We study a vertical market with an upsteam supplier and multiple downstream retailers. Demand uncertainty falls to the supplier who acts first and sets a uniform wholesale price before the retailers observe the realized demand and engage in retail competition. Our focus is on the supplier's optimal pricing decision. We express the price elasticity of expected demand in terms of the mean residual demand (MRD) function of the demand distribution. This allows for a closed form characterization of the points of unitary elasticity that maximize the supplier's profits and the derivation of a mild unimodality condition for the supplier's objective function that generalizes the widely used increasing generalized failure rate (IGFR) condition. A direct implication is that optimal prices between different markets can be ordered if the markets can be stochastically ordered according to their MRD functions or equivalently to their elasticities. Based on this, we apply the theory of stochastic orders to study the response of the supplier's optimal price to various features of the demand distribution. Our findings challenge previously established economic insights about the effects of market size, demand transformations and demand variability on wholesale prices and indicate that the conclusions largely depend on the exact notion that will be employed. We then turn to measure market performance and derive a distribution free and tight bound on the probability of no trade between the supplier and the retailers. If trade takes place, our findings indicate that ovarall performance depends on the interplay between demand uncertainty and level of retail competition.

中文翻译：

---

需求不确定的垂直市场中的垄断定价

我们研究了一个具有上游供应商和多个下游零售商的垂直市场。在零售商观察已实现的需求并参与零售竞争之前，首先采取行动并设定统一批发价格的供应商会产生需求不确定性。我们的重点是供应商的最优定价决策。我们用需求分布的平均剩余需求 (MRD) 函数来表达预期需求的价格弹性。这允许对最大化供应商利润的单一弹性点进行封闭形式表征，并为供应商的目标函数推导温和的单峰条件，该条件概括了广泛使用的增加广义失效率 (IGFR) 条件。一个直接的含义是，如果市场可以根据其 MRD 函数或等效于其弹性进行随机排序，则可以对不同市场之间的最优价格进行排序。在此基础上，我们应用随机订单理论来研究供应商的最优价格对需求分布的各种特征的响应。我们的研究结果挑战了先前关于市场规模、需求转变和需求变化对批发价格的影响的经济见解，并表明结论在很大程度上取决于将采用的确切概念。然后，我们转向衡量市场表现，并在供应商和零售商之间没有交易的概率上得出一个免费且严格的分布。如果发生贸易，