Abstract
In practice, selling goods through a powerful retailer such as Wal-Mart enables the supplier to access the retailer’s ERP for accurate demand information (e.g., Wal-Mart’s Retail Link). However, in the recent years, we observe the suppliers are suffering from longer and longer average account period when they contract with powerful retailers. Therefore, whether partnering with a powerful retailer at the cost of a longer account period becomes the supplier’s strategic decision. In this paper, we formulate the supplier’s tradeoffs among the information advantage, payment disadvantage, and channel competition when it makes retailing decisions. We study the supplier’s two representative strategies: (1) relying on a small retailer that does not accumulate much information but can settle accounts immediately (referred to as Real-time Payment Retailing) or (2) relying on a powerful retailer that shares accurate demand information but incurs deferred payment (referred to as Deferred Payment Retailing). We built game-theoretical models and found that, interestingly, the supplier will prefer Deferred Payment Retailing when the supplier’s cash opportunity cost is high. We identify three interactive forces, namely, the pricing power effect, the demand size effect, and the information value, to interpret the rationality of the supplier’s preferences over Real-time and Deferred Payment Retailing strategies.
Similar content being viewed by others
References
Amrouche, N., Martinherran, G., & Zaccour, G. (2008). Feedback Stackelberg equilibrium strategies when the private label competes with the national brand. Annals of Operations Research, 164(1), 79–95.
Appel. (2009). How will losing Wal-Mart exclusivity impact Cott Corp? Retrieved November, 2019, from https://seekingalpha.com/article/116960-how-will-losing-wal-mart-exclusivity-impact-cott-corp.
Business Wire. (2019). CE China 2019: Suning launches new Biu products. Retrieved December, 2019, from https://www.businesswire.com/news/home/20190923005490/en/CE-China-2019-Suning-Launches-New-Biu.
Chen, X. F., Cai, G. S., & Song, J. S. (2019). The cash flow advantages of 3PLs as supply chain orchestrators. Manufacturing & Service Operations Management, 21(2), 435–451.
Chen, L., Gilbert, S. M., & Xia, Y. (2011). Private labels: Facilitators or impediments to supply chain coordination. Decision Sciences, 42(3), 689–720.
Chen, L., Kök, A. G., & Tong, J. D. (2013). The effect of payment schemes on inventory decisions: The role of mental accounting. Management Science, 59(2), 436–451.
Chiu, C. H., Choi, T. M., Li, X., & Yiu, C. K.-F. (2016). Coordinating supply chains with a general price-dependent demand function: Impacts of channel leadership and information asymmetry. IEEE Transactions on Engineering Management, 63(4), 390–402.
Cho, S. H., Fang, X., Tayur, S., & Xu, Y. (2019). Combating child labor: Incentives and information disclosure in global supply chains. Manufacturing & Service Operations Management, 21(3), 556–570.
Commercial Capital LLC. (2020). Financing options for Wal-Mart suppliers. Retrieved January, 2020, from https://www.comcapfactoring.com/blog/Wal-Mart-vendor-financing/.
Construction Payments Report. (2019). 2019 National construction payments report. Retrieved October, 2019, from https://www.levelset.com/blog/2019-national-construction-payments-report/.
Consumer Goods. (2009). Cott to phase out supply agreement with Wal-Mart. Retrieved October, 2019, from https://consumergoods.com/cott-phase-out-supply-agreement-wal-mart.
Coughlan, A. T. (1985). Competition and cooperation in marketing channel choice: Theory and application. Marketing Science, 4(2), 110–129.
Cui, Q., Chiu, C. H., Dai, X., & Li, Z. (2016). Store brand introduction in a two-echelon logistics system with a risk-averse retailer. Transportation Research Part E: Logistics and Transportation Review, 90, 69–89.
Dong, C. W., Yang, Y. P., & Zhao, M. (2018). Dynamic selling strategy for a firm under asymmetric information: Direct selling vs. agent selling. International Journal of Production Economics, 204, 204–213.
Forbes. (2017). Really big data at Wal-Mart: Real-time insights from their 40 + petabyte data cloud. Retrieved January, 2017, from https://www.forbes.com/sites/bernardmarr/2017/01/23/really-big-data-at-Wal-Mart-real-time-insights-from-their-40-petabyte-data-cloud.
