Vehicles manufacturer's pricing strategies considering service level of car-sharing modes

Abstract

As the era of sharing economy is coming, more and more auto manufacturers begin offering consumers cars for sales and for sharing simultaneously, but consumers have lower acceptance of sharing-cars in China. The key reason is that the availability of sharing-cars, as a service level, has much effect on consumers' time for searching a sharing-car. This study investigates the optimal pricing policies for sharing-cars and cars-for-sale, and reveals how service level affects the manufacturer's optimal decision and modes selection. We find that the hybrid mode should be adopted when consumers' unit time value is in an interval and service level is high. The combined optimal sharing price and sales price can be obtained and they are interralated with each other by key costs and service level in the hybrid mode. And the manufacturer should not introduce sharing-car programs when service level is less than a critical value. Furthermore, observations in numerical analysis show that the sales mode will be completely cannibalized if the sharing service level is much bigger and the optimal prices both first decreases and then increases with a hybrid mode.

Appendix

1.1 A1: Proof of Proposition 1

According to Eq. (4) and Eq. (5), we have

$$\tau (\theta )=\frac{v-{p}_{r}}{n\omega }{\lambda }_{1},$$

(A.1)

and

$$\tau \left(\theta \right)=\frac{{p}_{b}-{\varepsilon }_{b}-{\lambda }_{2}{p}_{r}}{n\omega }.$$

(A.2)

Based on the previous derivation, we have $v-{p}_{r}>0$, we can know that the searching time $\tau \left(\theta \right)$ is increasing in the marginal renter ${\lambda }_{1}$ and decreasing in the marginal buyer ${\lambda }_{2}$ when the prices are fixed from Equation (A-1)--(A-2).

Because $\frac{\partial {\lambda }_{1}}{\partial \omega }=\frac{n\tau (\theta )}{v-{p}_{r}}>0$, and $\frac{\partial {\lambda }_{2}}{\partial \omega }=\frac{-n\tau (\theta )}{{p}_{r}}<0$, $\frac{\partial {Q}_{r}}{\partial \omega }<0$ and $\frac{\partial {Q}_{b}}{\partial \omega }>0$.

1.2 A2: Proof of Proposition 2

(a) Taking the first derivative of $\Pi ({\lambda }_{1})$ with respect to${\lambda }_{1}$, we have $\frac{{\partial \prod \lambda_{1} }}{{\partial \lambda_{1} }}{ = }\frac{kn\omega }{{2\theta \lambda_{1}^{2} }} + \frac{kn\omega }{{2\theta }} - v\lambda_{1} + \theta (c_{r} - \varepsilon_{r} )$. We find that the derivative has at most one zero. When ${\lambda }_{1}$=0, the derivative is greater than zero. In order to ensure that the derivative has a point equal to zero, we must have: when ${\lambda }_{1}$=1, the derivative is less than zero. Therefore, there is a unique $0<{\lambda }_{1}^{*}<1$ optimizing Problem (11) when ${\theta }^{2}\left({c}_{r}-{\varepsilon }_{r}\right)+kn\omega -v\theta <0$, under the sharing strategy, we have maximum profit ${\pi }^{*}$ and optimal marginal renter${\lambda }_{1}^{*}$. Therefore, the service level must satisfies$\theta >\frac{-v+\sqrt{{v}^{2}+4kn\omega ({\varepsilon }_{r}-{c}_{r})}}{2({\varepsilon }_{r}-{c}_{r})}$.

(b) From Eq. (11), the first order condition is $\frac{{\partial \Pi \left( {\lambda_{1}^{ * } } \right)}}{\partial \theta } = - \left( {c_{r} - \varepsilon_{r} } \right)\left( {1 - \lambda_{1}^{ * } } \right) + \frac{{kn\omega \left( {1 - \lambda_{1}^{ * 2} } \right)}}{{2\theta^{2} \lambda_{1}^{ * } }} = 0$. The second order condition is $\frac{{\partial^{2} \Pi \left( {\lambda_{1}^{ * } } \right)}}{{\partial \theta^{2} }} = - \frac{{kn\omega \left( {1 - \lambda_{1}^{ * 2} } \right)}}{{\theta^{3} \lambda_{1}^{ * } }} < 0$. Hence, the optimal service level is $\theta^{ * } = \sqrt {\frac{{kn\omega \left( {1 + \lambda_{1}^{ * } } \right)}}{{2\lambda_{1}^{ * } \left( {c_{r} - \varepsilon_{r} } \right)}}}$.

