# American Institute of Mathematical Sciences

July  2020, 16(4): 1999-2027. doi: 10.3934/jimo.2019040

## Cost-sharing strategy for carbon emission reduction and sales effort: A nash game with government subsidy

 1 Department of Information Management and Decision Sciences, School of Business Administration, Northeastern University, Shenyang 110169, China 2 State Key Laboratory of Synthetical Automation for Process Industries, Northeastern University, Shenyang 110819, China 3 School of Management, Guangzhou University, Guangzhou 510006, China

* Corresponding author: Bing-Bing Cao

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

Fund Project: The first author is supported by the National Natural Science Foundation of China (Project No. 71871049) and the 111 Project (B16009)

We investigate the cost-sharing strategies of a retailer and a manufacturer in a Nash game considering government subsidy, consumers' green preference and retailer's sales effort. We provide a function to describe the demand for green products considering the effect of green preference of consumers and the sales effort of the retailer. Next, we construct profit functions of the manufacturer and the retailer considering government subsidy for four scenarios: no sharing of cost (NSC), sharing of carbon emission reduction cost (SCERC), sharing of sales effort cost (SSEC), and sharing both carbon emission reduction cost and sales effort cost (SBC). Furthermore, we determine the optimal policies of price, sales effort level, wholesale price and carbon emission reduction effort level for the four scenarios by maximizing the profits of the manufacturer and the retailer in the Nash game. We find that the sales effort cost-sharing ratio and the carbon emission reduction cost-sharing ratio can affect the optimal policies of the manufacturer and the retailer, and the trends and extent of effects may be different. Our results show that it is advantageous for the manufacturer and the retailer to consider the cost-sharing effects of sales effort and carbon emission reduction effort, and the optimal policies of the retailer and the manufacturer are different for different scenarios.

