Start-up remanufacturing firms’ capacity investment: a two-period model

Abstract

Time decisions of capacity investment are critical for start-up third party remanufacturers (3PR), especially when they are involved in downstream competition from their original equipment manufacturer (OEM). Considering a two-period model, we analyze 3PRs’ capacity investment timing issues by assuming their objective to be the maximization of the probability of survival. 3PRs may invest early to enjoy the first-mover advantage, or late to achieve coordination with the OEM and obtain the time value. In this paper, we first consider single 3PR setting and find that whether the 3PR invests early or late is affected by the fixed investment cost, the unit capacity cost and the recognition of the remanufactured products. However, in two competing 3PRs setting, there are three equilibriums: (1) One 3PR invests early, and the other invests late, that is, two symmetric 3PRs lead to an asymmetric equilibrium. (2) Both 3PRs invest early. (3) Both 3PRs invest late. Interestingly, we find that both 3PRs’ investing early may result in “Prisoner’s Dilemma”, although they can be better off by investing late simultaneously. Besides, we note that, the quantity of new products in the first period may limit the end-of-use (EOU) products which are collected for remanufacturing by the 3PRs. This motives the 3PR investing early sometimes takes first-mover the advantage to set a high capacity so as to force the other 3PR to leave the market.

Appendix

See Table 

Table 4 Results in model SF and SS

4,

Table 5 Results in model CFF and CSS

5,

Table 6 Results in model CFS and CSF

6,

Table 7 Results in model SF and SS consider price elasticity of consumer surplus

7,

Table 8 Results in model CFF and CSS consider price elasticity of consumer surplus

8 and

Table 9 Results in model CFS and CSF consider price elasticity of consumer surplus

9.

1.1 The Proof of Table 4

In model SF, to obtain the equilibrium decisions, the OEM’s profit and the 3PR’s survival probability, we use the backward induction to resolve the dynamic game.

In the second period, the OEM sets $Q_{2N}$. Because ${{\partial^{2} \Pi_{{\text{OEM - SF}}} } \mathord{\left/ {\vphantom {{\partial^{2} \Pi_{{\text{OEM - SF}}} } {\partial Q_{2N}^{2} }}} \right. \kern-\nulldelimiterspace} {\partial Q_{2N}^{2} }} = - 2 < 0$, the profit of the OEM is a concave function with $Q_{2N}$. From the first order condition (5) with respective to $Q_{2N}$, we get the optimal reaction function

$$ Q_{2N} \left( \kappa \right) = \frac{{1 - \delta \kappa - c_{N} }}{2} $$

(A1)

Then, substitute $Q_{2N} \left( \kappa \right)$ into the OEM’s and the 3PR’s decision problems as follows:

$$ \Pi_{{\text{OEM - SF}}} = \mathop {\max }\limits_{{Q_{1N} }} \left( {1 - Q_{1N} - c_{N} } \right)Q_{1N} + \frac{{\left( {1 - \delta \kappa - c_{N} } \right)^{2} }}{4} $$

(A2)

$$ \psi_{{{\text{SF}}}} = \mathop {\max }\limits_{\kappa \ge 0} \Pr \left( {\left( {\frac{{\delta - \delta \left( {2 - \delta } \right)\kappa + \delta c_{N} }}{2} + \theta } \right)\kappa \ge c_{F} \kappa + B} \right) $$

(A3)

Obviously, maximizing (A3) is equivalent to

$$ \psi_{{{\text{SF}}}} = \mathop {\max }\limits_{\kappa \ge 0} \Pr \left( {\theta \ge \frac{B}{\kappa } + \frac{{\delta \left( {2 - \delta } \right)\kappa }}{2} - \frac{{\delta \left( {1 + c_{N} } \right)}}{2} + c_{F} } \right) $$

Consequently, maximizing the survival probability (A3) equivalent to minimizing

$$ \varphi \left( \kappa \right) = \frac{B}{\kappa } + \frac{{\delta \left( {2 - \delta } \right)\kappa }}{2} $$

(A4)

In the first period, the OEM and the 3PR decide $Q_{1N}$ and $\kappa$, respectively. From (A2) and (A4), we find that $Q_{1N}$ and $\kappa$ are independent without constraint $\kappa \le Q_{1N}$. Therefore, we can easy obtain the optimal decision by solve (A2) and (A4) as follows:

$$ \left\{ \begin{gathered} Q_{{{\text{1N}}}}^{{{\text{SF}}}} = \frac{{1 - c_{N} }}{2} \hfill \\ \kappa^{{{\text{SF}}}} = \sqrt {\frac{2B}{{\delta \left( {2 - \delta } \right)}}} \hfill \\ \end{gathered} \right. $$

(A5)

Then, we substitute $\kappa^{{{\text{SF}}}}$ back into the OEM’s optimal reaction function, the optimal quantity of new products in the second is obtained as follows:

$$ Q_{{{\text{2N}}}}^{{{\text{SF}}}} = \frac{{1 - c_{N} + \theta }}{2} - \sqrt {\frac{\delta B}{{2\left( {2 - \delta } \right)}}} $$

(A6)

Substituting (A5) into (A2) and (A3), the equilibrium profit of the OEM and the survival probability of the 3PR can be obtained as follows:

$$ \Pi_{{\text{OEM - SF}}}^{ * } = \frac{{\left( {1 - c_{N} } \right)^{2} }}{4} + \left( {\frac{{1 - c_{N} }}{2} - \sqrt {\frac{\delta B}{{2\left( {2 - \delta } \right)}}} } \right)^{2} $$