Since the profit is a concave function of service level, let the profit in Eq. (11) equals to the opportunity cost $\mathrm{C}$. If $\frac{2C}{{1 - \lambda_{1}^{ * 2} }} < v < \frac{{2\left[ {C + \left( {c_{r} - \varepsilon_{r} } \right)\Upsilon_{11} } \right]}}{{1 - \lambda_{1}^{ * 2} }}$ and $k < k^{\prime}$, we obtain the range of the service level $\left[ {\underline {\theta } ,\overline{\theta }} \right]$ and $0 < \underline {\theta } < \overline{\theta } < 1$, where

$$ \underline {\theta } = \frac{1}{4}\left[ {\frac{{2C + v\left( { - 1 + \lambda_{1}^{ * 2} } \right)}}{{\left( {c_{r} - \varepsilon_{r} } \right)\left( { - \Upsilon_{11} } \right)}} - \sqrt {\frac{{ - 8kn\omega \Upsilon_{11}^{2} \left( {\Upsilon_{12} } \right)\left( {c_{r} - \varepsilon_{r} } \right) + \lambda_{1}^{ * } \left[ {2c + v\left( { - 1 + \lambda_{1}^{ * 2} } \right)} \right]^{2} }}{{\left( {c_{r} - \varepsilon_{r} } \right)^{2} \Upsilon_{11}^{2} \lambda_{1}^{ * } }}} } \right] > 0. $$

(A.3)

and

$$ \overline{\theta } = \frac{1}{4}\left[ {\frac{{2C + v\left( { - 1 + \lambda_{1}^{ * 2} } \right)}}{{\left( {c_{r} - \varepsilon_{r} } \right)\left( { - \Upsilon_{11} } \right)}} + \sqrt {\frac{{ - 8kn\omega \Upsilon_{11}^{2} \Upsilon_{12} \left( {c_{r} - \varepsilon_{r} } \right) + \lambda_{1}^{ * } \left[ {2c + v\left( { - 1 + \lambda_{1}^{ * 2} } \right)} \right]^{2} }}{{\left( {c_{r} - \varepsilon_{r} } \right)^{2} \Upsilon_{11}^{2} \lambda_{1}^{ * } }}} } \right]. $$

(A.4)

1.3 A3: Proof of Corollary 1

$$ \left( {\text{a}} \right)\quad \frac{{\partial \theta^{ * } }}{\partial \omega } = \frac{{kn\left( {1 + \lambda_{1}^{ * } } \right)}}{{2\sqrt 2 \left( {c_{r} - \varepsilon_{r} } \right)\lambda_{1}^{ * } \sqrt {\frac{{kn\omega \left( {1 + \lambda_{1}^{ * } } \right)}}{{\lambda_{1}^{ * } \left( {c_{r} - \varepsilon_{r} } \right)}}} }} > 0. $$

(A.5)

$$ \left( {\text{b}} \right)\quad \frac{{\partial \Pi^{ * } }}{\partial \omega } = \frac{{kn\left( { - 1 + \lambda_{1}^{ * 2} } \right)}}{{\sqrt 2 \lambda_{1}^{ * } \sqrt {\frac{{kn\omega \left( {1 + \lambda_{1}^{ * } } \right)}}{{\lambda_{1}^{ * } \left( {c_{r} - \varepsilon_{r} } \right)}}} }} < 0. $$

(A.6)

1.4 A4: Proof of Proposition 3

Taking the first derivative of $\Pi ({\lambda }_{2})$ with respect to ${\lambda }_{2}$, we have ${\lambda }_{2}^{*}=\frac{1}{2}-\frac{{\varepsilon }_{b}-{c}_{b}}{2v}$, and substitute into the Eq. (5), we receive the optimal sales price ${{p}_{b}}^{*}=\frac{1}{2}(v+{\varepsilon }_{b}+{c}_{b})$, substituting into the Eq. (12), we have ${\Pi }^{*}=\frac{{{(v+\varepsilon }_{b}-{c}_{b})}^{2}}{4v}$.

1.5 A5: Proof of Proposition 4

• (a) In this case, the manufacturer's abandonment of any one mode will cause his profits to fall. First, taking the first derivative of $\Pi ({\lambda }_{1},{\lambda }_{2})$ with respect to ${\lambda }_{2}$, we have:

  $$ \frac{{\partial \prod (\lambda_{1} ,\lambda_{2} )}}{{\partial \lambda_{2} }} = \frac{kn\omega }{{\theta \lambda_{1} }}(\lambda_{2} - {1}) + \theta (\varepsilon_{r} - c_{r} ) + (c_{b} - \varepsilon_{b} ){ + (}1 - \lambda_{2} )v - \frac{kn\omega }{\theta }. $$

  (A.7)