Citation: Xue-Yan Wu, Zhi-Ping Fan, Bing-Bing Cao. Cost-sharing strategy for carbon emission reduction and sales effort: A nash game with government subsidy. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1999-2027. doi: 10.3934/jimo.2019040
##### References:
 [1] A. Arya and B. Mittendorf, Supply chain consequences of subsidies for corporate social responsibility, Prod. Oper. Manag., 24 (2015), 1346-1357. [2] Q. Bai, M. Chen and L. Xu, Revenue and promotional cost-sharing contract versus two-part tariff contract in coordinating sustainable supply chain systems with deteriorating items, Int. J. Prod. Econ., 187 (2017), 85-101. [3] B. B. Cao and Z. P. Fan, Ordering and sales effort investment for temperature-sensitive products considering retailer's disappointment aversion and elation seeking, Int. J. Prod. Res., 56 (2018), 2411-2436. [4] B. B. Cao, Z. P. Fan, H. Li and T. H. You, Joint inventory, pricing, and advertising decisions with surplus and stockout loss aversions, DISCRETE DYN. NAT. SOC., 2016 (2016), Art. ID 1907680, 14 pp. doi: 10.1155/2016/1907680. [5] K. Chen, J. Shen and M. Feng, Disruptions management of a supply chain under strategic subsidy policy for the demand-stimulating inventory, Comput. Ind. Eng., 76 (2014), 169-182. [6] T. Chernonog, T. Avinadav and T. Ben-Zvi, Pricing and sales-effort investment under bi-criteria in a supply chain of virtual products involving risk, Eur. J. Oper. Res., 246 (2015), 471-475.  doi: 10.1016/j.ejor.2015.05.024. [7] R. Dai, J. Zhang and W. Tang, Cartelization or cost-sharing? Comparison of cooperation modes in a green supply chain, J. Clean Prod., 156 (2017), 159-173. [8] T. Deleire, A. Chappel, K. Finegold and E. Gee, Do individuals respond to cost-sharing subsidies in their selections of marketplace health insurance plans?, J. Health Econ., 56 (2017), 71-86. [9] T. Elkin, Intel inside at 10, Advertising Age, 72 (2001), 4-5. [10] A. N. Elmachtoub and R. Levi, From cost sharing mechanisms to online selection problems, Math. Oper. Res., 40 (2015), 542-557.  doi: 10.1287/moor.2014.0684. [11] H. Fu, B. Dan and X. Sun, Joint optimal pricing and ordering decisions for seasonal products with weather-sensitive demand, DISCRETE DYN. NAT. SOC., 2014 (2014), Art. ID 105098, 8 pp. doi: 10.1155/2014/105098. [12] D. Ghosh and J. Shah, Supply chain analysis under green sensitive consumer demand and cost sharing contract, Int. J. Prod. Econ., 164 (2015), 319-329. [13] V. Gkatzelis, K. Kollias and T. Roughgarden, Optimal cost-sharing in general resource selection games, Oper. Res., 64 (2016), 1230-1238.  doi: 10.1287/opre.2016.1512. [14] R. Gopalakrishnan, J. R. Marden and A. Wierman, Potential games are necessary to ensure pure Nash equilibria in cost sharing games, Math. Oper. Res., 39 (2014), 1252-1296.  doi: 10.1287/moor.2014.0651. [15] H. Gurnani and Y. Xu, Resale price maintenance contracts with retailer sales effort: Effect of flexibility and competition, Nav. Res. Logist., 53 (2010), 448-463.  doi: 10.1002/nav.20152. [16] M. B. Jamali and M. Rasti-Barzoki, A game theoretic approach for green and non-green product pricing in chain-to-chain competitive sustainable and regular dual-channel supply chains, J. Clean Prod., 165 (2018), 1029-1043. [17] F. Jaramillo and J. P. Mulki, Sales effort: The intertwined roles of the leader, customers, and the salesperson, J. Pers. Sel., 28 (2008), 37-51. [18] J. Y. Lee and R. K. Cho, Contracting for vendor-managed inventory with consignment stock and stockout-cost sharing, Int. J. Prod. Econ., 151 (2014), 158-173. [19] J. Y. Lee, R. K. Cho and S. K. Paik, Supply chain coordination in vendor-managed inventory systems with stockout-cost sharing under limited storage capacity, Eur. J. Oper. Res., 248 (2016), 95-106.  doi: 10.1016/j.ejor.2015.06.080. [20] S. Li, V. Jayaraman, A. Paulra and K. Shang, Proactive environmental strategies and performance: Role of green supply chain processes and green product design in the Chinese high-tech industry, Int. J. Prod. Res., 258 (2017), 165-180.  doi: 10.1016/j.ejor.2016.08.053. [21] B. Li, W. Chen, C. Xu and P. Hou, Impacts of government subsidies for environmental-friendly products in a dual-channel supply chain, J. Clean Prod., 171 (2018), 1558-1576. [22] B. Li, M. Zhu, Y. Jiang and Z. Li, Pricing policies of a competitive dual-channel green supply chain, J. Clean Prod., 112 (2016), 2029-2042. [23] X. Li, Z. Xiao and B. Cao, Joint decision on pricing and waste emission level in industrial symbiosis chain, J. Ind. Manag. Optim., 14 (2018), 135-164.  