Properties of Nash Equilibrium Retail Prices in Contract Model with a Supplier, Multiple Retailers and Price-Dependent Demand
ABSTRACT
Recently, price contract models between suppliers and retailers, with stochastic demand have been analyzed based on well-known newsvendor problems. In Bernstein and Federgruen [6], they have analyzed a contract model with single supplier and multiples retailers and price dependent demand, where retailers compete on retail prices. Each retailer decides a number of products he procures from the supplier and his retail price to maximize his own profit. This is achieved after giving the wholesale and buy-back prices, which are determined by the supplier as the supplier’s profit is maximized. Bernstein and Federgruen have proved that the retail prices become a unique Nash equilibrium solution under weak conditions on the price dependent distribution of demand. The authors, however, have not mentioned the numerical values and proprieties on these retail prices, the number of products and their individual and overall profits. In this paper, we analyze the model numerically. We first indicate some numerical problems with respect to theorem of Nash equilibrium solutions, which Bernstein and Federgruen proved, and we show their modified results. Then, we compute numerically Nash equilibrium prices, optimal wholesale and buy-back prices for the supplier’s and retailers’ profits, and supply chain optimal retailers’ prices. We also discuss properties on relation between these values and the demand distribution.
Recently, price contract models between suppliers and retailers, with stochastic demand have been analyzed based on well-known newsvendor problems. In Bernstein and Federgruen [6], they have analyzed a contract model with single supplier and multiples retailers and price dependent demand, where retailers compete on retail prices. Each retailer decides a number of products he procures from the supplier and his retail price to maximize his own profit. This is achieved after giving the wholesale and buy-back prices, which are determined by the supplier as the supplier’s profit is maximized. Bernstein and Federgruen have proved that the retail prices become a unique Nash equilibrium solution under weak conditions on the price dependent distribution of demand. The authors, however, have not mentioned the numerical values and proprieties on these retail prices, the number of products and their individual and overall profits. In this paper, we analyze the model numerically. We first indicate some numerical problems with respect to theorem of Nash equilibrium solutions, which Bernstein and Federgruen proved, and we show their modified results. Then, we compute numerically Nash equilibrium prices, optimal wholesale and buy-back prices for the supplier’s and retailers’ profits, and supply chain optimal retailers’ prices. We also discuss properties on relation between these values and the demand distribution.
Cite this paper
nullK. NAKADE, S. TSUBOUCHI and I. SEDIRI, "Properties of Nash Equilibrium Retail Prices in Contract Model with a Supplier, Multiple Retailers and Price-Dependent Demand," Journal of Software Engineering and Applications, Vol. 3 No. 1, 2010, pp. 27-33. doi: 10.4236/jsea.2010.31003.
nullK. NAKADE, S. TSUBOUCHI and I. SEDIRI, "Properties of Nash Equilibrium Retail Prices in Contract Model with a Supplier, Multiple Retailers and Price-Dependent Demand," Journal of Software Engineering and Applications, Vol. 3 No. 1, 2010, pp. 27-33. doi: 10.4236/jsea.2010.31003.
References
[1] G. P. Cachon, “Supply chain coordination with contracts, in supply chain management, design, coordination and operation,” Elsevier, Vol. 11, Amsterdam, pp. 229–339. [2] C. A. Yano and S. M. Gilbert, “Coordinated pricing and production/procurement decisions: A review,” A. Chak-ravarty, J. Eliashberg, eds. “Managing business interfaces: marketing, engineering and manufacturing perspectives,” Kluwer Academic Publishers, Boston, MA, 2003. [3] Y. Wang, J. Li, and Z. Shen., “Channel performance un-der consignment contract with revenue sharing,” Man-agement Sci. Vol. 50, No. 1, pp. 34–47, January 2004. [4] N. C. Petruzzi, “Newsvendor pricing, purchasing and consignment: Supply chain modeling implications and in-sights,” Working Paper, College of Business, University of Illinois, Urbana-Champaign, IL, 2004. [5] Y. Song, S. Ray, and S. Li, “Structural properties of buy-back contracts for price-setting newsvendors,” MSOM, Vol. 10, pp. 1–18, January 2007. [6] F. Bernstein and A. Federgruen, “Decentralized supply chains with competing retailers under demand uncer-tainty,” Management Science, Vol. 51, pp. 18–29, Janu-ary 2005.
[1] G. P. Cachon, “Supply chain coordination with contracts, in supply chain management, design, coordination and operation,” Elsevier, Vol. 11, Amsterdam, pp. 229–339. [2] C. A. Yano and S. M. Gilbert, “Coordinated pricing and production/procurement decisions: A review,” A. Chak-ravarty, J. Eliashberg, eds. “Managing business interfaces: marketing, engineering and manufacturing perspectives,” Kluwer Academic Publishers, Boston, MA, 2003. [3] Y. Wang, J. Li, and Z. Shen., “Channel performance un-der consignment contract with revenue sharing,” Man-agement Sci. Vol. 50, No. 1, pp. 34–47, January 2004. [4] N. C. Petruzzi, “Newsvendor pricing, purchasing and consignment: Supply chain modeling implications and in-sights,” Working Paper, College of Business, University of Illinois, Urbana-Champaign, IL, 2004. [5] Y. Song, S. Ray, and S. Li, “Structural properties of buy-back contracts for price-setting newsvendors,” MSOM, Vol. 10, pp. 1–18, January 2007. [6] F. Bernstein and A. Federgruen, “Decentralized supply chains with competing retailers under demand uncer-tainty,” Management Science, Vol. 51, pp. 18–29, Janu-ary 2005.