Effects of demand uncertainty on equilibrium of prices and quantities in a competitive market

Competition has been identified by researchers as one of the understudied topics in stochastic inventory theory. Competitive decisions of firms have been mainly analyzed for two strategic variables in deterministic markets: Quantity and price. In this study, we address the effects of stochastic demand on market equilibrium for firms competing over quantity and price. First, we consider quantity competition in a duopoly where the aggregate production level dictates the market price. In the presence of an auctioneer, the aggregate production quantity in a duopoly exceeds the production quantity of a monopolist. In this case, the duopoly price and total profits are lower than the monopoly. If there is no auctioneer in the market, however, the firms choose the same price and compete on quantities, the duopoly price may be higher than the monopoly price under certain conditions. In a competitive market with stochastic demand, we first analyze price competition for three different demand structures: additive demand, multiplicative demand, and demand with random reservation price. We consider two different backorder cost structures: (i) when the spill-over demand can cause additional backorder penalty for competitive firms, (ii) when it does not. Subsequently, we analyze price competition with a general demand distribution. We find that the existence of market equilibrium depends on the backorder cost structure among other things. If the firms are penalized for backorder due to spill-over demand from the competitive firm, there is no pure strategy Nash equilibrium. Also, if the firms make the decisions collectively, the price is higher than the monopoly price, however, the total profits are lower than the monopoly profits. Next, we consider a market where the firms subcontract their shortages, and show that at the Nash equilibrium the expected profits of the firms are zero. Finally, we present conditions for a pure strategy Nash equilibrium, when the firms do not incur additional backorder penalty due to spill-over demand from the other firm.