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November  2020, 16(6): 2723-2741. doi: 10.3934/jimo.2019077

## Bargaining equilibrium in a two-echelon supply chain with a capital-constrained retailer

 1 School of Business Administration, Hunan University, Changsha, Hunan Province 410082, China 2 School of Business, State University of New York at Oswego, Oswego, NY 13126, USA 3 School of Marketing and Logistics Management, Nanjing University of Finance and Economics, Nanjing, Jiangsu 210023, China

* Corresponding author: ottoyang@126.com (Honglin Yang)

Received  August 2018 Revised  March 2019 Published  November 2020 Early access  July 2019

Fund Project: This work was supported by the National Natural Science Foundation of China under Grant Nos. 71571065, 71790593 and 71521061

We investigate the bargaining equilibrium in a two-echelon supply chain consisting of a supplier and a capital-constrained retailer. The newsvendor-like retailer can borrow from a bank or use the supplier's trade credit to fund his business. In the presence of bankruptcy risk for both the supplier and retailer, with a wholesale price contract, we model the player's strategic interactions under the Nash and Rubinstein bargaining games. In both financing schemes, the Nash bargaining game overcomes the double marginalization effect under the Stackelberg game and achieves supply chain coordination. The Rubinstein bargaining game realizes the Pareto improvement of the supply chain. The player with stronger bargaining power always prefers to initially offer a contract under the Rubinstein bargaining game to obtain greater expected profit. Furthermore, we characterize the conditions under which bargaining power and discount factor affect the bargaining equilibrium. We numerically verify our theoretical results.

Citation: Honglin Yang, Qiang Yan, Hong Wan, Wenyan Zhuo. Bargaining equilibrium in a two-echelon supply chain with a capital-constrained retailer. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2723-2741. doi: 10.3934/jimo.2019077
##### References:

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##### References:
Mobike’s financing and operation
$\pi_{Bj}^{R*}$ and $\delta_j$
$w$ and $\beta$
Relationship of $\pi_{is}^{N*}$ and $\pi_{is}^{S*}$ with $\beta$ with BCF and TCF
Relationship of $\pi_{ir}^{N*}$ and $\pi_{is}^{S*}$ with $\beta$ with BCF and TCF
Notations and explanations
 Notation Explanation $D$ Uncertain demand. $p$ Retailer's retail price per unit, where $p=1$. $c$ Supplier's production cost per unit, where $0  Notation Explanation$ D $Uncertain demand.$ p $Retailer's retail price per unit, where$ p=1 $.$ c $Supplier's production cost per unit, where$ 0
$\pi_{is}^{R*}$ and $\delta_j$ given $c = 0.2$
 $\delta_s \backslash \delta_r$ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 B 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 T 47.35 47.35 47.35 47.35 47.35 47.35 47.35 47.35 47.35 0.2 B 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 T 46.89 46.89 46.89 46.89 46.89 46.89 46.89 46.89 46.89 0.3 B 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 T 46.43 46.43 46.43 46.43 46.43 46.43 46.43 46.43 46.43 0.4 B 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 T 45.97 45.97 45.97 45.97 45.97 45.97 45.97 45.97 45.97 0.5 B 34.38 34.38 34.38 34.38 34.38 34.38 34.76 34.76 34.76 T 45.51 45.51 45.51 45.51 45.51 45.51 45.51 45.51 45.51 0.6 B 31.70 31.70 31.70 31.70 31.70 31.70 32.97 34.76 34.76 T 45.05 45.05 45.05 45.05 45.05 45.05 45.05 45.05 45.05 0.7 B 29.01 29.01 29.01 29.01 29.01 29.01 29.01 32.60 34.76 T 44.59 44.59 44.59 44.59 44.59 44.59 44.59 44.59 44.59 0.8 B 26.33 26.33 26.33 26.33 26.33 26.33 26.33 26.56 34.15 T 44.13 44.13 44.13 44.13 44.13 44.13 44.13 44.13 44.13 0.9 B 23.64 23.64 23.64 23.64 23.64 23.64 23.64 23.64 25.16 T 43.67 43.67 43.67 43.67 43.67 43.67 43.67 43.67 43.67 Note:The symbols "B" and "T" denote "BCF" and "TCF", respectively.
 $\delta_s \backslash \delta_r$ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 B 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 T 47.35 47.35 47.35 47.35 47.35 47.35 47.35 47.35 47.35 0.2 B 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 T 46.89 46.89 46.89 46.89 46.89 46.89 46.89 46.89 46.89 0.3 B 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 T 46.43 46.43 46.43 46.43 46.43 46.43 46.43 46.43 46.43 0.4 B 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 34.76 T 45.97 45.97 45.97 45.97 45.97 45.97 45.97 45.97 45.97 0.5 B 34.38 34.38 34.38 34.38 34.38 34.38 34.76 34.76 34.76 T 45.51 45.51 45.51 45.51 45.51 45.51 45.51 45.51 45.51 0.6 B 31.70 31.70 31.70 31.70 31.70 31.70 32.97 34.76 34.76 T 45.05 45.05 45.05 45.05 45.05 45.05 45.05 45.05 45.05 0.7 B 29.01 29.01 29.01 29.01 29.01 29.01 29.01 32.60 34.76 T 44.59 44.59 44.59 44.59 44.59 44.59 44.59 44.59 44.59 0.8 B 26.33 26.33 26.33 26.33 26.33 26.33 26.33 26.56 34.15 T 44.13 44.13 44.13 44.13 44.13 44.13 44.13 44.13 44.13 0.9 B 23.64 23.64 23.64 23.64 23.64 23.64 23.64 23.64 25.16 T 43.67 43.67 43.67 43.67 43.67 43.67 43.67 43.67 43.67 Note:The symbols "B" and "T" denote "BCF" and "TCF", respectively.
$\pi_{is}^{R*}$ and $\delta_j$ given $c = 0.65$
 $\delta_s \backslash \delta_r$ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 T 6.36 6.43 6.49 6.56 6.63 6.7 6.77 6.85 6.92 0.2 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 T 5.71 5.83 5.96 6.09 6.22 6.36 6.51 6.67 6.83 0.3 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 T 5.05 5.21 5.38 5.57 5.76 5.97 6.2 6.45 6.71 0.4 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 T 4.37 4.56 4.77 5 5.25 5.53 5.83 6.18 6.56 0.5 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 T 3.68 3.89 4.12 4.37 4.67 5 5.38 5.83 6.36 0.6 B 4.82 4.82 4.82 4.82 4.82 4.82 4.83 5.19 5.19 T 2.98 3.18 3.41 3.68 4 4.37 4.83 5.38 6.09 0.7 B 4.46 4.46 4.46 4.46 4.46 4.46 4.46 4.77 5.19 T 2.26 2.44 2.66 2.92 3.23 3.62 4.12 4.77 5.67 0.8 B 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.1 5 T -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 0.9 B 3.74 3.74 3.74 3.74 3.74 3.74 3.74 3.74 3.74 T -0.91 -0.91 -0.91 -0.91 -0.91 -0.91 -0.91 -0.91 -0.91
 $\delta_s \backslash \delta_r$ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 T 6.36 6.43 6.49 6.56 6.63 6.7 6.77 6.85 6.92 0.2 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 T 5.71 5.83 5.96 6.09 6.22 6.36 6.51 6.67 6.83 0.3 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 T 5.05 5.21 5.38 5.57 5.76 5.97 6.2 6.45 6.71 0.4 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 T 4.37 4.56 4.77 5 5.25 5.53 5.83 6.18 6.56 0.5 B 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 5.19 T 3.68 3.89 4.12 4.37 4.67 5 5.38 5.83 6.36 0.6 B 4.82 4.82 4.82 4.82 4.82 4.82 4.83 5.19 5.19 T 2.98 3.18 3.41 3.68 4 4.37 4.83 5.38 6.09 0.7 B 4.46 4.46 4.46 4.46 4.46 4.46 4.46 4.77 5.19 T 2.26 2.44 2.66 2.92 3.23 3.62 4.12 4.77 5.67 0.8 B 4.1 4.1 4.1 4.1 4.1 4.1 4.1 4.1 5 T -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 0.9 B 3.74 3.74 3.74 3.74 3.74 3.74 3.74 3.74 3.74 T -0.91 -0.91 -0.91 -0.91 -0.91 -0.91 -0.91 -0.91 -0.91
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