# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021194
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## Retail outsourcing strategy in Cournot & Bertrand retail competitions with economies of scale

 1 Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 211100, China 2 College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 211100, China 3 School of Business, The College of New Jersey, Ewing, NJ 08618, USA

* Corresponding author: Kebing Chen

Received  April 2021 Revised  August 2021 Early access November 2021

This paper investigates a manufacturer's retail outsourcing strategies under different competition modes with economies of scale. We focus on the effects of market competition modes, economies of scale and competitor's behavior on manufacturer's retail outsourcing decisions, and then we develop four game models under three competition modes. Firstly, we find the channel structure where both manufacturers choose retail outsourcing cannot be an equilibrium structure under the Cournot competition. The Cournot competition mode is less profitable to the firm than the Bertrand competition when the products are complements. Secondly, under the hybrid Cournot-Bertrand competition mode, there is only one equilibrium supply chain structure where neither manufacturer chooses retail outsourcing, when the substitutability and complementarity levels are not sufficiently high. In addition, setting price (quantity) contracts as the strategic variables is the dominant strategy for the direct-sale manufacturer who provides complementary (substitutable) products. Thirdly, both competitive firms will benefit from the situation where they choose the same competition mode. When the products are substitutes (complements), both of them choose the Cournot (Bertrand) competition mode. Finally, we show that the economies of scale have little impact on the equilibrium of the outsourcing structure but a great impact on the competition mode equilibrium.

Citation: Mingxia Li, Kebing Chen, Shengbin Wang. Retail outsourcing strategy in Cournot & Bertrand retail competitions with economies of scale. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021194
##### References:
 [1] S. M. Ali, M. H. Rahman, T. J. Tumpa, A. A. M. Rifat and S. K. Paul, Examining price and service competition among retailers in a supply chain under potential demand disruption, Journal of Retailing and Consumer Services, 40 (2018), 40-47.  doi: 10.1016/j.jretconser.2017.08.025. [2] A. Arya, B. Mittendorf and D. E. M. Sappington, Outsourcing, vertical integration, and price vs. quantity competition, International Journal of Industrial Organization, 26 (2008), 1-16.  doi: 10.1016/j.ijindorg.2006.10.006. [3] A. Arya, B. Mittendorf and D. E. M. Sappington, The make-or-buy decision in the presence of a rival: Strategic outsourcing to a common supplier, Management Science, 54 (2008), 1747-1758.  doi: 10.1287/mnsc.1080.0896. [4] D. Atkins and L. Liang, A note on competitive supply chains with generalized supply costs, European Journal of Operational Research, 207 (2010), 1316-1320.  doi: 10.1016/j.ejor.2010.07.012. [5] P. Bajec and M. Zanne, The current status of the Slovenian logistics outsourcing market, its ability and potential measures to improve the pursuit of global trends, International Journal of Logistics Systems & Management, 18 (2014), 436-448.  doi: 10.1504/IJLSM.2014.063979. [6] J. Bian, K. K. Lai, Z. Hua, X. Zhao and G. Zhou, Bertrand vs. Cournot competition in distribution channels with upstream collusion, International Journal of Production Economics, 204 (2018), 278-289.  doi: 10.1016/j.ijpe.2018.08.007. [7] G. P. Cachon and P. T. Harker, Competition and outsourcing with scale economies, Management Science, 48 (2002), 1314-1333.  doi: 10.1287/mnsc.48.10.1314.271. [8] K. Chen and T. Xiao, Outsourcing strategy and production disruption of supply chain with demand and capacity allocation uncertainties, International Journal of Production Economics, 170 (2015), 243-257.  doi: 10.1016/j.ijpe.2015.09.028. [9] K. Chen, R. Xu and H. Fang, Information disclosure model under supply chain competition with asymmetric demand disruption, Asia-Pacific Journal of Operational Research, 33 (2016), 1650043, 35 pp. doi: 10.1142/S0217595916500433. [10] J. Chen and S.-H. Lee., Cournot-bertrand comparison under R & D competition: Output versus R & D subsidies, Cogent Business & Management, (2021), https://mpra.ub.uni-muenchen.de/107949/. [11] L. K. Cheng, Comparing Bertrand and Cournot equilibria: A geometric approach, The Rand Journal of Economics, 16 (1985), 146-152.  doi: 10.2307/2555596. [12] H.-R. Din and C.-H. Sun, Welfare improving licensing with endogenous choice of prices versus quantities, The North American Journal of Economics & Finance, 51 (2020), 100859.  doi: 10.1016/j.najef.2018.10.007. [13] Y. Fang and B. Shou, Managing supply uncertainty under supply chain Cournot competition, European Journal of Operational Research, 243 (2015), 156-176.  doi: 10.1016/j.ejor.2014.11.038. [14] L. Fanti and M. Scrimitore, How to competer Cournot versus Bertrand in a vertical structure with an integrated input supplier, Southern Economic Journal, 85 (2019), 796-820.  doi: 10.1002/soej.12324. [15] A. Farahat and G. Perakis, A comparison of Bertrand and Cournot profits in oligopolies with differentiated products, Operations Research, 59 (2011), 507-513.  doi: 10.1287/opre.1100.0900. [16] E. Garaventa and T. Tellefsen, Outsourcing: The hidden costs, Review of Business, 22 (2001), 28-31. [17] A. Ghosh and M. Mitra, Comparing Bertrand and Cournot in mixed markets, Economics Letters, 109 (2010), 72-74.  doi: 10.1016/j.econlet.2010.08.021. [18] B. C. Giri and B. R. Sarker, Improving performance by coordinating a supply chain with third party logistics outsourcing under production disruption, Computers & Industrial Engineering, 103 (2017), 168-177.  