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

Inspection of (5) and (9) implies that ρ < v. It is useful to rewrite condition (9) as

1 _ [¹____dy____> c.

0_o 1 + bNyN^-1 v

This equation gives the set of parameters for which an equilibrium where buyers search once for
sure exists. For future reference, let us denote the left-hand-side of equation (10) as Φ(1;μ; N).4
We note that 0 < Φ(1; μ; N) < 1 for all values of the parameters.

Proposition 1 Let 0 < v ≤ Φ(1; μ, N). Then a search equilibrium with full consumer participation
exists where buyers search for one price and firms charge prices from the set [(1 _ μ)ρ/(μN + (1 _
μ)),ρ] according to the price distribution (3) where p = ρ and ρ solves equation (8).

From (5) and (9) and the continuity of the expression of the expected price, it follows that
the reservation price ρ converges to v when c/v approaches Φ(1; μ, N). The question then arises
what happens when search costs are high, in particular when C > Φ(1; μ,N)? In what follows we
show that an equilibrium with partial consumer participation arises. This equilibrium is new in
the sequential search literature and its properties are interesting.

Case b: Equilibrium with partial consumer participation

Now assume that non-shoppers randomize between searching for one price quotation and not search-
ing at all, i.e., θ0 > 0, θ0 + θ1 = 1. The expected payoff to firm i is

∏i(Pi, F(Pi)) = Pi (1 N)θ¹ + μ(1 _ F(Pi))N ¹

The economic interpretation of equation (11) is analogous to that of equation (1), except that there
are now (1 _ μ)θ ₁ non-shoppers active, rather than 1 _ μ. A similar analysis as above yields the
following equilibrium price distribution:
with support [p,p] where p = (1 _ μ)θ ₁ p/(μN + (1 _ μ)θ ₁). We now notice that the upper bound is
no longer equal to ρ, but equal to v . To see this, note that non-shoppers should now be indifferent

⁴The number 1 in the arguments of Φ(∙) stands for θ ₁=1.

<<   5   6   7   8   9   10   11   >>

More intriguing information