when the number of firms increases without bound. This is due to the fact that when the fraction
of shoppers becomes very small, or when the number of firms increases beyond a critical value, the
economy turns into a partial participation equilibrium and in such an equilibrium expected price
is insensitive to changes in these parameters.
The rest of this note is organized as follows. Section 2 presents the model. A full characterization
and an overview of the two types of equilibria is given in Section 3. Section 4 presents the different
comparative statics results and Section 5 concludes.
2 The Model
We examine the model of oligopolistic competition and sequential consumer search presented in
Stahl (1989), but we assume that all price quotations are costly to acquire for non-shoppers. The
features of the model are as follows. There are N firms that produce a homogeneous good at
constant returns to scale. Their identical unit cost can be normalized to zero and prices can be
interpreted as price-to-cost margins. There is a unit mass of buyers and we assume that buyers
hold inelastic demands.3 A consumer wishes to purchase at most a single unit of the good and
his/her valuation for the item is v > 0. A proportion μ ∈ (0, 1) of the consumers has negligible
opportunity cost of time and therefore searches for prices costlessly. These consumers are referred
to as shoppers. The other 1 — μ percent of the buyers, referred to as non-shoppers, must pay search
cost c > 0 to observe every price quotation they get, including the first one. The non-shoppers
search sequentially, i.e., a buyer first decides whether to sample a first firm or not and then, upon
observation of the price of the first firm, decides to search for a second price or not, and so on. We
assume that v > c.
Firms and buyers play the following game. An individual firm chooses its price taking price
choices of the rivals as well as consumers’ search behavior as given. Likewise, an individual buyer
forms conjectures about the distribution of prices in the market and decides on his/her optimal
search strategy. We restrict the analysis to symmetric Nash equilibria. The distribution of prices
charged by a firm is denoted by F (p).
3Stahl (1989) considers a more general specification of the demand function. The assumption of inelastic demand
allows us to compute explicitly the reservation price and give a full characterization of which equilibrium exists for
which configurations of parameters. The qualitative results do not depend on this assumption though.
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