doi: 10.1016/j.cie.2016.11.022. [19] A. Goli and B. Malmir, A covering tour approach for disaster relief locating and routing with fuzzy demand, International Journal of Intelligent Transportation Systems Research, 18 (2020), 140-152.  doi: 10.1007/s13177-019-00185-2. [20] A. Goli, E. B. Tirkolaee and N. S. Aydin, Fuzzy integrated cell formation and production scheduling considering automated guided vehicles and human factors, IEEE Transactions on Fuzzy Systems, (2021). doi: 10.1109/TFUZZ.2021.3053838. [21] A. Goli, H. K. Zare, R. T. Moghaddam and A. Sadeghieh, A comprehensive model of demand prediction based on hybrid artificial intelligence and metaheuristic algorithms: A case study in dairy industry, Journal of Industrial and Systems Engineering, 11 (2018), 190-203. [22] J. Haraguchi and T. Matsumura, Cournot-Bertrand comparison in a mixed oligopoly, Journal of Economics, 117 (2016), 117-136.  doi: 10.1007/s00712-015-0452-6. [23] M. Huang, J. Tu, X. Chao and D. Jin, Quality risk in logistics outsourcing: A fourth party logistics perspective, European Journal of Operational Research, 276 (2019), 855-879.  doi: 10.1016/j.ejor.2019.01.049. [24] B. Jiang, J. A. Belohlav and S. T. Young, Outsourcing impact on manufacturing firms' value: Evidence from Japan, Journal of Operations Management, 25 (2007), 885-900.  doi: 10.1016/j.jom.2006.12.002. [25] M. Kaya and Ö. Özer, Quality risk in outsourcing: Noncontractible product quality and private quality cost information, Naval Research Logistics, 56 (2009), 669-685.  doi: 10.1002/nav.20372. [26] T. Kremic, O. I. Tukel and W. O. Rom, Outsourcing decision support: A survey of benefits, risks, and decision factors, Supply Chain Management, 11 (2006), 467-482.  doi: 10.1108/13598540610703864. [27] J. R. Kroes and S. Ghosh, Outsourcing congruence with competitive priorities: Impact on supply chain and firm performance, Journal of Operations Management, 28 (2010), 124-143.  doi: 10.1016/j.jom.2009.09.004. [28] Y. J. Lin, Oligopoly and vertical integration: Note, The American Economic Review, 78 (1988), 251-254. [29] Z. Liu and A. Nagurney, Supply chain outsourcing under exchange rate risk and competition, Omega: International Journal of Management Science, 39 (2011), 539-549.  doi: 10.1016/j.omega.2010.11.003. [30] T. Matsumura and A. Ogawa, Price versus quantity in a mixed duopoly, Economics Letters, 116 (2012), 174-177.  doi: 10.1016/j.econlet.2012.02.012. [31] T. W. McGuire and R. Staelin, An industry equilibrium analysis of downstream vertical integration, Marketing Science, 2 (1983), 161-191.  doi: 10.1287/mksc.2.2.161. [32] M. Morris, M. Schindehutte and J. Allen, The entrepreneur's business model: Toward a unified perspective, Journal of Business Research, 58 (2005), 726-735.  doi: 10.1016/j.jbusres.2003.11.001. [33] T. Nariu, D. Flath and M. Okamura, A vertical oligopoly in which entry increases every firm's profit, Journal of Economics & Management Strategy, 30 (2021), 684-694.  doi: 10.1111/jems.12426. [34] S. M. Pahlevan, S. Hosseini and A. Goli, Sustainable supply chain network design using products' life cycle in the aluminum industry, Environmental Science and Pollution Research, (2021). doi: 10.1007/s11356-020-12150-8. [35] N. Singh and X. Vives, Price and quantity competition in a differentiated duopoly, The Rand Journal of Economics, 15 (1984), 546-554.  doi: 10.2307/2555525. [36] S. Sinha and S. P. Sarmah, Supply chain coordination model with insufficient production capacity and option for outsourcing, Mathematical and Computer Modelling, 46 (2007), 1442-1452.  doi: 10.1016/j.mcm.2007.03.014. [37] H. Sun, Y. Wan, Y. Li, L. L. Zhang and Z. Zhou, Competition in a dual-channel supply chain considering duopolistic retailers with different behaviours, Journal of Industrial & Management Optimization, 17 (2021), 601-631.  doi: 10.3934/jimo.2019125. [38] V. J. Tremblay, C. H. Tremblay and K. Isariyawongse, Cournot and Bertrand competition when advertising rotates demand: The case of Honda and Scion, International Journal of the Economics of Business, 20 (2013), 125-141.  doi: 10.1080/13571516.2012.750045. [39] L. Wang, H. Song, D. Zhang and H. Yang, Pricing decisions for complementary products in a fuzzy dual-channel supply chain, Journal of Industrial & Management Optimization, 15 (2019), 343-364.  doi: 10.3934/jimo.2018046. [40] C. Y. Wong and N. Karia, Explaining the competitive advantage of logistics service providers: A resource-based view approach, International Journal of Production Economics, 128 (2010), 51-67.  doi: 10.1016/j.ijpe.2009.08.026. [41] T. Xiao, Y. Xia and G. P. Zhang, Strategic outsourcing decisions for manufacturers competing on product quality, IIE Transactions, 46 (2014), 313-329.  doi: 10.1080/0740817X.2012.761368. [42] T. Xiao, Y. Xia and G. P. Zhang, Strategic outsourcing decisions for manufacturers that produce partially substitutable products in a quantity-setting duopoly situation, Decision Sciences, 38 (2007), 81-106.  doi: 10.1111/j.1540-5915.2007.00149.x. [43] Q. Yang, X. Zhao, H. Y. J. Yeung and Y. Liu, Improving logistics outsourcing performance through transactional and relational mechanisms under transaction uncertainties: Evidence from China, International Journal of Production Economics, 175 (2016), 12-23.  doi: 10.1016/j.ijpe.2016.01.022. [44] J. Zhao, X. Hou, Y. Guo and J. Wei, Pricing policies for complementary products in a dual-channel supply chain, Applied Mathematical Modelling, 49 (2017), 437-451.  doi: 10.1016/j.apm.2017.04.023. [45] W. Zhu, S. C. H. Ng, Z. Wang and X. Zhao, The role of outsourcing management process in improving the effectiveness of logistics outsourcing, International Journal of Production Economics, 188 (2017), 29-40.  doi: 10.1016/j.ijpe.2017.03.004.

