Revenue sharing contracts for horizontal capacity sharing under competition

Abstract

The capacity-sharing strategy is a widely used strategy to alleviate the mismatch between supply and demand. To investigate the performance of the capacity-sharing strategy, we consider two firms competing for business in a single market in this paper. Both firms can choose to join the horizontal capacity-sharing strategy with a revenue-sharing contract. By comparing the equilibrium solutions of analytical models where firms share capacities and firms don’t share capacities respectively, we find that both firms raise their prices with a low revenue sharing rate when the total desired demand can be satisfied by the total capacities; But the change of prices depends on the combined effect of competition intensity and the revenue sharing rate if the capacity sharing can’t satisfy the total desired demand. In addition, we find that the profit of the firm with insufficient capacity increases if capacity sharing is present. However, the profit of the firm with underutilized capacity increases in the revenue sharing rate when it is small, but decreases in it when it becomes relatively large. Thus, capacity sharing is not always better for the firm with underutilized capacity.

Appendix

1.1 Proof of Proposition 1

Based on the benchmark model, we have

$$ \left\{ \begin{array}{l} \pi_{1}^{0} (p_{1} ) = (p_{1} - c_{1} )(a_{1} - p_{1} + \beta p_{2} ) \hfill \\ \pi_{2}^{0} (p_{2} ) = (p_{2} - c_{2} )(a_{2} - p_{2} + \beta p_{1} ) \hfill \\ \end{array} \right.. $$

We can conclude that the second derivatives of the profits with respect to the price are as follows:

$$ \left\{ \begin{array}{l} \frac{{\partial^{2} \pi_{1}^{0} }}{{\partial p_{1}^{2} }} = - 2 < 0 \hfill \\ \frac{{\partial^{2} \pi_{2}^{0} }}{{\partial p_{2}^{2} }} = - 2 < 0 \hfill \\ \end{array} \right.. $$

Thus, the firms’ profits are concave with the retailing prices. The optimal pricing strategies $\left( {p_{1}^{0*} ,p_{2}^{0*} } \right)$ exist.

Assuming $\left\{ \begin{array}{l} \partial \pi_{1}^{0} /\partial p_{1} = 0 \hfill \\ \partial \pi_{2}^{0} /\partial p_{2} = 0 \hfill \\ \end{array} \right.$, we can thus obtain

$$ \left\{ \begin{array}{l} p_{1}^{0*} = \frac{1}{{4 - \beta^{2} }}\left( {2\left( {a_{1} + c_{1} } \right) + \beta \left( {a_{2} + c_{2} } \right)} \right) \hfill \\ p_{2}^{0*} = \frac{1}{{4 - \beta^{2} }}\left( {2\left( {a_{2} + c_{2} } \right) + \beta \left( {a_{1} + c_{1} } \right)} \right) \hfill \\ \end{array} \right.. $$

Therefore, we have $\left( {\pi_{1}^{0*} (p_{1}^{0*} ,p_{2}^{0*} ),\pi_{2}^{0*} (p_{1}^{0*} ,p_{2}^{0*} )} \right)$ and $\left( {d_{1}^{0*} (p_{1}^{0*} ,p_{2}^{0*} ),d_{2}^{0*} (p_{1}^{0*} ,p_{2}^{0*} )} \right)$ as follows:

$$ \begin{aligned} & \left\{ \begin{array}{l} \pi_{1}^{0*} (p_{1}^{0*} ,p_{2}^{0*} ) = \frac{1}{{\left( {4 - \beta^{2} } \right)^{2} }}\left( {2a_{1} + \beta \left( {a_{2} + c_{2} } \right) - \left( {2 - \beta^{2} } \right)c_{1} } \right)^{2} \hfill \\ \pi_{2}^{0*} (p_{1}^{0*} ,p_{2}^{0*} )=\frac{1}{{\left( {4 - \beta^{2} } \right)^{2} }}\left( {2a_{2} + \beta \left( {a_{1} + c_{1} } \right) - \left( {2 - \beta^{2} } \right)c_{2} } \right)^{2} \hfill \\ \end{array} \right.; \\ & \left\{ \begin{array}{l} d_{1} (p_{1}^{0*} ,p_{2}^{0*} ) = \frac{1}{{4 - \beta^{2} }}\left( {2a_{1} + \beta \left( {a_{2} + c_{2} } \right) - \left( {2 - \beta^{2} } \right)c_{1} } \right) \hfill \\ d_{2} (p_{1}^{0*} ,p_{2}^{0*} ) = \frac{1}{{4 - \beta^{2} }}\left( {2a_{2} + \beta \left( {a_{1} + c_{1} } \right) - \left( {2 - \beta^{2} } \right)c_{2} } \right) \hfill \\ \end{array} \right.. \\ \end{aligned} $$

1.2 Proof of Corollary 1

The proof is straightforward and omitted.

1.3 Proof of Proposition 2

If $K_{1} < d_{1} (p_{1}^{0*} ,p_{2}^{0*} )$ and $K_{2} \ge d_{2} (p_{1}^{0*} ,p_{2}^{0*} )$, then $\left( {K_{1} ,K_{2} } \right)$ satisfies

$$ \left\{ \begin{array}{l} a_{1} > 2K_{1} + c_{1} \hfill \\ 2a_{2} \le 3K_{2} + c_{2} - a_{1} - c_{1} \hfill \\ \end{array} \right.. $$

If $K_{1} < d_{1} (p_{1}^{0*} ,p_{2}^{0*} )$ and $K_{2} \ge d_{2} (p_{1}^{0*} ,p_{2}^{0*} )$, the capacity of Firm 1 is limited and that of Firm 2 is not limited. The benchmark model becomes Case 4-1 as follows:

$$ \left\{ \begin{array}{l} \pi_{1}^{1} = (p_{1} - c_{1} )K_{1} \hfill \\ \pi_{2}^{1} = (p_{2} - c_{2} )(a_{2} - p_{2} + \beta p_{1} ) \hfill \\ \end{array} \right.. $$

Thus, the first derivatives of Firm 1’s profit and the second derivatives of Firm 2’s profit with respect to the price are as follows:

$$ \left\{ \begin{array}{l} \partial \pi_{1}^{1} /\partial p_{1} = K_{1} > 0 \hfill \\ \partial^{1} \pi_{2}^{2} /\partial p_{2}^{2} = - 2 < 0 \hfill \\ \end{array} \right.. $$

The profit of Firm 1 increases along with the price and Firm 2 has optimal pricing strategies.

