An analysis of product strategy in cloud transition considering SaaS customization

Abstract

When traditional enterprise software vendors adapt to software as a service (SaaS) practices and evolving to cloud service models, there is a major change occurring in the enterprise: a new hybrid product strategy consists of on-premises software and competitive customized SaaS. For the first time, we build a stylized model to reveal the influence of SaaS customization on the decisions of monopoly software vendor in the transition period. Increasing the customization efficiency of SaaS results in two possible structural regimes in the market. One is single on-premises software dominate the market if SaaS is customized at a low level and the other is hybrid products segment the market if SaaS is moderate-level customized. Surprisingly, software vendors with high customization proficiency should not allow SaaS products to dominate the market. It would benefit more from offering a competitive hybrid product strategy. Therefore, this paper does not recommend traditional software vendors to transform into pure cloud service providers. This key findings remain valid in the extended analysis of other customization technologies. Besides, the extension models show that both configuration and personalization customization technologies outperform the modification customization technology.

Appendices

Appendix

Proof of Proposition 1

The optimal price of on-premises software before transition is $p_{1}^{^{\prime}} = \frac{{c_{1}^{2} + \left( {\left( {\theta + 1} \right)c_{2} - 2{\mkern 1mu} u} \right)c_{1} + 2{\mkern 1mu} \theta {\mkern 1mu} c_{2}^{2} - u\left( {\theta + 1} \right)c_{2} + u^{2} }}{{2{\mkern 1mu} u - 2{\mkern 1mu} c_{1} }}$ and the market share of traditional software is ${\mkern 1mu} 1 - d^{*} = \frac{{c_{1}^{2} + \left( {\left( {\theta + 1} \right)c_{2} - 2{\mkern 1mu} u} \right)c_{1} + 2{\mkern 1mu} \theta {\mkern 1mu} c_{2}^{2} - u\left( {\theta + 1} \right)c_{2} + u^{2} }}{{2\left( {u - c_{1} - c_{2} } \right)\left( {u - c_{1} } \right)}}$, then we derive the profit of vendor by $\Pi_{1}^{*} = (1 - d^{*} )p_{1}^{^{\prime}} = \frac{{\left( {c_{1}^{2} + \left( {\left( {\theta + 1} \right)c_{2} - 2{\mkern 1mu} u} \right)c_{1} + 2{\mkern 1mu} \theta {\mkern 1mu} c_{2}^{2} - u\left( {\theta + 1} \right)c_{2} + u^{2} } \right)^{2} }}{{4\left( {u - c_{1} } \right)^{2} \left( {u - c_{1} - c_{2} } \right)}}$.

Then we take the derivative of those with respect to $\theta$, and we get.

$\frac{{\delta \Pi_{1} }}{\delta \theta } = - \frac{{\left( {2{\mkern 1mu} \theta {\mkern 1mu} c_{2}^{2} - \left( {\theta + 1} \right)\left( {u - c_{1} } \right)c_{2} + \left( {u - c_{1} } \right)^{2} } \right)c_{2} {\mkern 1mu} \left( { - c_{1} - 2{\mkern 1mu} c_{2} + u} \right)}}{{2\left( {u - c_{1} } \right)^{2} \left( {u - c_{1} - c_{2} } \right)}}$ $\frac{{\delta (1 - d^{*} )}}{\delta \theta } = {\mkern 1mu} \frac{{c_{2} ({\mkern 1mu} c_{1} + 2{\mkern 1mu} c_{2} - u)}}{{2\left( {u - c_{1} - c_{2} } \right)\left( {u - c_{1} } \right)}}$; $\frac{{\delta p_{1}^{^{\prime}} }}{\delta \theta } = \frac{{c_{2} {\mkern 1mu} \left( {c_{1} + 2{\mkern 1mu} c_{2} - u} \right)}}{{2{\mkern 1mu} u - 2{\mkern 1mu} c1}}$. Recall that $u > c_{1} + c_{2}$, then if $c_{1} + c_{2} < u < c_{1} + 2c_{2}$, we have $\frac{{\delta \Pi_{1}^{*} }}{\delta \theta } > 0$, $\frac{{\delta (1 - d^{*} )}}{\delta \theta } > 0$, $\frac{{\delta p_{1}^{^{\prime}} }}{\delta \theta } > 0$ and vice versa.