Forbes. (2019). Wal-Mart: Big data analytics at the world’s biggest retailer. Retrieved 2019, from https://www.bernardmarr.com/default.asp?contentID=690.
Groznik, A., & Heese, H. S. (2010). Supply chain interactions due to store-brand introductions: The impact of retail competition. European Journal of Operational Research, 203(3), 575–582.
GuruFocus. (2020). JD.com Days Payable: 56.43 (As of Mar. 2020). Retrieved July, 2020, from https://www.gurufocus.com/term/DaysPayable/NAS:JD/Days-Payable/JDcom.
Ha, A. Y., & Tong, S. (2008). Contracting and information sharing under supply chain competition. Management Science, 54(4), 701–715.
Huang, S., Guan, X., & Chen, Y. J. (2018). Retailer information sharing with supplier encroachment. Production and Operations Management, 27(6), 1133–1147.
Jin, Y. N., Wu, X. L., & Hu, Q. Y. (2017). Interaction between channel strategy and store brand decisions. European Journal of Operational Research, 256(3), 911–923.
Kurata, H., Yao, D. Q., & Liu, J. J. (2007). Pricing policies under direct vs. indirect channel competition and national vs. store brand competition. European Journal of Operational Research, 180(1), 262–281.
Lan, Y. F., Peng, J., Wang, F. W., & Gao, C. S. (2018). Quality disclosure with information value under competition. International Journal of Machine Learning and Cybernetics, 9(9), 1489–1503.
Lan, Y. F., Yan, H. K., Ren, D., & Guo, R. (2019). Merger strategies in a supply chain with asymmetric capital-constrained retailers upon market power dependent trade credit. OMEGA-The International Journal of Management Science, 83, 299–318.
Li, Z. L., Ryan, J. K., Shao, L. S., & Sun, D. W. (2015). Supply contract design for competing heterogeneous suppliers under asymmetric information. Production and Operations Management, 24(5), 791–807.
Li, Y., Zhen, X., & Cai, X. (2016). Trade credit insurance, capital constraint, and the behavior of manufacturers and banks. Annals of Operations Research, 240(2), 395–414.
Luo, R. (2018). Store brands and retail grocery competition. Journal of Economics & Management Strategy, 27(4), 653–668.
Lus, B., & Muriel, A. (2009). Measuring the impact of increased product substitution on pricing and capacity decisions under linear demand models. Production and Operations Management, 18(1), 95–108.
Mattel. (2009). Mattel Inc. annual report 2008. Retrieved December, 2009, from https://mattel.gcs-web.com/static-files/913b4c95-74bb-401c-8241-c17a26b012ef.
McGuire, T. W., & Staelin, R. (1983). An industry equilibrium analysis of downstream vertical integration. Marketing Science, 2(2), 161–191.
Moorthy, K. S. (1985). Using game theory to model competition. Journal of Marketing Research, 22(3), 262–282.
Nielsen. (2018). The rise and rise again of private label. Retrieved January, 2020, from https://www.nielsen.com/wp-content/uploads/sites/3/2019/04/global-private-label-report.pdf.
Niu, B., & Xie, F. (2020). Incentive alignment of brand-owner and remanufacturer towards quality certification to refurbished products. Journal of Cleaner Production, 242, 118314.
Niu, B. Z., Chen, K. L., Fang, X., Yue, X. H., & Wang, X. (2019). Technology specifications and production timing in a co-opetitive supply chain. Production and Operations Management, 28(8), 1990–2007.
Raju, J., Sethuraman, R., & Dhar, S. K. (1995). The introduction and performance of store brands. Management Science, 41(6), 957–978.
Sayman, S., & Raju, J. S. (2004). How category characteristics affect the number of store brands offered by the retailer: A model and empirical analysis. Journal of Retailing, 80(4), 279–287.
Shamir, N., & Shin, H. (2016). Public forecast information sharing in a market with competing supply chains. Management Science, 62(10), 2994–3022.
Shao, L., Wu, X., & Zhang, F. (2020). Sourcing competition under cost uncertainty and information asymmetry. Production and Operations Management, 29(2), 447–461.
Spiegel, Y. (1993). Horizontal subcontracting. The Rand Journal of Economics, 24(4), 570–590.