Then, taking the first derivative of $\Pi ({\lambda }_{1},{\lambda }_{2})$ with respect to ${\lambda }_{1}$, we have:

$$ \frac{{\partial \prod (\lambda_{1} ,\lambda_{2} )}}{{\partial \lambda_{1} }}{ = }\frac{kn\omega }{{\theta \lambda_{1}^{2} }}\lambda_{2} - \frac{kn\omega }{{2\theta \lambda_{1}^{2} }}\lambda_{2}^{2} + \frac{kn\omega }{{2\theta }} - v\lambda_{1} + \theta (c_{r} - \varepsilon_{r} ). $$

(A.8)

Furthermore, if

$$ \begin{gathered} \frac{{\partial^{2} \Pi \left( {\lambda_{1} ,\lambda_{2} } \right)}}{{\partial \lambda_{1}^{2} }} \cdot \frac{{\partial^{2} \Pi \left( {\lambda_{1} ,\lambda_{2} } \right)}}{{\partial \lambda_{2}^{2} }} - \left( {\frac{{\partial^{2} \Pi \left( {\lambda_{1} ,\lambda_{2} } \right)}}{{\partial \lambda_{1} \partial \lambda_{2} }}} \right)^{2} \hfill \\ = \left( {\frac{{ - v\theta \lambda_{1}^{3} + kn\left( {\lambda_{2} - 2} \right)\lambda_{2} \omega }}{{\theta \lambda_{1}^{3} }}} \right) \cdot \left( { - v + \frac{kn\omega }{{\theta \lambda_{1} }}} \right) - \left( {\frac{{kn\left( {2 - 2\lambda_{2} } \right)\omega }}{{2\theta \lambda_{1}^{2} }}} \right)^{2} > 0, \hfill \\ \end{gathered} $$

(A.9)

that is, $\theta < \frac{1}{2}\left( {G - H} \right)$ or $\theta > \frac{1}{2}\left( {G + H} \right)$, a manufacturer can obtain the optimal marginal renter ${\uplambda }_{1}^{*}$ and ${\uplambda }_{2}^{*}$ simultaneously (where $G{ = }\frac{kn}{{v\lambda_{1}^{3} }}\left( {\lambda_{1}^{2} + \left( {\lambda_{2} - 2} \right)\lambda_{2} \omega } \right)$ and $H{ = }\frac{kn\omega }{{v\lambda_{1}^{3} }}\sqrt {\left( {\lambda_{1}^{2} + \left( {\lambda_{2} - 2} \right)^{2} } \right)\left( {\lambda_{1}^{2} + \lambda_{2}^{2} } \right)}$).

Furthermore, if $G - H < 0$, $\omega > \frac{{\lambda_{1}^{2} }}{{\lambda_{2} \left( {2 - \lambda_{2} } \right) + \sqrt M }}$, where $M = \left( {\lambda_{1}^{2} + \left( {\lambda_{2} - 2} \right)^{2} } \right)\left( {\lambda_{1}^{2} + \lambda_{2}^{2} } \right)$. Sinceθ $\in$(0, 1), $\frac{1}{2}\left( {G + H} \right)$ has to be less than 1. Hence, $\omega < \frac{{{{2v\lambda_{1}^{3} } \mathord{\left/ {\vphantom {{2v\lambda_{1}^{3} } {kn}}} \right. \kern-\nulldelimiterspace} {kn}} - \lambda_{1}^{2} }}{{\lambda_{2} \left( {\lambda_{2} - 2} \right) + \sqrt M }}$. Therefore, when $\frac{{\lambda_{1}^{2} }}{{\lambda_{2} \left( {2 - \lambda_{2} } \right) + \sqrt M }} < \omega < \frac{{{{\left( {2v\lambda_{1}^{3} } \right)} \mathord{\left/ {\vphantom {{\left( {2v\lambda_{1}^{3} } \right)} {\left( {kn} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {kn} \right)}} - \lambda_{1}^{2} }}{{\lambda_{2} \left( {\lambda_{2} - 2} \right) + \sqrt M }}$, i.e., consumers’ unit time value is in an interval, a manufacturer can obtain the optimal marginal renter $\lambda_{1}^{ * }$ and $\lambda_{2}^{ * }$ simultaneously if $\theta > \frac{1}{2}\left( {G + H} \right)$.

(b) From (a), we know that the optimal marginal renter ${\lambda }_{1}^{*}$ and ${\lambda }_{2}^{*}$ exist. And from Eq. (4) and (5), we obtain that the combined optimal price $p_{b}^{ * }$ and $p_{r}^{ * }$ exist. Furthermore, through the first order condition, we have $p_{b}^{ * } = p_{r}^{ * } + c_{b} + \theta \left( {\varepsilon_{r} - c_{r} } \right)$.

