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

Since F is a distribution function there must be some p for which F(p) = 0. Solving for p one
obtains the lower bound of the price distribution p = (1 — μ)p/(μN + (1 — μ)).

The price distribution (3) represents optimal firm pricing. We now turn to discuss optimal
consumer behavior. Consider a buyer who has observed a given price p. This consumer will
continue searching if the expected benefits from continued search exceed the search costs. We can
define the reservation price ρ as the price that makes a consumer indifferent between searching once
more and accepting the price at hand; this price satisfies:

(ρ —p)f(p)dp = c.

^p

No firm will charge a price above ρ since this will lead to continued search (Stahl, 1989). As a
result the upper bound p = ρ. We now derive an expression for ρ. Integrating by parts in (4) gives

ρ — E [p] — c = 0.

To calculate E[p] we solve equation (3) for p, which gives

________^ρ________

1 + bN (1 — F ) N^{-1 ,}
where b = μ/(1 — μ) > 0. We note now that E[p] = ρ — pρ F(p)dp. By changing variables we can
write E[p] = ^R₀¹ pdy. Plugging p from equation (6) gives, after rewriting,

E^[p]= ^P 0_o 1 + bNyN^-1 ∙                              ⁽⁷⁾

Equation (7) can be plugged into equation (5) to solve for ρ:

P =-----r~~—√----,                                   (8)

R R-C¹ dy                                          ^v

¹ Jo 1+bNyN^-1

We note that the reservation price ρ increases in c and in N, decreases in μ and is insensitive to v.

It must be the case that ρ ≤ v. In addition, non-shoppers must find it profitable to search once,
rather than not searching at all, i.e,

v — E[p] — c > 0∙                                       (9)

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