(A7)

$$ \psi_{{{\text{SF}}}}^{ * } = 1 - F\left( {\frac{{2\sqrt {2\delta \left( {2 - \delta } \right)B} + 2c_{F} - \delta \left( {1 + c_{N} } \right)}}{2}} \right) $$

(A8)

In model SS, the OEM first decides $Q_{1N}$ in the first period, and then she and the 3PR choose respectively $Q_{{{2}N}}$ and $\kappa$ in the second period. Similarly, we use the backward induction to resolve this two-stage game. In the second period, the OEM and the 3PR set $Q_{2N}$ and $\kappa$ simultaneously. From the two first order conditions (7) and (8) with respective to $Q_{2N}$ and $\kappa$ respectively, we get the two optimal reaction functions

$$ Q_{2N} \left( \kappa \right) = \frac{{1 - \delta \kappa - c_{N} }}{2} $$

(A9)

$$ \kappa \left( {Q_{2N} } \right) = \frac{{\delta \left( {1 - Q_{2N} } \right) + \theta - c_{S} }}{2\delta } $$

(A10)

Thus, the optimal decisions can be obtained as follows:

$$ Q_{{{\text{2N}}}}^{{{\text{SS}}}} = \frac{{2 - \delta - \theta + c_{S} - 2c_{N} }}{4 - \delta } $$

(A11)

$$ \kappa^{{{\text{SS}}}} = \frac{{\delta \left( {1 + c_{N} } \right) + 2\theta - 2c_{S} }}{{\delta \left( {4 - \delta } \right)}} $$

(A12)

Substituting (A11) and (A12) into the OEM’s and 3PR’s profits, we can rewrite them as follows:

$$ \Pi_{{\text{OEM - SS}}} = \mathop {\max }\limits_{{Q_{1N} }} \left( {1 - Q_{1N} - c} \right)Q_{1N} + \frac{{\left( {2 - \delta + c_{S} - 2c_{N} } \right)^{2} + \sigma^{2} }}{{\left( {4 - \delta } \right)^{2} }} $$

(A13)

$$ \psi_{{{\text{SS}}}} = \Pr \left( {\frac{{\left( {\delta \left( {1 + c_{N} } \right) + 2\theta - 2c_{S} } \right)^{2} }}{{\delta \left( {4 - \delta } \right)^{2} }} \ge B} \right) $$

(A14)

In the first period, the OEM decides $Q_{1N}$, then we get the optimal quantity in the first period as follows:

$$ Q_{{{\text{1N}}}} = \frac{{1 - c_{N} }}{2} $$

(A15)

Then, the equilibrium profit of the OEM and the survival probability of the 3PR can be obtained as follows:

$$ \Pi_{{\text{OEM - SF}}}^{ * } = \frac{{\left( {1 - c_{N} } \right)^{2} }}{4} + \frac{{\left( {2 - \delta + c_{S} - 2c_{N} } \right)^{2} + \sigma^{2} }}{{\left( {4 - \delta } \right)^{2} }} $$

(A16)

$$ \psi_{{{\text{SS}}}}^{*} = 1 - F\left( {\frac{{\left( {4 - \delta } \right)\sqrt {\delta B} + 2c_{S} - \delta \left( {1 + c_{N} } \right)}}{2}} \right) $$

(A17)

Then, we obtain these results in Table 4

1.2 The Proofs of Table 5 and Table 6

Because the proof processes of Table 5 and Table 6 are similar to that of the Table 4, we omit them.

1.3 The Proof of Proposition 1

Substituting (A11) and (A12) into inverse demand functions

$$ p_{{{\text{2N}}}} = 1 - Q_{{{\text{2N}}}} - \delta Q_{{\text{R}}} $$

$$ p_{{\text{R}}} = \delta \left( {1 - Q_{{{\text{2N}}}} - Q_{{\text{R}}} } \right) + \theta $$

we obtain the corresponding retail prices as follows:

$$ p_{{{\text{2N}}}}^{{{\text{SF}}}} = \frac{{1 + c_{N} }}{2} - \sqrt {\frac{\delta B}{{2\left( {2 - \delta } \right)}}} $$

(A18)

$$ p_{{\text{R}}}^{{{\text{SF}}}} = \frac{{\delta \left( {1 + c_{N} } \right)}}{2} - \sqrt {\frac{{\delta \left( {2 - \delta } \right)B}}{2}} $$

(A19)

And then, we take $Q_{{{\text{2N}}}}^{{{\text{SF}}}}$, $p_{{{\text{2N}}}}^{{{\text{SF}}}}$, $\kappa^{{{\text{SF}}}}$ and $p_{{\text{R}}}^{{{\text{SF}}}}$ respect to B,

$\frac{{\partial Q_{{{\text{2N}}}}^{{{\text{SF}}}} }}{\partial B} = - \frac{1}{2}\sqrt {\frac{\delta }{{2\left( {2 - \delta } \right)B}}} < 0$,

$\frac{{\partial p_{{{\text{2N}}}}^{{{\text{SF}}}} }}{\partial B} = - \frac{1}{2}\sqrt {\frac{\delta }{{2\left( {2 - \delta } \right)B}}} < 0$,

$\frac{{\partial \kappa^{{{\text{SF}}}} }}{\partial B} = \frac{1}{2}\sqrt {\frac{2}{{\delta \left( {2 - \delta } \right)B}}} > 0$,

$\frac{{\partial p_{{\text{R}}}^{{{\text{SF}}}} }}{\partial B} = - \frac{1}{2}\sqrt {\frac{{\delta \left( {2 - \delta } \right)}}{2B}} < 0$,

So, we can obtain the results of Proposition 1.

