A Note on Costly Sequential Search and Oligopoly Pricing (new title: Truly Costly Sequential Search and Oligopolistic Pricing,)

3 Equilibria

We first derive some auxiliary results.

Lemma 1 An equilibrium where non-shoppers do not search at all does not exist.

Proof. Suppose non-shoppers did not search. Then the only consumers left in the market
would be the shoppers. Therefore, competition between stores would drive prices down to marginal
cost. But then, as v - c > 0, the non-shoppers would gain by deviating and searching once. ¥

Lemma 1 reveals that existence of equilibrium requires the non-shoppers to be active in the
market with strictly positive probability. The next result is provided by Stahl (1989).

Lemma 2 In equilibrium non-shoppers will not search beyond the first firm.

Proof. See Lemma 2 of Stahl (1989). ¥

The idea behind Lemma 2 is that pricing above consumers’ reservation price is never optimal
for firms since buyers will continue searching; as a result, the price buyers will find at the first store
they encounter will always be accepted and no further search will take place.

Let us introduce the following notation. Let θ1 be the probability with which a non-shopper
searches for a price quotation. Lemmas 1 and 2 together imply that only two candidates for
equilibrium exist: either (a) θ1 = 1, or (b) 0 < θ1 < 1. The first case is analyzed in Stahl
(1989). We shall refer to this equilibrium as an equilibrium with full consumer participation. This
is because if all consumers sample one firm they will all buy. This contrasts with case (b) where
consumers mix between not searching at all and searching once so not all consumers will conduct
a transaction. Let θ0 denote the probability with which a non-shopper does not search; then an
equilibrium with partial consumer participation is characterized by θ0 + θ1 = 1, 0 < θ1 < 1.

The next remark is that, since θ1 > 0 in any equilibrium, the equilibrium price distribution
must be atomless:

Lemma 3 Irrespective of the search behavior of non-shoppers, if F (p) is an equilibrium price dis-
tribution, then it is atomless. Hence, there is no pure strategy equilibrium.

Proof. See Lemma 1 of Stahl (1989). The proof extends straightforwardly to the case of partial
consumer participation. ¥

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