show all references

##### References:
 [1] S. M. Ali, M. H. Rahman, T. J. Tumpa, A. A. M. Rifat and S. K. Paul, Examining price and service competition among retailers in a supply chain under potential demand disruption, Journal of Retailing and Consumer Services, 40 (2018), 40-47.  doi: 10.1016/j.jretconser.2017.08.025. [2] A. Arya, B. Mittendorf and D. E. M. Sappington, Outsourcing, vertical integration, and price vs. quantity competition, International Journal of Industrial Organization, 26 (2008), 1-16.  doi: 10.1016/j.ijindorg.2006.10.006. [3] A. Arya, B. Mittendorf and D. E. M. Sappington, The make-or-buy decision in the presence of a rival: Strategic outsourcing to a common supplier, Management Science, 54 (2008), 1747-1758.  doi: 10.1287/mnsc.1080.0896. [4] D. Atkins and L. Liang, A note on competitive supply chains with generalized supply costs, European Journal of Operational Research, 207 (2010), 1316-1320.  doi: 10.1016/j.ejor.2010.07.012. [5] P. Bajec and M. Zanne, The current status of the Slovenian logistics outsourcing market, its ability and potential measures to improve the pursuit of global trends, International Journal of Logistics Systems & Management, 18 (2014), 436-448.  doi: 10.1504/IJLSM.2014.063979. [6] J. Bian, K. K. Lai, Z. Hua, X. Zhao and G. Zhou, Bertrand vs. Cournot competition in distribution channels with upstream collusion, International Journal of Production Economics, 204 (2018), 278-289.  doi: 10.1016/j.ijpe.2018.08.007. [7] G. P. Cachon and P. T. Harker, Competition and outsourcing with scale economies, Management Science, 48 (2002), 1314-1333.  doi: 10.1287/mnsc.48.10.1314.271. [8] K. Chen and T. Xiao, Outsourcing strategy and production disruption of supply chain with demand and capacity allocation uncertainties, International Journal of Production Economics, 170 (2015), 243-257.  doi: 10.1016/j.ijpe.2015.09.028. [9] K. Chen, R. Xu and H. Fang, Information disclosure model under supply chain competition with asymmetric demand disruption, Asia-Pacific Journal of Operational Research, 33 (2016), 1650043, 35 pp. doi: 10.1142/S0217595916500433. [10] J. Chen and S.-H. Lee., Cournot-bertrand comparison under R & D competition: Output versus R & D subsidies, Cogent Business & Management, (2021), https://mpra.ub.uni-muenchen.de/107949/. [11] L. K. Cheng, Comparing Bertrand and Cournot equilibria: A geometric approach, The Rand Journal of Economics, 16 (1985), 146-152.  doi: 10.2307/2555596. [12] H.-R. Din and C.-H. Sun, Welfare improving licensing with endogenous choice of prices versus quantities, The North American Journal of Economics & Finance, 51 (2020), 100859.  doi: 10.1016/j.najef.2018.10.007. [13] Y. Fang and B. Shou, Managing supply uncertainty under supply chain Cournot competition, European Journal of Operational Research, 243 (2015), 156-176.  doi: 10.1016/j.ejor.2014.11.038. [14] L. Fanti and M. Scrimitore, How to competer Cournot versus Bertrand in a vertical structure with an integrated input supplier, Southern Economic Journal, 85 (2019), 796-820.  doi: 10.1002/soej.12324. [15] A. Farahat and G. Perakis, A comparison of Bertrand and Cournot profits in oligopolies with differentiated products, Operations Research, 59 (2011), 507-513.  doi: 10.1287/opre.1100.0900. [16] E. Garaventa and T. Tellefsen, Outsourcing: The hidden costs, Review of Business, 22 (2001), 28-31. [17] A. Ghosh and M. Mitra, Comparing Bertrand and Cournot in mixed markets, Economics Letters, 109 (2010), 72-74.  doi: 10.1016/j.econlet.2010.08.021. [18] B. C. Giri and B. R. Sarker, Improving performance by coordinating a supply chain with third party logistics outsourcing under production disruption, Computers & Industrial Engineering, 103 (2017), 168-177.  doi: 10.1016/j.cie.2016.11.022. [19] A. Goli and B. Malmir, A covering tour approach for disaster relief locating and routing with fuzzy demand, International Journal of Intelligent Transportation Systems Research, 18 (2020), 140-152.  doi: 10.1007/s13177-019-00185-2. [20] A. Goli, E. B. Tirkolaee and N. S. Aydin, Fuzzy integrated cell formation and production scheduling considering automated guided vehicles and human factors, IEEE Transactions on Fuzzy Systems, (2021). doi: 10.1109/TFUZZ.2021.3053838. [21] A. Goli, H. K. Zare, R. T. Moghaddam and A. Sadeghieh, A comprehensive model of demand prediction based on hybrid artificial intelligence and metaheuristic algorithms: A case study in dairy industry, Journal of Industrial and Systems Engineering, 11 (2018), 190-203. [22] J. Haraguchi and T. Matsumura, Cournot-Bertrand comparison in a mixed oligopoly, Journal of Economics, 117 (2016), 117-136.  