Suning. (2019). Suning to present at CES Asia with its latest retail technology for a smarter life. Retrieved June, 2019, from http://www.suningholdings.com/cms/corporateNewsJson/23966.htm.
Taylor, T. A., & Xiao, W. Q. (2010). Does a supplier benefit from selling to a better-forecasting retailer? Management Science, 56(9), 1584–1598.
Tong, J., DeCroix, G., & Song, J. S. (2020). Modeling payment timing in multiechelon inventory systems with applications to supply chain coordination. Manufacturing & Service Operations Management, 22(2), 346–363.
Tunca, T. I., & Zhu, W. M. (2018). Buyer intermediation in supplier finance. Management Science, 64(12), 5631–5650.
Ulusoy, G., Sivrikayaşerifoglu, F., & Şahin, Ş. (2001). Four payment models for the multi-mode resource constrained project scheduling problem with discounted cash flows. Annals of Operations Research, 102(1), 237–261.
Vives, X. (1984). Duopoly information equilibrium: Cournot and Bertrand. Journal of Economic Theory, 34(1), 71–94.
Wal-Mart. (2019). Private brand. Retrieved December, 2019, from http://www.wal-martchina.com/english/promotion/pb/sb.htm.
Wal-Mart. (2020). Apply to be a supplier. Retrieved February, 2020, from https://corporate.walmart.com/suppliers/apply-to-be-a-supplier.
Wang, J. C., Lau, A. H. L., & Lau, H. S. (2009). When should a supplier share truthful manufacturing cost information with a dominant retailer? European Journal of Operational Research, 1(16), 266–286.
Wang, J. C., Lau, A. H. L., & Lau, H. S. (2013). Dollar vs. percentage markup pricing schemes under a dominant retailer. European Journal of Operational Research, 227(3), 471–482.
Wang, Y., Niu, B., Guo, P., & Song, J. S. (2020). Direct sourcing or agent sourcing? Contract negotiation in procurement outsourcing. Manufacturing & Service Operations Management. https://doi.org/10.1287/msom.2019.0843.
Wang, Y. Y., Sun, J., & Wang, J. C. (2016). Equilibrium markup pricing strategies for the dominant retailers under supply chain to chain competition. International Journal of Production Research, 54(7), 2075–2092.
Wang, J. C., Wang, Y. Y., & Lai, F. (2019). Impact of power structure on supply chain performance and consumer surplus. International Transactions in Operational Research, 26(5), 1752–1785.
Wu, X. L., & Zhang, F. Q. (2014). Home or overseas? An analysis of sourcing strategies under competition. Management Science, 60(5), 1223–1240.
Yang, S. A., Birge, J. R., & Parker, R. P. (2015). The supply chain effects of bankruptcy. Management Science, 61(10), 2320–2338.
Yoo, W. S., & Lee, E. (2011). Internet channel entry: A strategic analysis of mixed channel structures. Marketing Science, 30(1), 29–41.
Zhang, M., Tse, Y. K., Dai, J., & Chan, H. K. (2017). Examining green supply chain management and financial performance: roles of social control and environmental dynamism. IEEE Transactions on Engineering Management, 66(1), 20–34.
Zhao, L., & Huchzermeier, A. (2019). Managing supplier financial distress with advance payment discount and purchase order financing. OMEGA—The International Journal of Management Science, 88, 77–90.
Zhao, D., & Li, Z. (2018). The impact of manufacturer’s encroachment and nonlinear production cost on retailer’s information sharing decisions. Annals of Operations Research, 264(1), 499–539.
Zhu, X. (2017). Outsourcing management under various demand Information Sharing scenarios. Annals of Operations Research, 257(1), 449–467.
