_{s}^{(M)}$ yields $f^{(MI)} = \frac{{(\varOmega - \alpha J)(\kappa + \upsilon )^{2} + 2\alpha H\varOmega J}}{{\alpha \left( {(\kappa + \upsilon )^{2} - 2H\varOmega } \right)}}$, which is less than $J$ due to $\phi < Q_{2}$.

1.9.2 Case 2: pure cloud model

Solving $\pi_{s}^{(CI)} = \pi_{s}^{(M)}$ yields $f^{(CI)} = \upsilon - \sigma \phi - \beta \kappa - \frac{2}{\alpha }\left( {\frac{{(\kappa + \upsilon )^{2} - 2\kappa \varOmega }}{2\varOmega } - H} \right)$, which is the maximal service fee the intermediary may charge in the business model.

Results

Solving $f^{(MI)} < \frac{(1 - \alpha \beta )\kappa - \alpha \sigma \phi - (1 - \alpha )\upsilon }{\alpha }$ yields $\hat{H} < \frac{{(\kappa + \upsilon ) \cdot \left( {(1 - \alpha \beta )\kappa - (1 - \alpha )\upsilon - \alpha \phi } \right)}}{2\varOmega }$. However, $Q_{3} < \phi$ may imply $(1 - \alpha \beta )\kappa - (1 - \alpha )\upsilon < \alpha \phi$. Therefore, $\hat{H} < 0$ is a contradiction so that the ISV always adopts the mixed cloud model.

1.10 Proof of Lemma 7

1.10.1 Case 1: mixed cloud model

Note that $J$ is the intermediary’s optimal service fee without any constraints. Solving $\pi_{s}^{(MI)} = \pi_{s}^{(B)}$ yields $f^{(MI)} = \upsilon - \sigma \phi - \beta \kappa$, which is less than $J$.

1.10.2 Case 2: pure cloud model

Solving $\pi_{s}^{(CI)} = \pi_{s}^{(B)}$ yields $f^{(CI)} = \frac{{2\alpha \upsilon (\upsilon - \sigma \phi - \beta \kappa ) - (\upsilon - \kappa )^{2} }}{2\alpha \upsilon }$, which is the maximal service fee the intermediary may charge in the business model.

Results

According to Lemma 4, the ISV always adopts the mixed cloud model because $f^{(CI)} > \frac{(1 - \alpha \beta )\kappa - \alpha \sigma \phi - (1 - \alpha )\upsilon }{\alpha }$. However, the ISV still keeps the baseline model when $\phi \ge {{(\upsilon - \beta \kappa )} \mathord{\left/ {\vphantom {{(\upsilon - \beta \kappa )} \sigma }} \right. \kern-0pt} \sigma }$ because cloud sales cannot increase revenue for the ISV partnering with the intermediary.

1.11 Proof of Proposition 2

Combining Lemma 5-7 leads to the result.

1.12 Proof of Proposition 3

• (1) In case of $\phi < Min\{ Q_{1} ,Q_{3} \}$

  Note that $\phi > \hat{\phi } \Leftrightarrow (\upsilon + \kappa ) - \left( {\alpha \cdot (\upsilon - \phi - \beta \kappa ) + 2\kappa - 2H} \right) > 0$. Thus, we have

  $$\frac{{\partial f^{ * } }}{\partial \phi } = \left\{ {\begin{array}{*{20}l} {\frac{{(\upsilon + \kappa )^{2} - \sigma \cdot \left( {\alpha \cdot (\upsilon - \phi - \beta \kappa ) + 2\kappa - 2H} \right)^{2} }}{{\left( {\alpha \cdot (\upsilon - \phi - \beta \kappa ) + 2\kappa - 2H} \right)^{2} }} > 0,} & {\quad \phi > \hat{\phi }} \\ {1 - \sigma > 0,} & {\quad \phi < \hat{\phi }} \\ \end{array} } \right.$$