1.4 The Proof of Proposition 2

Because

$$ \psi_{{{\text{SF}}}}^{ * } = 1 - F\left( {\frac{{2\sqrt {2\delta \left( {2 - \delta } \right)B} + 2c_{F} - \delta \left( {1 + c_{N} } \right)}}{2}} \right) $$

$$ \psi_{{{\text{SS}}}}^{ * } = 1 - F\left( {\frac{{\left( {4 - \delta } \right)\sqrt {\delta B} + 2c_{S} - \delta \left( {1 + c_{N} } \right)}}{2}} \right) $$

$\psi_{{{\text{SF}}}}^{ * } > \psi_{{{\text{SS}}}}^{ * }$ if and only if $c_{S} - c_{F} > {{\left( {2\sqrt {2\left( {2 - \delta } \right)} - \left( {4 - \delta } \right)} \right)\sqrt {\delta B} } \mathord{\left/ {\vphantom {{\left( {2\sqrt {2\left( {2 - \delta } \right)} - \left( {4 - \delta } \right)} \right)\sqrt {\delta B} } 2}} \right. \kern-\nulldelimiterspace} 2}$.

1.5 The Proof of Proposition 3

From Tables 5 and 6, the survival probabilities of the 3PRs are as follows

$$ \psi_{{\text{CFF - A}}}^{*} = \psi_{{\text{CFF - B}}}^{*} = 1 - F\left( {3\sqrt {\frac{{\delta \left( {2 - \delta } \right)B}}{2}} - \frac{{\delta + \delta c_{N} }}{2} + c_{F} } \right) $$

$$ \psi_{{\text{CSS - A}}}^{*} = \psi_{{\text{CSS - B}}}^{*} = 1 - F\left( {\frac{{2\left( {3 - \delta } \right)\sqrt {\delta B} + 2c_{S} - \left( {\delta + \delta c_{N} } \right)}}{2}} \right) $$

$$ \psi_{{\text{CFS - A}}}^{*} = \psi_{{\text{CSF - B}}}^{*} = 1 - F\left( {\frac{{2\sqrt {\delta \left( {2 - \delta } \right)\left( {4 - \delta } \right)B} - \left( {\delta + \delta c_{N} + \left( {2 - \delta } \right)c_{S} - \left( {4 - \delta } \right)c_{F} } \right)}}{2}} \right) $$

$$ \psi_{{\text{CFS - B}}}^{*} = \psi_{{\text{CSF - A}}}^{*} = 1 - F\left( {\frac{{\left( {4 - \delta } \right)\sqrt {\delta B} - \left( {\delta + \delta c_{N} } \right) + 2c_{S} + \sqrt {\delta \left( {2 - \delta } \right)\left( {4 - \delta } \right)B} }}{2}} \right) $$

Obviously, $\psi_{{\text{CSF - A}}}^{*} = \psi_{{\text{CFS - B}}}^{*} > \psi_{{\text{CFF - A}}}^{*} = \psi_{{\text{CFF - B}}}^{*}$ holds if and only if $c_{F} - c_{S} >$ $\frac{{\sqrt {\delta B} }}{2}\left( {\left( {4 - \delta } \right) + \sqrt {\left( {2 - \delta } \right)\left( {4 - \delta } \right)} - 3\sqrt {2\left( {2 - \delta } \right)} } \right)$. We denote.

$\Delta_{3} = \frac{{\sqrt {\delta B} }}{2}\left( {\left( {4 - \delta } \right) + \sqrt {\left( {2 - \delta } \right)\left( {4 - \delta } \right)} - 3\sqrt {2\left( {2 - \delta } \right)} } \right)$.

In the same way, we can prove $\psi_{{\text{CFS - A}}}^{*} = \psi_{{\text{CSF - B}}}^{*} > \psi_{{\text{CSS - A}}}^{*} = \psi_{{\text{CSS - B}}}^{*}$ if and only if $c_{F} - c_{S} < \frac{{2\sqrt {\delta B} }}{4 - \delta }\left( {\left( {3 - \delta } \right) - \sqrt {\left( {2 - \delta } \right)\left( {4 - \delta } \right)} } \right)$. We denote.

$\Delta_{2} = \frac{{2\sqrt {\delta B} }}{4 - \delta }\left( {\left( {3 - \delta } \right) - \sqrt {\left( {2 - \delta } \right)\left( {4 - \delta } \right)} } \right)$.

In addition, we have that $\psi_{{\text{CFF - A}}}^{*} = \psi_{{\text{CFF - B}}}^{*} > \psi_{{\text{CSS - A}}}^{*} = \psi_{{\text{CSS - B}}}^{*}$ if and only if.

$c_{F} - c_{S} < \frac{{\sqrt {\delta B} }}{2}\left( {2\left( {3 - \delta } \right) - 3\sqrt {2\left( {2 - \delta } \right)} } \right)$. We denote.

$\Delta_{1} = \frac{{\sqrt {\delta B} }}{2}\left( {2\left( {3 - \delta } \right) - 3\sqrt {2\left( {2 - \delta } \right)} } \right)$.

Obviously, $\Delta_{1} < 0$, $\Delta_{{2}} > 0$ and $\Delta_{3} > 0$ always hold. In addition, we find that.

$\Delta_{3} - \Delta_{2} = \frac{{\sqrt {\delta B} }}{2}\left( {\frac{{\left( {2 - \delta } \right)^{2} }}{4 - \delta } + \frac{8 - \delta }{{4 - \delta }}\sqrt {\left( {2 - \delta } \right)\left( {4 - \delta } \right)} - 3\sqrt {2\left( {2 - \delta } \right)} } \right) > 0$.