doi: 10.3934/jimo.2017040. [24] H. Liu, M. Lei, H. Deng, G. K. Leong and T. Huang, A dual channel, quality-based price competition model for the WEEE recycling market with government subsidy, Omega-Int. J. Manage. Sci., 59 (2016), 290-302. [25] P. Liu and S. Yi, Pricing policies of green supply chain considering targeted advertising and product green degree in the big data environment, J. Clean Prod., 164 (2017), 1614-1622. [26] W. M. Ma, Z. Zhao and H. Ke, Dual-channel closed-loop supply chain with government consumption-subsidy, Eur. J. Oper. Res., 226 (2013), 221-227.  doi: 10.1016/j.ejor.2012.10.033. [27] J. Neff, Walmart ups the ante with brand co-op ads-in more ways than one, Advertising Age, 80 (2009), 4-5. [28] I. Nouira, R. Hammami, Y. Frein and C. Temponi, Design of forward supply chains: Impact of a carbon emissions-sensitive demand, Int. J. Prod. Econ., 173 (2016), 80-98. [29] J. Pan, W. Shi and X. Tang, Pricing and ordering strategies of supply chain with selling gift cards, J. Ind. Manag. Optim., 14 (2018), 349-369.  doi: 10.3934/jimo.2017050. [30] S. Panda, N. M. Modak and M. Basu, Disposal cost sharing and bargaining for coordination and profit division in a three-echelon supply chain, INT. J. Manage. Sci., 9 (2014), 276-285. [31] A. A. Taleizadeh and H. Heydaryan, Pricing strategy and return policy of one-echelon green supply chain under both green and hybrid productions, J. Ind. Sys. Eng., 9 (2016), 17-29. [32] D. Tang, Q. Wang and I. Ullah, Optimisation of product configuration in consideration of customer satisfaction and low carbon, Int. J. Prod. Res., 55 (2016), 3349-3373. [33] C. S. Tapiero and J. U. Farley, Optimal control of sales force effort in time, Manage. Sci., 21 (1975), 976-985. [34] Y. C. Tsao and G. J. Sheen, Effects of promotion cost sharing policy with the sales learning curve on supply chain coordination, Comput. Oper. Res., 39 (2012), 1872-1878.  doi: 10.1016/j.cor.2011.07.009. [35] X. J. Wang and S. H. Choi, Impacts of carbon emission reduction mechanisms on uncertain make-to-order manufacturing, Int. J. Prod. Res., 54 (2015), 1-18. [36] K. Wang, Y. Zhao, Y. Cheng and T. M. Choi, Cooperation or competition? Channel choice for a remanufacturing fashion supply chain with government subsidy, Sustainability, 6 (2014), 7292-7310. [37] G. Xie, W. Yue and S. Wang, Optimal selection of cleaner products in a green supply chain with risk aversion, J. Ind. Manag. Optim., 11 (2015), 515-528.  doi: 10.3934/jimo.2015.11.515. [38] D. Xing and T. Liu, Sales effort free riding and coordination with price match and channel rebate, Eur. J. Oper. Res., 219 (2012), 264-271.  doi: 10.1016/j.ejor.2011.11.029. [39] L. Yang, J. Ji and K. Chen, Advertising games on national brand and store brand in a dual-channel supply chain, J. Ind. Manag. Optim., 14 (2018), 105-134.  doi: 10.3934/jimo.2017039. [40] D. Yang and T. Xiao, Pricing and green level decisions of a green supply chain with governmental interventions under fuzzy uncertainties, J. Clean Prod., 149 (2017), 1174-1187. [41] L. Yang, R. Tang and K. Chen, Call, put and bidirectional option contracts in agricultural supply chains with sales effort, Appl. Math. Model., 47 (2017), 1-16.  doi: 10.1016/j.apm.2017.03.002. [42] Y. Zeng, L. Zhang, X. Cai and J. Li, Cost sharing for capacity transfer in cooperating queueing systems, Prod. Oper. Manag., 27 (2018), 644-662. [43] C. T. Zhang, H. X. Wang and M. L. Ren, Research on pricing and coordination strategy of green supply chain under hybrid production mode, Comput. Ind. Eng., 72 (2014), 24-31. [44] C. Zhang, J. W. H. Jr and S. Viswanathan, Reciprocity between buyer cost sharing and supplier technology sharing, Int. J. Prod. Econ., 163 (2015), 61-70. [45] S. Zhao, Q. Zhu and L. Cui, A decision-making model for remanufacturers: Considering both consumers' environmental preference and the government subsidy policy, Resour. Conserv. Recycl., 128 (2018), 176-186. [46] Y. Zhou, M. Bao, X. Chen and X. Xu, Co-op advertising and emission reduction cost sharing contracts and coordination in low-carbon supply chain based on fairness concerns, J. Clean Prod., 133 (2016), 402-413.