Proof of position 2

The software vendor’s decision problem under the hybrid product strategy is

$$\max_{{p_{1} ,p_{2} }} \Pi_{2} = \int_{0}^{{d^{*} }} {(p_{2} - c_{1} )xdx - twd^{*} Q_{{}}^{2*} + (1 - d^{*} )} p_{1}$$

$${\text{S.T.}}\;c_{1} \le p_{2} \le u - a(1 - \eta );\;p_{1} \ge 0;\;0 \le d^{*} \le 1$$

After simplifying the constraints we can write the Lagrange equation as

$$\begin{gathered} \Pi _{2} = \int_{0}^{{d^{*} }} {(p_{2} - c_{1} )xdx - twd^{*} Q^{{2*}} - \eta c_{3} + (1 - d^{*} )} p_{1} + \lambda _{1} {\mkern 1mu} \left( {p_{2} - c_{1} } \right) + \lambda _{2} {\mkern 1mu} \left( {\frac{{\theta c_{1} {\mkern 1mu} c_{2} + 2{\mkern 1mu} \theta {\mkern 1mu} {\mkern 1mu} c_{2} ^{2} - \theta {\mkern 1mu} {\mkern 1mu} c_{2} {\mkern 1mu} u}}{{u - c_{1} }} - p_{1} } \right) \hfill \\ \;\;\;\;\;\; + \lambda _{3} {\mkern 1mu} \left( {\frac{{ac_{1} {\mkern 1mu} \eta - a\eta {\mkern 1mu} u + \theta {\mkern 1mu} c_{1} {\mkern 1mu} {\mkern 1mu} c_{2} + 2{\mkern 1mu} \theta {\mkern 1mu} {\mkern 1mu} c_{2} ^{2} - \theta {\mkern 1mu} c_{2} {\mkern 1mu} u - ac_{1} + au + c_{1} ^{2} + {\mkern 1mu} c_{2} {\mkern 1mu} c_{1} - c_{1} {\mkern 1mu} u - {\mkern 1mu} c_{2} {\mkern 1mu} u}}{{ - u + c_{1} }} - p_{2} + p_{1} } \right) + \lambda _{4} {\mkern 1mu} p_{1} \hfill \\ \end{gathered}$$

There are 16 combinations of lambda and we summary all in the following table. For each lambda, it has two states. The first state is $\lambda_{i} { = }0$, and in the second state is $\lambda_{i} > 0$($i = 1,2,3,4.$). We denote state 1 as 0 and state 2 as 1. In this case, $\lambda_{1} { = }0,\lambda_{2} { = }0,\lambda_{3} { = }0,\lambda_{4} { = }0$ can be denoted as (0, 0, 0, 0); and $\lambda_{1} > 0,\lambda_{2} { = }0,\lambda_{3} { = }0,\lambda_{4} { = }0$ is written as (1, 0, 0, 0) Table

Table 3 All the combination of $\lambda$

3 and

Table 4 The result of Lagrange problem

4.

We begin with combination (1,1), then we discuss combination (1,4) and finally we explain combination (4,4).

Conditional Solution 1

When $\lambda_{1} > 0,\lambda_{2} = 0,\lambda_{3} = 0,\lambda_{4} = 0$, there is demand of both on-premises software and SaaS. Thus we have.

$p_{1}^{*} = \frac{{A_{1} u^{2} + B_{1} u - 2{\mkern 1mu} w\theta {\mkern 1mu} \left( {\left( {\eta - 1} \right)a + c_{2} } \right)tc_{1}^{2} }}{{2{\mkern 1mu} \left( {\left( {\eta - 1} \right)a - tw + c_{2} } \right)\left( {u - c_{1} } \right)u}}$ and $p_{2}^{*} = c_{1}$

$$d_{t}^{2*} = \frac{{\left( {\left( {wt - c_{2} } \right)\theta + \left( {\eta - 1} \right)a + c_{2} } \right)u^{2} + \left( {\left( {\left( { - 3{\mkern 1mu} wt + c_{2} } \right)\theta - c_{2} + \left( { - \eta + 1} \right)a} \right){\mkern 1mu} c_{1} + 2{\mkern 1mu} \theta {\mkern 1mu} c_{2}^{2} } \right)u + 2{\mkern 1mu} c_{1}^{2} t\theta {\mkern 1mu} w}}{{2\left( {\left( {\eta - 1} \right)a - wt + c_{2} } \right)\left( {u - {\mkern 1mu} c_{1} } \right)u}}$$

$$A_{1} = - \left( {\eta - 1} \right)^{2} a^{2} - \left( {\left( {\theta + 2} \right)c_{2} + \theta {\mkern 1mu} tw} \right)\left( {\eta - 1} \right)a + \left( {\left( { - \theta - 1} \right)c_{2} + \theta {\mkern 1mu} tw} \right)c_{2}$$

$$\begin{gathered} B_{1} = c_{1} {\mkern 1mu} {\mkern 1mu} \left( {\eta - 1} \right)^{2} a^{2} + 3{\mkern 1mu} \left( {\frac{2}{3}\theta {\mkern 1mu} c_{2}^{2} + \frac{1}{3}c_{1} {\mkern 1mu} {\mkern 1mu} \left( {\theta + 2} \right)c_{2} + c_{1} t\theta {\mkern 1mu} w} \right)\left( {\eta - 1} \right)a \hfill \\ \qquad + \left( {2{\mkern 1mu} \theta {\mkern 1mu} c_{2} {\mkern 1mu}^{2} + \left( {\left( { - 4{\mkern 1mu} tw + c_{1} {\mkern 1mu} } \right)\theta + c_{1} {\mkern 1mu} } \right)c_{2} + c_{1} {\mkern 1mu} t\theta {\mkern 1mu} w} \right)c_{2} \hfill \\ \end{gathered}$$