doi: 10.1007/s00712-015-0452-6. [23] M. Huang, J. Tu, X. Chao and D. Jin, Quality risk in logistics outsourcing: A fourth party logistics perspective, European Journal of Operational Research, 276 (2019), 855-879.  doi: 10.1016/j.ejor.2019.01.049. [24] B. Jiang, J. A. Belohlav and S. T. Young, Outsourcing impact on manufacturing firms' value: Evidence from Japan, Journal of Operations Management, 25 (2007), 885-900.  doi: 10.1016/j.jom.2006.12.002. [25] M. Kaya and Ö. Özer, Quality risk in outsourcing: Noncontractible product quality and private quality cost information, Naval Research Logistics, 56 (2009), 669-685.  doi: 10.1002/nav.20372. [26] T. Kremic, O. I. Tukel and W. O. Rom, Outsourcing decision support: A survey of benefits, risks, and decision factors, Supply Chain Management, 11 (2006), 467-482.  doi: 10.1108/13598540610703864. [27] J. R. Kroes and S. Ghosh, Outsourcing congruence with competitive priorities: Impact on supply chain and firm performance, Journal of Operations Management, 28 (2010), 124-143.  doi: 10.1016/j.jom.2009.09.004. [28] Y. J. Lin, Oligopoly and vertical integration: Note, The American Economic Review, 78 (1988), 251-254. [29] Z. Liu and A. Nagurney, Supply chain outsourcing under exchange rate risk and competition, Omega: International Journal of Management Science, 39 (2011), 539-549.  doi: 10.1016/j.omega.2010.11.003. [30] T. Matsumura and A. Ogawa, Price versus quantity in a mixed duopoly, Economics Letters, 116 (2012), 174-177.  doi: 10.1016/j.econlet.2012.02.012. [31] T. W. McGuire and R. Staelin, An industry equilibrium analysis of downstream vertical integration, Marketing Science, 2 (1983), 161-191.  doi: 10.1287/mksc.2.2.161. [32] M. Morris, M. Schindehutte and J. Allen, The entrepreneur's business model: Toward a unified perspective, Journal of Business Research, 58 (2005), 726-735.  doi: 10.1016/j.jbusres.2003.11.001. [33] T. Nariu, D. Flath and M. Okamura, A vertical oligopoly in which entry increases every firm's profit, Journal of Economics & Management Strategy, 30 (2021), 684-694.  doi: 10.1111/jems.12426. [34] S. M. Pahlevan, S. Hosseini and A. Goli, Sustainable supply chain network design using products' life cycle in the aluminum industry, Environmental Science and Pollution Research, (2021). doi: 10.1007/s11356-020-12150-8. [35] N. Singh and X. Vives, Price and quantity competition in a differentiated duopoly, The Rand Journal of Economics, 15 (1984), 546-554.  doi: 10.2307/2555525. [36] S. Sinha and S. P. Sarmah, Supply chain coordination model with insufficient production capacity and option for outsourcing, Mathematical and Computer Modelling, 46 (2007), 1442-1452.  doi: 10.1016/j.mcm.2007.03.014. [37] H. Sun, Y. Wan, Y. Li, L. L. Zhang and Z. Zhou, Competition in a dual-channel supply chain considering duopolistic retailers with different behaviours, Journal of Industrial & Management Optimization, 17 (2021), 601-631.  doi: 10.3934/jimo.2019125. [38] V. J. Tremblay, C. H. Tremblay and K. Isariyawongse, Cournot and Bertrand competition when advertising rotates demand: The case of Honda and Scion, International Journal of the Economics of Business, 20 (2013), 125-141.  doi: 10.1080/13571516.2012.750045. [39] L. Wang, H. Song, D. Zhang and H. Yang, Pricing decisions for complementary products in a fuzzy dual-channel supply chain, Journal of Industrial & Management Optimization, 15 (2019), 343-364.  doi: 10.3934/jimo.2018046. [40] C. Y. Wong and N. Karia, Explaining the competitive advantage of logistics service providers: A resource-based view approach, International Journal of Production Economics, 128 (2010), 51-67.  doi: 10.1016/j.ijpe.2009.08.026. [41] T. Xiao, Y. Xia and G. P. Zhang, Strategic outsourcing decisions for manufacturers competing on product quality, IIE Transactions, 46 (2014), 313-329.  doi: 10.1080/0740817X.2012.761368. [42] T. Xiao, Y. Xia and G. P. Zhang, Strategic outsourcing decisions for manufacturers that produce partially substitutable products in a quantity-setting duopoly situation, Decision Sciences, 38 (2007), 81-106.  doi: 10.1111/j.1540-5915.2007.00149.x. [43] Q. Yang, X. Zhao, H. Y. J. Yeung and Y. Liu, Improving logistics outsourcing performance through transactional and relational mechanisms under transaction uncertainties: Evidence from China, International Journal of Production Economics, 175 (2016), 12-23.  doi: 10.1016/j.ijpe.2016.01.022. [44] J. Zhao, X. Hou, Y. Guo and J. Wei, Pricing policies for complementary products in a dual-channel supply chain, Applied Mathematical Modelling, 49 (2017), 437-451.  doi: 10.1016/j.apm.2017.04.023. [45] W. Zhu, S. C. H. Ng, Z. Wang and X. Zhao, The role of outsourcing management process in improving the effectiveness of logistics outsourcing, International Journal of Production Economics, 188 (2017), 29-40.  doi: 10.1016/j.ijpe.2017.03.004.
