IJE TRANSACTIONS C: Aspects Vol. 30, No. 6, (June 2017) 859-866 Please cite this article as: S. Melnikov, Bertrand N - ash Equilibrium in the Retail Duopoly Model under Asymmetric Costs, International Journal of Engineering (IJE), TRANSACTIONS C: Aspects Vol. 30, No. 6, (June 2017) 859-866 International Journal of Engineering Journal Homepage: www.ije.ir Bertrand N - ash Equilibrium in the Retail Duopoly Model under Asymmetric Costs S. Melnikov Faculty of Economics and Management, Odessa National Maritime University, Odessa, Ukraine PAPER INFO Paper history: Received 26 October 2016 Received in revised form 10 March 2017 Accepted 21 April 2017 Keywords: Economic Order Quantity Model Retail Duopoly Model Bertrand-Nash Equilibrium Market Potential Bifurcation Parameter Return on Logistics Costs Barrier to Entry ABSTRACT In this paper, the Bertrand's price competition in the retail duopoly with asymmetric costs is analyzed. Retailers sell substitute products in the framework of the classical economic order quantity (EOQ) model with linear demand function. The market potential and competitor price are considered to be the bifurcation parameters of retailers. Levels of the barriers to market penetration depending on the bifurcation parameters are analyzed. The conditions of Bertrand-Nash equilibrium in parametric and trigonometric forms are found. doi: 10.5829/idosi.ije.2017.30.06c.06 1. INTRODUCTION 1 Integrating operations and marketing decisions are an important objective for retailers in today’s competitive environment. An obvious problem common to the retail industry is the joint optimization of the lot size of a product to be stocked and the selling price in order to maximize profit. Pricing and inventory control strategies are closely related to each other. Pricing decisions alter demand forecasts, which are used by inventory control systems. We note that in the classical economic order quantity model, the demand is assumed as constant. A lot of important results have been obtained in this field of study. There are many studies which modify this condition [1], in particular with linear demand function [2, 3]. Further studies are related to the complexity of market structures, in particular, considering the horizontal competition between retailers [4]. In this article, we developed the obtained results from literature [4] for model duopoly retailers with the Bertrand's price competition in the case of substitute products. Along with trigonometric solution, parametric *Corresponding Author’s Email: nfn333@ukr.net (S. Melnikov) solution is received as well, where the parameter is return on logistics costs. Parametric solution allows to determine the sufficient conditions for the existence of equilibrium with the asymmetry retailers cost. The market potential and price of competitor are considered to be bifurcation parameters of retailers. Depending on the values of the bifurcation parameters, barriers to entry for retailers are analyzed. This article is organized as follows. Section 2 reviews the related literature. Section 3 describes the problem statement. In section 4, we analyze barriers to entry depending on the bifurcation parameters. In section 5, we present the analytical solutions of the problem in parametric and trigonometric forms. Section 6 presents the numerical example and sensitivity analysis. Finally, section 7 summarizes the results. 2. LITERATURE REVIEW Whitin [5] was the first researcher who indicated the fundamental connection between price theory and inventory control. He extended the basic EOQ model by considering the selling price in addition to the order quantity as the decision variables.