doi: 10.3934/jimo.2022085
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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

Received  July 2021 Revised  March 2022 Early access June 2022

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.

Citation: Fuli Zhang, Yangyang Peng, Xiaolin Xu, Xing Yin, Lianmin Zhang. Cooperative and noncooperative R&D in duopoly manufacturers with a common supplier. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022085
References:
[1]

R. Amir and J. Wooders, Cooperation vs. competition in R&D: The role of stability of equilibrium, Journal of Economics, 1 (1998), 63-73. 

[2]

R. Amir, Modelling imperfectly appropriable R&D via spillovers, International Journal of Industrial Organization, 7 (2000), 1013-1032. 

[3] K. Arrow, Economic Welfare and the Allocation of Resources for Invention, Princeton University Press, Princeton, NJ, 1962. 
[4]

G. Atallah, Vertical R&D spillovers, cooperation, market structure, and innovation, Economics of Innovation and New Technology, 3 (2002), 179-209. 

[5]

C. Atiyeh, BMW and Mercedes-Benz will stop fighting and join forces to make a better autonomous car, Car and Driver, https://www.caranddriver.com/news/a26572775/bmw-and-mercedes-benz-joint-autonomous-car/, (2019).

[6]

K. Brockhoff, R&D cooperation between firms-A perceived transaction cost perspective, Management Science, 4 (1992), 514-524. 

[7]

P. G. CanbolatB. GolanyI. Mund and U. G. Rothblum, A stochastic competitive R&D race where "winner takes all", Operations Research, 3 (2012), 700-715.  doi: 10.1287/opre.1120.1055.

[8]

R. Cellini and L. Lambertini, A differential game approach to investment in product differentiation, Journal of Economic Dynamics and Control, 1 (2002), 51-62.  doi: 10.1016/S0165-1889(01)00026-4.

[9]

R. Cellini and L. Lambertini, Dynamic R&D with spillovers: Competition vs cooperation, Journal of Economic Dynamics and Control, 33 (2009), 568-582.  doi: 10.1016/j.jedc.2008.08.006.

[10]

Chinadaily, Vivo and Samsung make a joint push into 5G smartphone chip, Chinadaily, https://www.chinadaily.com.cn/a/201911/08/WS5dc50472a310cf3e355763a3.html, (2019).

[11]

S. C. Choi, Price competition in a channel structure with a common retailer, Marketing Science, 4 (1991), 271-296. 

[12]

X. ChenX. Wang and M. Zhou, Firms' green R&D cooperation behaviour in a supply chain: Technological spillover, power and coordination, International Journal of Production Economics, 219 (2019), 118-134. 

[13]

C. PascaleB. De Reyck and N. Taneri, Licensing contracts: Control rights, options, and timing, Management Science, 4 (2017), 1131-1149. 

[14]

C. D'Aspremont and A. Jacquemin, Cooperative and noncooperative R&D in duopoly with spillovers, American Economic Review, 5 (1988), 1133-1137. 

[15]

M. Del GiudiceV. ScuottoA. Garcia-Perez and A. M. Petruzzelli, Shifting wealth II in Chinese economy. The effect of the horizontal technology spillover for SMEs for international growth, Technological Forecasting and Social Change, 145 (2019), 307-316.  doi: 10.1016/j.techfore.2018.03.013.

[16]

Ericsson, Ericsson and OPPO sign initial patent license agreement, Ericsson, https://www.ericsson.com/en/press-releases/2019/2/ericsson-and-oppo-sign-initial-patent-license-agreement, (2019).

[17]

N. Erkal and D. Piccinin, Cooperative R&D under uncertainty with free entry, International Journal of Industrial Organization, 1 (2010), 74-85. 

[18]

R. Falvey and K. T. Teerasuwannajak, Competitive and harmonized R&D policies for international R&D alliances involving asymmetric firms, Review of international economics, 2 (2016), 302-329. 

[19]

Z. GeQ. Hu and Y. Xia, Firms' R&D cooperation behavior in a supply chain, Production and Operations Management, 4 (2014), 599-609. 

[20]

I. Henriques, Cooperative and noncooperative R&D in duopoly with spillovers: Comment, American Economic Review, 3 (1990), 638-640. 

[21]

A. Ishii, Cooperative R&D between vertically related firms with spillovers, International Journal of Industrial Organization, 9 (2004), 1213-1235. 