Assuming $\left\{ \begin{array}{l} K_{1} = a_{1} - p_{1}^{1*} + \beta p_{2}^{1*} \hfill \\ \partial \pi_{2}^{2} /\partial p_{2} = 0 \hfill \\ \end{array} \right.$, we can thus obtain

$$ \left\{ \begin{array}{l} p_{1}^{1*} = \frac{1}{{2 - \beta^{2} }}\left( {2(a_{1} - K_{1} ) + \beta \left( {a_{2} + c_{2} } \right)} \right) \hfill \\ p_{2}^{1*} = \frac{1}{{2 - \beta^{2} }}\left( {a_{2} + c_{2} + \beta \left( {a_{1} - K_{1} } \right)} \right) \hfill \\ \end{array} \right.. $$

Given $(p_{1}^{1*} ,p_{2}^{1*} )$, we have $\left( {\pi_{1}^{1*} (p_{1}^{1*} ,p_{2}^{1*} ),\pi_{2}^{1*} (p_{1}^{1*} ,p_{2}^{1*} )} \right)$ and $\left( {d_{1}^{1*} (p_{1}^{1*} ,p_{2}^{1*} ),d_{2}^{1*} (p_{1}^{1*} ,p_{2}^{1*} )} \right)$ as follows:

$$ \begin{aligned} & \left\{ \begin{array}{l} \pi_{1}^{1*} = \frac{1}{{2 - \beta^{2} }}\left( {2(a_{1} K_{1} - K_{1}^{2} ) + \beta \left( {a_{2} K_{1} + c_{2} K_{1} } \right) - \left( {2 - \beta^{2} } \right)c_{1} K_{1} } \right) \hfill \\ \pi_{2}^{1*} = \frac{1}{{\left( {2 - \beta^{2} } \right)^{2} }}\left( {a_{2} + \beta (a_{1} - K_{1} ) - \left( {1 - \beta^{2} } \right)c_{2} } \right)^{2} \hfill \\ \end{array} \right.; \\ & \left\{ \begin{array}{l} d_{1} (p_{1}^{1*} ,p_{2}^{1*} ) = K_{1} \hfill \\ d_{2} (p_{1}^{1*} ,p_{2}^{1*} ) = \frac{1}{{2 - \beta^{2} }}\left( {a_{2} + \beta (a_{1} - K_{1} ) - \left( {1 - \beta^{2} } \right)c_{2} } \right) \hfill \\ \end{array} \right.. \\ \end{aligned} $$

1.4 Proof of Corollary 2

The proof is straightforward and then omitted.

1.5 Proof of Proposition 3

Based on the proof of Proposition 2, we have $\left\{ \begin{array}{l} a_{1} > 2K_{1} + c_{1} \hfill \\ 2a_{2} \le 3K_{2} + c_{2} - a_{1} - c_{1} \hfill \\ \end{array} \right.$.

We can now analyze that the prices and profits of Case 4-1 and the benchmark model are as follows:

$$ \left\{ \begin{array}{l} p_{1}^{0*} - p_{1}^{N * } = \frac{{ - 4a_{1} + \left( {4 - 2\beta^{2} } \right)c_{1} + \left( {8 - 2\beta^{2} } \right)K{}_{1} - 2\beta \left( {a_{2} + c_{2} } \right)}}{{\left( {4 - \beta^{2} } \right)\left( {2 - \beta^{2} } \right)}} \hfill \\ \quad < \frac{{ - 4\left( {2K_{1} + c_{1} } \right) + \left( {4 - 2\beta^{2} } \right)c_{1} + \left( {8 - 2\beta^{2} } \right)K_{1} - 2\beta \left( {a_{2} + c_{2} } \right)}}{{\left( {4 - \beta^{2} } \right)\left( {2 - \beta^{2} } \right)}} \le 0 \hfill \\ p_{2}^{0*} - p_{2}^{N * } = \frac{{ - 2\beta a_{1} + \left( {2\beta - \beta^{3} } \right)c_{1} + \left( {4\beta - 4\beta^{3} } \right)K{}_{1} - \beta^{2} \left( {a_{2} + c_{2} } \right)}}{{\left( {4 - \beta^{2} } \right)\left( {2 - \beta^{2} } \right)}} \hfill \\ \quad< \frac{{ - 2\beta \left( {2K_{1} + c_{1} } \right) + \left( {2\beta - \beta^{3} } \right)c_{1} + \left( {4\beta - 4\beta^{3} } \right)K{}_{1} - \beta^{2} \left( {a_{2} + c_{2} } \right)}}{{\left( {4 - \beta^{2} } \right)\left( {2 - \beta^{2} } \right)}} \le 0 \hfill \\ \end{array} \right.; $$

