# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021157
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## Equilibrium decisions on pricing and innovation that impact reference price dynamics

 1 School of Management, Hefei University of Technology, Hefei 230009, China 2 Naveen Jindal School of Management, University of Texas at Dallas, Dallas, TX 75080, USA 3 Belk College of Business, University of North Carolina at Charlotte, Charlotte, NC 28223, USA 4 Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China

* Corresponding authors: Xiuli He and Xianjin Du

Received  February 2021 Revised  June 2021 Early access September 2021

Fund Project: This work is supported by the National Natural Science Foundation of China (NSFC) under Grant 72071065

Previous studies have confirmed that reference prices play an essential role in consumer purchasing decisions, and some researchers have suggested that reference prices are positively influenced by innovation. Therefore, we construct an interactive effect of innovation and reference price to study their combined impact on supply chain decisions. We model a supply chain, where a manufacturer determines the innovation level and the wholesale price while the retailer controls the retail price, as a dynamic Stackelberg game. We show that the interactive effect causes the steady-state wholesale and retail prices to increase, thus motivating the manufacturer to increase innovation investment. We see that the retail price and the level of innovation increase in reference price effect whereas they decrease in consumer memory. The centralized firm has a higher steady-state innovation level and innovation/price ratio and lower steady-state retail price compared to the decentralized supply chain. Consumers also benefit from the interactive effect as well as from centralization. Finally, we use numerical analysis to demonstrate our results and offer some managerial implications.

Citation: Shaokun Tao, Xianjin Du, Suresh P. Sethi, Xiuli He, Yu Li. Equilibrium decisions on pricing and innovation that impact reference price dynamics. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021157
##### References:
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Duan, Joint production and pricing inventory system under stochastic reference price effect, Computers & Industrial Engineering, 143 (2020), 106411.  doi: 10.1016/j.cie.2020.106411. [7] T. Chakraborty, S. S. Chauhan and M. Ouhimmou, Cost-sharing mechanism for product quality improvement in a supply chain under competition, International Journal of Production Economics, 208 (2019), 566-587.  doi: 10.1016/j.ijpe.2018.12.015. [8] K. Chen, Y. Zha, L. C. Alwan and L. Zhang, Dynamic pricing in the presence of reference price effect and consumer strategic behaviour, International Journal of Production Research, 58 (2020), 546-561.  doi: 10.1080/00207543.2019.1598592. [9] X. Chen, P. Hu and Z. Hu, Efficient algorithms for the dynamic pricing problem with reference price effect, Management Science, 63 (2016), 4389-4408.  doi: 10.1287/mnsc.2016.2554. [10] X. Chen, P. Hu, S. Shum and Y. Zhang, Dynamic stochastic inventory management with reference price effects, Oper. Res., 64 (2016), 1529-1536.  doi: 10.1287/opre.2016.1524. [11] E. J. Dockner, S. Jorgensen, N. Van Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, Cambridge, UK, 2000.  doi: 10.1017/CBO9780511805127. [12] C.-Y. Dye, C.-T. Yang and C.-C. Wu, Joint dynamic pricing and preservation technology investment for an integrated supply chain with reference price effects, Journal of the Operational Research Society, 69 (2018), 811-824. [13] G. Fibich, A. Gavious and O. Lowengart, Explicit solutions of optimization models and differential games with nonsmooth (asymmetric) reference-price effects, Oper. Res., 51 (2003), 721-734.  doi: 10.1287/opre.51.5.721.16758. [14] H. Fu, K. Li and W. Fu, Investing in suppliers with capacity constraints in a decentralized assembly system, Computers & Industrial Engineering, 142 (2020), 106332.  doi: 10.1016/j.cie.2020.106332. [15] H. Fu, M. Liu and B. 