[22]

N. Ishikawa and T. Shibata, Market competition, R&D spillovers, and firms' cost asymmetry, Economics of Innovation and New Technology, (2019), 1–19.

[23]

P. T. Joanna, Equilibrium and optimal size of a research joint venture in an oligopoly with spillovers, Journal of Industrial Economics, 2 (1995), 209-226. 

[24]

T. Kabiraj and S. Chattopadhyay, Cooperative vs. non-cooperative R&D incentives under incomplete information, Economics of Innovation and New Technology, 6 (2015), 624-632. 

[25]

M. I. KamienE. Muller and I. Zang, Research joint ventures and R&D cartels, American Economic Review, 5 (1992), 1293-1306. 

[26]

P. S. KohD. M. Reeb and W. Zhao, CEO confidence and unreported R&D, Management Science, 12 (2018), 5725-5747. 

[27]

J. L. Krieger, Trials and terminations: Learning from competitors' R&D failures, Management Science, first online, (2021).

[28]

S. Marjit, Incentives for cooperative and non-cooperative R&D in duopoly, Economics Letters, 2 (1991), 187-191. 

[29]

J. A. Martin and K. M. Eisenhardt, Rewiring: Cross-Business-Unit collaborations in multibusiness organizations, Academy of Management Journal, 2 (2010), 265-301. 

[30]

M. Motta, Cooperative R&D and vertical product differentiation, International Journal of Industrial Organization, 4 (1992), 643-661. 

[31]

J. Ning and V. Babich, R&D investments in the presence of knowledge spillover and debt financing: Can risk shifting cure free riding?, Manufacturing & Service Operations Management, 1 (2018), 97-112. 

[32]

C. PennetierK. Girotra and J. Mihm, R&D spending: Dynamic or persistent?, Manufacturing & Service Operations Management, 3 (2019), 636-657. 

[33]

G. Smrkolj and F. Wagener, Dynamic R&D with spillovers: A comment, Journal of Economic Dynamics and Control, 2 (2016), 453-457.  doi: 10.1016/j.jedc.2016.10.011.

[34]

M. Spence, Cost reduction, competition, and industry performance, Econometrica, 1 (1984), 101-121. 

[35]

L. Suzumura, Cooperative and noncooperative R&D in an oligopoly with spillovers, American Economic Review, 5 (1992), 1307-1320. 

[36]

Y. Taba, Optimal product R&D policies with endogenous quality choices and unilateral spillover, Journal of Economic Analysis & Policy, 1 (2016), 365-391. 

[37]

L. XuD. LiangZ. Duan and X. Xiao, Stability analysis of R&D cooperation in a supply chain, Mathematical Problems in Engineering, 8 (2015), 1-10.  doi: 10.1155/2015/409286.

[38]

M. Xing, The optimal risk choice of cooperative and noncooperative R&D in duopoly with spillovers, Bulletin of Economic Research, 4 (2017), 173-185.  doi: 10.1111/boer.12109.

[39]

D. ZengL. Xu and X. Bi, Effects of asymmetric knowledge spillovers on the stability of horizontal and vertical R&D cooperation, Computational and Mathematical Organization Theory, 1 (2017), 32-60. 

[40]

S. K. ByunJ. Min and H. Xia, Incremental vs. Breakthrough innovation: The role of technology spillovers, Management Science, 3 (2021), 1779-1802. 

[41]

N. BloomM. Schankerman and J. Van Reenen, Identifying technology spillovers and product market rivalry, Econometrica, 4 (2013), 1347-1393.  doi: 10.3982/ECTA9466.

[42]

J. Qiu and C. Wan, Technology spillovers and corporate cash holdings, Journal of Financial Economics, 3 (2015), 558-573. 

[43]

X. PanM. LiM. WangJ. Chu and H. Bo, The effects of outward foreign direct investment and reverse technology spillover on China's carbon productivity, Energy Policy, 145 (2020), 111730. 

[44]

Y. Hu, K. Fisher-Vanden and B. Su, Technological spillover through industrial and regional linkages: Firm-level evidence from China, Economic Modelling, 89 (2020).

[45]

Y. GaoS. B. TsaiX. XueT. RenX. DuQ. Chen and et al., An empirical study on green innovation efficiency in the green institutional environment, Sustainability, 3 (2018), 724. 

[46]

A. Ll, A. Zz, A. Mz, A. Cz and B. Dz, The effects of environmental regulation on outward foreign direct investment's reverse green technology spillover: Crowding out or facilitation?, Journal of Cleaner Production, 2 (2020).