We denote the following:

$$ \begin{aligned} L_{1} \left( {K_{1} } \right) & = 2\left( {4 - \beta^{2} } \right)^{2} K_{1}^{2} - \left( {4 - \beta^{2} } \right)^{2} \left( {2a_{1} - 2c_{1} + a_{2} \beta + \beta c_{2} + \beta^{2} c_{1} } \right)K_{1} \\ & \quad + \,\left( {2 - \beta^{2} } \right)\left( {2a_{1} - 2c_{1} + a_{2} \beta + \beta c_{2} + \beta^{2} c_{1} } \right)^{2} , \\ \end{aligned} $$

$$ \begin{aligned} L_{2} \left( {K_{1} } \right) & = - \beta^{2} \left( {4 - \beta^{2} } \right)^{2} K_{1}^{2} - 2\beta \left( {4 - \beta^{2} } \right)^{2} \left( {a_{2} - c_{2} + a_{1} \beta + \beta^{2} c_{2} } \right)K_{1} \\ & \quad+ \left( {2a_{1} - 2c_{1} + a_{2} \beta + \beta c_{2} + \beta^{2} c_{1} } \right) \left(2\beta^{4} c_{2} + \beta^{3} \left( {2a_{1} + c_{1} } \right) \right.\\ &\quad \left.+\, 3\beta^{2} \left( {a_{2} - 3c_{2} } \right)- 2\beta \left( {3a_{1} + c_{1} } \right) - 8\left( {a_{2} - c_{2} } \right) \right) \\ \end{aligned} $$

We have

$$ \pi_{1}^{0 * } - \pi_{1}^{N * } = \frac{1}{{\left( {4 - \beta^{2} } \right)^{2} \left( {2 - \beta^{2} } \right)}}\left( {L_{1} \left( {K_{1} } \right)} \right),\quad \pi_{2}^{0 * } - \pi_{2}^{N * } = \frac{1}{{\left( {4 - \beta^{2} } \right)^{2} \left( {2 - \beta^{2} } \right)^{2} }}\left( {L_{2} \left( {K_{1} } \right)} \right). $$

• If $L_{1} \ge 0$, then $\pi_{1}^{0*} \ge \pi_{1}^{N*}$. If $L_{1} < 0$, then $\pi_{1}^{0*} < \pi_{1}^{N*}$.

• If $L_{2} \ge 0$, then $\pi_{2}^{0*} \ge \pi_{2}^{N*}$. If $L_{2} < 0$, then $\pi_{2}^{0*} < \pi_{2}^{N*}$.

1.6 Proof of Proposition 4

Based on Model S:

$$ \left\{ \begin{array}{l} \pi_{1}^{s} (p_{1} ) = (p_{1} - c_{1} )K_{1} + (\lambda p_{1} - c_{2} )(a_{1} - p_{1} + \beta p_{2} - K_{1} ) \hfill \\ \pi_{2}^{s} (p_{2} ) = (p_{2} - c_{2} )(a_{1} - p_{2} + \beta p_{1} ) + (1 - \lambda )p_{1} (a_{1} - p_{1} + \beta p_{2} - K_{1} ) \hfill \\ \end{array} \right., $$

we can conclude that the second derivatives of the profits with respect to the price as follows:

$$ \left\{ \begin{array}{l} \partial^{2} \pi_{1}^{s} /\partial p_{1}^{2} = - 2\lambda < 0 \hfill \\ \partial^{2} \pi_{2}^{s} /\partial p_{2}^{2} = - 2 < 0 \hfill \\ \end{array} \right.. $$

The optimal pricing strategies $\left( {p_{1}^{s*} ,p_{2}^{s*} } \right)$ exist.

Assuming $\left\{ \begin{array}{l} \partial \pi_{1}^{s} /\partial p_{1} = 0 \hfill \\ \partial \pi_{2}^{s} /\partial p_{2} = 0 \hfill \\ \end{array} \right.$, we can thus obtain

$$ \left\{ \begin{array}{l} p_{1}^{s1 * } = p_{1}^{s * } = \frac{1}{{\lambda \left( {\lambda \beta^{2} - 2\beta^{2} + 4} \right)}}\left( {2a_{1} \lambda + a_{2} \beta \lambda + \left( {2 + \lambda \beta } \right)c_{2} + 2\left( {1 - \lambda } \right)K_{1} } \right) \hfill \\ p_{2}^{s2 * } = p_{2}^{s * } = \frac{1}{{\lambda \left( {\lambda \beta^{2} - 2\beta^{2} + 4} \right)}}\left( {\left( {2 - \lambda } \right)\lambda \beta a_{1} + \left( {1 - \lambda } \right)\left( {2 - \lambda } \right)K_{1} \beta + 2a_{2} \lambda + c_{2} \left( {2\beta + 2\lambda - \lambda \beta } \right)} \right) \hfill \\ \end{array} \right.. $$

Then, we obtain $\left( {d_{1} (p_{1}^{s1*} ,p_{2}^{s1*} ),d_{2} (p_{1}^{s1*} ,p_{2}^{s1*} )} \right)$ and $\left( {\pi_{1}^{{{\text{s1}}*}} (p_{1}^{s1*} ,p_{2}^{s1*} ),\pi_{2}^{s1*} (p_{1}^{s1*} ,p_{2}^{s1*} )} \right)$ as follows:

$$ \begin{aligned} & \left\{ \begin{array}{l} d_{1} (p_{1}^{s * } ,p_{2}^{s * } ) = \frac{1}{{\lambda \left( {\lambda \beta^{2} - 2\beta^{2} + 4} \right)}}\left( \begin{array}{l} K_{1} \lambda^{2} \beta^{2} + \lambda \left( {\beta^{2} \left( { - c_{2} - 3K_{1} } \right) + \beta \left( {a_{2} + c_{2} } \right) + 2a_{1} + 2K_{1} } \right) \hfill \\ \quad- \left( {2 - 2\beta^{2} } \right)\left( {c_{2} + K_{1} } \right) \hfill \\ \end{array} \right) \hfill \\ d_{2} (p_{1}^{s * } ,p_{2}^{s * } ) = \frac{1}{{\lambda \beta^{2} - 2\beta^{2} + 4}}\left( {\lambda \left( {a_{2} \beta^{2} + \left( {a_{1} - K_{1} } \right)\beta } \right) + \left( {2 - \beta^{2} } \right)\left( {a_{2} - c_{2} } \right) + \left( {c_{2} + K_{1} } \right)\beta } \right) \hfill \\ \end{array} \right. \\ & \left\{ {\begin{array}{l} {\pi_{1}^{s1*} \, (p_{1}^{s1 * } ) = (p_{1}^{s1 * } - c_{1} )K_{1} + (\lambda p_{1}^{s1 * } - c_{2} )(a_{1} - p_{1}^{s1 * } + \beta p_{2}^{s1 * } - K_{1} )} \\ {\pi_{2}^{s1*} (p_{2}^{s1 * } ) = (p_{2}^{s1 * } - c_{2} )(a_{1} - p_{2}^{s1 * } + \beta p_{1}^{s1 * } ) + (1 - \lambda )p_{1}^{s1 * } (a_{1} - p_{1}^{s1 * } + \beta p_{2}^{s1 * } - K_{1} )} \\ \end{array} } \right.. \\ \end{aligned} $$