Possible regions for equilibrium structures in the pure Cournot competition
Possible regions for the more profitable competition mode
Possible regions for equilibrium structure in pure Bertrand competition
Possible regions for equilibrium structures in the Cournot-Bertrand competition
Possible regions for the equilibrium competition mode
The effect of the retail outsourcing strategy on $M_{1}$'s profit when $M_{2}$ uses direct sales
The effect of the retail outsourcing strategy on $M_{1}$'s profit when $M_{2}$ uses retail outsourcing
The effect of the retail outsourcing strategy on $M_{1}$'s profit when $M_{2}$ uses direct sales
The effect of the retail outsourcing strategy on $M_{1}$'s profit when $M_{2}$ uses retail outsourcing
Compare and contrast our model with the extant literature
 Article Model Market competition mode Economies of scale Product complementation Outsourcing decision Structure equilbrium Competition mode equilbrium Bertrand Cournot Chen & Lee [10] Price vs. quantity under R&D competition √ √ Nariu et al [33] Product-differentiated Cournot competition √ √ Arya et al [2] Price vs. quantity competition √ √ √ Farahat & Perakis [15] Equilibrium profits of price & quantity competition √ √ Matsumura & Ogawa [30] Choice of a price or a quantity contract in a mixed duopoly √ √ √ Fang & Shou [13] Cournot competition between two supply chains √ √ Zhao et al [44] Pricing of complementary products in dual channel chain √ √ √ Atkins & Liang [4] Competitive supply chains with generalised supply costs √ √ √ Haraguchi & Matsumura [22] Comparing price and quantity competition √ √ √ Din & Sun [12] Choice of prices versusquantities with patent licensing √ √ √ Wang et al [39] pricing problem of complementary products √ √ √ Our paper Equilibrium analysis under the supply chain competition √ √ √ √ √ √ √
 Article Model Market competition mode Economies of scale Product complementation Outsourcing decision Structure equilbrium Competition mode equilbrium Bertrand Cournot Chen & Lee [10] Price vs. quantity under R&D competition √ √ Nariu et al [33] Product-differentiated Cournot competition √ √ Arya et al [2] Price vs. quantity competition √ √ √ Farahat & Perakis [15] Equilibrium profits of price & quantity competition √ √ Matsumura & Ogawa [30] Choice of a price or a quantity contract in a mixed duopoly √ √ √ Fang & Shou [13] Cournot competition between two supply chains √ √ Zhao et al [44] Pricing of complementary products in dual channel chain √ √ √ Atkins & Liang [4] Competitive supply chains with generalised supply costs √ √ √ Haraguchi & Matsumura [22] Comparing price and quantity competition √ √ √ Din & Sun [12] Choice of prices versusquantities with patent licensing √ √ √ Wang et al [39] pricing problem of complementary products √ √ √ Our paper Equilibrium analysis under the supply chain competition √ √ √ √ √ √ √
Equilibrium solutions for the four different structures
 X= CC CD(DC) DD Chain i Chain 1(2) Chain 2(1) Chain i $q_{{X_i}}^{*QQ}$ $\frac{{a - {c_0}}}{{A + b}}$ $\frac{{({A^2} + 2A - {b^2} - Ab)(a - {c_0})}}{{A({A^2} + 2A - 2{b^2})}}$ $\frac{{(A - b)(a - {c_0})}}{{{A^2} + 2A - 2{b^2}}}$ $\frac{{2({c_0} - a)}}{{(b - 4)(b + 2) + 4\theta }}$ $\pi _{{X_{Mi}}}^{*QQ}$ $(1 - \theta ){(q_{C{C_i}}^{QQ})^2}$ $(1 - \theta ){(q_{C{D_1}}^{QQ})^2}$ $\frac{{(2 - \theta )A - {b^2}}}{A}{(q_{C{D_2}}^{QQ})^2}$ $\frac{{(4 - 2\theta - {b^2})}}{2}{(q_{D{D_i}}^{QQ})^2}$