Then, we consider four situations: $c_{F} - c_{S} < \Delta_{1}$, $\Delta_{1} < c_{F} - c_{S} < \Delta_{2}$, $\Delta_{2} < c_{F} - c_{S} < \Delta_{3}$ and $c_{F} - c_{S} > \Delta_{3}$, and discuss the equilibrium in each situation. Therefore, we can obtain the results of Proposition 3.

1.6 The Proof of Proposition 4

Because the more quantity of remanufactured products is remanufactured, the more environment-friendliness is, we can use the quantity of remanufactured products to measure the degree of environment-friendliness. In model SF and SS, the expected quantities of remanufactured products are as follows:

$$ Q_{{\text{R}}}^{{{\text{SF}}}} = \sqrt {\frac{2B}{{\delta \left( {2 - \delta } \right)}}} $$

$$ Q_{{\text{R}}}^{{{\text{SS}}}} = \frac{{\delta \left( {1 + c_{N} } \right) - 2c_{S} }}{{\delta \left( {4 - \delta } \right)}} $$

Thus, we have

$$ Q_{{\text{R}}}^{{{\text{SF}}}} - Q_{{\text{R}}}^{{{\text{SS}}}} = \sqrt {\frac{2B}{{\delta \left( {2 - \delta } \right)}}} - \frac{{\delta \left( {1 + c_{N} } \right) - 2c_{S} }}{{\delta \left( {4 - \delta } \right)}} $$

Obviously, if $B > \frac{{\left( {2 - \delta } \right)\left( {\delta \left( {1 + c_{N} } \right) - 2c_{S} } \right)}}{{2\delta \left( {4 - \delta } \right)^{2} }}$, $Q_{{\text{R}}}^{{{\text{SF}}}} > Q_{{\text{R}}}^{{{\text{SS}}}}$ holds, and vice versa.

1.7 The Proof of Proposition 5

Similar to Proposition 4, we can use the quantity of remanufactured products to measure the degree of environment-friendliness in each situation. The expected quantities of remanufactured products in model CFF, CSS, CFS and CSF are as follows

$$ Q_{{\text{R}}}^{{{\text{CFF}}}} = 2\sqrt {\frac{2B}{{2 - \delta }}} $$

$$ Q_{{\text{R}}}^{{{\text{CSS}}}} = \frac{{\delta \left( {1 + c_{N} } \right) - 2c_{S} }}{{\delta \left( {3 - \delta } \right)}} $$

$$ Q_{{\text{R}}}^{{{\text{CFS}}}} = Q_{{\text{R}}}^{{{\text{CSF}}}} = 2\sqrt {\frac{B}{{\delta \left( {2 - \delta } \right)\left( {4 - \delta } \right)}}} + \frac{{\delta \left( {1 + c_{N} } \right) - 2c_{S} }}{{\delta \left( {4 - \delta } \right)}} $$

Thus, we have

$$ Q_{{\text{R}}}^{{{\text{CFF}}}} > Q_{{\text{R}}}^{{{\text{CSS}}}} \Leftrightarrow B > \frac{{\left( {2 - \delta } \right)\left( {\delta \left( {1 + c_{N} } \right) - 2c_{S} } \right)^{2} }}{{8\delta^{2} \left( {3 - \delta } \right)^{2} }} $$

$$ Q_{{\text{R}}}^{{{\text{CSS}}}} > Q_{{\text{R}}}^{{{\text{CFS}}}} = Q_{{\text{R}}}^{{{\text{CSF}}}} \Leftrightarrow B < \frac{{\left( {2 - \delta } \right)\left( {\delta \left( {1 + c_{N} } \right) - 2c_{S} } \right)^{2} }}{{4\delta \left( {4 - \delta } \right)\left( {3 - \delta } \right)^{2} }} $$

$$ Q_{{\text{R}}}^{{{\text{CFF}}}} > Q_{{\text{R}}}^{{{\text{CFS}}}} = Q_{{\text{R}}}^{{{\text{CSF}}}} \Leftrightarrow \left( {\sqrt {2\delta \left( {4 - \delta } \right)} - 1} \right)\sqrt B > \frac{{\sqrt {2 - \delta } \left( {\delta \left( {1 + c_{N} } \right) - 2c_{S} } \right)}}{{2\sqrt {\delta \left( {4 - \delta } \right)} }} $$

Obviously, we have $\sqrt {2\delta \left( {4 - \delta } \right)} - 1 = \left\{ \begin{gathered} > 0,{\text{ if }}2 - {{\sqrt {14} } \mathord{\left/ {\vphantom {{\sqrt {14} } 2}} \right. \kern-\nulldelimiterspace} 2} < \delta \le 1 \hfill \\ \le 0,{\text{ if 0}} < \delta \le 2 - {{\sqrt {14} } \mathord{\left/ {\vphantom {{\sqrt {14} } 2}} \right. \kern-\nulldelimiterspace} 2} \hfill \\ \end{gathered} \right.$. For simplicity, we denote

$$ B_{1} = \frac{{\left( {2 - \delta } \right)\left( {\delta + \delta c_{N} - 2c_{S} } \right)^{2} }}{{4\delta \left( {4 - \delta } \right)\left( {3 - \delta } \right)^{2} }} $$

$$ B_{2} = \frac{{\left( {2 - \delta } \right)\left( {\delta + \delta c_{N} - 2c_{S} } \right)^{2} }}{{8\delta^{2} \left( {3 - \delta } \right)^{2} }} $$

$$ B_{3} = \frac{{\left( {2 - \delta } \right)\left( {\delta + \delta c_{N} - 2c_{S} } \right)^{2} }}{{4\delta \left( {4 - \delta } \right)\left( {\sqrt {2\delta \left( {4 - \delta } \right)} - 1} \right)^{2} }} $$

when $2 - {{\sqrt {14} } \mathord{\left/ {\vphantom {{\sqrt {14} } 2}} \right. \kern-\nulldelimiterspace} 2} < \delta \le 1$, we have $B_{1} < B_{2} < B_{3}$, and we can obtain the results of Proposition 5d–g. When ${0} < \delta \le 2 - {{\sqrt {14} } \mathord{\left/ {\vphantom {{\sqrt {14} } 2}} \right. \kern-\nulldelimiterspace} 2}$, $Q_{{\text{R}}}^{{{\text{CFF}}}} < Q_{{\text{R}}}^{{{\text{CFS}}}} = Q_{{\text{R}}}^{{{\text{CSF}}}}$ always hold, and the results of Proposition 5a–c can be obtained.