Acknowledgements
The authors are grateful to the editors and reviewers for their helpful comments. This work was supported by NSFC Excellent Young Scientists Fund (No. 71822202), Chang Jiang Scholars Program (Niu Baozhuang 2017), and the Fundamental Research Funds for the Central Universities.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix 1: The derivation of outcomes in each strategies
Under Strategy R, the supplier relies on the small retailer for product selling. There is a chain-to-chain competition where the supplier does not have demand information. We solve this game using backward induction. Since the powerful retailer has big data and perfect demand forecast, she determines her optimal quantity according to the perfectly forecasted ε. The small retailer has no big data, so it determines its optimal quantity according to E[ε]. The expected profit of the small retailer is
The profit of the powerful retailer is
For any given ε, the first-order conditions of \( \frac{{\partial E\left[ {\pi_{SR} } \right]}}{{\partial q_{SR} }} \) and \( \frac{{\partial \pi_{PR} }}{{\partial q_{PR} }} \) yield
Solving the equation sets, we have
The supplier determines his expected wholesale price to maximize his expected profit. Substituting Eq. (4) into Eq. (7) yields
The first-order condition \( \frac{{\partial E\left[ {\pi_{S} } \right]}}{\partial w} \) yields his optimal wholesale price
Substituting Eq. (8) to the response functions (5) and (6) and profit functions (1), (2) and (7), we obtain the equilibrium outcomes under Strategy R in Table 9.
Next, we analyze the outcomes under Strategy D where the supplier sells products through the powerful retailer. Under Strategy D, the supplier has perfect demand forecast because of the sharing of the powerful retailer (e.g. Wal-Mart’s Retail Link). Their decisions are based on accurate forecast \( \epsilon \). Thus, the supplier and the powerful retailer’s profit functions are
The powerful retailer’s profit function is strictly concave in qS and qPR, so the first-order conditions yield the best response functions
Solving Eqs. (11) and (12), we have
Substituting Eqs. (13) and (14) into Eq. (9), the supplier’s optimal wholesale price can be derived by letting \( \frac{{\partial \pi_{S} }}{\partial w} = 0. \) We have
Substituting Eq. (15) to Eqs. (13) and (14), we obtain the outcomes under Strategy D in Table 10.
Next, we prove the propositions and lemmas. Note that all the discussions are in the feasible set of the parameters \( \left\{ {\left( {a,b,c,r, \sigma} \right) \left |c < a< {2c,c} > 0, 0 < b< {1,\sigma}\,\right. \,> 0,0 < r < \frac{a - c}{c}} \right\} \).
Appendix 2: Proofs of propositions
Proof of Proposition 1
It’s easy to show that \( \frac{{\partial w^{R} }}{\partial r} = - \frac{{a\left( {2 - b} \right)}}{{4\left( {1 + r} \right)^{2} }} < 0 \) and \( \left\{ {\left( {a,b,c,r, \sigma}\right) \left |c < a< {2c,c} > 0, 0 < b< {1,\sigma}\,\right. \,> 0,0 < r < \frac{a - c}{c}} \right\} \).
Therefore, the difference between the wD and wR is
The condition that \( w^{D} > w^{R} \) equals to the condition that \( f_{1} \left( r \right) = 2c\left( {1 + b} \right)r^{2} + \left( {2a - 2ab + 2c + 3bc} \right)r + bc - ab > 0 \). Next, we analyze \( f_{1} \left( r \right) \).
The discriminant of the quadratic function \( f_{1} \left( r \right) \) is \( \Delta_{1} = 8b\left( {1 + b} \right)\left( {a - c} \right)c + \left( { - 2a\left( { - 1 + b} \right) + \left( {2 + 3b} \right)c} \right)^{2} > 0 \), which indicates \( f_{1} \left( r \right) \) has two different roots.
Solving \( f_{1} \left( r \right) = 0 \), we have
It becomes immediate that \( \frac{a - c}{c} > r_{w} > 0 \) and \( r_{v} < 0 \).
Since \( 2c\left( {1 + b} \right) > 0 \), the quadratic function \( f_{1} \left( r \right) \) opens up. Therefore, \( f_{1} \left( r \right) > 0 \) equals to \( r > r_{w} \). Proposition 1 is proven.
Proof of Proposition 2
-
(a)
It is straightforward to show that \( q_{S}^{D} - q_{S}^{R} = - \frac{{b\left( {a - c - cr} \right)}}{{4\left( {1 + b} \right)\left( {2 + b} \right)}} < 0 \).
-
(b)
Obviously, we have \( \frac{{\partial q_{S}^{D} }}{\partial r} = - \frac{c}{{4\left( {1 + b} \right)}} < 0,\frac{{\partial q_{S}^{R} }}{\partial r} = - \frac{c}{{2\left( {2 + b} \right)}} < 0,\frac{{\partial q_{S}^{D} }}{\partial r} - \frac{{\partial q_{S}^{R} }}{\partial r} = \frac{bc}{{4\left( {2 + 3b + b^{2} } \right)}} > 0 \).