• (2) In case of $Q_{3} < \phi < Q_{2}$

  Note that $f^{ * } = \frac{{(\varOmega - \alpha J)(\kappa + \upsilon )^{2} + 2\alpha H\varOmega J}}{{\alpha \left( {(\kappa + \upsilon )^{2} - 2H\varOmega } \right)}} > 0 \Rightarrow (\kappa + \upsilon )^{2} > 2H\varOmega$. Thus, we have

  $$\frac{{\partial f^{ * } }}{\partial \phi } = \frac{{4H\varOmega \sigma \left( {(\upsilon + \kappa )^{2} - H\varOmega } \right) + (1 - \sigma )(\upsilon + \kappa )^{4} }}{{\left( {(\upsilon + \kappa )^{2} - 2H\varOmega } \right)^{2} }} > 0$$

• (3) In case of $\phi > Max\{ Q_{1} ,Q_{2} \}$

  $${{\partial f^{ * } } \mathord{\left/ {\vphantom {{\partial f^{ * } } {\partial \phi }}} \right. \kern-0pt} {\partial \phi }} = - \sigma < 0$$

1.13 Proof of Proposition 4

• (1) In case of $\phi < Min\{ Q_{1} ,Q_{3} \}$

  $$\frac{{\partial \pi_{I}^{ * } }}{\partial \phi } = \left\{ {\begin{array}{*{20}c} {\varPsi ,} & {\quad \phi > \hat{\phi }} \\ {{{\alpha ( 1- \sigma )} \mathord{\left/ {\vphantom {{\alpha ( 1- \sigma )} 2}} \right. \kern-0pt} 2} > 0,} & {\quad \phi < \hat{\phi }} \\ \end{array} }, \right.$$

  where

  $$\varPsi \equiv - \frac{{\alpha \cdot \left\{ \begin{aligned} (H - \kappa )8\sigma \alpha \phi - 2(2 + \sigma )\alpha^{2} \upsilon \beta \kappa + 2\alpha^{2} \phi (\beta \kappa - \upsilon )(1 + 2\sigma ) + 4(\sigma + \beta )\upsilon \alpha \kappa + \sigma \alpha^{2} \upsilon^{2} \hfill \\ - 4\upsilon (1 + \sigma )(\alpha H + \kappa ) + (4\upsilon + 3\sigma \alpha \phi )\alpha \phi + \left( {(2 + \sigma )\alpha \beta - 4(1 + \sigma )} \right)\alpha \beta \kappa^{2} + 2(\alpha - 2)\alpha \upsilon^{2} \hfill \\ + 8(\upsilon - \sigma \kappa )H + (\upsilon - \kappa )^{2} + 4\sigma (\kappa^{2} + H^{2} ) + 4\alpha \beta \kappa H(1 + \sigma ) + 4(\alpha + \sigma ) \cdot \kappa \upsilon \hfill \\ \end{aligned} \right\}}}{{2(\upsilon + \kappa )^{2} }}$$

  Note that $\phi < Q_{3} \Leftrightarrow \alpha \phi < (1 - \alpha \beta )\kappa - (1 - \alpha )\upsilon$. Thus, we have

  $$\frac{{\partial \pi_{I}^{(MI)} }}{\partial \phi } = \left\{ {\begin{array}{*{20}c} {\alpha \cdot \frac{(3 - 2\alpha )\upsilon + 2\alpha \beta \kappa - \kappa }{2(\upsilon + \kappa )} > 0,} & {if\;\sigma = 0\;and\;\phi = \hat{\phi }} \\ {2\alpha \cdot \frac{(1 - \alpha )\upsilon - (1 - \alpha \beta )\kappa + \alpha \phi }{2(\upsilon + \kappa )} < 0\;,} & {if\;\sigma = 1\;and\;\phi = \hat{\phi }} \\ \end{array} } \right.$$

  That is, the sign of ${{\partial \pi_{I}^{ * } } \mathord{\left/ {\vphantom {{\partial \pi_{I}^{ * } } {\partial \phi }}} \right. \kern-0pt} {\partial \phi }}$ is uncertain when $\phi < Min\{ Q_{1} ,Q_{3} \}$ and $\phi > \hat{\phi }$.