(c) From Eq. (10), the first order condition is $\frac{{\partial \Pi \left( {\lambda_{1}^{ * } ,\lambda_{2}^{ * } } \right)}}{\partial \theta } = \frac{{2\left( {c_{r} - \varepsilon_{r} } \right)\theta^{2} \left( { - 1 + \lambda_{1}^{ * } } \right)\lambda_{1}^{ * } - kn\omega \left[ { - 1 + \lambda_{1}^{ * 2} - 2\left( { - 1 + \lambda_{2}^{ * } } \right)\left( {\lambda_{1}^{ * } - \lambda_{2}^{ * } } \right)} \right]}}{{2\theta^{2} \lambda_{1}^{ * } }} = 0$. The second order condition is $\frac{{\partial^{2} \Pi \left( {\lambda_{1}^{ * } ,\lambda_{2}^{ * } } \right)}}{{\partial \theta^{2} }} = \frac{{kn\omega \left[ { - 1 + \lambda_{1}^{ * 2} + 2\left( {1 - \lambda_{2}^{ * } } \right)\left( {\lambda_{1}^{ * } - \lambda_{2}^{ * } } \right)} \right]}}{{\theta^{3} \lambda_{1}^{ * } }} < 0$. Hence, the optimal service level is $\theta^{\prime * } = \sqrt {\frac{{kn\omega \left( { - 1 + \lambda_{1}^{ * 2} + 2\lambda_{1}^{ * } \Upsilon_{11} - 2\Upsilon_{21} \lambda_{2}^{ * } } \right)}}{{2\lambda_{1}^{ * } \left( {\varepsilon_{r} - c_{r} } \right)\Upsilon_{11} }}}$.

1.6 A6: Proof of Corollary 2

• (a) $\frac{{\partial \theta^{\prime * } }}{\partial \omega } = \frac{1}{2\sqrt 2 \omega }\sqrt {\frac{{kn\omega \left[ {1 - \lambda_{1}^{ * 2} + 2\left( {1 - \lambda_{2}^{ * } } \right)\left( {\lambda_{2}^{ * } - \lambda_{1}^{ * } } \right)} \right]}}{{\lambda_{1}^{ * } \left( {c_{r} - \varepsilon_{r} } \right)\left( {1 - \lambda_{1}^{ * } } \right)}}} > 0$,

Given $\lambda_{1}^{ * }$, $\frac{{\partial \theta^{\prime * } }}{{\partial \lambda_{2}^{ * } }} = \frac{{kn\omega \left( {1 + \lambda_{1}^{ * } - 2\lambda_{2}^{ * } } \right)}}{{\sqrt 2 \lambda_{1}^{ * } \left( {c_{r} - \varepsilon_{r} } \right)\left( {1 - \lambda_{1}^{ * } } \right)\sqrt {\frac{{kn\omega \left[ {1 - \lambda_{1}^{ * 2} + 2\left( {1 - \lambda_{2}^{ * } } \right)\left( {\lambda_{2}^{ * } - \lambda_{1}^{ * } } \right)} \right]}}{{\lambda_{1}^{ * } \left( {c_{r} - \varepsilon_{r} } \right)\left( {1 - \lambda_{1}^{ * } } \right)}}} }}$.

If $1 + \lambda_{1}^{ * } - 2\lambda_{2}^{ * } = \left( {1 - \lambda_{2}^{ * } } \right) - \left( {\lambda_{2}^{ * } - \lambda_{1}^{ * } } \right) > 0$, $\frac{{\partial \theta^{\prime * } }}{{\partial \lambda_{2}^{ * } }} > 0$;

if $1 + \lambda_{1}^{ * } - 2\lambda_{2}^{ * } = \left( {1 - \lambda_{2}^{ * } } \right) - \left( {\lambda_{2}^{ * } - \lambda_{1}^{ * } } \right) < 0$, $\frac{{\partial \theta^{\prime * } }}{{\partial \lambda_{2}^{ * } }} < 0$.

• (c) $\frac{{\partial \Pi^{ * } }}{\partial \omega } = \frac{{\left( {c_{r} - \varepsilon_{r} } \right)\left( { - 1 + \lambda_{1}^{ * } } \right)\sqrt {\frac{{kn\omega \left[ {1 - \lambda_{1}^{ * 2} + 2\left( {1 - \lambda_{2}^{ * } } \right)\left( {\lambda_{2}^{ * } - \lambda_{1}^{ * } } \right)} \right]}}{{\lambda_{1}^{ * } \left( {c_{r} - \varepsilon_{r} } \right)\left( {1 - \lambda_{1}^{ * } } \right)}}} }}{\sqrt 2 \omega } < 0$.