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Shah, A comparative analysis of greening policies across supply chain structures, International Journal of Production Economics, 135 (2012), 568-583.  doi: 10.1016/j.ijpe.2011.05.027. [21] D. Ghosh and J. Shah, Supply chain analysis under green sensitive consumer demand and cost sharing contract, International Journal of Production Economics, 164 (2015), 319-329.  doi: 10.1016/j.ijpe.2014.11.005. [22] S. M. Gilbert and V. Cvsa, Strategic commitment to price to stimulate downstream innovation in a supply chain, European Journal of Operational Research, 150 (2003), 617-639.  doi: 10.1016/S0377-2217(02)00590-8. [23] B. G. Hardie, E. J. Johnson and P. S. Fader, Modeling loss aversion and reference dependence effects on brand choice, Marketing Science, 12 (1993), 378-394.  doi: 10.1287/mksc.12.4.378. [24] Y. He, Q. Xu, B. Xu and P. Wu, Supply chain coordination in quality improvement with reference effects, Journal of the Operational Research Society, 67 (2017), 1158-1168.  doi: 10.1057/jors.2016.10. [25] S. L. Holak, Determinants of innovative durables adoption an empirical study with implications for early product screening, Journal of Product Innovation Management, 5 (1988), 50-69.  doi: 10.1111/1540-5885.510050. [26] S. L. Holak and D. R. Lehmann, Purchase intentions and the dimensions of innovation: An exploratory model, Journal of Product Innovation Management, 7 (1990), 59-73.  doi: 10.1111/1540-5885.710059. [27] T.-P. Hsieh and C.-Y. Dye, Optimal dynamic pricing for deteriorating items with reference price effects when inventories stimulate demand, European J. Oper. Res., 262 (2017), 136-150.  doi: 10.1016/j.ejor.2017.03.038. [28] R. F. Hurley and G. T. M. Hult, Innovation, market orientation, and organizational learning: An integration and empirical examination, Journal of Marketing, 62 (1998), 42-54. [29] L. Jia and L. Tong, Renewables and storage in distribution systems: Centralized vs. Decentralized integration, IEEE Journal on Selected Areas in Communications, 34 (2016), 665-674.  doi: 10.1109/JSAC.2016.2525638. [30] G. Kalyanaram and R. S. Winer, Empirical generalizations from reference price research, Marketing Science, 14 (1995), 161-169.  doi: 10.1287/mksc.14.3.G161. [31] M. Karlsson, L. Trygg and B.-O. Elfström, Measuring R & D productivity: Complementing the picture by focusing on research activities, Technovation, 24 (2004), 179-186.  doi: 10.1016/S0166-4972(03)00111-1. [32] B. Kim, Coordinating an innovation in supply chain management, European Journal of Operational Research, 123 (2000), 568-584.  doi: 10.1016/S0377-2217(99)00113-7. [33] P. K. Kopalle, A. G. Rao and J. L. Assuncao, Asymmetric reference price effects and dynamic pricing policies, Marketing Science, 15 (1996), 60-85.  doi: 10.1287/mksc.15.1.60. [34] P. K. Kopalle and R. S. Winer, A dynamic model of reference price and expected quality, Marketing Letters, 7 (1996), 41-52.  doi: 10.1007/BF00557310. [35] J. M. Lattin and R. E. Bucklin, Reference effects of price and promotion on brand choice behavior, Journal of Marketing Research, 26 (1989), 299-310. [36] G. Li, M. Reimann and W. Zhang, When remanufacturing meets product quality improvement: The impact of production cost, European J. Oper. Res., 271 (2018), 913-925.  doi: 10.1016/j.ejor.2018.05.060. [37] Z. Lin, Price promotion with reference price effects in supply chain, Transportation Research Part E: Logistics and Transportation Review, 85 (2016), 52-68.  doi: 10.1016/j.tre.2015.11.002. [38] G. Liu, S. P. Sethi and J. Zhang, Myopic vs. far-sighted behaviours in a revenue-sharing supply chain with reference quality effects, International Journal of Production Research, 54 (2016), 1334-1357.  doi: 10.1080/00207543.2015.1068962. [39] L. Lu, Q. Gou, W. Tang and J. Zhang, Joint pricing and advertising strategy with reference price effect, International Journal of Production Research, 54 (2016), 5250-5270.  doi: 10.1080/00207543.2016.1165878. [40] G. Martín-Herrán, S. Taboubi and G. Zaccour, Dual role of price and myopia in a marketing channel, European J. Oper. Res., 219 (2012), 284-295.  