show all references

##### References:
 [1] A. Arya and B. Mittendorf, Supply chain consequences of subsidies for corporate social responsibility, Prod. Oper. Manag., 24 (2015), 1346-1357. [2] Q. Bai, M. Chen and L. Xu, Revenue and promotional cost-sharing contract versus two-part tariff contract in coordinating sustainable supply chain systems with deteriorating items, Int. J. Prod. Econ., 187 (2017), 85-101. [3] B. B. Cao and Z. P. Fan, Ordering and sales effort investment for temperature-sensitive products considering retailer's disappointment aversion and elation seeking, Int. J. Prod. Res., 56 (2018), 2411-2436. [4] B. B. Cao, Z. P. Fan, H. Li and T. H. You, Joint inventory, pricing, and advertising decisions with surplus and stockout loss aversions, DISCRETE DYN. NAT. SOC., 2016 (2016), Art. ID 1907680, 14 pp. doi: 10.1155/2016/1907680. [5] K. Chen, J. Shen and M. Feng, Disruptions management of a supply chain under strategic subsidy policy for the demand-stimulating inventory, Comput. Ind. Eng., 76 (2014), 169-182. [6] T. Chernonog, T. Avinadav and T. Ben-Zvi, Pricing and sales-effort investment under bi-criteria in a supply chain of virtual products involving risk, Eur. J. Oper. Res., 246 (2015), 471-475.  doi: 10.1016/j.ejor.2015.05.024. [7] R. Dai, J. Zhang and W. Tang, Cartelization or cost-sharing? Comparison of cooperation modes in a green supply chain, J. Clean Prod., 156 (2017), 159-173. [8] T. Deleire, A. Chappel, K. Finegold and E. Gee, Do individuals respond to cost-sharing subsidies in their selections of marketplace health insurance plans?, J. Health Econ., 56 (2017), 71-86. [9] T. Elkin, Intel inside at 10, Advertising Age, 72 (2001), 4-5. [10] A. N. Elmachtoub and R. Levi, From cost sharing mechanisms to online selection problems, Math. Oper. Res., 40 (2015), 542-557.  doi: 10.1287/moor.2014.0684. [11] H. Fu, B. Dan and X. Sun, Joint optimal pricing and ordering decisions for seasonal products with weather-sensitive demand, DISCRETE DYN. NAT. SOC., 2014 (2014), Art. ID 105098, 8 pp. doi: 10.1155/2014/105098. [12] D. Ghosh and J. Shah, Supply chain analysis under green sensitive consumer demand and cost sharing contract, Int. J. Prod. Econ., 164 (2015), 319-329. [13] V. Gkatzelis, K. Kollias and T. Roughgarden, Optimal cost-sharing in general resource selection games, Oper. Res., 64 (2016), 1230-1238.  doi: 10.1287/opre.2016.1512. [14] R. Gopalakrishnan, J. R. Marden and A. Wierman, Potential games are necessary to ensure pure Nash equilibria in cost sharing games, Math. Oper. Res., 39 (2014), 1252-1296.  doi: 10.1287/moor.2014.0651. [15] H. Gurnani and Y. Xu, Resale price maintenance contracts with retailer sales effort: Effect of flexibility and competition, Nav. Res. Logist., 53 (2010), 448-463.  doi: 10.1002/nav.20152. [16] M. B. Jamali and M. Rasti-Barzoki, A game theoretic approach for green and non-green product pricing in chain-to-chain competitive sustainable and regular dual-channel supply chains, J. Clean Prod., 165 (2018), 1029-1043. [17] F. Jaramillo and J. P. Mulki, Sales effort: The intertwined roles of the leader, customers, and the salesperson, J. Pers. Sel., 28 (2008), 37-51. [18] J. Y. Lee and R. K. Cho, Contracting for vendor-managed inventory with consignment stock and stockout-cost sharing, Int. J. Prod. Econ., 151 (2014), 158-173. [19] J. Y. Lee, R. K. Cho and S. K. Paik, Supply chain coordination in vendor-managed inventory systems with stockout-cost sharing under limited storage capacity, Eur. J. Oper. Res., 248 (2016), 95-106.  doi: 10.1016/j.ejor.2015.06.080. [20] S. Li, V. Jayaraman, A. Paulra and K. Shang, Proactive environmental strategies and performance: Role of green supply chain processes and green product design in the Chinese high-tech industry, Int. J. Prod. Res., 258 (2017), 165-180.  doi: 10.1016/j.ejor.2016.08.053. [21] B. Li, W. Chen, C. Xu and P. Hou, Impacts of government subsidies for environmental-friendly products in a dual-channel supply chain, J. Clean Prod., 171 (2018), 1558-1576. [22] B. Li, M. Zhu, Y. Jiang and Z. Li, Pricing policies of a competitive dual-channel green supply chain, J. Clean Prod., 112 (2016), 2029-2042. [23] X. Li, Z. Xiao and B. Cao, Joint decision on pricing and waste emission level in industrial symbiosis chain, J. Ind. Manag. Optim., 14 (2018), 135-164.  doi: 10.3934/jimo.2017040. [24] H. Liu, M. Lei, H. Deng, G. K. Leong and T. Huang, A dual channel, quality-based price competition model for the WEEE recycling market with government subsidy, Omega-Int. J. Manage. Sci., 59 (2016), 290-302. [25] P. Liu and S. Yi, Pricing policies of green supply chain considering targeted advertising and product green degree in the big data environment, J. Clean Prod., 164 (2017), 1614-1622. [26] W. M. Ma, Z. Zhao and H. Ke, Dual-channel closed-loop supply chain with government consumption-subsidy, Eur. J. Oper. Res., 226 (2013), 221-227.  