If $\eta > \eta_{1} = \frac{{\left( {\left( { - tw + c_{2} } \right)\theta + a - c_{2} } \right)u^{2} + \left( {\left( {\left( {3{\mkern 1mu} tw - c_{2} } \right)c_{1} - 2{\mkern 1mu} c_{2}^{2} } \right)\theta - c_{1} {\mkern 1mu} \left( {a - c_{2} } \right)} \right)u - 2{\mkern 1mu} \theta {\mkern 1mu} c_{1}^{2} wt}}{{au\left( {u - c_{1} } \right)}}$, we have $\lambda_{1} > 0$.

Conditional solution 2

(See combination (1,4)) when $\lambda_{1} = 0,\lambda_{2} = 0,\lambda_{3} = 0,\lambda_{4} > 0$ there is only demand of SaaS.

If $\eta > \eta_{2} = \frac{{4{\mkern 1mu} w\left( {u - c_{1} } \right)\left( {u/2 + a - c_{1} - c_{2} } \right)t - uc_{2} {\mkern 1mu} \left( {u - c_{1} - 2{\mkern 1mu} c_{2} } \right)c_{2} }}{{4wa\left( {u - c_{1} } \right)t}}$, we have $\lambda_{4} > 0$. And the prices are.

$p_{2}^{*} = \frac{{A_{2} u^{2} + B_{2} u + 4{\mkern 1mu} wtc_{1} \left( {c_{1} + \left( {\eta - 1} \right)a + c_{2} } \right)^{2} }}{{ - 4{\mkern 1mu} tw\left( {u - c_{1} } \right)\left( {\frac{ - u}{2} + \left( {\eta - 1} \right)a + c_{2} + c_{1} } \right) - uc{}_{2}{\mkern 1mu} \left( {u - c_{1} - 2{\mkern 1mu} c_{2} } \right)}}$ and $p_{1}^{*} = 0$.

Then $d_{t}^{3*} = - 2{\mkern 1mu} \frac{{\left( {w\left( { - u/2 + \left( {\eta - 1} \right)a + c_{2} + c_{1} } \right)\left( {u - c_{1} } \right)t + 1/4{\mkern 1mu} uc_{2} \left( {u - c_{1} - 2{\mkern 1mu} c_{2} } \right)} \right)\theta }}{{u\left( { - 2{\mkern 1mu} tw + c_{2} + \left( {\eta - 1} \right)a} \right)\left( {u - c_{1} } \right)}}$

$$A_{2} = 2{\mkern 1mu} \left( {\left( {\eta - 1} \right)a + c_{1} - c_{2} } \right)wt + \left( { - c_{1} + \left( {\eta - 1} \right)a + c_{2} } \right)c_{2}$$

$$B_{2} = - 4{\mkern 1mu} w\left( {\left( {\eta - 1} \right)^{2} a^{2} + \frac{2}{5}{\mkern 1mu} \left( {\eta - 1} \right)\left( {c_{1} + \frac{4}{5}{\mkern 1mu} c_{2} } \right)a + \frac{3}{2}{\mkern 1mu} c_{1}^{2} + \frac{3}{2}{\mkern 1mu} c_{1} c_{2} - c_{2}^{2} } \right)t - \left( { - c_{1} + \left( {\eta - 1} \right)a + c_{2} } \right)c_{2} {\mkern 1mu} \left( {c_{1} + 2{\mkern 1mu} c_{2} } \right)$$

Conditional Solution 3

When $\lambda_{1} = 0,\lambda_{2} = 0,\lambda_{3} = 0,\lambda_{4} = 0$ we derive a conditional solution with only demand of on-premises software (see combination (4, 4)). And we have $p_{2}^{*} = c_{1}$ and $p_{1}^{*} = - \frac{{c_{2} {\mkern 1mu} \theta {\mkern 1mu} \left( {c_{1} + 2c_{2} - u} \right)}}{{ - u + c_{1} }}$, $d_{2}^{1*} = 0$. Then $Eu_{2} = (u - c_{1} - a(1 - \eta ))d_{i} < 0$, thus $\eta < \eta_{3} = 1 - \frac{{u - c_{1} }}{a}$, then we have $\eta_{3} > \eta_{1}$. Thus when $\eta < \eta_{3}$,$Eu_{2} = (u - c_{1} - a(1 - \eta ))d_{i} < 0$ and there is only demand of on-premises software. So if $\eta < \eta_{1}$ there is only on-premises software dominated the market and if $\eta > \eta_{2}$ we have only demand of SaaS software, then if $\eta > \eta_{1}$ there is both demand of on-premises software and SaaS.