doi: 10.1016/j.ejor.2011.12.015. [41] T. Mazumdar and P. Papatla, An investigation of reference price segments, Journal of Marketing Research, 37 (2000), 246-259.  doi: 10.1509/jmkr.37.2.246.18727. [42] T. Mazumdar, S. P. Raj and I. Sinha, Reference price research: Review and Propositions, Journal of Marketing, 69 (2005), 84-102. [43] S. K. Mukhopadhyay and R. Setaputra, A dynamic model for optimal design quality and return policies, European Journal of Operational Research, 180 (2007), 1144-1154.  doi: 10.1016/j.ejor.2006.05.016. [44] J. Ni and J. Zhao, A dynamic analysis of the investment in product innovation in a supply chain under reference price effect: Competition vs cooperation, International Journal of Systems Science: Operations & Logistics, 8 (2021), 1-13.  doi: 10.1080/23302674.2019.1629040. [45] M. Nouri, S.-M. Hosseini-Motlagh, M. Nematollahi and B. R. Sarker, Coordinating manufacturer's innovation and retailer's promotion and replenishment using a compensation-based wholesale price contract, International Journal of Production Economics, 198 (2018), 11-24.  doi: 10.1016/j.ijpe.2018.01.023. [46] NPD, Apple Owners Nearly 40 Percent More Interested in the iPad than Non-Apple Owners, Technical report, Port Washington, New York, 2010. [47] I. Popescu and Y. Wu, Dynamic pricing strategies with reference effects, Ope. Res., 55 (2007), 413-429.  doi: 10.1287/opre.1070.0393. [48] M. Porter and V. Linde, Green and competitive, Harvard Business Review, 73 (1995), 120-134. [49] D. Pujari, Eco-innovation and new product development: Understanding the influences on market performance, Technovation, 26 (2006), 76-85.  doi: 10.1016/j.technovation.2004.07.006. [50] D. S. Putler, Incorporating reference price effects into a theory of consumer choice, Marketing Science, 11 (1992), 287-309.  doi: 10.1287/mksc.11.3.287. [51] S. P. Sethi and T. Li, A review of dynamic Stackelberg game models, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 125-159.  doi: 10.3934/dcdsb.2017007. [52] T. A. Suter and D. M. Hardesty, Maximizing earnings and price fairness perceptions in online consumer-to-consumer auctions, Journal of Retailing, 81 (2005), 307-317.  doi: 10.1016/j.jretai.2005.01.007. [53] K. Uğurlu, Dynamic optimal contract under parameter uncertainty with risk-averse agent and principal, Turkish J. Math., 42 (2018), 977-992.  doi: 10.3906/mat-1703-102. [54] J. Wang and H. Shin, The impact of contracts and competition on upstream innovation in a supply chain, Production and Operations Management, 24 (2015), 134-146.  doi: 10.1111/poms.12218. [55] J. Wei and C. Wang, Improving interaction mechanism of carbon reduction technology innovation between supply chain enterprises and government by means of differential game, Journal of Cleaner Production, 296 (2021), 126578. [56] D. T. Wong and E. W. Ngai, Critical review of supply chain innovation research (1999–2016), Industrial Marketing Management, 82 (2019), 158-187.  doi: 10.1016/j.indmarman.2019.01.017. [57] A. Yenipazarli, To collaborate or not to collaborate: Prompting upstream eco-efficient innovation in a supply chain, European J. Oper. Res., 260 (2017), 571-587.  doi: 10.1016/j.ejor.2016.12.035. [58] J. Zhang, W.-Y. K. Chiang and L. Liang, Strategic pricing with reference effects in a competitive supply chain, Omega, 44 (2014), 126-135.  doi: 10.1016/j.omega.2013.07.002. [59] J. Zhang, Q. Gou, L. Liang and Z. Huang, Supply chain coordination through cooperative advertising with reference price effect, Omega, 41 (2013), 345-353.  doi: 10.1016/j.omega.2012.03.009. [60] J. Zhang, Q. Gou, J. Zhang and L. Liang, Supply chain pricing decisions with price reduction during the selling season, International Journal of Production Research, 52 (2014), 165-187.  doi: 10.1080/00207543.2013.831189. [61] Q. Zhang, J. Zhang and W. Tang, Coordinating a supply chain with green innovation in a dynamic setting, 4OR, 15 (2017), 133-162.  doi: 10.1007/s10288-016-0327-x. [62] S. Zhang, J. Zhang, J. Shen and W. Tang, A joint dynamic pricing and production model with asymmetric reference price effect, J. Ind. Manag. Optim., 15 (2019), 667-688.  doi: 10.3934/jimo.2018064.