[47]

Z. Cheng and B. R. Nault, Relative industry concentration and customer-driven it spillovers, Information Systems Research, 2 (2012), 340-355. 

[48]

P. Tambe and L. M. Hitt, Measuring information technology spillovers, Information Systems Research, 1 (2013), 53-71. 

[49]

P. Tambe and L. M. Hitt, Job hopping, information technology spillovers, and productivity growth, Management Science, 2 (2013), 338-355. 

[50]

X. QinD. Du and M. P. Kwan, Spatial spillovers and value chain spillovers: Evaluating regional R&D efficiency and its spillover effects in China, Scientometrics, 119 (2019), 721-747. 

[51]

N. Ishikawa and T. Shibata, R&D competition and cooperation with asymmetric spillovers in an oligopoly market, International Review of Economics & Finance, 72 (2020), 624-642.  doi: 10.1016/j.iref.2020.12.016.

[52]

J. NiH. HuangP. Wang and W. Zhou, Capacity investment and green R&D in a dynamic oligopoly under the potential shift in environmental damage, Economic Modelling, 7 (2019), 312-319. 

[53]

C. LiP. Zhou and Y. Li, Managerial overconfidence, overinvestment, and R&D spillover, Managerial and Decision Economics, 40 (2019), 1-4. 

[54]

D. Zhang, H. Dai, L. Dong, F. Qi, N. Zhang, X. Liu, et al., The long-term and spillover effects of price promotions on retailing platforms: Evidence from a large randomized experiment on Alibaba, Management Science, 6 (2020), 2589–2609. doi: 10.1109/TSP.2021.3083988.

[55]

D. Y. Joe and F. D. Oh, Spillover effects within business groups: The case of Korean chaebols, Management Science, 3 (2018), 1396-1412. 

[56]

A. HavivY. Huang and N. Li, Intertemporal demand spillover effects on video game platforms, Management Science, 10 (2020), 4788-4807. 

show all references

References:
[1]

R. Amir and J. Wooders, Cooperation vs. competition in R&D: The role of stability of equilibrium, Journal of Economics, 1 (1998), 63-73. 

[2]

R. Amir, Modelling imperfectly appropriable R&D via spillovers, International Journal of Industrial Organization, 7 (2000), 1013-1032. 

[3] K. Arrow, Economic Welfare and the Allocation of Resources for Invention, Princeton University Press, Princeton, NJ, 1962. 
[4]

G. Atallah, Vertical R&D spillovers, cooperation, market structure, and innovation, Economics of Innovation and New Technology, 3 (2002), 179-209. 

[5]

C. Atiyeh, BMW and Mercedes-Benz will stop fighting and join forces to make a better autonomous car, Car and Driver, https://www.caranddriver.com/news/a26572775/bmw-and-mercedes-benz-joint-autonomous-car/, (2019).

[6]

K. Brockhoff, R&D cooperation between firms-A perceived transaction cost perspective, Management Science, 4 (1992), 514-524. 

[7]

P. G. CanbolatB. GolanyI. Mund and U. G. Rothblum, A stochastic competitive R&D race where "winner takes all", Operations Research, 3 (2012), 700-715.  doi: 10.1287/opre.1120.1055.

[8]

R. Cellini and L. Lambertini, A differential game approach to investment in product differentiation, Journal of Economic Dynamics and Control, 1 (2002), 51-62.  doi: 10.1016/S0165-1889(01)00026-4.

[9]

R. Cellini and L. Lambertini, Dynamic R&D with spillovers: Competition vs cooperation, Journal of Economic Dynamics and Control, 33 (2009), 568-582.  doi: 10.1016/j.jedc.2008.08.006.

[10]

Chinadaily, Vivo and Samsung make a joint push into 5G smartphone chip, Chinadaily, https://www.chinadaily.com.cn/a/201911/08/WS5dc50472a310cf3e355763a3.html, (2019).

[11]

S. C. Choi, Price competition in a channel structure with a common retailer, Marketing Science, 4 (1991), 271-296. 

[12]

X. ChenX. Wang and M. Zhou, Firms' green R&D cooperation behaviour in a supply chain: Technological spillover, power and coordination, International Journal of Production Economics, 219 (2019), 118-134. 

[13]

C. PascaleB. De Reyck and N. Taneri, Licensing contracts: Control rights, options, and timing, Management Science, 4 (2017), 1131-1149. 