1.8 The Proof of Proposition 6

Now, we consider constraint $\kappa \le Q_{1N}$. In model SF, the OEM’s decision in second period remain unchanged from those in the benchmark case. Thus, we focus on the decisions making in the first period. Then, the OEM’s and the 3PR’s decision problems in the first period are as follows:

$$ \begin{gathered} \Pi_{{\text{OEM - SF}}} = \mathop {\max }\limits_{{Q_{1N} }} \left( {1 - Q_{1N} - c_{N} } \right)Q_{1N} + \frac{{\left( {1 - \delta \kappa - c_{N} } \right)^{2} }}{4} \hfill \\ {\text{s.t.}}\kappa \le Q_{{{\text{1N}}}} \hfill \\ \end{gathered} $$

(A20)

$$ \begin{gathered} \psi_{{{\text{SF}}}} = \mathop {\max }\limits_{\kappa \ge 0} \Pr \left( {\left( {\frac{{\delta - \delta \left( {2 - \delta } \right)\kappa + \delta c_{N} }}{2} + \theta } \right)\kappa \ge c_{F} \kappa + B} \right) \hfill \\ {\text{s.t.}}\kappa \le Q_{{{\text{1N}}}} \hfill \\ \end{gathered} $$

(A21)

The OEM’s optimal response function is $Q_{{{\text{1N}}}} = \left\{ \begin{gathered} {{\left( {1 - c_{N} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - c_{N} } \right)} {2}}} \right. \kern-\nulldelimiterspace} {2}},{\text{ if }}{{\left( {1 - c_{N} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - c_{N} } \right)} {2}}} \right. \kern-\nulldelimiterspace} {2}} \ge \kappa \hfill \\ {{\left( {1 - c_{N} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - c_{N} } \right)} {\left( {{2} + \delta } \right),{\text{ if }}{{\left( {1 - c_{N} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - c_{N} } \right)} {2}}} \right. \kern-\nulldelimiterspace} {2}} < \kappa }}} \right. \kern-\nulldelimiterspace} {\left( {{2} + \delta } \right),{\text{ if }}{{\left( {1 - c_{N} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - c_{N} } \right)} {2}}} \right. \kern-\nulldelimiterspace} {2}} < \kappa }} \hfill \\ \end{gathered} \right.$, and the 3PR’s optimal reaction function is $\kappa = \min \left( {\sqrt {\frac{2B}{{\delta \left( {2 - \delta } \right)}}} ,Q_{{{\text{1N}}}} } \right)$. Therefore, we can easily obtain the optimal decision as follows:

$$ Q_{{{\text{IN}}}}^{{{\text{SF}}}} = \left\{ \begin{gathered} {{\left( {1 - c_{N} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - c_{N} } \right)} {2}}} \right. \kern-\nulldelimiterspace} {2}},{\text{ if }}c_{{\text{N}}} < c_{{\text{N - SF}}} \hfill \\ {{\left( {1 - c_{N} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - c_{N} } \right)} {\left( {{2} + \delta } \right),{\text{ if }}c_{{\text{N}}} \ge c_{{\text{N - SF}}} }}} \right. \kern-\nulldelimiterspace} {\left( {{2} + \delta } \right),{\text{ if }}c_{{\text{N}}} \ge c_{{\text{N - SF}}} }} \hfill \\ \end{gathered} \right. $$

(A22)

$$ \kappa^{{{\text{SF}}}} = \left\{ \begin{gathered} \sqrt {\frac{2B}{{\delta \left( {2 - \delta } \right)}}} , \, c_{{\text{N}}} < c_{{\text{N - SF}}} \hfill \\ \frac{{1 - c_{N} }}{2}, \, c_{{\text{N}}} \ge c_{{\text{N - SF}}} \hfill \\ \end{gathered} \right. $$

(A23)

where $c_{{\text{N-SF}}} = 1 - {2}\sqrt {\frac{2B}{{\delta \left( {2 - \delta } \right)}}}$.

In model SS, the OEM’s and the 3PR’s decision problems in the second period are as follows:

$$ \Pi_{{\text{OEM-SS}}} = \mathop {\max }\limits_{{Q_{2N} }} \left( {1 - Q_{1N} - c_{N} } \right)Q_{1N} + \left( {1 - Q_{2N} - \delta \kappa - c_{N} } \right)Q_{2N} $$

(A24)

$$ \begin{gathered} \psi_{{{\text{SS}}}} = \Pr \left( {\mathop {\max }\limits_{\kappa \ge 0} \left( {\left( {\delta \left( {1 - Q_{2N} - \kappa } \right) + \theta } \right)\kappa } \right) \ge c_{S} \kappa + B} \right) \hfill \\ {\text{s.t.}}\kappa \le Q_{{{\text{1N}}}} \hfill \\ \end{gathered} $$

(A25)