-
(c)
The difference between \( \frac{{q_{S}^{D} }}{{q_{S}^{D} + q_{PR}^{D} }} \) and \( \frac{{q_{S}^{R} }}{{q_{S}^{R} + q_{PR}^{R} }} \) is \( \frac{{q_{S}^{D} }}{{q_{S}^{D} + q_{PR}^{D} }} - \frac{{q_{S}^{R} }}{{q_{S}^{R} + q_{PR}^{R} }} = - \frac{b}{{18 + 9b + b^{2} }} < 0 \).
-
(d)
The difference between \( \frac{{q_{PR}^{D} }}{{q_{PR}^{D} + q_{S}^{D} }} \) and \( \frac{{q_{PR}^{R} }}{{q_{PR}^{R} + q_{S}^{R} }} \) is \( \frac{{q_{PR}^{D} }}{{q_{PR}^{D} + q_{S}^{D} }} - \frac{{q_{PR}^{R} }}{{q_{PR}^{R} + q_{S}^{R} }} = \frac{b}{{18 + 9b + b^{2} }} > 0 \).
Hereby, we prove Proposition 2.
Proof of Proposition 3
The difference between \( E\left[ {\pi_{S}^{D} } \right] \) and \( E\left[ {\pi_{S}^{R} } \right] \) is
The condition that \( E\left[ {\pi_{S}^{D} } \right] > E\left[ {\pi_{S}^{R} } \right] \) equals to the condition \( f_{2} \left( r \right) = - 2bc^{2} r^{2} + 4bc\left( {a - c} \right)r - 2b\left( {a - c} \right)^{2} + \left( {2 - b - b^{2} } \right)\sigma^{2} \). Next, we analyze \( f_{2} \left( r \right) \).
The discriminant of the quadratic function \( f_{2} \left( r \right) \) is \( \Delta_{2} = 8b\left( {2 - b - b^{2} } \right)c^{2} \sigma^{2} > 0 \), which indicates \( f_{2} \left( r \right) \) has two different roots.
Solving \( f_{2} \left( r \right) = 0 \), we have
Then we can show that \( \frac{a - c}{c} > r_{1} \) and \( r_{a} > \frac{a - c}{c} \).
-
1.
If \( r_{1} > 0 \), we have \( \sigma \le \sigma_{1} = \frac{{\left( {a - c} \right)\sqrt {2b} }}{{\sqrt {\left( {1 - b} \right)\left( {2 + b} \right)} }} \). Since \( - 2bc^{2} < 0 \), the quadratic function \( f_{2} \left( r \right) \) opens down, therefore, we find that \( f_{2} \left( r \right) > 0 \) equals to \( r > r_{1} \). It then can be shown that \( \sigma \le \sigma_{1} = \frac{{\left( {a - c} \right)\sqrt {2b} }}{{\sqrt {\left( {1 - b} \right)\left( {2 + b} \right)} }} \) and \( r > r_{1} = \frac{a - c}{c} - \frac{{\sqrt {\left( {1 - b} \right)\left( {2 + b} \right)} }}{{c\sqrt {2b} }}\sigma \).
-
2.
If \( r_{1} < 0 \), we have \( \sigma \ge \sigma_{1} = \frac{{\left( {a - c} \right)\sqrt {2b} }}{{\sqrt {\left( {1 - b} \right)\left( {2 + b} \right)} }} \). It can be shown that \( f_{2} \left( r \right) > 0 \) holds in the interval \( \frac{a - c}{c} > r > 0 \). Therefore, we have \( \sigma > \sigma_{1} = \frac{{\left( {a - c} \right)\sqrt {2b} }}{{\sqrt {\left( {1 - b} \right)\left( {2 + b} \right)} }} \).
Proposition 3 is proven.
Proof of Proposition 4
The difference between \( E\left[ {\pi^{D} } \right] \) and \( E\left[ {\pi^{R} } \right] \) is
The condition that \( E\left[ {\pi^{D} } \right] > E\left[ {\pi^{R} } \right] \) equals to \( f_{3} \left( r \right) = - 2b\left( {2 - 2b - b^{2} } \right)c^{2} r^{2} + 4b\left( {2 - 2b - b^{2} } \right)\left( {a - c} \right)cr - 2b\left( {2 - 2b - b^{2} } \right)\left( {a - c} \right)^{2} + 3\left( {1 - b} \right)\left( {2 + b} \right)^{2} \sigma^{2} > 0 \), which indicates \( f_{3} \left( r \right) \) has two different roots.
