# American Institute of Mathematical Sciences

October  2016, 12(4): 1587-1611. doi: 10.3934/jimo.2016.12.1587

## Optimal rebate strategies in a two-echelon supply chain with nonlinear and linear multiplicative demands

 1 School of Management, Huazhong University of Science and Technology, Wuhan, China 2 Faculty of Business, University of Macau, Macau, China

Received  February 2014 Revised  July 2015 Published  January 2016

We examine the pure rebate strategies in a two-echelon supply chain under stochastic demand with multiplicative error. Given exogenous wholesale price and retail price, we characterize the unique Nash equilibrium when both manufacturer and retailer provide rebate policy to consumers under nonlinear and linear price-dependent demand functions, including iso-elastic multiplicative demand function (EMDF) and linear multiplicative demand function (LMDF). Based on a game theoretical framework, we prove that there still exists a unique equilibrium when the price elasticity is rather small with constraint conditions in the former case. We also find that in this case the retailer(manufacturer) may increase its rebate value in reaction to the manufacturer's(retailer's) rebate value in order to stimulate sales, which is contrary to the conventional wisdom that the retailer(manufacturer) will shrink its rebate value to gain an extra advantage" unfairly. As a result, both parties share the same profit at equilibrium. Further, we compare the expected profit outcomes at equilibrium among joint-rebate game, single-party rebate game and no-rebate game by using numerical examples. It is shown that the joint-rebate policy is not always dominates the others unless the price elasticity is sufficiently flexible.
Citation: Jianbin Li, Niu Yu, Zhixue Liu, Lianjie Shu. Optimal rebate strategies in a two-echelon supply chain with nonlinear and linear multiplicative demands. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1587-1611. doi: 10.3934/jimo.2016.12.1587
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