show all references

##### References:
 [1] A. Atasu and G. C. Souza, How does product recovery affect quality choice?, Production and Operations Management, 22 (2013), 991-1010.  doi: 10.1111/j.1937-5956.2011.01290.x. [2] T. Başar, A. Bensoussan and S. P. Sethi, Differential games with mixed leadership: The open-loop solution, Appl. Math. Comput., 217 (2010), 972-979.  doi: 10.1016/j.amc.2010.01.048. [3] T. Başar and G. J. Olsder, Dynamic Noncooperative Game Theory, PA, Philadelphia: SIAM, 1999. [4] H. Benchekroun, G. Martín-Herrán and S. Taboubi, Could myopic pricing be a strategic choice in marketing channels? A game theoretic analysis, J. Econom. Dynam. Control, 33 (2009), 1699-1718.  doi: 10.1016/j.jedc.2009.03.005. [5] A. Bensoussan, S. Chen and S. Sethi, Linear quadratic differential games with mixed leadership: The open-loop solution, Numer. Algebra Control Optim., 3 (2013), 95-108.  doi: 10.3934/naco.2013.3.95. [6] Y. Cao and Y. Duan, Joint production and pricing inventory system under stochastic reference price effect, Computers & Industrial Engineering, 143 (2020), 106411.  doi: 10.1016/j.cie.2020.106411. [7] T. Chakraborty, S. S. Chauhan and M. Ouhimmou, Cost-sharing mechanism for product quality improvement in a supply chain under competition, International Journal of Production Economics, 208 (2019), 566-587.  doi: 10.1016/j.ijpe.2018.12.015. [8] K. Chen, Y. Zha, L. C. Alwan and L. Zhang, Dynamic pricing in the presence of reference price effect and consumer strategic behaviour, International Journal of Production Research, 58 (2020), 546-561.  doi: 10.1080/00207543.2019.1598592. [9] X. Chen, P. Hu and Z. Hu, Efficient algorithms for the dynamic pricing problem with reference price effect, Management Science, 63 (2016), 4389-4408.  doi: 10.1287/mnsc.2016.2554. [10] X. Chen, P. Hu, S. Shum and Y. Zhang, Dynamic stochastic inventory management with reference price effects, Oper. Res., 64 (2016), 1529-1536.  doi: 10.1287/opre.2016.1524. [11] E. J. Dockner, S. Jorgensen, N. Van Long and G. Sorger, Differential Games in Economics and Management Science, Cambridge University Press, Cambridge, UK, 2000.  doi: 10.1017/CBO9780511805127. [12] C.-Y. Dye, C.-T. Yang and C.-C. Wu, Joint dynamic pricing and preservation technology investment for an integrated supply chain with reference price effects, Journal of the Operational Research Society, 69 (2018), 811-824. [13] G. Fibich, A. Gavious and O. Lowengart, Explicit solutions of optimization models and differential games with nonsmooth (asymmetric) reference-price effects, Oper. Res., 51 (2003), 721-734.  doi: 10.1287/opre.51.5.721.16758. [14] H. Fu, K. Li and W. Fu, Investing in suppliers with capacity constraints in a decentralized assembly system, Computers & Industrial Engineering, 142 (2020), 106332.  doi: 10.1016/j.cie.2020.106332. [15] H. Fu, M. Liu and B. Chen, Supplier's investment in manufacturer's quality improvement with equity holding, J. Ind. Manag. Optim., 17 (2021), 649-688.  doi: 10.3934/jimo.2019127. [16] P. D. Giovanni, Quality improvement vs. advertising support: Which strategy works better for a manufacturer?, European J. Oper. Res., 208 (2011), 119-130.  doi: 10.1016/j.ejor.2010.08.003. [17] P. D. Giovanni and G. Zaccour, Optimal quality improvements and pricing strategies with active and passive product returns, Omega, 88 (2019), 248-262.  doi: 10.1016/j.omega.2018.09.007. [18] M. R. Galbreth, T. Boyacı and V. Verter, Product reuse in innovative industries, Production and Operations Management, 22 (2013), 1011-1033.  doi: 10.1111/j.1937-5956.2012.01330.x. [19] Z. Ge, Q. Hu and Y. Xia, Firms' R & D cooperation behavior in a supply chain, Production and Operations Management, 23 (2014), 599-609.  doi: 10.1111/poms.12037. [20] D. Ghosh and J. Shah, A comparative analysis of greening policies across supply chain structures, International Journal of Production Economics, 135 (2012), 568-583.  