doi: 10.1016/j.ejor.2012.10.033. [27] J. Neff, Walmart ups the ante with brand co-op ads-in more ways than one, Advertising Age, 80 (2009), 4-5. [28] I. Nouira, R. Hammami, Y. Frein and C. Temponi, Design of forward supply chains: Impact of a carbon emissions-sensitive demand, Int. J. Prod. Econ., 173 (2016), 80-98. [29] J. Pan, W. Shi and X. Tang, Pricing and ordering strategies of supply chain with selling gift cards, J. Ind. Manag. Optim., 14 (2018), 349-369.  doi: 10.3934/jimo.2017050. [30] S. Panda, N. M. Modak and M. Basu, Disposal cost sharing and bargaining for coordination and profit division in a three-echelon supply chain, INT. J. Manage. Sci., 9 (2014), 276-285. [31] A. A. Taleizadeh and H. Heydaryan, Pricing strategy and return policy of one-echelon green supply chain under both green and hybrid productions, J. Ind. Sys. Eng., 9 (2016), 17-29. [32] D. Tang, Q. Wang and I. Ullah, Optimisation of product configuration in consideration of customer satisfaction and low carbon, Int. J. Prod. Res., 55 (2016), 3349-3373. [33] C. S. Tapiero and J. U. Farley, Optimal control of sales force effort in time, Manage. Sci., 21 (1975), 976-985. [34] Y. C. Tsao and G. J. Sheen, Effects of promotion cost sharing policy with the sales learning curve on supply chain coordination, Comput. Oper. Res., 39 (2012), 1872-1878.  doi: 10.1016/j.cor.2011.07.009. [35] X. J. Wang and S. H. Choi, Impacts of carbon emission reduction mechanisms on uncertain make-to-order manufacturing, Int. J. Prod. Res., 54 (2015), 1-18. [36] K. Wang, Y. Zhao, Y. Cheng and T. M. Choi, Cooperation or competition? Channel choice for a remanufacturing fashion supply chain with government subsidy, Sustainability, 6 (2014), 7292-7310. [37] G. Xie, W. Yue and S. Wang, Optimal selection of cleaner products in a green supply chain with risk aversion, J. Ind. Manag. Optim., 11 (2015), 515-528.  doi: 10.3934/jimo.2015.11.515. [38] D. Xing and T. Liu, Sales effort free riding and coordination with price match and channel rebate, Eur. J. Oper. Res., 219 (2012), 264-271.  doi: 10.1016/j.ejor.2011.11.029. [39] L. Yang, J. Ji and K. Chen, Advertising games on national brand and store brand in a dual-channel supply chain, J. Ind. Manag. Optim., 14 (2018), 105-134.  doi: 10.3934/jimo.2017039. [40] D. Yang and T. Xiao, Pricing and green level decisions of a green supply chain with governmental interventions under fuzzy uncertainties, J. Clean Prod., 149 (2017), 1174-1187. [41] L. Yang, R. Tang and K. Chen, Call, put and bidirectional option contracts in agricultural supply chains with sales effort, Appl. Math. Model., 47 (2017), 1-16.  doi: 10.1016/j.apm.2017.03.002. [42] Y. Zeng, L. Zhang, X. Cai and J. Li, Cost sharing for capacity transfer in cooperating queueing systems, Prod. Oper. Manag., 27 (2018), 644-662. [43] C. T. Zhang, H. X. Wang and M. L. Ren, Research on pricing and coordination strategy of green supply chain under hybrid production mode, Comput. Ind. Eng., 72 (2014), 24-31. [44] C. Zhang, J. W. H. Jr and S. Viswanathan, Reciprocity between buyer cost sharing and supplier technology sharing, Int. J. Prod. Econ., 163 (2015), 61-70. [45] S. Zhao, Q. Zhu and L. Cui, A decision-making model for remanufacturers: Considering both consumers' environmental preference and the government subsidy policy, Resour. Conserv. Recycl., 128 (2018), 176-186. [46] Y. Zhou, M. Bao, X. Chen and X. Xu, Co-op advertising and emission reduction cost sharing contracts and coordination in low-carbon supply chain based on fairness concerns, J. Clean Prod., 133 (2016), 402-413.
The structure and communication process of a green supply chain
The green sensitivity demand function

The carbon emission reduction effort level

The impacts of sharing ratios ${\gamma _A}$ and ${\gamma _e}$ on price $p$
The impacts of sharing ratios ${\gamma _A}$ and ${\gamma _e}$ on sales effort level $A$
The impacts of sharing ratios ${\gamma _A}$ and ${\gamma _e}$ on wholesale price $w$
The impacts of sharing ratios ${\gamma _A}$ and ${\gamma _e}$ on carbon emission reduction effort level $e$
The impacts of sharing ratios ${\gamma _A}$ and ${\gamma _e}$ on the retailer's profit $\pi {_R^{4*} }$
The impacts of sharing ratios ${\gamma _A}$ and ${\gamma _e}$ on the manufacturer's profit ${\pi _M}$
The impacts of sharing ratios ${\gamma _A}$ and ${\gamma _e}$ on profit ${\pi _{SC}}$ of the supply chain
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