Publication Date: March 2020
Consider a market with many identical ﬁrms oﬀering a homogeneous good. A consumer obtains price quotes from a subset of ﬁrms and buys from the ﬁrm oﬀering the lowest price. The “price count” is the number of ﬁrms from which the consumer obtains a quote. For any given ex ante distribution of the price count, we obtain a tight upper bound (under ﬁrst-order stochastic dominance) on the equilibrium distribution of sale prices. The bound holds across all models of ﬁrms’ common-prior higher-order beliefs about the price count, including the extreme cases of complete information ( ﬁrms know the price count exactly) and no information ( ﬁrms only know the ex ante distribution of the price count). A qualitative implication of our results is that even a small ex ante probability that the price count is one can lead to dramatic increases in the expected price. The bound also applies in a wide class of models where the price count distribution is endogenized, including models of simultaneous and sequential consumer search.
Keywords: Search, Price Competition, Bertrand Competition, "Law of One Price", Price Count, Price Quote, Information Structure, Bayes Correlated Equilibrium
JEL Classification Codes: D41, D42, D43, D83CFDP 2224R