In Case 5-1, when $\lambda = 0$, we have the following:

$$ \left\{ \begin{array}{l} \pi_{1}^{s1} (p_{1} ) = (p_{1} - c_{1} )K_{1} - c_{2} \left( {d_{1} - K_{1} } \right) \hfill \\ \pi_{2}^{s1} (p_{2} ) = (p_{2} - c_{2} )d_{2} + p_{1} \left( {d_{1} - K_{1} } \right) \hfill \\ \end{array} \right.. $$

We can conclude that the first derivatives of Firm 1’s profit and the second derivatives of Firm 2’s profit with respect to the price are as follows:

$$ \left\{ \begin{array}{l} \partial \pi_{1}^{s1} /\partial p_{1} = K_{1} + c_{2} > 0 \hfill \\ \partial^{2} \pi_{2}^{s1} /\partial p_{2}^{2} = - 2 < 0 \hfill \\ \end{array} \right.. $$

Assuming $\left\{ \begin{array}{l} K_{1} = a_{1} - p_{1}^{s1*} + \beta p_{2}^{s1*} \hfill \\ \partial \pi_{2}^{s1} /\partial p_{2} = 0 \hfill \\ \end{array} \right.$, then we can obtain

$$ \left\{ \begin{array}{l} p_{1}^{s1 * } = \frac{1}{{2\left( {1 - \beta^{2} } \right)}}\left( {2\left( {a_{1} - K_{1} } \right) + \beta \left( {a_{2} + c_{2} } \right)} \right) \hfill \\ p_{2}^{s1 * } = \frac{1}{{2\left( {1 - \beta^{2} } \right)}}\left( {2\beta \left( {a_{1} - K_{1} } \right) + \left( {a_{2} + c_{2} } \right)} \right) \hfill \\ \end{array} \right.. $$

1.7 Proof of Corollary 4

The proof is straightforward and omitted.

1.8 Proof of Proposition 5

Based on the proof of Proposition 4 and $d_{2} \left( {p_{1}^{s*} ,p_{2}^{s*} } \right) < K_{2}$, $d_{1} \left( {p_{1}^{s*} ,p_{2}^{s*} } \right) \le K_{1} + K_{2} - d_{2} \left( {p_{1}^{s*} ,p_{2}^{s*} } \right)$, we have $a_{1} > 2K_{1} + c_{2}$.

We can analyze that the price and profit of Model N and 5-1 with $\lambda \ne 0$ are as follows:

$$ \left\{ \begin{array}{l} p_{1}^{s1*} - p_{1}^{N*} = \frac{{\left( { - \beta^{2} \left( {2a_{1} - 2K_{1} + a_{2} \beta + {\text{c}}_{ 2} \beta } \right)\lambda^{2} - \left( {2 - \beta^{2} } \right)\left( {2a_{1} - 2K_{1} + a_{2} \beta + {\text{c}}_{ 2} \beta } \right)\lambda + 2\left( {2 - \beta^{2} } \right)\left( {c_{2} + K_{1} } \right)} \right)}}{{\lambda \left( {\lambda \beta^{2} - 2\beta^{2} + 4} \right)\left( {2 - \beta^{2} } \right)}} \hfill \\ p_{2}^{s1*} - p_{2}^{N*} = \frac{{\left( { - \beta \left( {2a_{1} - 2K_{1} + a_{2} \beta + {\text{c}}_{ 2} \beta } \right)\lambda^{2} - \left( {2 - \beta^{2} } \right)\left( {c_{2} + K_{1} } \right)\lambda + 2\beta \left( {2 - \beta^{2} } \right)\left( {c_{2} + K_{1} } \right)} \right)}}{{\lambda \left( {\lambda \beta^{2} - 2\beta^{2} + 4} \right)\left( {2 - \beta^{2} } \right)}} \hfill \\ \end{array} \right. $$

We denote

$$ \left\{ \begin{array}{l} \lambda_{1}^{s1} = \frac{\begin{array}{l} \left( {2 - \beta^{2} } \right)\left( {2a_{1} - 2K_{1} + a_{2} \beta + {\text{c}}_{ 2} \beta } \right) - \hfill \\ \sqrt {\left( {2 - \beta^{2} } \right)^{2} \left( {2a_{1} - 2K_{1} + a_{2} \beta + {\text{c}}_{ 2} \beta } \right)^{2} + 8\beta^{2} \left( {2 - \beta^{2} } \right)\left( {c_{2} + K_{1} } \right)\left( {2a_{1} - 2K_{1} + a_{2} \beta + {\text{c}}_{ 2} \beta } \right)} \hfill \\ \end{array} }{{ - 2\beta^{2} \left( {2a_{1} - 2K_{1} + a_{2} \beta + {\text{c}}_{ 2} \beta } \right)}} \hfill \\ \lambda_{2}^{s1} = \frac{{\left( {2 - \beta^{2} } \right)\left( {c_{2} + K_{1} } \right) - \sqrt {\left( {2 - \beta^{2} } \right)^{2} \left( {c_{2} + K_{1} } \right)^{2} + 8\beta^{2} \left( {2 - \beta^{2} } \right)\left( {c_{2} + K_{1} } \right)\left( {2a_{1} - 2K_{1} + a_{2} \beta + {\text{c}}_{ 2} \beta } \right)} }}{{ - 2\beta \left( {2a_{1} - 2K_{1} + a_{2} \beta + {\text{c}}_{ 2} \beta } \right)}} \hfill \\ \end{array} \right.. $$

