We note that firms have an incentive to charge low prices in order to attract all the shoppers
but at the same time they also have an incentive to charge high prices to extract income from the
consumers who do not compare prices. These two forces are balanced when firms randomize their
prices. Lemma 3 shows that equilibria must necessarily exhibit price dispersion, and that firm
pricing is always characterized by atomless price distributions. In what follows we shall examine
the characterization and the existence of the different equilibria.
Case a: Equilibrium with full consumer participation (Stahl, 1989)
Assume that non-shoppers search for one price with probability 1, i.e., θ1 = 1. This is the case
analyzed by Stahl (1989) with two modifications. First, as Stahl considers a more general demand
structure, an explicit expression for the reservation price cannot be obtained. Second, as Stahl
(1989) assumes the first price quotation to be for free, this full participation equilibrium exists for
all values of the parameters in his model, but not in ours. We will explicitly define the parameter
space for which the full participation equilibrium exists in our case. These two modifications deserve
a slightly extended analysis.
Under full participation, the expected payoff to firm i from charging price pi when its rivals
choose a random pricing strategy according to the cumulative distribution F(∙) is
πi (pi, F(pi)) = pi
1N + μ(1 - F(p.))N--
(1)
This profit expression follows from noting that expected demand faced by a firm stems from the
two different groups of consumers. Firm i attracts the μ shoppers when it charges a price that is
lower than its rivals’ prices, which happens with probability (1 - F(pi))N-1 . The firm also serves
the 1 — μ non-shoppers whenever they they visit its store, which occurs with probability 1 /N.
In equilibrium, a firm must be indifferent between charging any price in the support of F(∙).
Let us denote the upper bound of F(∙) by p. Any price in the support of F(∙) must then satisfy
∏i(Pi, F(∙)) = ∏i(p), i∙e.,
Pi N + μ(1 - F(Pi))N- 1
(1 - μ ) p
N
(2)
Solving this equation for the price distribution yields
F(P) = 1 -
μ (1 - μ)(p - p) ∖ *-1
∖ Nμp . ’
(3)