Solving the equation \( f_{3} \left( r \right) = 0 \), we have
It can be shown that \( r_{2} < \frac{a - c}{c} \) and \( r_{b} > \frac{a - c}{c} \).
-
1.
If \( r_{2} > 0, \) we have \( \sigma \le \sigma_{2} = \frac{{\left( {a - c} \right)\sqrt {2b\left( {2 - 2b - b^{2} } \right)} }}{{\left( {2 + b} \right)\sqrt {3\left( {1 - b} \right)} }} \). Since \( - 2b\left( {2 - 2b - b^{2} } \right)c^{2} < 0 \), the quadratic function \( f_{3} \left( r \right) \) opens down. Therefore, it can be shown that \( f_{3} \left( r \right) > 0 \) equals to \( r > r_{2} \). Thus, we find that \( \sigma \le \sigma_{2} = \frac{{\left( {a - c} \right)\sqrt {2b\left( {2 - 2b - b^{2} } \right)} }}{{\left( {2 + b} \right)\sqrt {3\left( {1 - b} \right)} }} \) and \( r > r_{2} = \frac{a - c}{c} - \frac{{\left( {2 + b} \right)\sqrt {3\left( {1 - b} \right)} }}{{c\sqrt {2b\left( {2 - 2b - b^{2} } \right)} }}\sigma \).
-
2.
If \( r_{2} < 0, \) we have \( \sigma \ge \sigma_{2} = \frac{{\left( {a - c} \right)\sqrt {2b\left( {2 - 2b - b^{2} } \right)} }}{{\left( {2 + b} \right)\sqrt {3\left( {1 - b} \right)} }} \). It can be shown that \( f_{3} \left( r \right) > 0 \) holds in the interval \( \frac{a - c}{c} > r > 0 \). So, we find that \( \sigma \ge \sigma_{2} = \frac{{\left( {a - c} \right)\sqrt {2b\left( {2 - 2b - b^{2} } \right)} }}{{\left( {2 + b} \right)\sqrt {3\left( {1 - b} \right)} }} \).
Proposition 4 is proven.
Proof of Proposition 5
The difference between \( \frac{{E\left[ {\pi_{S}^{D} } \right]}}{{E\left[ {\pi^{D} } \right]}} \) and \( \frac{{E\left[ {\pi_{S}^{R} } \right]}}{{E\left[ {\pi^{R} } \right]}} \) is
We find that the condition that \( \frac{{E\left[ {\pi_{S}^{D} } \right]}}{{E\left[ {\pi^{D} } \right]}} > \frac{{E\left[ {\pi_{S}^{R} } \right]}}{{E\left[ {\pi^{R} } \right]}} \) equals to \( f_{4} \left( r \right) = - \left( {4 - b} \right)b\left( {3 + b} \right)c^{2} r^{2} + 2\left( {4 - b} \right)b\left( {3 + b} \right)c\left( {a - c} \right)r + 8\sigma^{2} - b\left( {3 + b} \right)\left[ {\left( {4 - b} \right)\left( {a - c} \right)^{2} + 2b\sigma^{2} } \right] > 0 \), indicating \( f_{4} \left( r \right) \) has two different roots.
Solving the equation \( f_{4} \left( r \right) = 0 \), we have\( r_{3} = \frac{a - c}{c} - \frac{{\left( {2 + b} \right)\sqrt {2\left( {1 - b} \right)} }}{{c\sqrt {b\left( {4 - b} \right)\left( {3 + b} \right)} }}\sigma ,\quad r_{c} = \frac{a - c}{c} + \frac{{\left( {2 + b} \right)\sqrt {2\left( {1 - b} \right)} }}{{c\sqrt {b\left( {4 - b} \right)\left( {3 + b} \right)} }}\sigma . \).Therefore, it is shown that \( r_{3} < \frac{a - c}{c} \) and \( r_{c} > \frac{a - c}{c} \).
-
1.