The OEM’s optimal reaction function is $Q_{{{\text{2N}}}} = {{\left( {1 - \delta \kappa - c_{N} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \delta \kappa - c_{N} } \right)} {2}}} \right. \kern-\nulldelimiterspace} {2}}$, and the 3PR’s optimal reaction function is $\kappa = \min \left( {\frac{{\delta \left( {1 - Q_{{{\text{2N}}}} } \right) + \theta - c_{S} }}{2\delta },Q_{{{\text{1N}}}} } \right)$. Therefore, we can easily obtain the optimal decision as follows:

$$ \kappa = \left\{ \begin{gathered} \frac{{\delta \left( {1 + c_{N} } \right) + 2\theta }}{{\delta \left( {4 - \delta } \right)}},\frac{{\delta \left( {1 + c_{N} } \right) + 2\theta }}{{\delta \left( {4 - \delta } \right)}} \le Q_{1N} \hfill \\ Q_{1N} ,\frac{{\delta \left( {1 + c_{N} } \right) + 2\theta }}{{\delta \left( {4 - \delta } \right)}} > Q_{1N} \hfill \\ \end{gathered} \right. $$

(A26)

$$ Q_{2N} = \left\{ \begin{gathered} \frac{{2 - \delta - \theta - 2c_{N} }}{4 - \delta },\frac{{\delta \left( {1 + c_{N} } \right) + 2\theta }}{{\delta \left( {4 - \delta } \right)}} \le Q_{1N} \hfill \\ \frac{{1 - \delta Q_{1N} - c_{N} }}{2},\frac{{\delta \left( {1 + c_{N} } \right) + 2\theta }}{{\delta \left( {4 - \delta } \right)}} > Q_{1N} \hfill \\ \end{gathered} \right. $$

(A27)

In the first period, the OEM can anticipate the 3PR’s response but can’t observe the value of $\theta$. Therefore, from the perspective of the expectation, the OEM infer the 3PR’s and its response functions as follows:

$$ E[\kappa ] = \left\{ \begin{gathered} \frac{{\delta \left( {1 + c_{N} } \right)}}{{\delta \left( {4 - \delta } \right)}},\frac{{1 + c_{N} }}{4 - \delta } \le Q_{{{\text{1N}}}} \hfill \\ Q_{{{\text{1N}}}} ,\frac{{1 + c_{N} }}{4 - \delta } > Q_{{{\text{1N}}}} \hfill \\ \end{gathered} \right. $$

(A28)

$$ E[Q_{{{\text{2N}}}} ] = \left\{ \begin{gathered} \frac{{2 - \delta - 2c_{N} }}{4 - \delta },\frac{{1 + c_{N} }}{4 - \delta } \le Q_{{{\text{1N}}}} \hfill \\ \frac{{1 - \delta Q_{1N} - c_{N} }}{2},\frac{{1 + c_{N} }}{4 - \delta } > Q_{{{\text{1N}}}} \hfill \\ \end{gathered} \right. $$

(A29)

Substituting (A28) into (A29) and (A24), then we get the optimal quantity in the first period as follows:

$$ Q_{{{\text{IN}}}}^{{{\text{SF}}}} = \left\{ \begin{gathered} {{\left( {1 - c_{N} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - c_{N} } \right)} {2}}} \right. \kern-\nulldelimiterspace} {2}},{\text{ if }}c_{{\text{N}}} < c_{{\text{N - SS}}} \hfill \\ {{\left( {1 - c_{N} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - c_{N} } \right)} {\left( {{2} + \delta } \right),{\text{ if }}c_{{\text{N}}} \ge c_{{\text{N - SS}}} }}} \right. \kern-\nulldelimiterspace} {\left( {{2} + \delta } \right),{\text{ if }}c_{{\text{N}}} \ge c_{{\text{N - SS}}} }} \hfill \\ \end{gathered} \right. $$

(A30)

where $c_{{\text{N - SS}}} = \frac{2 - \delta }{{6 - \delta }}$.

1.9 The Proof of Corollary 1

Because $c_{{\text{N - SF}}} - c_{{\text{N - SS}}} = \frac{{4}}{6 - \delta } - {2}\sqrt {\frac{2B}{{\delta \left( {2 - \delta } \right)}}}$, $c_{{\text{N - SF}}} > c_{{\text{N - SS}}}$ holds if and only if $B < \frac{{2\delta \left( {2 - \delta } \right)}}{{\left( {6 - \delta } \right)^{2} }}$. Thus, we can obtain Corollary 1.

1.10 The Proof of Proposition 6

The proof process of CFF and CSS similar to that of SF and SS, so we omit them. In model CFS, the OEM’s and the 3PR-B’s decision problems in the second period are as follows:

$$ \Pi_{{\text{OEM - CFS}}} = {}_{{Q_{2N} }}^{\max } \left( {1 - Q_{1N} - c_{N} } \right)Q_{1N} + \left( {1 - Q_{2N} - \delta \left( {\kappa_{A} + \kappa_{B} } \right) - c_{2N} } \right)Q_{2N} $$

(A31)

$$ \begin{gathered} \psi_{{\text{CFS - B}}} = \Pr \left( {{}_{{\kappa_{B \ge 0} }}^{\max } \left( {\left( {\delta \left( {1 - Q_{2N} - \left( {\kappa_{A} + \kappa_{B} } \right)} \right) + \theta - c_{s} } \right)\kappa_{b} } \right) \ge B} \right) \hfill \\ {\text{s}}.{\text{t}}\,\kappa_{B} \le Q_{1N} - \kappa_{A} \hfill \\ \end{gathered} $$

(A32)