 X= CC CD(DC) DD Chain i Chain 1(2) Chain 2(1) Chain i $q_{{X_i}}^{*QQ}$ $\frac{{a - {c_0}}}{{A + b}}$ $\frac{{({A^2} + 2A - {b^2} - Ab)(a - {c_0})}}{{A({A^2} + 2A - 2{b^2})}}$ $\frac{{(A - b)(a - {c_0})}}{{{A^2} + 2A - 2{b^2}}}$ $\frac{{2({c_0} - a)}}{{(b - 4)(b + 2) + 4\theta }}$ $\pi _{{X_{Mi}}}^{*QQ}$ $(1 - \theta ){(q_{C{C_i}}^{QQ})^2}$ $(1 - \theta ){(q_{C{D_1}}^{QQ})^2}$ $\frac{{(2 - \theta )A - {b^2}}}{A}{(q_{C{D_2}}^{QQ})^2}$ $\frac{{(4 - 2\theta - {b^2})}}{2}{(q_{D{D_i}}^{QQ})^2}$
Equilibrium solutions for the four different structures
 X chain $w_{{X_i}}^{*PP}$ $q_{{X_i}}^{*PP}$ $\pi _{{X_{Mi}}}^{*PP}$ CC i NA $\frac{{a - {c_0}}}{{(2 - b)(1 + b) - 2\theta }}$ $(1 - {b^2} - \theta ){(q_{C{C_i}}^{*PP})^2}$ CD 1(2) NA $\frac{{(1 - b)(2 + b)a + ({b^2} - 2){c_0} + bw_{C{D_2}}^{*PP}}}{S}$ $(1 - {b^2} - \theta ){(q_{C{D_1}}^{*PP})^2}$ (DC) 2(1) $\frac{{(S + 2\theta T)(T + b)a + (ST - Sb - 2\theta Tb){c_0}}}{{2T(S + \theta T)}}$ $\frac{{ - (T + b)a + b{c_0} + Tw_{C{D_2}}^{*PP}}}{S}$ $\frac{{(4 - {b^2})(1 - {b^2})}}{{2 - {b^2}}}{(q_{C{D_2}}^{*PP})^2}$ DD i $\frac{{Sa - (1 + b)(2 - b)({b^2} - 2){c_0}}}{{S - (1 + b)(2 - b)({b^2} - 2)}}$ $\frac{{a - w_{D{D_i}}^{*PP}}}{{(1 + b)(2 - b)}}$ $[\frac{{(4 - {b^2})(1 - {b^2})}}{{2 - {b^2}}} - \theta ]{(q_{D{D_i}}^{*PP})^2}$
 X chain $w_{{X_i}}^{*PP}$ $q_{{X_i}}^{*PP}$ $\pi _{{X_{Mi}}}^{*PP}$ CC i NA $\frac{{a - {c_0}}}{{(2 - b)(1 + b) - 2\theta }}$ $(1 - {b^2} - \theta ){(q_{C{C_i}}^{*PP})^2}$ CD 1(2) NA $\frac{{(1 - b)(2 + b)a + ({b^2} - 2){c_0} + bw_{C{D_2}}^{*PP}}}{S}$ $(1 - {b^2} - \theta ){(q_{C{D_1}}^{*PP})^2}$ (DC) 2(1) $\frac{{(S + 2\theta T)(T + b)a + (ST - Sb - 2\theta Tb){c_0}}}{{2T(S + \theta T)}}$ $\frac{{ - (T + b)a + b{c_0} + Tw_{C{D_2}}^{*PP}}}{S}$ $\frac{{(4 - {b^2})(1 - {b^2})}}{{2 - {b^2}}}{(q_{C{D_2}}^{*PP})^2}$ DD i $\frac{{Sa - (1 + b)(2 - b)({b^2} - 2){c_0}}}{{S - (1 + b)(2 - b)({b^2} - 2)}}$ $\frac{{a - w_{D{D_i}}^{*PP}}}{{(1 + b)(2 - b)}}$ $[\frac{{(4 - {b^2})(1 - {b^2})}}{{2 - {b^2}}} - \theta ]{(q_{D{D_i}}^{*PP})^2}$
Equilibrium solutions for the four supply chain structures
 Chain i $w_{{X_i}}^{*QP}$ $p_{{X_i}}^{*QP}$ $q_{{X_i}}^{*QP}$ $\pi _{{X_{Mi}}}^{*QP}$ $\pi _{{X_{Ri}}}^{*QP}$ CC 1 NA $\frac{{({b^2} - b - 2)(1 - b)(a - {c_0}) + (3{b^2} - 4){c_0}}}{{3{b^2} - 4}}$ $\frac{{(b - 2)(a - {c_0})}}{{3{b^2} - 4}}$ $(1 - {b^2}){(q_{C{C_1}}^{*QP})^2}$ NA 2 NA $\frac{{({b^2} + b - 2)(a - {c_0}) + (3{b^2} - 4){c_0}}}{{3{b^2} - 4}}$ $\frac{{({b^2} + b - 2)(a - {c_0})}}{{3{b^2} - 4}}$ ${(q_{C{C_2}}^{*QP})^2}$ NA CD 1 NA $\frac{{(1 - {b^2})(2 - b)a + (2 - {b^2}){c_0} + b(1 - {b^2})w_{C{D_2}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{(2 - b)a - 2{c_0} + bw_{C{D_2}}^{*QP}}}{{4 - 3{b^2}}}$ $(1 - {b^2}){(q_{C{D_1}}^{*QP})^2}$ NA 2 $\frac{{(1 - b)(2 + b)a - (1 + b)(b - 2){c_0}}}{{2(2 - {b^2})}}$ $\frac{{(1 - b)(2 + b)a + b{c_0} + 2(1 - {b^2})w_ {C{D_2}}^ {*QP} }}{{4 - 3{b^2}}}$ $\frac{{(2 - {b^2} - b)a + b{c_0} + ({b^2} - 2)w_{C{D_2}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{4 - 3{b^2}}}{{2 - {b^2} }} {(q_{C{D_2}}^{*QP})^2}$ ${(q_{C{D_2}}^{*QP})^2}$ DC 1 $\frac{{(2 - b)a + (2 + b){c_0}}}{4}$ $\frac{{(1 - {b^2})(2 - b)a + b(1 - {b^2}){c_0} + (2 - {b^2})w_{D{C_1}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{(2 - b)a + b{c_0} - 2w_{D{C_1}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{4 - 3{b^2}}}{2}{(q_{D{C_1}}^{*QP})^2}$ $(1 - {b^2}){(q_{D{C_1}}^{*QP})^2}$ 2 NA $\frac{{(1 - b)(2 + b)a + 2(1 - {b^2}){c_0} + bw_{D{C_1}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{(2 - {b^2} - b)a + ({b^2} - 2){c_0} + bw_{D{C_1}}^{*QP}}}{{4 - 