When is competition price-increasing? The impact of expected competition on prices

We examine the effect of expected competition on markups in a random utility model where the number of competing firms may differ across consumers. Firms observe consumers' utility shocks and set prices using personalized pricing. We derive a precise condition under which the expected markup across consumers can be represented by a simple expression involving consumers' expected utility and the expected demand. This delivers a general condition under which greater expected competition is price-increasing. Whether this condition holds depends on the distribution of utility shocks, consumers' outside option, the expected number of competing firms, and the distribution of the number of firms competing for each consumer.