• If $0 < \lambda_{1}^{s1} < 1$, $\lambda \in \left( {0,\lambda_{1}^{s1} } \right]$, we have $p_{1}^{s1*} \ge p_{1}^{N*}$; $\lambda \in \left( {\lambda_{1}^{s1} ,1} \right]$, we have$p_{1}^{s1*} < p_{1}^{N*}$.

• If $\lambda_{1}^{s1} \ge 1$, $\lambda \in \left( {0,1} \right]$, we have $p_{1}^{s1*} \ge p_{1}^{N*}$.

• If $0 < \lambda_{2}^{s1} < 1$, $\lambda \in \left( {0,\lambda_{2}^{s1} } \right]$, $p_{2}^{s1*} \ge p_{2}^{N*}$; $\lambda \in \left( {\lambda_{2}^{s1} ,1} \right]$, we have$p_{2}^{s1*} < p_{2}^{N*}$.

• If $\lambda_{2}^{s1} \ge 1$, $\lambda \in \left( {0,1} \right]$, we have $p_{2}^{s1*} \ge p_{2}^{N*}$.

1.9 Proof of Proposition 6

In Case 5-2, based on $d_{2} (p_{1}^{s*} ,p_{2}^{s*} ) < K_{2}$ and $d_{1} (p_{1}^{s*} ,p_{2}^{s*} ) \ge K_{1} + K_{2} - d_{2} (p_{1}^{s*} ,p_{2}^{s*} )$, we have $\frac{{a_{2} + a_{1} - K_{1} - K_{2} - p_{2} + \beta p_{2} }}{1 - \beta } \le p_{1} .$

The optimal pricing strategies of Firm 1 and Firm 2 can be concluded by

$$ \left\{ {\begin{array}{l} {a_{1} - p_{1} + \beta p_{2} - K_{1} = K_{2} - (a_{2} - p_{2} + \beta p_{1} )} \\ {\partial \pi_{2}^{s} /\partial p_{2} = - (p_{2} - c_{2} ) + (a_{2} - p_{2} + \beta p_{1} ) + (1 - \lambda )\beta p_{1} = 0} \\ \end{array} } \right.. $$

We can obtain two optimal solutions as follows:

$$ \begin{aligned} & \left\{ \begin{array}{l} p_{1}^{s2 * } =p_{1}^{s * } = \frac{{2a_{1} + a_{2} - c_{2} - 2K_{1} - 2K_{2} + a_{2} \beta + c_{2} \beta }}{{(1 - \beta )\left( {2 + 2\beta - \lambda \beta } \right)}} \hfill \\ p_{2}^{s2 * } =p_{2}^{s * } = \frac{{a_{2} + c_{2} + \left( {2a_{1} + a_{2} - c_{2} - 2K_{1} - 2K_{2} } \right)\beta - \left( {a_{1} + a_{2} - K_{1} - K_{2} } \right)\beta \lambda }}{{(1 - \beta )\left( {2 + 2\beta - \lambda \beta } \right)}} \hfill \\ \end{array} \right., \\ & \left\{ {\begin{array}{l} {\pi_{1}^{s2*} \, (p_{1}^{s2 * } ) = (p_{1}^{s2 * } - c_{1} )K_{1} + (\lambda p_{1}^{s2 * } - c_{2} )(a_{1} - p_{1}^{s2*} + \beta p_{2}^{s2*} - K_{1} )} \\ {\pi_{2}^{s2*} (p_{2}^{s2 * } ) = (p_{2}^{s2 * } - c_{2} )(a_{1} - p_{2}^{s2 * } + \beta p_{1}^{s2 * } ) + (1 - \lambda )p_{1}^{s2 * } (a_{1} - p_{1}^{s2*} + \beta p_{2}^{s2*} - K_{1} )} \\ \end{array} } \right.. \\ \end{aligned} $$

The shared capacity quantity is $(a_{1} - p_{1}^{s2*} + \beta p_{2}^{s2*} - K_{1} ) = (K_{2} - a_{2} - p_{2}^{s2*} + \beta p_{1}^{s2*} ).$

1.10 Proof of Corollary 5

The proof is straightforward and omitted.

1.11 Proof of Proposition 7

Based on $d_{2} (p_{1}^{s*} ,p_{2}^{s*} ) < K_{2}$ and $d_{1} (p_{1}^{s*} ,p_{2}^{s*} ) \ge K_{1} + K_{2} - d_{2} (p_{1}^{s*} ,p_{2}^{s*} )$, we have

$$ \left\{ \begin{array}{l} a_{1} > 2K_{1} + c_{2} \hfill \\ a_{2} < 2K_{2} + c_{2} \hfill \\ 2a_{1} + a_{2} > 2\left( {K_{1} + K_{2} } \right) + c_{2} \hfill \\ \end{array} \right.. $$