Proof of Lemma 1

According to proposition 2 if $\eta < \eta_{1}$ there is only on-premises software dominated the market.

Proof of Lemma 2

According to proposition 2 if $\eta_{1} < \eta < \eta_{2}$ we have both demand of SaaS software.

$$d_{t}^{2*} = \frac{{\left( {\left( {wt - c_{2} } \right)\theta + \left( {\eta - 1} \right)a + c_{2} } \right)u^{2} + \left( {\left( {\left( { - 3{\mkern 1mu} wt + c_{2} } \right)\theta - c_{2} + \left( { - \eta + 1} \right)a} \right){\mkern 1mu} c_{1} + 2{\mkern 1mu} \theta {\mkern 1mu} c_{2}^{2} } \right)u + 2{\mkern 1mu} c_{1}^{2} t\theta {\mkern 1mu} w}}{{2\left( {\left( {\eta - 1} \right)a - wt + c_{2} } \right)\left( {u - {\mkern 1mu} c_{1} } \right)u}}$$

$$\frac{{\delta d_{t}^{2*} }}{\delta \eta } = - \frac{{a\left( {\left( {\left( {tw - c_{2} } \right)\theta + tw} \right)\left( {c_{1} + 2{\mkern 1mu} c_{2} } \right)^{2} + \left( {\left( { - 3{\mkern 1mu} twc_{1} + c_{2} {\mkern 1mu} \left( {c_{1} + 2{\mkern 1mu} c_{2} } \right)} \right)\theta - twc_{1} } \right)\left( {c_{1} + 2c_{2} } \right) + 2{\mkern 1mu} c_{1}^{2} t\theta {\mkern 1mu} w} \right)}}{{4c_{2} {\mkern 1mu} \left( {\left( {\eta - 1} \right)a - tw + c_{2} } \right)^{2} \left( {c_{1} + 2{\mkern 1mu} c_{2} } \right)c_{2} }} > 0$$

Thus the demand of SaaS increases in customization proficiency.

Proof of Proposition 3

Recall the proof of proposition 2, we can see the prices of on-premises software and SaaS.

$$\left( {p_{1}^{*} ,\;p_{2}^{*} } \right) = \left( {\frac{{A_{1} u^{2} + B_{1} u - 2{\mkern 1mu} w\theta {\mkern 1mu} \left( {\left( {\eta - 1} \right)a + c_{2} } \right)tc_{1}^{2} }}{{2{\mkern 1mu} \left( {\left( {\eta - 1} \right)a - tw + c_{2} } \right)\left( {u - c_{1} } \right)u}};c_{1} } \right)$$

$$A_{1} = - \left( {\eta - 1} \right)^{2} a^{2} - \left( {\left( {\theta + 2} \right)c_{2} + \theta {\mkern 1mu} tw} \right)\left( {\eta - 1} \right)a + \left( {\left( { - \theta - 1} \right)c_{2} + \theta {\mkern 1mu} tw} \right)c_{2}$$

$$\begin{gathered} B_{1} = c_{1} {\mkern 1mu} {\mkern 1mu} \left( {\eta - 1} \right)^{2} a^{2} + 3{\mkern 1mu} \left( {\frac{2}{3}\theta {\mkern 1mu} c_{2} ^{2} + \frac{1}{3}c_{1} {\mkern 1mu} {\mkern 1mu} \left( {\theta + 2} \right)c_{2} + c_{1} t\theta {\mkern 1mu} w} \right)\left( {\eta - 1} \right)a \hfill \\ \;\;\;\;\; \quad + \left( {2{\mkern 1mu} \theta {\mkern 1mu} c_{2} {\mkern 1mu} ^{2} + \left( {\left( { - 4{\mkern 1mu} tw + c_{1} {\mkern 1mu} } \right)\theta + c_{1} {\mkern 1mu} } \right)c_{2} + c_{1} {\mkern 1mu} t\theta {\mkern 1mu} w} \right)c_{2} \hfill \\ \end{gathered}$$

Thus, the SaaS product line has an effect on on-premises product line by impacting the prices through customization proficiency $\eta$.

$$\eta^{*} \left| {_{{\frac{{\delta d_{t}^{2*} }}{\delta t} = 0}} } \right. = \frac{{\left( {a\theta + a - c_{2} } \right)u^{2} + \left( {\left( {\left( { - 3{\mkern 1mu} a + 2{\mkern 1mu} c_{2} } \right)c_{1} - 2{\mkern 1mu} c_{2}^{2} } \right)\theta - c_{1} {\mkern 1mu} \left( {a - c_{2} } \right)} \right)u + 2c_{1}^{2} \theta {\mkern 1mu} \left( {a - c_{2} } \right)}}{{\left( {u - c_{1} } \right)\left( {\left( {\theta + 1} \right)u - 2{\mkern 1mu} c_{1} {\mkern 1mu} \theta } \right)a}}$$

