Replenishment quantities of rental stores for the newly released product in the duopoly environment

This paper considers the replenishment problem of the rental stores in a duopoly environment. Specifically, when a new product such as video tapes, DVDs, books, etc., is released to the market, stores determine the replenishment quantities, respectively, in order to satisfy the demand in the retention time. The demand for each store is continuous and decreasing in time. Proportion of the unsatisfied demand of one store will turn to the other store. The store has to take the demand spill over effect into consideration when he makes the replenishment decision. We prove, in this duopoly environment, the replenishment quantity of each store consists a unique Nash equilibrium. And there exists a time window in the retention time in which the store with the higher penalty cost could earn extra profit due to the spill over effect.