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.