doi: 10.1016/j.ijpe.2011.05.027. [21] D. Ghosh and J. Shah, Supply chain analysis under green sensitive consumer demand and cost sharing contract, International Journal of Production Economics, 164 (2015), 319-329.  doi: 10.1016/j.ijpe.2014.11.005. [22] S. M. Gilbert and V. Cvsa, Strategic commitment to price to stimulate downstream innovation in a supply chain, European Journal of Operational Research, 150 (2003), 617-639.  doi: 10.1016/S0377-2217(02)00590-8. [23] B. G. Hardie, E. J. Johnson and P. S. Fader, Modeling loss aversion and reference dependence effects on brand choice, Marketing Science, 12 (1993), 378-394.  doi: 10.1287/mksc.12.4.378. [24] Y. He, Q. Xu, B. Xu and P. Wu, Supply chain coordination in quality improvement with reference effects, Journal of the Operational Research Society, 67 (2017), 1158-1168.  doi: 10.1057/jors.2016.10. [25] S. L. Holak, Determinants of innovative durables adoption an empirical study with implications for early product screening, Journal of Product Innovation Management, 5 (1988), 50-69.  doi: 10.1111/1540-5885.510050. [26] S. L. Holak and D. R. Lehmann, Purchase intentions and the dimensions of innovation: An exploratory model, Journal of Product Innovation Management, 7 (1990), 59-73.  doi: 10.1111/1540-5885.710059. [27] T.-P. Hsieh and C.-Y. Dye, Optimal dynamic pricing for deteriorating items with reference price effects when inventories stimulate demand, European J. Oper. Res., 262 (2017), 136-150.  doi: 10.1016/j.ejor.2017.03.038. [28] R. F. Hurley and G. T. M. Hult, Innovation, market orientation, and organizational learning: An integration and empirical examination, Journal of Marketing, 62 (1998), 42-54. [29] L. Jia and L. Tong, Renewables and storage in distribution systems: Centralized vs. Decentralized integration, IEEE Journal on Selected Areas in Communications, 34 (2016), 665-674.  doi: 10.1109/JSAC.2016.2525638. [30] G. Kalyanaram and R. S. Winer, Empirical generalizations from reference price research, Marketing Science, 14 (1995), 161-169.  doi: 10.1287/mksc.14.3.G161. [31] M. Karlsson, L. Trygg and B.-O. Elfström, Measuring R & D productivity: Complementing the picture by focusing on research activities, Technovation, 24 (2004), 179-186.  doi: 10.1016/S0166-4972(03)00111-1. [32] B. Kim, Coordinating an innovation in supply chain management, European Journal of Operational Research, 123 (2000), 568-584.  doi: 10.1016/S0377-2217(99)00113-7. [33] P. K. Kopalle, A. G. Rao and J. L. Assuncao, Asymmetric reference price effects and dynamic pricing policies, Marketing Science, 15 (1996), 60-85.  doi: 10.1287/mksc.15.1.60. [34] P. K. Kopalle and R. S. Winer, A dynamic model of reference price and expected quality, Marketing Letters, 7 (1996), 41-52.  doi: 10.1007/BF00557310. [35] J. M. Lattin and R. E. Bucklin, Reference effects of price and promotion on brand choice behavior, Journal of Marketing Research, 26 (1989), 299-310. [36] G. Li, M. Reimann and W. Zhang, When remanufacturing meets product quality improvement: The impact of production cost, European J. Oper. Res., 271 (2018), 913-925.  doi: 10.1016/j.ejor.2018.05.060. [37] Z. Lin, Price promotion with reference price effects in supply chain, Transportation Research Part E: Logistics and Transportation Review, 85 (2016), 52-68.  doi: 10.1016/j.tre.2015.11.002. [38] G. Liu, S. P. Sethi and J. Zhang, Myopic vs. far-sighted behaviours in a revenue-sharing supply chain with reference quality effects, International Journal of Production Research, 54 (2016), 1334-1357.  doi: 10.1080/00207543.2015.1068962. [39] L. Lu, Q. Gou, W. Tang and J. Zhang, Joint pricing and advertising strategy with reference price effect, International Journal of Production Research, 54 (2016), 5250-5270.  doi: 10.1080/00207543.2016.1165878. [40] G. Martín-Herrán, S. Taboubi and G. Zaccour, Dual role of price and myopia in a marketing channel, European J. Oper. Res., 219 (2012), 284-295.  doi: 10.1016/j.ejor.2011.12.015. [41] T. Mazumdar and P. Papatla, An investigation of reference price segments, Journal of Marketing Research, 37 (2000), 246-259.  