Stochastic comparisons in a price-quantity setting firm with uncertain demand and emergency procurement

We consider a newsvendor problem with price-dependent demand, in either additive or multiplicative format. The newsvendor has two modes of purchasing: regular ordering at the beginning of the selling season and emergency ordering (if the realized demand exceeds the initial order quantity) at the end of the selling season. By stochastic comparisons, we systematically investigate the effects of demand magnitude and demand randomness on pricing and ordering quantity decisions as well as expected profit of the newsvendor, under both usual stochastic order (first order stochastic dominance) and convex order (less variable). Our key findings include: (i) in contrary to the case where price is exogenous, a stochastically larger demand shock may even lead to a lower order quantity; (ii) a stochastically larger demand shock leads to a higher price for the additive demand case, but may lead to a lower price for the multiplicative demand case; (iii) a stochastically larger demand shock leads to a higher expected profit for both demand models; (iv) in general, a less variable demand leads to a higher expected profit for both demand models; and furthermore, a less variable demand shock has no effect on price for the additive demand model, but leads to a higher price for the multiplicative demand model. The implications of all these findings for pricing and order quantity are discussed in detail.