Cooperative and noncooperative R&amp;D in duopoly manufacturers with a common supplier

Fuli Zhang; Yangyang Peng; Xiaolin Xu; Xing Yin; Lianmin Zhang

doi:10.3934/jimo.2022085

Article Contents

2023, Volume 19, Issue 5: 3230-3254. Doi: 10.3934/jimo.2022085

This issue Previous Article An approach to solve local and global optimization problems based on exact objective filled penalty functions Next Article Strategic service investment by retailers confronted by manufacturer encroachment

Cooperative and noncooperative R&D in duopoly manufacturers with a common supplier

1.
School of Business, Nanjing Audit University, Nanjing, 211815, China
2.
School of Data Science, The Chinese University of Hong Kong, Shenzhen, School of Management and Economics, University of Electronic Science and Technology, Shenzhen, 518172, China; Chengdu, 610054, China
3.
School of Business, Nanjing University, Nanjing, 210093, China
4.
Shenzhen Research Institute of Big Data, Shenzhen, 518172, China

^*Corresponding author: Yangyang Peng
^*Corresponding author: Yangyang Peng

Received: July 2021

Revised: March 2022

Early access: June 2022

Published: May 2023

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Abstract

We consider the R&D strategy of firms under competitive environments from the supply chain perspective. Specifically, we investigate a supply chain consisting of one upstream component supplier and two downstream manufacturers, who however are the Stackelberg leader(s). At the early stage (R&D stage), the two manufacturers decide on whether to cooperate or not in the R&D activities and how much to invest in R&D accordingly. At the late stage (market stage), the component supplier decides on the uniform wholesale price and the manufacturers decide on the production quantities. Our main findings include: (ⅰ) Cooperative R&D strategy will be adopted when the technology spillover effect is either too large or too small and in contrast non-cooperative strategy will be accepted when the spillover effect is moderate. However, the underlying driving forces for coordination are different when the spillover effect is small or large, i.e., cost reduction effect and sales increasing effect. (ⅱ) Cooperative R&D could increase the social welfare when both the technology spillover effect and the (initial) unit production cost are high. (ⅲ) As the equilibrium under the cooperative R&D strategy is unstable, we give a coordination mechanism, to guarantee the stability of cooperative R&D investments.

Keywords:

Mathematics Subject Classification: Primary: 90B06, 90B50; Secondary: 91A80.

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Figure 1. The Four-Stage Game between Upstream Supplier and Downstream Manufacturers

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Figure 2. Optimal R&D Strategies for Downstream Manufacturers

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Figure 3. The R&D Decision Matrix

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Figure 4. The Effect of Supplier's Willingness to Cooperate

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Table 1. The Equilibrium Outcomes in Noncooperative R&D Strategy

Scenario	$ c_n(\beta)<c_d<a-c_u $	$ 0<c_d \le c_n(\beta) $
$ x_i^N $	$ \frac{{(7 - 5\beta )k}}{{29 - 2\beta + 5{\beta ^2}}} $	$ \frac{{{c_d}}}{{1 + \beta }} $
$ w^N $	$ \frac{{18(a - {c_d}) + (11 - 2\beta + 5{\beta ^2}){c_u}}}{{29 - 2\beta + 5{\beta ^2}}} $	$ \frac{{a + {c_u}}}{2} $
$ p^N $	$ \frac{{a(17 - 2\beta + 5{\beta ^2}) + 12({c_u} + {c_d})}}{{29 - 2\beta + 5{\beta ^2}}} $	$ \frac{{2a + {c_u}}}{3} $
$ q_i^N $	$ \frac{{6k}}{{29 - 2\beta + 5{\beta ^2}}} $	$ \frac{{a - {c_u}}}{6} $
$ \pi_u^N $	$ \frac{{216{k^2}}}{{{{(29 - 2\beta + 5{\beta ^2})}^2}}} $	$ \frac{{{{(a - {c_u})}^2}}}{6} $
$ \pi_i^N $	$ \frac{{(23 + 70\beta - 25{\beta ^2}){k^2}}}{{2{{(29 - 2\beta + 5{\beta ^2})}^2}}} $	$ \frac{{{{(a - {c_u})}^2}{{(1 + \beta )}^2} - 18c_d^2}}{{36{{(1 + \beta )}^2}}} $

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Table 2. The Equilibrium Outcomes in Cooperative R&D Strategy