Now, we can analyze that the prices and profits of Model N and 5-2 are as follows:

$$ \left\{ \begin{aligned} p_{1}^{s2*} - p_{1}^{N*} &= \frac{\begin{array}{l} [\beta \left( {1 - \beta } \right)\left( {2a_{1} - 2K_{1} + a_{2} \beta + {\text{c}}_{ 2} \beta } \right)\lambda + \left( {a_{2} + {\text{c}}_{ 2} } \right)\beta^{3} \hfill \\ + \left( {2a_{1} - a_{2} + c_{2} - 2K_{1} + 2K_{2} } \right)\beta^{2} + 2a_{2} - 2c_{ 2} - 4K_{2} ] \hfill \\ \end{array} }{{\left( {1 - \beta } \right)\left( {2 - \beta^{2} } \right)\left( {2\beta - \lambda \beta + 2} \right)}} \hfill \\ p_{2}^{s2*} - p_{2}^{N*} &= \frac{\begin{array}{l} [ - \beta \left( {2a_{1} + a{}_{2} - c_{2} - 2K_{1} - 2K_{2} - \left( {a_{1} - a_{2} - c_{ 2} - K_{1} } \right)\beta - a_{2} \beta^{2} + K_{2} \beta^{2} } \right)\lambda \hfill \\ + \beta \left( {2a_{1} + 2a{}_{2} - 2c_{2} - 2K_{1} - 4K_{2} + \left( {a_{2} + {\text{c}}_{ 2} } \right)\beta - \left( {a_{2} - c_{2} - 2K_{2} } \right)\beta^{2} } \right)] \hfill \\ \end{array} }{{\left( {1 - \beta } \right)\left( {2 - \beta^{2} } \right)\left( {2\beta - \lambda \beta + 2} \right)}} \hfill \\ \end{aligned} \right. $$

We denote the following:

$$ \begin{aligned} & L_{4} \left( \beta \right) = \left( {a_{2} + {\text{c}}_{ 2} } \right)\beta^{3} + \left( {2a_{1} - a_{2} + c_{2} - 2K_{1} + 2K_{2} } \right)\beta^{2} + 2a_{2} - 2c_{ 2} - 4K_{2} ; \\ & \left\{ \begin{array}{l} \lambda_{1}^{\text{s2}} = \frac{{ - \left( {\left( {a_{2} + {\text{c}}_{ 2} } \right)\beta^{3} + \left( {2a_{1} - a_{2} + c_{2} - 2K_{1} + 2K_{2} } \right)\beta^{2} + 2a_{2} - 2c_{ 2} - 4K_{2} } \right)}}{{\beta \left( {1 - \beta } \right)\left( {2a_{1} - 2K_{1} + a_{2} \beta + {\text{c}}_{ 2} \beta } \right)}} \hfill \\ \lambda_{2}^{s2} = \frac{{2a_{1} + 2a{}_{2} - 2c_{2} - 2K_{1} - 4K_{2} + \left( {a_{2} + {\text{c}}_{ 2} } \right)\beta - \left( {a_{2} - c_{2} - 2K_{2} } \right)\beta^{2} }}{{2a_{1} + a{}_{2} - c_{2} - 2K_{1} - 2K_{2} - \left( {a_{1} - a_{2} - c_{ 2} - K_{1} } \right)\beta - a_{2} \beta^{2} + K_{2} \beta^{2} }} \hfill \\ \end{array} \right.; \\ & \beta_{2} = \frac{{ - a_{2} - c_{ 2} + \sqrt {\hbox{max} \left( {0,\left( {a_{2} + {\text{c}}_{ 2} } \right)^{2} + 8\left( {a_{1} + a{}_{2} - c_{2} - K_{1} - 2K_{2} } \right)\left( {a_{2} - c_{2} - 2K_{2} } \right)} \right)} }}{{2c_{2} + 4K_{2} - 2a_{2} }} < 1. \\ \end{aligned} $$

From $L_{4} \left( \beta \right)$, we have $L_{4}^{\prime } \left( \beta \right)$, $L_{4} \left( \beta \right)\left| {_{\beta = 0} < 0}, \right.L_{4} \left( \beta \right)\left| {_{\beta \to 1} > 0} \right.$, $\beta_{1} \in \left[ {0,1} \right)$, $L_{4} \left( \beta \right)=0$. $\beta_{1} \in [0,1)$ is solved by $L_{4} \left( \beta \right)=0$.

We can obtain

• If $\beta \in \left[ {0,\beta_{1} } \right)$, $\lambda_{1}^{s2} > 0$, $\lambda \in \left[ {0,\hbox{min} \left( {\lambda_{1}^{s2} ,1} \right)} \right)$, $p_{1}^{s2*} < p_{1}^{N*}$, $\lambda \in \left[ {\hbox{min} \left( {\lambda_{1}^{s2} ,1} \right),1} \right]$, $p_{1}^{s2*} \ge p_{1}^{N*}$;

• If $\beta \in \left[ {\beta_{1} ,1} \right)$, $p_{1}^{s2*} \ge p_{1}^{N*}$.

• If $2a_{1} + 2a{}_{2} - 2c_{2} - 2K_{1} - 4K_{2} \ge 0$, $\lambda_{2}^{s2} > 0$, $\lambda \in \left[ {0,\hbox{min} \left( {\lambda_{2}^{s2} ,1} \right)} \right)$, $p_{2}^{s2*} \ge p_{2}^{N*}$, $\lambda \in \left[ {\hbox{min} \left( {\lambda_{2}^{s2} ,1} \right),1} \right]$, $p_{2}^{s2*} \le p_{2}^{N*}$;

• If $2a_{1} + 2a{}_{2} - 2c_{2} - 2K_{1} - 4K_{2} < 0$, $\beta \in \left[ {0,\hbox{max} (0,\beta_{2} )} \right)$, $p_{2}^{s2*} \le p_{2}^{N*}$; $\beta \in \left[ {\hbox{max} (0,\beta_{2} ),1} \right)$, $\lambda_{2}^{s2} > 0$, $\lambda \in \left[ {0,\hbox{min} \left( {\lambda_{2}^{s2} ,1} \right)} \right)$, $p_{2}^{s2*} \ge p_{2}^{N*}$, $\lambda \in \left[ {\hbox{min} \left( {\lambda_{2}^{s2} ,1} \right),1} \right]$, $p_{2}^{s2*} \le p_{2}^{N*}$;