[14]

C. D'Aspremont and A. Jacquemin, Cooperative and noncooperative R&D in duopoly with spillovers, American Economic Review, 5 (1988), 1133-1137. 

[15]

M. Del GiudiceV. ScuottoA. Garcia-Perez and A. M. Petruzzelli, Shifting wealth II in Chinese economy. The effect of the horizontal technology spillover for SMEs for international growth, Technological Forecasting and Social Change, 145 (2019), 307-316.  doi: 10.1016/j.techfore.2018.03.013.

[16]

Ericsson, Ericsson and OPPO sign initial patent license agreement, Ericsson, https://www.ericsson.com/en/press-releases/2019/2/ericsson-and-oppo-sign-initial-patent-license-agreement, (2019).

[17]

N. Erkal and D. Piccinin, Cooperative R&D under uncertainty with free entry, International Journal of Industrial Organization, 1 (2010), 74-85. 

[18]

R. Falvey and K. T. Teerasuwannajak, Competitive and harmonized R&D policies for international R&D alliances involving asymmetric firms, Review of international economics, 2 (2016), 302-329. 

[19]

Z. GeQ. Hu and Y. Xia, Firms' R&D cooperation behavior in a supply chain, Production and Operations Management, 4 (2014), 599-609. 

[20]

I. Henriques, Cooperative and noncooperative R&D in duopoly with spillovers: Comment, American Economic Review, 3 (1990), 638-640. 

[21]

A. Ishii, Cooperative R&D between vertically related firms with spillovers, International Journal of Industrial Organization, 9 (2004), 1213-1235. 

[22]

N. Ishikawa and T. Shibata, Market competition, R&D spillovers, and firms' cost asymmetry, Economics of Innovation and New Technology, (2019), 1–19.

[23]

P. T. Joanna, Equilibrium and optimal size of a research joint venture in an oligopoly with spillovers, Journal of Industrial Economics, 2 (1995), 209-226. 

[24]

T. Kabiraj and S. Chattopadhyay, Cooperative vs. non-cooperative R&D incentives under incomplete information, Economics of Innovation and New Technology, 6 (2015), 624-632. 

[25]

M. I. KamienE. Muller and I. Zang, Research joint ventures and R&D cartels, American Economic Review, 5 (1992), 1293-1306. 

[26]

P. S. KohD. M. Reeb and W. Zhao, CEO confidence and unreported R&D, Management Science, 12 (2018), 5725-5747. 

[27]

J. L. Krieger, Trials and terminations: Learning from competitors' R&D failures, Management Science, first online, (2021).

[28]

S. Marjit, Incentives for cooperative and non-cooperative R&D in duopoly, Economics Letters, 2 (1991), 187-191. 

[29]

J. A. Martin and K. M. Eisenhardt, Rewiring: Cross-Business-Unit collaborations in multibusiness organizations, Academy of Management Journal, 2 (2010), 265-301. 

[30]

M. Motta, Cooperative R&D and vertical product differentiation, International Journal of Industrial Organization, 4 (1992), 643-661. 

[31]

J. Ning and V. Babich, R&D investments in the presence of knowledge spillover and debt financing: Can risk shifting cure free riding?, Manufacturing & Service Operations Management, 1 (2018), 97-112. 

[32]

C. PennetierK. Girotra and J. Mihm, R&D spending: Dynamic or persistent?, Manufacturing & Service Operations Management, 3 (2019), 636-657. 

[33]

G. Smrkolj and F. Wagener, Dynamic R&D with spillovers: A comment, Journal of Economic Dynamics and Control, 2 (2016), 453-457.  doi: 10.1016/j.jedc.2016.10.011.

[34]

M. Spence, Cost reduction, competition, and industry performance, Econometrica, 1 (1984), 101-121. 

[35]

L. Suzumura, Cooperative and noncooperative R&D in an oligopoly with spillovers, American Economic Review, 5 (1992), 1307-1320. 

[36]

Y. Taba, Optimal product R&D policies with endogenous quality choices and unilateral spillover, Journal of Economic Analysis & Policy, 1 (2016), 365-391. 

[37]

L. XuD. LiangZ. Duan and X. Xiao, Stability analysis of R&D cooperation in a supply chain, Mathematical Problems in Engineering, 8 (2015), 1-10.  doi: 10.1155/2015/409286.