The OEM’s optimal reaction function is $Q_{{2{\text{N}}}} = {{\left( {1 - \delta \left( {\kappa_{A} + \kappa_{B} } \right) - c_{N} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \delta \left( {\kappa_{A} + \kappa_{B} } \right) - c_{N} } \right)} 2}} \right. \kern-\nulldelimiterspace} 2}$, and the 3PR-B’s optimal reaction function is $\kappa_{B} = \min \left( {\frac{{\delta \left( {1 - Q_{{2{\text{N}}}} - \kappa_{A} } \right) + \theta - c_{s} }}{2\delta },Q_{{1{\text{N}}}} - \kappa_{A} } \right)$. Therefore, under certain condition, the 3PR-B will set $\kappa_{B} = Q_{{{\text{1N}}}} - \kappa_{A}$. Next, we will prove in this situation, the 3PR-A will set $\kappa_{A} = Q_{{{\text{1N}}}}$. Substituting $\kappa_{A} = Q_{{{\text{1N}}}}$ and $Q_{{{\text{2N}}}} = {{\left( {1 - \delta Q_{{{\text{1N}}}} - c_{N} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \delta Q_{{{\text{1N}}}} - c_{N} } \right)} {2}}} \right. \kern-\nulldelimiterspace} {2}}$ into the 3PR-A’s decision problem as follows:

$$ \psi_{{\text{CFS - A}}} = \mathop {\max }\limits_{{\kappa_{A} \ge 0}} \Pr \left( {\left( {\delta \left( {1 - Q_{{{\text{2N}}}} - Q_{{{\text{1N}}}} } \right) + \theta - c_{F} } \right)\kappa_{A} \ge B} \right) $$

(A33)

Obviously, maximizing the survival probability (A33) equivalents to minimizing

$$ \frac{B}{{\kappa_{A} }} - \delta \left( {1 - Q_{{{\text{2N}}}} - Q_{{{\text{1N}}}} } \right) + c_{F} $$

(A34)

Therefore, the 3PR-A will choose $\kappa_{{\text{A}}}^{{{\text{CFS}}}} = Q_{{{\text{1N}}}}$. The two 3PRs coexist in the market if and only if $\kappa_{A} + \kappa_{B} < Q_{{{\text{1N}}}}$, i.e., $c_{N} < \frac{2 - \delta }{{6 - \delta }} + \frac{{4c_{S} }}{{\delta \left( {6 - \delta } \right)}} - \frac{4}{{\left( {6 - \delta } \right)}}\sqrt {\frac{{\left( {4 - \delta } \right)B}}{{\delta \left( {2 - \delta } \right)}}}$. Thus, we have $c_{{\text{N - CFS}}} = \frac{2 - \delta }{{6 - \delta }} + \frac{{4c_{S} }}{{\delta \left( {6 - \delta } \right)}} - \frac{4}{{\left( {6 - \delta } \right)}}\sqrt {\frac{{\left( {4 - \delta } \right)B}}{{\delta \left( {2 - \delta } \right)}}}$. The proof process of model CSF is same as that of model CFS, so we omit it.

1.11 The Proof of Corollary 2

Because $c_{{\text{N - CFF}}} > c_{{\text{N - CSS}}}$ if and only if $B < B_{3} = \frac{{\left( {2 - \delta } \right)\left( {\delta - c_{S} } \right)^{2} }}{{8\delta^{2} \left( {4 - \delta } \right)^{2} }}$, $c_{{\text{N - CFF}}} > c_{{\text{N - CFS}}}$ if and only if $B < B_{5} = \frac{{\left( {2 - \delta } \right)\left( {\delta - c_{S} } \right)^{2} }}{{\delta^{2} }}\left( {\left( {6 - \delta } \right)\sqrt 2 - \sqrt {\frac{{\left( {4 - \delta } \right)}}{\delta }} } \right)^{ - 2}$, and $c_{{\text{N - CSS}}} > c_{{\text{N - CFS}}}$ always hold, we can easy obtain the results of Corollary 2.

1.12 The Proof of Proposition 8

Because

$$ \begin{gathered} \psi_{{{\text{SF}}}}^{ * } = 1 - F\left( {c_{{\text{F}}} + \sqrt {\frac{{2\delta \left( {2 - \delta } \right)B}}{{b_{2} }}} - \frac{{\delta \left( {1 + b_{1} c_{{\text{N}}} } \right)}}{{2b_{2} }}} \right) \hfill \\ \psi_{{{\text{SS}}}}^{ * } = 1 - F\left( {\frac{{\left( {4 - \delta } \right)\sqrt {\delta b_{2} B} - \delta \left( {1 + b_{1} c_{N} } \right) + 2b_{2} c_{{\text{S}}} }}{{2b_{2} }}} \right) \hfill \\ \end{gathered} $$

$\psi_{{{\text{SF}}}}^{ * } > \psi_{{{\text{SS}}}}^{ * }$ if and only if $c_{S} - c_{F} > {{\left( {\left( {4 - \delta } \right) - \sqrt {8\left( {2 - \delta } \right)} } \right)\sqrt {\delta B} } \mathord{\left/ {\vphantom {{\left( {\left( {4 - \delta } \right) - \sqrt {8\left( {2 - \delta } \right)} } \right)\sqrt {\delta B} } {\left( {2\sqrt {b_{2} } } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2\sqrt {b_{2} } } \right)}}$.