3{b^2}}}$ ${(q_{D{C_2}}^{*QP})^2}$ NA DD 1 $\frac{{({b^3} - 5{b^2} - 2b + 8)a - ({b^3} + 4{b^2} - 2b - 8){c_0}}}{{16 - 9{b^2}}}$ $\frac{{(1 - {b^2})(2 - b)a + (2 - {b^2})w_{D{D_1}}^{*QP} + b(1 - {b^2})w_{D{D_2}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{2({b^3} - 5{b^2} - 2b + 8)(a - {c_0})}}{{(4 - 3{b^2})(16 - 9{b^2})}}$ $\frac{{4 - 3{b^2}}}{2}{(q_{D{D_1}}^{*QP})^2}$ $(1 - {b^2}){(q_{D{D_1}}^{*QP})^2}$ 2 $\frac{{(8 - 5{b^2} - 2b)a - (4{b^2} - 2b - 8){c_0}}}{{16 - 9{b^2}}}$ $\frac{{(1 - b)(2 + b)a + 2(1 - {b^2})w_{D{D_2}}^{*QP} + bw_{D{D_1}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{(5{b^2} + 2b - 8)({b^2} - 2)(a - {c_0})}}{{(4 - 3{b^2})(16 - 9{b^2})}}$ $\frac{{4 - 3{b^2}}}{{2 - {b^2}}}{(q_{D{D_2}}^{*QP})^2}$ ${(q_{D{D_2}}^{*QP})^2}$
 Chain i $w_{{X_i}}^{*QP}$ $p_{{X_i}}^{*QP}$ $q_{{X_i}}^{*QP}$ $\pi _{{X_{Mi}}}^{*QP}$ $\pi _{{X_{Ri}}}^{*QP}$ CC 1 NA $\frac{{({b^2} - b - 2)(1 - b)(a - {c_0}) + (3{b^2} - 4){c_0}}}{{3{b^2} - 4}}$ $\frac{{(b - 2)(a - {c_0})}}{{3{b^2} - 4}}$ $(1 - {b^2}){(q_{C{C_1}}^{*QP})^2}$ NA 2 NA $\frac{{({b^2} + b - 2)(a - {c_0}) + (3{b^2} - 4){c_0}}}{{3{b^2} - 4}}$ $\frac{{({b^2} + b - 2)(a - {c_0})}}{{3{b^2} - 4}}$ ${(q_{C{C_2}}^{*QP})^2}$ NA CD 1 NA $\frac{{(1 - {b^2})(2 - b)a + (2 - {b^2}){c_0} + b(1 - {b^2})w_{C{D_2}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{(2 - b)a - 2{c_0} + bw_{C{D_2}}^{*QP}}}{{4 - 3{b^2}}}$ $(1 - {b^2}){(q_{C{D_1}}^{*QP})^2}$ NA 2 $\frac{{(1 - b)(2 + b)a - (1 + b)(b - 2){c_0}}}{{2(2 - {b^2})}}$ $\frac{{(1 - b)(2 + b)a + b{c_0} + 2(1 - {b^2})w_ {C{D_2}}^ {*QP} }}{{4 - 3{b^2}}}$ $\frac{{(2 - {b^2} - b)a + b{c_0} + ({b^2} - 2)w_{C{D_2}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{4 - 3{b^2}}}{{2 - {b^2} }} {(q_{C{D_2}}^{*QP})^2}$ ${(q_{C{D_2}}^{*QP})^2}$ DC 1 $\frac{{(2 - b)a + (2 + b){c_0}}}{4}$ $\frac{{(1 - {b^2})(2 - b)a + b(1 - {b^2}){c_0} + (2 - {b^2})w_{D{C_1}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{(2 - b)a + b{c_0} - 2w_{D{C_1}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{4 - 3{b^2}}}{2}{(q_{D{C_1}}^{*QP})^2}$ $(1 - {b^2}){(q_{D{C_1}}^{*QP})^2}$ 2 NA $\frac{{(1 - b)(2 + b)a + 2(1 - {b^2}){c_0} + bw_{D{C_1}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{(2 - {b^2} - b)a + ({b^2} - 2){c_0} + bw_{D{C_1}}^{*QP}}}{{4 - 3{b^2}}}$ ${(q_{D{C_2}}^{*QP})^2}$ NA DD 1 $\frac{{({b^3} - 5{b^2} - 2b + 8)a - ({b^3} + 4{b^2} - 2b - 8){c_0}}}{{16 - 9{b^2}}}$ $\frac{{(1 - {b^2})(2 - b)a + (2 - {b^2})w_{D{D_1}}^{*QP} + b(1 - {b^2})w_{D{D_2}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{2({b^3} - 5{b^2} - 2b + 8)(a - {c_0})}}{{(4 - 3{b^2})(16 - 9{b^2})}}$ $\frac{{4 - 3{b^2}}}{2}{(q_{D{D_1}}^{*QP})^2}$ $(1 - {b^2}){(q_{D{D_1}}^{*QP})^2}$ 2 $\frac{{(8 - 5{b^2} - 2b)a - (4{b^2} - 2b - 8){c_0}}}{{16 - 9{b^2}}}$ $\frac{{(1 - b)(2 + b)a + 2(1 - {b^2})w_{D{D_2}}^{*QP} + bw_{D{D_1}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{(5{b^2} + 2b - 8)({b^2} - 2)(a - {c_0})}}{{(4 - 3{b^2})(16 - 9{b^2})}}$ $\frac{{4 - 3{b^2}}}{{2 - {b^2}}}{(q_{D{D_2}}^{*QP})^2}$ ${(q_{D{D_2}}^{*QP})^2}$
Payoff table for the strategic competition choice of the X supply chain structure
 Supply chain 2 Cournot Bertrand Supply chain 1 Cournot $(\pi _{{X_{M1}}}^{*QQ},\pi _{{X_{M2}}}^{*QQ})$ $(\pi _{{X_{M1}}}^{*QP},\pi _{{X_{M2}}}^{*QP})$ Bertrand $(\pi _{{X_{M1}}}^{*PQ},\pi _{{X_{M2}}}^{*PQ})$ $(\pi _{{X_{M1}}}^{*PP},\pi _{{X_{M2}}}^{*PP})$
 Supply chain 2 Cournot Bertrand Supply chain 1 Cournot $(\pi _{{X_{M1}}}^{*QQ},\pi _{{X_{M2}}}^{*QQ})$ $(\pi _{{X_{M1}}}^{*QP},\pi _{{X_{M2}}}^{*QP})$ Bertrand $(\pi _{{X_{M1}}}^{*PQ},\pi _{{X_{M2}}}^{*PQ})$ $(\pi _{{X_{M1}}}^{*PP},\pi _{{X_{M2}}}^{*PP})$