[38]

M. Xing, The optimal risk choice of cooperative and noncooperative R&D in duopoly with spillovers, Bulletin of Economic Research, 4 (2017), 173-185.  doi: 10.1111/boer.12109.

[39]

D. ZengL. Xu and X. Bi, Effects of asymmetric knowledge spillovers on the stability of horizontal and vertical R&D cooperation, Computational and Mathematical Organization Theory, 1 (2017), 32-60. 

[40]

S. K. ByunJ. Min and H. Xia, Incremental vs. Breakthrough innovation: The role of technology spillovers, Management Science, 3 (2021), 1779-1802. 

[41]

N. BloomM. Schankerman and J. Van Reenen, Identifying technology spillovers and product market rivalry, Econometrica, 4 (2013), 1347-1393.  doi: 10.3982/ECTA9466.

[42]

J. Qiu and C. Wan, Technology spillovers and corporate cash holdings, Journal of Financial Economics, 3 (2015), 558-573. 

[43]

X. PanM. LiM. WangJ. Chu and H. Bo, The effects of outward foreign direct investment and reverse technology spillover on China's carbon productivity, Energy Policy, 145 (2020), 111730. 

[44]

Y. Hu, K. Fisher-Vanden and B. Su, Technological spillover through industrial and regional linkages: Firm-level evidence from China, Economic Modelling, 89 (2020).

[45]

Y. GaoS. B. TsaiX. XueT. RenX. DuQ. Chen and et al., An empirical study on green innovation efficiency in the green institutional environment, Sustainability, 3 (2018), 724. 

[46]

A. Ll, A. Zz, A. Mz, A. Cz and B. Dz, The effects of environmental regulation on outward foreign direct investment's reverse green technology spillover: Crowding out or facilitation?, Journal of Cleaner Production, 2 (2020).

[47]

Z. Cheng and B. R. Nault, Relative industry concentration and customer-driven it spillovers, Information Systems Research, 2 (2012), 340-355. 

[48]

P. Tambe and L. M. Hitt, Measuring information technology spillovers, Information Systems Research, 1 (2013), 53-71. 

[49]

P. Tambe and L. M. Hitt, Job hopping, information technology spillovers, and productivity growth, Management Science, 2 (2013), 338-355. 

[50]

X. QinD. Du and M. P. Kwan, Spatial spillovers and value chain spillovers: Evaluating regional R&D efficiency and its spillover effects in China, Scientometrics, 119 (2019), 721-747. 

[51]

N. Ishikawa and T. Shibata, R&D competition and cooperation with asymmetric spillovers in an oligopoly market, International Review of Economics & Finance, 72 (2020), 624-642.  doi: 10.1016/j.iref.2020.12.016.

[52]

J. NiH. HuangP. Wang and W. Zhou, Capacity investment and green R&D in a dynamic oligopoly under the potential shift in environmental damage, Economic Modelling, 7 (2019), 312-319. 

[53]

C. LiP. Zhou and Y. Li, Managerial overconfidence, overinvestment, and R&D spillover, Managerial and Decision Economics, 40 (2019), 1-4. 

[54]

D. Zhang, H. Dai, L. Dong, F. Qi, N. Zhang, X. Liu, et al., The long-term and spillover effects of price promotions on retailing platforms: Evidence from a large randomized experiment on Alibaba, Management Science, 6 (2020), 2589–2609. doi: 10.1109/TSP.2021.3083988.

[55]

D. Y. Joe and F. D. Oh, Spillover effects within business groups: The case of Korean chaebols, Management Science, 3 (2018), 1396-1412. 

[56]

A. HavivY. Huang and N. Li, Intertemporal demand spillover effects on video game platforms, Management Science, 10 (2020), 4788-4807. 

Figure 1.  The Four-Stage Game between Upstream Supplier and Downstream Manufacturers
Figure 2.  Optimal R&D Strategies for Downstream Manufacturers
Figure 3.  The R&D Decision Matrix
Figure 4.  The Effect of Supplier's Willingness to Cooperate
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}}} $
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}}} $
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}}} $
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}}} $
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}}} $
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}}} $
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}}} $
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}}} $
[1]

Yanfang Zhang. Social responsibility and R&D investments: Implications for a retailer and competitive manufacturers. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022082

[2]

Saeed Assani, Jianlin Jiang, Ahmad Assani, Feng Yang. Scale efficiency of China's regional R & D value chain: A double frontier network DEA approach. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1357-1382. doi: 10.3934/jimo.2020025