If $c_{1} + 2c_{2} > u > c_{1} + c_{2}$, we have $\eta_{1} < \eta^{*} < \eta_{2}$; when $\eta_{1} < \eta < \eta^{*}$, thus $\frac{{\delta d_{t}^{2*} }}{\delta t} > 0$; and when $\eta^{*} \le \eta < \eta_{2}$, we have $\frac{{\delta d_{t}^{2*} }}{\delta t} \le 0$, $\frac{{\delta d_{t}^{2*} }}{\delta t\delta \eta } < 0$.

Proof of Lemma 3

$\eta_{2} \le \eta < 1$, the optimal pricing strategy is.

$\left( {p_{1}^{*} ,p_{2}^{*} } \right) = \left( {0,\frac{{A_{2} u^{2} + B_{2} u + 4{\mkern 1mu} wtc_{1} \left( {c_{1} + \left( {\eta - 1} \right)a + c_{2} } \right)^{2} }}{{ - 4{\mkern 1mu} tw\left( {u - c_{1} } \right)\left( {\frac{ - u}{2} + \left( {\eta - 1} \right)a + c_{2} + c_{1} } \right) - uc{}_{2}{\mkern 1mu} \left( {u - c_{1} - 2{\mkern 1mu} c_{2} } \right)}}} \right)$

And the demand is.

$d^{*} = - 2{\mkern 1mu} \frac{{\left( {w\left( { - u/2 + \left( {\eta - 1} \right)a + c_{2} + c_{1} } \right)\left( {u - c_{1} } \right)t + 1/4{\mkern 1mu} uc_{2} \left( {u - c_{1} - 2{\mkern 1mu} c_{2} } \right)} \right)\theta }}{{u\left( { - 2{\mkern 1mu} tw + c_{2} + \left( {\eta - 1} \right)a} \right)\left( {u - c_{1} } \right)}}$.

$$A_{2} = 2{\mkern 1mu} \left( {\left( {\eta - 1} \right)a + c_{1} - c_{2} } \right)wt + \left( { - c_{1} + \left( {\eta - 1} \right)a + c_{2} } \right)c_{2}$$

$$B_{2} = - 4{\mkern 1mu} w\left( {\left( {\eta - 1} \right)^{2} a^{2} + \frac{2}{5}{\mkern 1mu} \left( {\eta - 1} \right)\left( {c_{1} + \frac{4}{5}{\mkern 1mu} c_{2} } \right)a + \frac{3}{2}{\mkern 1mu} c_{1}^{2} + \frac{3}{2}{\mkern 1mu} c_{1} c_{2} - c_{2}^{2} } \right)t - \left( { - c_{1} + \left( {\eta - 1} \right)a + c_{2} } \right)c_{2} {\mkern 1mu} \left( {c_{1} + 2{\mkern 1mu} c_{2} } \right)$$

Proof of proposition 5

The profit function of the vendor is:

$$\max_{{p_{1}^{c} ,p_{2}^{c} }} \Pi_{t}^{c} = \int_{0}^{{d^{c*} }} {(p_{2}^{c} - c_{1} )xdx - twd^{c*} Q_{{}}^{2*} - \eta^{c} c_{4} - \frac{1}{2}k(\eta^{c} )^{2} - m + (1 - d^{c*} )} p_{1}^{c}$$

S.T.$c_{1} \le p_{2}^{c} \le u - a(1 - \eta^{c} )$; $p_{1}^{c} \ge 0$; $0 \le d^{c*} \le 1$.

After simplifying the constraints we can write the Lagrange equation as

$$\begin{gathered} \Pi_{t}^{c} = \int_{0}^{{d^{c*} }} {(p_{2}^{c} - c_{1} )xdx - twd^{c*} Q^{2*} - \eta^{c} c_{3} + (1 - d^{c*} )} p_{1} + \lambda_{1} {\mkern 1mu} \left( {p_{2}^{c} - c_{1} } \right) + \lambda_{2} {\mkern 1mu} \left( {\frac{{\theta c_{1} {\mkern 1mu} c_{2} + 2{\mkern 1mu} \theta {\mkern 1mu} {\mkern 1mu} c_{2}^{2} - \theta {\mkern 1mu} {\mkern 1mu} c_{2} {\mkern 1mu} u}}{{u - c_{1} }} - p_{1}^{c} } \right) \hfill \\ \quad + \lambda_{3} {\mkern 1mu} \left( {\frac{{ac_{1} {\mkern 1mu} \eta - a\eta {\mkern 1mu} u + \theta {\mkern 1mu} c_{1} {\mkern 1mu} {\mkern 1mu} c_{2} + 2{\mkern 1mu} \theta {\mkern 1mu} {\mkern 1mu} c_{2}^{2} - \theta {\mkern 1mu} c_{2} {\mkern 1mu} u - ac_{1} + au + c_{1}^{2} + {\mkern 1mu} c_{2} {\mkern 1mu} c_{1} - c_{1} {\mkern 1mu} u - {\mkern 1mu} c_{2} {\mkern 1mu} u}}{{ - u + c_{1} }} - p_{2}^{c} + p_{1}^{c} } \right) + \lambda_{4} {\mkern 1mu} p_{1}^{c} \hfill \\ \end{gathered}$$