If \( r_{3} > 0, \) we have \( \sigma \le \sigma_{3} = \frac{{\left( {a - c} \right)\sqrt {b\left( {4 - b} \right)\left( {3 + b} \right)} }}{{\left( {2 + b} \right)\sqrt {2\left( {1 - b} \right)} }} \). Since \( - \left( {4 - b} \right)b\left( {3 + b} \right)c^{2} < 0 \), the quadratic function \( f_{4} \left( r \right) \) opens down. Therefore, it can be shown that \( f_{4} \left( r \right) > 0 \) equals to \( r > r_{3} \). We find that \( \sigma \le \sigma_{3} = \frac{{\left( {a - c} \right)\sqrt {b\left( {4 - b} \right)\left( {3 + b} \right)} }}{{\left( {2 + b} \right)\sqrt {2\left( {1 - b} \right)} }} \) and \( r > r_{3} = \frac{a - c}{c} - \frac{{\left( {2 + b} \right)\sqrt {2\left( {1 - b} \right)} }}{{c\sqrt {b\left( {4 - b} \right)\left( {3 + b} \right)} }}\sigma \).
-
2.
If \( r_{3} < 0, \) we have \( \sigma \ge \sigma_{3} = \frac{{\left( {a - c} \right)\sqrt {b\left( {4 - b} \right)\left( {3 + b} \right)} }}{{\left( {2 + b} \right)\sqrt {2\left( {1 - b} \right)} }} \). It is obviously that \( f_{4} \left( r \right) > 0 \) holds in the region \( \frac{a - c}{c} > r > 0 \). So, we show that \( \sigma \ge \sigma_{3} = \frac{{\left( {a - c} \right)\sqrt {b\left( {4 - b} \right)\left( {3 + b} \right)} }}{{\left( {2 + b} \right)\sqrt {2\left( {1 - b} \right)} }} \).
Proposition 5 becomes immediate.
Appendix 3: Powerful retailer has no incentive to take the first-mover advantage
In this subsection, we investigate the powerful retailer’s incentive to take the first-mover advantage under Strategy D, which means the powerful retailer determines her order quantity qPR before the supplier determines the wholesale price w. Therefore, the event sequence under Strategy D is revised. See Fig. 12.
We first derive the equilibrium outcomes when the powerful retailer is the first-mover under Strategy D, and then investigate whether she has incentives to be the first-mover.
The supplier and the powerful retailer’s profit functions are the same as Eqs. (9) and (10). We solve the game by backward induction. First, given qPR and w, we derive the best response function of the powerful retailer with respect to qS as
Then, substituting Eq. (16) into Eq. (9), we obtain the supplier’s optimal wholesale price
Substituting Eq. (17) into Eq. (16), we have
Then the powerful retailer’s optimal quantity of her self-branded product qPR can be derived as
Substituting Eq. (19) back to Eqs. (17) and (18), we obtain the supplier’s equilibrium wholesale price w and the powerful retailer’s equilibrium order quantity qS
Similarly, we obtain the expected profits of the supplier and the powerful retailer. All the outcomes under Strategy D are in Table 11. For ease of expression, we use Strategy D3 to denote the case in which the powerful retailer is the first-mover.
Next, we investigate the powerful retailer’s incentive to be the first-mover by comparing her profits under Strategy D and Strategy D3. Note that all the discussions require the parameters satisfy \( \left\{ {\left( {c,a,b,\sigma ,r} \right) |c < a\left\langle {2c,c} \right\rangle 0, 0 < b\left\langle {1,\sigma } \right\rangle 0,0 < r < \frac{a - c}{c}} \right\} \).
Proposition 9
The powerful retailer prefers Strategy D for any feasible parameters (i.e., \( E\left[ {\pi_{PR}^{{D_{3} }} } \right] < E\left[ {\pi_{PR}^{D} } \right] \)).
Intuitively, a player who moves first can acquire the first-mover advantage Cournot competition (Lus and Muriel 2009). However, Proposition 9 demonstrates that the powerful retailer has no incentive to be the first-mover. To explain the interesting finding, we derive the following three lemmas.
Lemma 1
If the powerful retailer is the first-mover, the supplier will determine a higher wholesale price (i.e., \( w^{{D_{3} }} > w^{D} \)).