1.12 Proof of Proposition 8

When $\lambda = 1$, with $d_{2} (p_{1}^{s*} ,p_{2}^{s*} ) < K_{2}$ and $d_{1} (p_{1}^{s*} ,p_{2}^{s*} ) \ge K_{1}$, we have

$$ \left\{ \begin{array}{l} a_{1} > 2K_{1} + c_{2} \hfill \\ a_{2} < 2K_{2} + c_{2} \hfill \\ \end{array} \right.. $$

$$ \left\{ \begin{array}{l} d_{1} (p_{1}^{s * } ,p_{2}^{s * } ) = \frac{1}{{4 - \beta^{2} }}\left( {2a_{1} + a_{2} \beta - \left( {2 - \beta - \beta^{2} } \right)c_{2} } \right) \hfill \\ d_{2} (p_{1}^{s * } ,p_{2}^{s * } ) = \frac{1}{{4 - \beta^{2} }}\left( {2a_{2} + a_{1} \beta - \left( {2 - \beta - \beta^{2} } \right)c_{2} } \right) \hfill \\ \end{array} \right., $$

$$ d_{1} (p_{1}^{s * } ,p_{2}^{s * } ) + d_{2} (p_{1}^{s * } ,p_{2}^{s * } ) - K_{1} - K_{2} = \frac{1}{2 - \beta }\left( {a_{1} + a_{2} - 2K_{1} - 2K_{2} - 2c_{2} + \beta \left( {K_{1} + K_{2} + 2c_{2} } \right)} \right). $$

We denote the following: $\beta_{3} =\frac{{ - \left( {a_{1} + a_{2} - 2K_{1} - 2K_{2} - 2c_{2} } \right)}}{{K_{1} + K_{2} + 2c_{2} }}.$

• (1) If $\beta \in [0,\hbox{max} \{ 0,\beta_{3} \} )$, the capacity shared schemes is under Case 5-1. We have$p_{1}^{N * } \ge p_{1}^{s1 * } ,p_{2}^{N * } \ge p_{2}^{s1 * } ;\pi_{1}^{N * } < \pi_{1}^{s1 * } ,\pi_{2}^{N * } \ge \pi_{2}^{s1 * } .$

• (2) If $\beta \in [\hbox{max} \{ 0,\beta_{3} \} ,1)$, the capacity shared schemes is under Case 5-2. We have

The optimal pricing strategies of Firm 1 and Firm 2 can be concluded by

$$ \left\{ {\begin{array}{l} {a_{1} - p_{1} + \beta p_{2} - K_{1} = K_{2} - (a_{2} - p_{2} + \beta p_{1} )} \\ {\partial \pi_{2}^{s} /\partial p_{2} = - (p_{2} - c_{2} ) + (a_{2} - p_{2} + \beta p_{1} ) = 0} \\ \end{array} } \right.. $$

We can obtain two optimal solutions as follows:

$$ \begin{aligned} & \left\{ \begin{array}{l} p_{1}^{s2 * } = \frac{{2a_{1} + a_{2} - c_{2} - 2K_{1} - 2K_{2} + a_{2} \beta + c_{2} \beta }}{{(1 - \beta )\left( {2 + \beta } \right)}} \hfill \\ p_{2}^{s2 * } = \frac{{2a_{1} + a_{2} + c_{2} + \left( {a_{1} - c_{2} - K_{1} - K_{2} } \right)\beta }}{{(1 - \beta )\left( {2 + \beta } \right)}} \hfill \\ \end{array} \right.; \\ & \left\{ {\begin{array}{l} {\pi_{1}^{s2*} \, (p_{1}^{s2 * } ) = (p_{1}^{s2 * } - c_{1} )K_{1} + (\lambda p_{1}^{s2 * } - c_{2} )(a_{1} - p_{1}^{s2*} + \beta p_{2}^{s2*} - K_{1} )} \\ {\pi_{2}^{s2*} (p_{2}^{s2 * } ) = (p_{2}^{s2 * } - c_{2} )(a_{1} - p_{2}^{s2 * } + \beta p_{1}^{s2 * } ) + (1 - \lambda )p_{1}^{s2 * } (a_{1} - p_{1}^{s2*} + \beta p_{2}^{s2*} - K_{1} )} \\ \end{array} } \right.. \\ \end{aligned} $$

Now, we can analyze that the prices and profits of Cases 4-1 and 5-1 are as follows:

$$ \left\{ {\begin{array}{l} {p_{1}^{s2 * } - p_{1}^{N * } = \frac{{2\left( {c_{2} + K_{2} } \right)\beta^{2} + 2\left( {a_{1} - K_{1} } \right)\beta + 2a_{2} - 2c_{2} - 4K_{2} }}{{(2 - \beta - \beta^{2} )\left( {2 - \beta^{2} } \right)}}} \\ {p_{2}^{s2 * } - p_{2}^{N * } = \frac{{\left( {c_{2} + K_{2} } \right)\beta^{3} + \left( {a_{1} - K_{1} } \right)\beta^{2} + \left( {a_{2} - c_{2} - 2K_{2} } \right)\beta }}{{(2 - \beta - \beta^{2} )\left( {2 - \beta^{2} } \right)}}=\frac{{\beta \left( {p_{1}^{s2 * } - p_{1}^{1 * } } \right)}}{2}} \\ \end{array} } \right.. $$

We denote the following:

$$ L_{5} \left( \beta \right) = 2\left( {c_{2} + K_{2} } \right)\beta^{2} + 2\left( {a_{1} - K_{1} } \right)\beta + 2a_{2} - 2c_{2} - 4K_{2} . $$

From $L_{5} \left( \beta \right)$, we have $2\left( {c_{2} + K_{2} } \right) > 0,\frac{{\left( {a_{1} - K_{1} } \right)}}{{ - 2\left( {c_{2} + K_{2} } \right)}} < 0$, $L_{5} \left( \beta \right)\left| {_{\beta = 0} < 0,} \right.L_{5} \left( \beta \right)\left| {_{\beta \to 1} > 0} \right.$,$\beta_{4} \in \left[ {0,1} \right)$, $L_{5} \left( {\beta_{4} } \right)=0$. $\beta_{4} = \frac{{ - a_{1} + K_{1} + \sqrt {\left( {a_{1} - K_{1} } \right)^{2} - 4\left( {c_{2} + K_{2} } \right)\left( {a_{2} - c_{2} - 2K_{2} } \right)} }}{{\left( {2c_{2} + 2K_{2} } \right)}}$.