Then we get $\eta_{1}^{c} = \frac{{D_{0} u^{2} + Du + 2{\mkern 1mu} twc_{1}^{2} \theta }}{{au(c_{1} - u)}}$, $\eta_{2}^{c} = \frac{{D_{1} u^{2} + D_{2} u + 8{\mkern 1mu} tw\left( {tw + a - c_{1} /4 - c_{2} } \right)\theta {\mkern 1mu} c_{1} }}{{a\left( {8{\mkern 1mu} c_{1} {\mkern 1mu} t\theta {\mkern 1mu} w - 8{\mkern 1mu} uw\theta {\mkern 1mu} t + 3{\mkern 1mu} c_{1} u - 3{\mkern 1mu} u^{2} } \right)}}$,

$\eta_{3}^{c} = \frac{{D_{3} u^{2} + D_{4} u + 12{\mkern 1mu} tw\left( {4/3{\mkern 1mu} tw + a - c_{1} /3 - c_{2} } \right)\theta {\mkern 1mu} c_{1} }}{{4{\mkern 1mu} a\left( {3{\mkern 1mu} c_{1} {\mkern 1mu} t\theta {\mkern 1mu} w - 3{\mkern 1mu} uw\theta {\mkern 1mu} t + c_{1} u - u^{2} } \right)}}$, $\eta_{4}^{c} = 1 + \frac{{2{\mkern 1mu} tw - c_{2} }}{a}$.

$$D_{0} = \left( {\left( {tw - c_{2} } \right)\theta - a + c_{2} } \right)$$

$$D = \left( {\left( {\left( { - 3{\mkern 1mu} tw + c_{2} } \right)c_{1} + 2{\mkern 1mu} c_{2}^{2} } \right)\theta + c_{1} \left( {a - c_{2} } \right)} \right)$$

$D_{1} = \left( {\left( { - tw + c_{2} } \right)\theta - 4{\mkern 1mu} tw - 3{\mkern 1mu} a + 3{\mkern 1mu} c_{2} } \right)$, $D_{2} = \left( {\left( { - 8{\mkern 1mu} t^{2} w^{2} - 8{\mkern 1mu} \left( {a - 3/8{\mkern 1mu} c_{1} - c_{2} } \right)wt - c_{2} {\mkern 1mu} \left( {c_{1} + 2{\mkern 1mu} c_{2} } \right)} \right)\theta + 3{\mkern 1mu} \left( {4/3{\mkern 1mu} tw + a - c_{2} } \right)c_{1} } \right)$

$$D_{3} = \left( {\left( { - 2{\mkern 1mu} tw + c_{2} } \right)\theta - 8{\mkern 1mu} tw - 4{\mkern 1mu} a + 4{\mkern 1mu} c_{2} } \right)$$

$$D_{4} = \left( {\left( { - 16{\mkern 1mu} t^{2} w^{2} - 12{\mkern 1mu} \left( {a - c_{1} /2 - c_{2} } \right)wt - c_{2} {\mkern 1mu} \left( {c_{1} + 2{\mkern 1mu} c_{2} } \right)} \right)\theta + 4{\mkern 1mu} c_{1} {\mkern 1mu} \left( {2{\mkern 1mu} tw + a - c_{2} } \right)} \right)$$

Conditional solution 1

$\eta_{1}^{c} < \eta < \eta_{2}^{c}$, there are demands from both SaaS and on-premises.

$$p_{2}^{c*} = c_{1}$$

$$p_{1}^{c*} = \frac{{A_{3} u^{2} + B_{3} u - 2{\mkern 1mu} tw\left( {c_{2} + \left( {\eta - 1} \right)a} \right)c_{1}^{2} \theta }}{{2(c_{1} - u)u(a(\eta - 1) - wt + c_{2} )}}; \quad A_{3} = - \left( {\eta - 1} \right)^{2} a^{2} - \left( {\eta - 1} \right)\left( {\left( {tw + c_{2} } \right)\theta + 2{\mkern 1mu} c_{2} } \right)a + c_{2} {\mkern 1mu} \left( {\left( {tw - c_{2} } \right)\theta - c_{2} } \right)$$

$$\begin{gathered} B_{3} = c1{\mkern 1mu} \left( {\eta - 1} \right)^{2} a^{2} + 3{\mkern 1mu} \left( {\eta - 1} \right)\left( {\left( {c_{1} {\mkern 1mu} tw + 1/3{\mkern 1mu} c_{1} {\mkern 1mu} c_{2} + 2/3{\mkern 1mu} c_{2}^{2} } \right)\theta + 2/3{\mkern 1mu} c_{1} c_{2} } \right)a \hfill \\ \quad + c_{2} {\mkern 1mu} \left( {\left( {2{\mkern 1mu} c_{2}^{2} + \left( { - 4{\mkern 1mu} tw + c_{1} } \right)c_{2} + c_{1} tw} \right)\theta + c_{1} c_{2} } \right) \hfill \\ \end{gathered}$$