doi: 10.1509/jmkr.37.2.246.18727. [42] T. Mazumdar, S. P. Raj and I. Sinha, Reference price research: Review and Propositions, Journal of Marketing, 69 (2005), 84-102. [43] S. K. Mukhopadhyay and R. Setaputra, A dynamic model for optimal design quality and return policies, European Journal of Operational Research, 180 (2007), 1144-1154.  doi: 10.1016/j.ejor.2006.05.016. [44] J. Ni and J. Zhao, A dynamic analysis of the investment in product innovation in a supply chain under reference price effect: Competition vs cooperation, International Journal of Systems Science: Operations & Logistics, 8 (2021), 1-13.  doi: 10.1080/23302674.2019.1629040. [45] M. Nouri, S.-M. Hosseini-Motlagh, M. Nematollahi and B. R. Sarker, Coordinating manufacturer's innovation and retailer's promotion and replenishment using a compensation-based wholesale price contract, International Journal of Production Economics, 198 (2018), 11-24.  doi: 10.1016/j.ijpe.2018.01.023. [46] NPD, Apple Owners Nearly 40 Percent More Interested in the iPad than Non-Apple Owners, Technical report, Port Washington, New York, 2010. [47] I. Popescu and Y. Wu, Dynamic pricing strategies with reference effects, Ope. Res., 55 (2007), 413-429.  doi: 10.1287/opre.1070.0393. [48] M. Porter and V. Linde, Green and competitive, Harvard Business Review, 73 (1995), 120-134. [49] D. Pujari, Eco-innovation and new product development: Understanding the influences on market performance, Technovation, 26 (2006), 76-85.  doi: 10.1016/j.technovation.2004.07.006. [50] D. S. Putler, Incorporating reference price effects into a theory of consumer choice, Marketing Science, 11 (1992), 287-309.  doi: 10.1287/mksc.11.3.287. [51] S. P. Sethi and T. Li, A review of dynamic Stackelberg game models, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 125-159.  doi: 10.3934/dcdsb.2017007. [52] T. A. Suter and D. M. Hardesty, Maximizing earnings and price fairness perceptions in online consumer-to-consumer auctions, Journal of Retailing, 81 (2005), 307-317.  doi: 10.1016/j.jretai.2005.01.007. [53] K. Uğurlu, Dynamic optimal contract under parameter uncertainty with risk-averse agent and principal, Turkish J. Math., 42 (2018), 977-992.  doi: 10.3906/mat-1703-102. [54] J. Wang and H. Shin, The impact of contracts and competition on upstream innovation in a supply chain, Production and Operations Management, 24 (2015), 134-146.  doi: 10.1111/poms.12218. [55] J. Wei and C. Wang, Improving interaction mechanism of carbon reduction technology innovation between supply chain enterprises and government by means of differential game, Journal of Cleaner Production, 296 (2021), 126578. [56] D. T. Wong and E. W. Ngai, Critical review of supply chain innovation research (1999–2016), Industrial Marketing Management, 82 (2019), 158-187.  doi: 10.1016/j.indmarman.2019.01.017. [57] A. Yenipazarli, To collaborate or not to collaborate: Prompting upstream eco-efficient innovation in a supply chain, European J. Oper. Res., 260 (2017), 571-587.  doi: 10.1016/j.ejor.2016.12.035. [58] J. Zhang, W.-Y. K. Chiang and L. Liang, Strategic pricing with reference effects in a competitive supply chain, Omega, 44 (2014), 126-135.  doi: 10.1016/j.omega.2013.07.002. [59] J. Zhang, Q. Gou, L. 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The trajectories of reference price and selling price in centralized decision-making
The comparison of $Q_{d-ss}^{IR}$ and $Q_{c}^{IN}$
The relationships between the steady-state profits and $\delta$, $\theta$, $\beta$
The comparisons of corresponding optimal profits under $'IR'$ and $'IN'$
Superscript and subscript annotations
 $IR$ the presence of an interactive effect $NR$ the absence of innovation $IN$ the absence of reference price c centralized decision-making d decentralized decision-making ss steady-state
 $IR$ the presence of an interactive effect $NR$ the absence of innovation $IN$ the absence of reference price c centralized decision-making d decentralized decision-making ss steady-state
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