• (2) In case of $Q_{3} < \phi < Q_{2}$

  The explicit value for ${{\partial \pi_{I}^{(MI)} } \mathord{\left/ {\vphantom {{\partial \pi_{I}^{(MI)} } {\partial \phi }}} \right. \kern-0pt} {\partial \phi }}$ is not reported because this involves a long expression, but this solution can be simplified when the parameter $\sigma$ is assigned as 0 or 1. Note that $f^{ * } > 0$ may imply $(\kappa + \upsilon )^{2} > 2H\varOmega$. Thus, we have

  $$\begin{aligned} \phi < Q_{2} \Leftrightarrow \alpha (\kappa + \upsilon )^{2} (\upsilon - \beta \kappa ) - 4H\upsilon \left( {(2 - \alpha )\upsilon + \alpha \beta \kappa } \right) > \phi \alpha \left( {(\kappa + \upsilon )^{2} + 4H\upsilon } \right) \Leftrightarrow \hfill \\ (\kappa + \upsilon )^{2} \left( {(2 - \alpha ) \cdot \upsilon - \alpha \phi + \alpha \beta \kappa } \right) - 4H\left( {(2 - \alpha )\upsilon + \alpha \beta \kappa + \phi \alpha } \right)\left( {(2 - \alpha ) \cdot \upsilon + \alpha \beta \kappa } \right) > \hfill \\ 2\left( {(\kappa + \upsilon )^{2} - 2H\varOmega } \right)((1 - \alpha ) \cdot \upsilon + \alpha \beta \kappa ) > 0 \hfill \\ \end{aligned}$$

  Consequently, ${{\partial \pi_{I}^{(MI)} } \mathord{\left/ {\vphantom {{\partial \pi_{I}^{(MI)} } {\partial \phi }}} \right. \kern-0pt} {\partial \phi }} = - {{(2\alpha H^{2} )} \mathord{\left/ {\vphantom {{(2\alpha H^{2} )} {(\upsilon + \kappa )^{2} }}} \right. \kern-0pt} {(\upsilon + \kappa )^{2} }} < 0$ if $\sigma = 1$;

  $$\frac{{\partial \pi_{I}^{(MI)} }}{\partial \phi } = \alpha \cdot \frac{{(\upsilon + \kappa )^{2} \cdot ((2 - \alpha ) \cdot \upsilon - \alpha \cdot \phi + \alpha \beta \kappa ) - 4H\left( {(2 - \alpha ) \cdot \upsilon + \alpha \phi + \alpha \beta \kappa } \right) \cdot \left( {(2 - \alpha ) \cdot \upsilon + \alpha \beta \kappa } \right)}}{{2\left( {(2 - \alpha ) \cdot \upsilon + \alpha \phi + \alpha \beta \kappa } \right)^{3} }} > 0$$

  if $\sigma = 0$. That is, the sign of ${{\partial \pi_{I}^{ * } } \mathord{\left/ {\vphantom {{\partial \pi_{I}^{ * } } {\partial \phi }}} \right. \kern-0pt} {\partial \phi }}$ is uncertain when $Q_{3} < \phi < Q_{2}$, but we can show how it links to the value of $\sigma$.

• (3) In case of $\phi > Max\{ Q_{1} ,Q_{2} \}$

  We have ${{\partial \pi_{I}^{(MI)} } \mathord{\left/ {\vphantom {{\partial \pi_{I}^{(MI)} } {\partial \phi }}} \right. \kern-0pt} {\partial \phi }} = {{ - \sigma \alpha (\kappa + \upsilon )^{2} } \mathord{\left/ {\vphantom {{ - \sigma \alpha (\kappa + \upsilon )^{2} } {(8\upsilon^{2} )}}} \right. \kern-0pt} {(8\upsilon^{2} )}} < 0$.