Conditional solution 2

$$\eta_{3}^{c} < \eta^{c} < \eta_{4}^{c}$$

There is only demand from SaaS.

$p_{1}^{c*} = 0$

$$p_{2}^{c*} = \frac{{A_{4} u^{2} + B_{4} u + 4{\mkern 1mu} wc_{1} {\mkern 1mu} t(c_{1} - c_{2} + a(\eta - 1))^{2} }}{{ - 4{\mkern 1mu} \left( {u - c_{1} } \right)\left( { - u/2 + \left( {\eta - 1} \right)a + c_{2} + c_{1} } \right)wt - uc_{2} {\mkern 1mu} \left( {u - c_{1} - 2{\mkern 1mu} c_{2} } \right)}}$$

$A_{4} = 2{\mkern 1mu} w\left( {\left( {\eta - 1} \right)a + c_{1} - c_{2} } \right)t + c_{2} {\mkern 1mu} \left( { - c_{1} + \left( {\eta - 1} \right)a + c_{2} } \right)$, $\begin{gathered} B_{4} = - 4{\mkern 1mu} w\left( {\left( {\eta - 1} \right)^{2} a^{2} + 5/2{\mkern 1mu} \left( {\eta - 1} \right)\left( {c_{1} + 4/5{\mkern 1mu} c_{2} } \right)a + 3/2{\mkern 1mu} c_{1}^{2} + 3/2{\mkern 1mu} c_{1} c_{2} - c_{2}^{2} } \right)t \hfill \\ \qquad - c_{2} {\mkern 1mu} \left( {c_{1} + 2{\mkern 1mu} c_{2} } \right)\left( { - c_{1} + \left( {\eta - 1} \right)a + c_{2} } \right) \hfill \\ \end{gathered}$.

Conditional Solution 3

When $\lambda_{1} = 0,\lambda_{2} = 0,\lambda_{3} = 0,\lambda_{4} = 0$ we derive a conditional solution with only demand of on-premises software. And we have.

$$p_{1}^{c*} = - \frac{{c_{2} {\mkern 1mu} \theta {\mkern 1mu} \left( {c_{1} + 2c_{2} - u} \right)}}{{ - u + c_{1} }};$$

$$p_{2}^{c*} = \frac{{u((c_{1} - u)(tw\theta - a(1 - \eta ) + c_{2} (1 - \theta ) + c_{1} ) - 2\theta c_{2}^{2} )}}{{(2tw\theta + u)(c_{1} - u)}}$$

We refer to the numerical simulation for the comparison of profits.

Proof of proposition 6

The profit of the software vendor is Table

Table 5 The result of lagrange problem

5.

$\max_{{p_{1}^{m} ,p_{2}^{m} }} \Pi_{t}^{m} = \int_{0}^{{d^{m*} }} {(p_{2}^{m} - c_{1} )xdx + (1 - d^{m*} )} p_{1}^{m} - twd^{m*} Q_{{}}^{2*} - d^{m*} \eta^{m} c_{3}$, S.T. $c_{1} \le p_{2} \le u$; $p_{1} \ge 0$; $0 \le d^{*} \le 1$. And the Lagrange equation is:

$$\begin{gathered} \Pi_{2} = \int_{0}^{{d^{*} }} {(p_{2} - c_{1} )xdx - wd^{*} Q^{2*} - d^{*} \eta c_{3} + (1 - d^{*} )} p_{1} + \lambda_{1} {\mkern 1mu} \left( {p_{2} - c_{1} } \right) + \lambda_{2} {\mkern 1mu} \left( {\frac{{\theta c_{1} {\mkern 1mu} c_{2} + 2{\mkern 1mu} \theta {\mkern 1mu} {\mkern 1mu} c_{2}^{2} - \theta {\mkern 1mu} {\mkern 1mu} c_{2} {\mkern 1mu} u}}{{u - c_{1} }} - p_{1} } \right) \hfill \\ + \lambda_{3} {\mkern 1mu} \left( {\left( { - u + c1} \right)p2 + p1{\mkern 1mu} \left( {u - c1} \right) + \left( {u - c1} \right)\left( {c1 + c2} \right) - \theta {\mkern 1mu} c2{\mkern 1mu} c1 - 2{\mkern 1mu} \theta {\mkern 1mu} c2^{2} + \theta {\mkern 1mu} c2{\mkern 1mu} u} \right) + \lambda_{4} {\mkern 1mu} p_{1} \hfill \\ \end{gathered}$$