Equilibrium solutions for the four structures with economies of scale
 Chain i $\tilde w_{{X_i}}^{*QP}$ $\tilde q_{{X_i}}^{*QP}$ $\tilde \pi _{{X_{Mi}}}^{*QP}$ CC 1 NA $\frac{{(T + b - {b^2})(a - {c_0})}}{{AT + {b^2}}}$ $(1 - {b^2} - \theta ){(\tilde q_{C{C_1}}^{*QP})^2}$ 2 NA $\frac{{(T + b)(a - {c_0})}}{{AT + {b^2}}}$ $(1 - \theta ){(\tilde q_{C{C_2}}^{*QP})^2}$ CD 1 NA $\frac{{(2 - b)a - 2{c_0} + b\tilde w_{C{D_2}}^{*QP}}}{{ - U}}$ $(1 - {b^2} - \theta ){(\tilde q_{C{D_1}}^{*QP})^2}$ 2 $\frac{{(U - 2\theta T)(T + b)a + [U(T - b) + 2\theta bT]{c_0}}}{{2T(U - \theta T)}}$ $\frac{{( - T - b)a + b{c_0} + T\tilde w_{C{D_2}}^{*QP}}}{{ - U}}$ $(\frac{U}{T} - \theta ){(\tilde q_{C{D_2}}^{*QP})^2}$ DC 1 $\frac{{(U - 2\theta T)(A - b)a + [U(b + A) - 2\theta b(T + Ab)]{c_0}}}{{2A[U + 2\theta (1 - {b^2} - \theta )]}}$ $\frac{{(A - b)a + b{c_0} - A\tilde w_{D{C_1}}^{*QP}}}{{2\theta {b^2} - U}}$ $\frac{{U + 2\theta (1 - {b^2} - \theta )}}{{ - A}}{(\tilde q_{D{C_1}}^{*QP})^2}$ 2 NA $\frac{{(2 + b)(1 - b)a + ({b^2} - 2){c_0} + b\tilde w_{D{C_1}}^{*QP}}}{{2\theta {b^2} - U}}$ $(1 - \theta ){(\tilde q_{D{C_2}}^{*QP})^2}$ DD 1 $\frac{{U[ - {b^3} + 5{b^2} + 2b - 8 + 2\theta (2 - {b^2})]a - (2 - {b^2})[5U + 4\theta (b - {b^2} - 3)]{c_0}}}{{(3{b^2} - 4)(9{b^2} - 16) + 2\theta (4 - {b^2})(5{b^2} - 8) + 8{\theta ^2}(2 - {b^2})}}$ $\frac{{2(\tilde w_{D{D_1}}^{*QP} - {c_0})}}{{ - U}}$ $(\frac{{4 - 3{b^2}}}{2} - \theta ){(\tilde q_{D{D_1}}^{*QP})^2}$ 2 $\frac{{U(b - 2)a - 2(U - 4\theta ){c_0} + 4(U - 2\theta )\tilde w_{D{D_1}}^{*QP}}}{{Ub}}$ $\frac{{(2 - {b^2})(\tilde w_{D{D_2}}^{*QP} - {c_0})}}{{2\theta {b^2} - U}}$ $(\frac{{4 - 3{b^2}}}{{2 - {b^2}}} - \theta ){(\tilde q_{D{D_2}}^{*QP})^2}$
 Chain i $\tilde w_{{X_i}}^{*QP}$ $\tilde q_{{X_i}}^{*QP}$ $\tilde \pi _{{X_{Mi}}}^{*QP}$ CC 1 NA $\frac{{(T + b - {b^2})(a - {c_0})}}{{AT + {b^2}}}$ $(1 - {b^2} - \theta ){(\tilde q_{C{C_1}}^{*QP})^2}$ 2 NA $\frac{{(T + b)(a - {c_0})}}{{AT + {b^2}}}$ $(1 - \theta ){(\tilde q_{C{C_2}}^{*QP})^2}$ CD 1 NA $\frac{{(2 - b)a - 2{c_0} + b\tilde w_{C{D_2}}^{*QP}}}{{ - U}}$ $(1 - {b^2} - \theta ){(\tilde q_{C{D_1}}^{*QP})^2}$ 2 $\frac{{(U - 2\theta T)(T + b)a + [U(T - b) + 2\theta bT]{c_0}}}{{2T(U - \theta T)}}$ $\frac{{( - T - b)a + b{c_0} + T\tilde w_{C{D_2}}^{*QP}}}{{ - U}}$ $(\frac{U}{T} - \theta ){(\tilde q_{C{D_2}}^{*QP})^2}$ DC 1 $\frac{{(U - 2\theta T)(A - b)a + [U(b + A) - 2\theta b(T + Ab)]{c_0}}}{{2A[U + 2\theta (1 - {b^2} - \theta )]}}$ $\frac{{(A - b)a + b{c_0} - A\tilde w_{D{C_1}}^{*QP}}}{{2\theta {b^2} - U}}$ $\frac{{U + 2\theta (1 - {b^2} - \theta )}}{{ - A}}{(\tilde q_{D{C_1}}^{*QP})^2}$ 2 NA $\frac{{(2 + b)(1 - b)a + ({b^2} - 2){c_0} + b\tilde w_{D{C_1}}^{*QP}}}{{2\theta {b^2} - U}}$ $(1 - \theta ){(\tilde q_{D{C_2}}^{*QP})^2}$ DD 1 $\frac{{U[ - {b^3} + 5{b^2} + 2b - 8 + 2\theta (2 - {b^2})]a - (2 - {b^2})[5U + 4\theta (b - {b^2} - 3)]{c_0}}}{{(3{b^2} - 4)(9{b^2} - 16) + 2\theta (4 - {b^2})(5{b^2} - 8) + 8{\theta ^2}(2 - {b^2})}}$ $\frac{{2(\tilde w_{D{D_1}}^{*QP} - {c_0})}}{{ - U}}$ $(\frac{{4 - 3{b^2}}}{2} - \theta ){(\tilde q_{D{D_1}}^{*QP})^2}$ 2 $\frac{{U(b - 2)a - 2(U - 4\theta ){c_0} + 4(U - 2\theta )\tilde w_{D{D_1}}^{*QP}}}{{Ub}}$ $\frac{{(2 - {b^2})(\tilde w_{D{D_2}}^{*QP} - {c_0})}}{{2\theta {b^2} - U}}$ $(\frac{{4 - 3{b^2}}}{{2 - {b^2}}} - \theta ){(\tilde q_{D{D_2}}^{*QP})^2}$
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