Lemma 2
If the powerful retailer is the first-mover, the powerful retailer will determine a larger quantity of her self-branded product and the supplier will receive a smaller order quantity from the powerful retailer (i.e., \( q_{PR}^{{D_{3} }} > q_{PR}^{D} \) and \( q_{S}^{{D_{3} }} < q_{S}^{D} \)).
Lemma 3
If the powerful retailer is the first-mover, she will obtain a higher marginal profit from her self-branded product and a lower marginal profit from supplier’s product (i.e., \( E\left[ {m_{PR}^{{D_{3} }} } \right] > E\left[ {m_{PR}^{D} } \right] \) and \( E\left[ {m_{S}^{{D_{3} }} } \right] < E\left[ {m_{S}^{D} } \right] \)). In addition, the marginal profit of the powerful retailer is hurt more when she is the first-mover (i.e., \( E\left[ {m_{PR}^{{D_{3} }} } \right] - E\left[ {m_{PR}^{D} } \right] < E\left[ {m_{S}^{D} } \right] - E\left[ {m_{S}^{{D_{3} }} } \right] \)).
Lemma 1 points out that the supplier will determine a higher wholesale price if the powerful retailer acts as the first-mover, which increases the powerful retailer’s procurement cost and hence, lowers her order quantity under Strategy D3. This motivates the powerful retailer to increase the quantity of self-branded product, as Lemma 2 shows. Therefore, the powerful retailer’s profit from the reselling business is hurt (\( E\left[ {m_{S}^{{D_{3} }} } \right] < E\left[ {m_{S}^{D} } \right] \)) but that from the self-brand business is increased (\( E\left[ {m_{PR}^{{D_{3} }} } \right] > E\left[ {m_{PR}^{D} } \right] \)). In equilibrium, we find that the marginal profit of the powerful retailer is hurt more when she is the first-mover, as Lemma 3 reveals that \( E\left[ {m_{PR}^{{D_{3} }} } \right] - E\left[ {m_{PR}^{D} } \right] < E\left[ {m_{S}^{D} } \right] - E\left[ {m_{S}^{{D_{3} }} } \right] \). Consequently, the powerful retailer has no incentive to be the first-mover.
We provide the proofs of Proposition 9, Lemmas 1, 2 and 3 as follows.
Proof of Proposition 9
Proposition 9 can be derived by comparing \( E\left[ {\pi_{PR}^{D} } \right] \) with \( E\left[ {\pi_{PR}^{{D_{3} }} } \right] \). We have
Clearly, we have \( 1 - b > 0 \) and \( 4 - b^{2} > 0 \) for any \( b \in \left( {0,1} \right) \). Therefore, it can be verified that \( E\left[ {\pi_{PR}^{D} } \right] - E\left[ {\pi_{PR}^{{D_{3} }} } \right] > 0 \) for any \( 0 < r < \frac{a - c}{c} \), \( 0 < b < 1 \) and \( \sigma > 0 \).
Thus, Proposition 9 is proven.
Proof of Lemma 1
Lemma 1 can be derived by comparing \( E\left[ {w^{{D_{3} }} } \right] \) with \( E\left[ {w^{D} } \right] \). We have
It can be verified that \( w^{{D_{3} }} - w^{D} > 0 \) for any \( 0 < r < \frac{a - c}{c} \) and \( 0 < b < 1 \).
Lemma 1 is proven.
Proof of Lemma 2
The proof of Lemma 2 is similar to the proof of Lemma 1. We therefore omit the proof here.
Proof of Lemma 3
Lemma 3 can be derived by comparing \( m_{PR}^{{D_{3} }} \) with \( m_{PR}^{D} ,m_{S}^{{D_{3} }} \) with \( m_{S}^{D} \) and \( m_{PR}^{D} - m_{PR}^{{D_{3} }} \) with \( m_{S}^{{D_{3} }} - m_{S}^{D} \), respectively. We have
for any \( 0 < r < \frac{a - c}{c} \) and \( 0 < b < 1 \).
Thus, Lemma 3 is proven.
Rights and permissions
About this article
Cite this article
Niu, B., Shen, Z. & Li, Q. Information advantage and payment disadvantage when selling goods through a powerful retailer. Ann Oper Res 331, 417–446 (2023). https://doi.org/10.1007/s10479-020-03889-x
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-020-03889-x