1.13 The Proof of Proposition 9

From Tables 8 and 9, the survival probabilities of the 3PRs are as follows

$$ \psi_{{{\text{CFF}} - A}}^{ * } = \psi_{{{\text{CFF}} - B}}^{ * } = 1 - F\left( {3\sqrt {\frac{{\delta \left( {2 - \delta } \right)B}}{{2b_{2} }}} - \frac{{\delta + \delta b_{1} c_{N} }}{{2b_{2} }} + c_{{\text{F}}} } \right) $$

$$ \psi_{{{\text{CSS}} - A}}^{ * } = \psi_{{{\text{CSS}} - B}}^{ * } = 1 - F\left( {\frac{{\left( {3 - \delta } \right)\sqrt {\delta b_{2} B} }}{{b_{2} }} - \frac{{\delta + \delta b_{1} c_{N} }}{{2b_{2} }} + c_{S} } \right) $$

$$ \psi_{{{\text{CFS}} - A}}^{ * } = \psi_{{{\text{CSF}} - B}}^{ * } = 1 - F\left( {\sqrt {\frac{{\delta \left( {2 - \delta } \right)\left( {4 - \delta } \right)B}}{{b_{2} }}} - \frac{{\delta + \delta b_{1} c_{N} }}{{2b_{2} }} - \frac{{\left( {2 - \delta } \right)c_{S} }}{2} + \frac{{\left( {4 - \delta } \right)c_{{\text{F}}} }}{2}} \right) $$

$$ \psi_{{{\text{CFS}} - B}}^{ * } = \psi_{{{\text{CSF}} - A}}^{ * } = 1 - F\left( {\frac{{\left( {4 - \delta } \right)\sqrt {b_{2} \delta B} }}{{2b_{2} }} + \frac{{\sqrt {\delta b_{2} \left( {2 - \delta } \right)\left( {4 - \delta } \right)B} }}{{2b_{2} }} - \frac{{\delta + \delta b_{1} c_{N} }}{{2b_{2} }} + c_{S} } \right) $$

Obviously, $\psi_{{\text{CSF - A}}}^{*} = \psi_{{\text{CFS - B}}}^{*} > \psi_{{\text{CFF - A}}}^{*} = \psi_{{\text{CFF - B}}}^{*}$ holds if and only if $c_{F} - c_{S} >$ $\frac{{\sqrt {\delta B} }}{{2\sqrt {b_{2} } }}\left( {\left( {4 - \delta } \right) + \sqrt {\left( {2 - \delta } \right)\left( {4 - \delta } \right)} - 3\sqrt {2\left( {2 - \delta } \right)} } \right)$. We denote.

$\Delta_{3} \left( {\delta ,b_{2} } \right) = \frac{{\sqrt {\delta B} }}{{2\sqrt {b_{2} } }}\left( {\left( {4 - \delta } \right) + \sqrt {\left( {2 - \delta } \right)\left( {4 - \delta } \right)} - 3\sqrt {2\left( {2 - \delta } \right)} } \right)$.

In the same way, we can prove $\psi_{{\text{CFS - A}}}^{*} = \psi_{{\text{CSF - B}}}^{*} > \psi_{{\text{CSS - A}}}^{*} = \psi_{{\text{CSS - B}}}^{*}$ if and only if $c_{F} - c_{S} < \frac{{2\sqrt {\delta B} }}{{\sqrt {b_{2} } \left( {4 - \delta } \right)}}\left( {\left( {3 - \delta } \right) - \sqrt {\left( {2 - \delta } \right)\left( {4 - \delta } \right)} } \right)$. We denote.

$\Delta_{2} \left( {\delta ,b_{2} } \right) = \frac{{2\sqrt {\delta B} }}{{\sqrt {b_{2} } \left( {4 - \delta } \right)}}\left( {\left( {3 - \delta } \right) - \sqrt {\left( {2 - \delta } \right)\left( {4 - \delta } \right)} } \right)$.

In addition, we have that $\psi_{{\text{CFF - A}}}^{*} = \psi_{{\text{CFF - B}}}^{*} > \psi_{{\text{CSS - A}}}^{*} = \psi_{{\text{CSS - B}}}^{*}$ if and only if.

$c_{F} - c_{S} < \frac{{\sqrt {\delta B} }}{{2\sqrt {b_{2} } }}\left( {2\left( {3 - \delta } \right) - 3\sqrt {2\left( {2 - \delta } \right)} } \right)$. We denote.

$\Delta_{1} \left( {\delta ,b_{2} } \right) = \frac{{\sqrt {\delta B} }}{{2\sqrt {b_{2} } }}\left( {2\left( {3 - \delta } \right) - 3\sqrt {2\left( {2 - \delta } \right)} } \right)$.

Obviously, $\Delta_{1} \left( {\delta ,b_{2} } \right) < 0$, $\Delta_{{2}} \left( {\delta ,b_{2} } \right) > 0$ and $\Delta_{3} \left( {\delta ,b_{2} } \right) > 0$ always hold. In addition we find that.

$\Delta_{3} \left( {\delta ,b_{2} } \right) - \Delta_{2} \left( {\delta ,b_{2} } \right) = \frac{{\sqrt {\delta B} }}{{2\sqrt {b_{2} } }}\left( {\frac{{\left( {2 - \delta } \right)^{2} }}{4 - \delta } + \frac{8 - \delta }{{4 - \delta }}\sqrt {\left( {2 - \delta } \right)\left( {4 - \delta } \right)} - 3\sqrt {2\left( {2 - \delta } \right)} } \right) > 0$.

Then, we consider four situations: $c_{F} - c_{S} < \Delta_{1} \left( {\delta ,b_{2} } \right)$, $\Delta_{1} \left( {\delta ,b_{2} } \right) < c_{F} - c_{S} < \Delta_{2} \left( {\delta ,b_{2} } \right)$, $\Delta_{2} \left( {\delta ,b_{2} } \right) < c_{F} - c_{S} < \Delta_{3} \left( {\delta ,b_{2} } \right)$ and $c_{F} - c_{S} > \Delta_{3} \left( {\delta ,b_{2} } \right)$, and discuss the equilibrium in each situation. Therefore, we can obtain the results of Proposition 9.