Scenario	$ c_o(\beta)<c_d<a-c_u $	$ 0<c_d \le c_o(\beta) $
$ x_i^C $	$ \frac{{k(1 + \beta )}}{{17 - 2\beta - {\beta ^2}}} $	$ \frac{{{c_d}}}{{1 + \beta }} $
$ w^C $	$ \frac{{9(a - {c_d}) + (8 - 2\beta - {\beta ^2}){c_u}}}{{17 - 2\beta - {\beta ^2}}} $	$ \frac{{a + {c_u}}}{2} $
$ p^C $	$ \frac{{a(11 - 2\beta - {\beta ^2}) + 6({c_u} + {c_d})}}{{17 - 2\beta - {\beta ^2}}} $	$ \frac{{2a + {c_u}}}{3} $
$ q_i^C $	$ \frac{{3k}}{{17 - 2\beta - {\beta ^2}}} $	$ \frac{{a - {c_u}}}{6} $
$ \pi_u^C $	$ \frac{{54{k^2}}}{{{{(17 - 2\beta - {\beta ^2})}^2}}} $	$ \frac{{{{(a - {c_u})}^2}}}{6} $
$ \pi_i^C $	$ \frac{{{k^2}}}{{2(17 - 2\beta - {\beta ^2})}} $	$ \frac{{{{(a - {c_u})}^2}{{(1 + \beta )}^2} - 18c_d^2}}{{36{{(1 + \beta )}^2}}} $

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Table 3. The Equilibrium Outcomes in Cooperative R&D Strategy

Scenario	$ c_o(\beta)<c_d<a-c_u $	$ 0<c_d \le c_o(\beta) $
$ x_i^C $	$ \frac{{k(1 + \beta(1+\theta) )}}{{17 - 2\beta(1+\theta) - {(\beta(1+\theta)) ^2}}} $	$ \frac{{{c_d}}}{{1 + \beta(1+\theta) }} $
$ w^C $	$ \frac{{9(a - {c_d}) + (8 - 2\beta(1+\theta) - {(\beta(1+\theta)) ^2}){c_u}}}{{17 - 2\beta(1+\theta) - {(\beta(1+\theta)) ^2}}} $	$ \frac{{a + {c_u}}}{2} $
$ p^C $	$ \frac{{a(11 - 2\beta(1+\theta) - {(\beta(1+\theta))^2}) + 6({c_u} + {c_d})}}{{17 - 2\beta(1+\theta) - {(\beta(1+\theta))^2}}} $	$ \frac{{2a + {c_u}}}{3} $
$ q_i^C $	$ \frac{{3k}}{{17 - 2\beta(1+\theta) - {(\beta(1+\theta)) ^2}}} $	$ \frac{{a - {c_u}}}{6} $
$ \pi_u^C $	$ \frac{{54{k^2}}}{{{{(17 - 2\beta(1+\theta) - {(\beta(1+\theta) )^2})}^2}}} $	$ \frac{{{{(a - {c_u})}^2}}}{6} $
$ \pi_i^C $	$ \frac{{{k^2}}}{{2(17 - 2\beta(1+\theta) - {(\beta(1+\theta) )^2})}} $	$ \frac{{{{(a - {c_u})}^2}{{(1 + \beta(1+\theta) )}^2} - 18c_d^2}}{{36{{(1 + \beta(1+\theta) )}^2}}} $

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Table 4. The Equilibrium Outcomes in Noncooperative R&D Strategy

Scenario	$ c_n(\beta)<c_d<a-c_u $	$ 0<c_d \le c_n(\beta) $
$ x_i^N $	$ \frac{{(7 - 5\beta(1+\theta) )k}}{{29 - 2\beta(1+\theta) + 5{(\beta(1+\theta) )^2}}} $	$ \frac{{{c_d}}}{{1 + \beta(1+\theta) }} $
$ w^N $	$ \frac{{18(a - {c_d}) + (11 - 2\beta(1+\theta) + 5{(\beta(1+\theta) )^2}){c_u}}}{{29 - 2\beta(1+\theta) + 5{(\beta(1+\theta) )^2}}} $	$ \frac{{a + {c_u}}}{2} $
$ p^N $	$ \frac{{a(17 - 2\beta(1+\theta) + 5{(\beta(1+\theta)) ^2}) + 12({c_u} + {c_d})}}{{29 - 2\beta(1+\theta) + 5{(\beta(1+\theta)) ^2}}} $	$ \frac{{2a + {c_u}}}{3} $
$ q_i^N $	$ \frac{{6k}}{{29 - 2\beta(1+\theta) + 5{(\beta(1+\theta))^2}}} $	$ \frac{{a - {c_u}}}{6} $
$ \pi_u^N $	$ \frac{{216{k^2}}}{{{{(29 - 2\beta(1+\theta) + 5{(\beta(1+\theta))^2})}^2}}} $	$ \frac{{{{(a - {c_u})}^2}}}{6} $
$ \pi_i^N $	$ \frac{{(23 + 70\beta(1+\theta) - 25{(\beta(1+\theta))^2}){k^2}}}{{2{{(29 - 2\beta(1+\theta) + 5{(\beta(1+\theta))^2})}^2}}} $	$ \frac{{{{(a - {c_u})}^2}{{(1 + \beta(1+\theta) )}^2} - 18c_d^2}}{{36{{(1 + \beta(1+\theta) )}^2}}} $

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