• If $\beta \in \left[ {0,\beta_{4} } \right)$, $p_{1}^{s2 * } < p_{1}^{N * }$; If $\beta \in \left[ {\beta_{4} ,1} \right)$, $p_{1}^{s2 * } \ge p_{1}^{N * }$;

• If $\beta \in \left[ {0,\beta_{4} } \right)$, $p_{2}^{s2 * } < p_{2}^{N * }$; If $\beta \in \left[ {\beta_{4} ,1} \right)$, $p_{2}^{s2 * } \ge p_{2}^{N * }$;

We also have

$$ \pi_{1}^{s2 * } - \pi_{1}^{N * } = \frac{{L_{6} }}{{\left( {2 - \beta - \beta^{2} } \right)^{2} \left( {2 - \beta^{2} } \right)}}. $$

We denote the following:

$$ \begin{aligned} L_{6} & = \left( { - 2\beta^{4} + 8\beta^{2} - 8} \right)K_{2}^{2} + \left( {2 - \beta^{2} } \right)\left( \begin{array}{l} 4a_{1} + 4a_{2} - 8c_{2} - 8K_{1} + 2a_{1} \beta + 2a_{2} \beta + 4c_{2} \beta - 2a_{1} \beta^{2} \hfill \\ - a_{2} \beta^{2} + 7c_{2} \beta^{2} + 4K_{1} \beta^{2} - a_{2} \beta^{3} - 2c_{2} \beta^{3} - c_{2} \beta^{4} \hfill \\ \end{array} \right)K_{2} \\ & \quad - \left( {a_{2} - c_{2} + a_{1} \beta - K_{1} \beta + c_{2} \beta^{2} } \right)\left( \begin{array}{l} 4a_{1} + 2a_{2} - 6c_{2} - 8K_{1} + 2a_{2} \beta + 4c_{2} \beta + 2K_{1} \beta - 2a_{1} \beta^{2} - a_{2} \beta^{2} \hfill \\ + 5c_{2} \beta^{2} + 4K_{1} \beta^{2} - a_{2} \beta^{3} - 2c_{2} \beta^{3} - c_{2} \beta^{4} \hfill \\ \end{array} \right). \\ \end{aligned} $$

If $L_{6} < 0$, $\pi_{1}^{1 * } > \pi_{1}^{s2 * }$; If $L_{6} \ge 0$, $\pi_{1}^{N * } \le \pi_{1}^{s2 * }$.

We denote the following:

$$ \left\{ \begin{aligned} K_{21}^{s2} = \frac{{\left( { - 2\beta \left( {2 - \beta^{2} } \right)^{2} + \sqrt {2\beta \left( {2 - \beta^{2} } \right)^{2} \left( {4\beta^{4} + 2\beta^{3} - 8\beta^{2} + 1} \right)} } \right)\left( {a_{2} - c_{2} + a_{1} \beta - K_{1} \beta + c_{2} \beta^{2} } \right)}}{{ - 2\beta^{6} + 8\beta^{4} - 8\beta^{2} }} \hfill \\ K_{22}^{s2} =\frac{{\left( { - 2\beta \left( {2 - \beta^{2} } \right)^{2} - \sqrt {2\beta (\left( {2 - \beta^{2} } \right)^{2} \left( {4\beta^{4} + 2\beta^{3} - 8\beta^{2} + 1} \right)} } \right)\left( {a_{2} - c_{2} + a_{1} \beta - K_{1} \beta + c_{2} \beta^{2} } \right)}}{{ - 2\beta^{6} + 8\beta^{4} - 8\beta^{2} }} \hfill \\ \end{aligned} \right.. $$

We also have

$$ \pi_{2}^{s2 * } - \pi_{2}^{N * } =\frac{\begin{array}{l} \left( { - \beta^{6} + 4\beta^{4} - 4\beta^{2} } \right)K_{2}^{2} + 2\beta \left( {2 - \beta^{2} } \right)^{2} \left( {a_{2} - c_{2} + a_{1} \beta - K_{1} \beta + c_{2} \beta^{2} } \right)K_{2} \hfill \\ + \beta \left( {2\beta^{2} + \beta - 4} \right)\left( {a_{2} - c_{2} + a_{1} \beta - K_{1} \beta + c_{2} \beta^{2} } \right)^{2} \hfill \\ \end{array} }{{\left( {2 - \beta - \beta^{2} } \right)^{2} \left( {2 - \beta^{2} } \right)^{2} }}. $$

Thus, when $\beta \in \left[ {0,0.39} \right]$, if $K_{2} \in \left( {K_{21}^{s2} ,K_{22}^{s2} } \right)$, $\pi_{2}^{s2 * } \ge \pi_{2}^{N * }$.

If $K_{2} \in \left[ {0,K_{21}^{s2} } \right] \cup \left[ {K_{22}^{s2} + \infty } \right)$, $\pi_{2}^{s2 * } \le \pi_{2}^{N * }$. When $\beta \in \left( {0.39,1} \right)$, $\pi_{2}^{s2 * } < \pi_{2}^{N * }$.