[3]

Ziyuan Zhang, Liying Yu. Research on optimal pricing decisions of the service supply chain oriented to strategic consumers. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022096

[4]

Wei Chen, Yongkai Ma, Weihao Hu. Electricity supply chain coordination with carbon abatement technology investment under the benchmarking mechanism. Journal of Industrial and Management Optimization, 2022, 18 (2) : 713-730. doi: 10.3934/jimo.2020175

[5]

Ying Dai, Yi Zhang, Han Song, Lin Zhou, Haiyan Li. Investment decision-making of closed-loop supply chain driven by big data technology. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022134

[6]

Min Li, Jiahua Zhang, Yifan Xu, Wei Wang. Effect of disruption risk on a supply chain with price-dependent demand. Journal of Industrial and Management Optimization, 2020, 16 (6) : 3083-3103. doi: 10.3934/jimo.2019095

[7]

Tinghai Ren, Kaifu Yuan, Dafei Wang, Nengmin Zeng. Effect of service quality on software sales and coordination mechanism in IT service supply chain. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021165

[8]

Ata Allah Taleizadeh, Leopoldo Eduardo Cárdenas-Barrón, Roya Sohani. Coordinating the supplier-retailer supply chain under noise effect with bundling and inventory strategies. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1701-1727. doi: 10.3934/jimo.2018118

[9]

Katherinne Salas Navarro, Jaime Acevedo Chedid, Whady F. Florez, Holman Ospina Mateus, Leopoldo Eduardo Cárdenas-Barrón, Shib Sankar Sana. A collaborative EPQ inventory model for a three-echelon supply chain with multiple products considering the effect of marketing effort on demand. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1613-1633. doi: 10.3934/jimo.2019020

[10]

Ziyuan Zhang, Liying Yu. Joint emission reduction dynamic optimization and coordination in the supply chain considering fairness concern and reference low-carbon effect. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021155

[11]

Honglin Yang, Siqi Zhao, Jiawu Peng. Optimal retail price and service level in a dual-channel supply chain with reference price effect. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022115

[12]

J. Becker, M. Ferreira, B.M.P.M. Oliveira, A.A. Pinto. R&d dynamics. Conference Publications, 2013, 2013 (special) : 61-68. doi: 10.3934/proc.2013.2013.61

[13]

Juliang Zhang, Jian Chen. Information sharing in a make-to-stock supply chain. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1169-1189. doi: 10.3934/jimo.2014.10.1169

[14]

Yeong-Cheng Liou, Siegfried Schaible, Jen-Chih Yao. Supply chain inventory management via a Stackelberg equilibrium. Journal of Industrial and Management Optimization, 2006, 2 (1) : 81-94. doi: 10.3934/jimo.2006.2.81

[15]

Feimin Zhong, Wei Zeng, Zhongbao Zhou. Mechanism design in a supply chain with ambiguity in private information. Journal of Industrial and Management Optimization, 2020, 16 (1) : 261-287. doi: 10.3934/jimo.2018151

[16]

Juliang Zhang. Coordination of supply chain with buyer's promotion. Journal of Industrial and Management Optimization, 2007, 3 (4) : 715-726. doi: 10.3934/jimo.2007.3.715

[17]

Liping Zhang. A nonlinear complementarity model for supply chain network equilibrium. Journal of Industrial and Management Optimization, 2007, 3 (4) : 727-737. doi: 10.3934/jimo.2007.3.727

[18]

Na Song, Ximin Huang, Yue Xie, Wai-Ki Ching, Tak-Kuen Siu. Impact of reorder option in supply chain coordination. Journal of Industrial and Management Optimization, 2017, 13 (1) : 449-475. doi: 10.3934/jimo.2016026

[19]

Joseph Geunes, Panos M. Pardalos. Introduction to the Special Issue on Supply Chain Optimization. Journal of Industrial and Management Optimization, 2007, 3 (1) : i-ii. doi: 10.3934/jimo.2007.3.1i

[20]

Jia Shu, Jie Sun. Designing the distribution network for an integrated supply chain. Journal of Industrial and Management Optimization, 2006, 2 (3) : 339-349. doi: 10.3934/jimo.2006.2.339

2021 Impact Factor: 1.411

Metrics

  • PDF downloads (127)
  • HTML views (105)
  • Cited by (0)

[Back to Top]