We get two key points of efficiency:

$$\eta_{1}^{m} = \frac{{D_{5} u^{2} + D_{6} u + {\mkern 1mu} c_{1} {\mkern 1mu} \left( {8w - 2c_{1} - c_{2} } \right)w\theta }}{{c_{3} u\left( {u - c_{1} } \right)}}$$

$$\eta_{2}^{m} = \frac{{2{\mkern 1mu} c_{1}^{2} w\theta + c_{1} {\mkern 1mu} c_{2} {\mkern 1mu} \theta {\mkern 1mu} u - 3{\mkern 1mu} c_{1} \theta {\mkern 1mu} uw + 2{\mkern 1mu} c_{2}^{2} \theta {\mkern 1mu} u - c_{2} {\mkern 1mu} \theta {\mkern 1mu} u^{2} + \theta {\mkern 1mu} u^{2} w - uc_{2} {\mkern 1mu} c_{1} + u^{2} c_{2} }}{{c_{3} {\mkern 1mu} u\left( { - u + c_{1} } \right)}}$$

$D_{5} = \left( {\left( { - w + c_{2} } \right)\theta - 4{\mkern 1mu} w + 3{\mkern 1mu} c_{2} } \right)$, $D_{6} = \left( {\left( { - 8{\mkern 1mu} w^{2} + \left( {3{\mkern 1mu} c_{1} + 8{\mkern 1mu} c_{2} } \right)w - c_{2} {\mkern 1mu} \left( {c_{1} + 2{\mkern 1mu} c_{2} } \right)} \right)\theta + \left( {4w - 3c_{2} } \right)c_{1} } \right)$.

Conditional Solution 1

When $\lambda_{1} = 0,\lambda_{2} = 0,\lambda_{3} = 0,\lambda_{4} = 0$ we derive a conditional solution with only demand of on-premises software. And the prices are $p_{1}^{m*} = \frac{{c_{2} \theta {\mkern 1mu} \left( {c_{1} + 2{\mkern 1mu} c_{2} - u} \right)}}{{u - c_{1} }}$, $p_{2}^{m*} = \frac{{u((c_{2} (1 - \theta ) + c_{3} \eta + tw\theta + c_{1} )(c_{1} - u) - 2\theta c_{2}^{2} )}}{{(u + 2tw\theta )(c_{1} - u)}}$. The demand of SaaS is $d_{1}^{m*} = 0$.

Conditional solution 2

$\eta_{1}^{m} < \eta^{m} < \eta_{2}^{m}$, on-premises software compete with the SaaS. The prices are $p_{1}^{m*} = \frac{{c_{2} {\mkern 1mu} \left( {A_{3} u^{2} + B_{3} u - 2{\mkern 1mu} c_{1}^{2} tw\theta } \right)}}{{ - 2u\left( {w - c_{2} } \right)\left( {u - c_{1} } \right)}},p_{2}^{m*} = c_{1}$

$A_{3} = \left( { - \left( {\theta + 1} \right)c_{2} + \theta {\mkern 1mu} w - c3_{3} \eta^{m} } \right)$, $B_{3} = \left( {2{\mkern 1mu} c_{2}^{2} \theta + \left( {\left( {c_{1} - 4{\mkern 1mu} w} \right)\theta + c_{1} } \right)c_{2} + c_{1} {\mkern 1mu} \left( {c_{3} {\mkern 1mu} \eta^{m} + \theta {\mkern 1mu} w} \right)} \right)$

Conditional solution 3

$\eta^{m} > \eta_{2}^{m}$, customized SaaS expel the traditional software. We have $p_{2}^{m*} = \frac{{A_{4} u^{2} + B_{4} u + 4c_{1} \theta wt(c_{1} + c_{2} )^{2} }}{{4c_{1} w\theta t(c_{1} + c_{2} ) + c_{2} \theta u(c_{1} + 2c_{2} - u) - 2u\theta wt(3c_{1} + 2c_{2} - u) + 2c_{3} u\eta (u - c_{1} )}}$.

$p_{1}^{m*} = 0$.

$$A_{4} = (c_{2} - 2w)(c_{2} - c_{1} )\theta + 2c_{3} \eta^{m} (c_{1} + c_{2} )$$

$$B_{4} = ((c_{2} (c_{1} - c_{2} ) - 2w(3c_{1} - c_{2} ))(c_{1} + 2c_{2} ) + 4c_{1} c_{2} w)\theta - 2c_{1} c_{3} (c_{1} + c_{2} )\eta^{m}$$

We refer to the numerical simulation for the comparison of profits.