between searching for one price and not searching all, i.e., it must be the case that
v - E[p] - c = 0
(13)
It is obvious that conditions (5) and (13) can only hold together if p = v.
It remains to check that the distribution function specified in equation (12) with support [p,v]
is consistent with optimal search behavior of the non-shoppers. As mentioned above, it must be the
case that non-shoppers are indifferent between searching and not searching, i.e., v - E[p] - c = 0.
This condition can be rewritten as:
dy
(14)
1 + θ1i bNyN-1
For future reference, denote the left-hand-side of equation (14) as Φ(θ 1 ,μ,N). Inspection of this
function reveals that Φ(0,μ,N) = 1 and that Φ(θ 1 ,μ,N) is monotonically decreasing in θ 1 and
increasing in μ. It can also be shown that Φ(θ 1 ,μ,N) decreases in N.5 Then:
Proposition 2 Let Φ(1 ,μ,N) < C < 1. Then a search equilibrium with partial consumer partici-
pation exists where firms set prices from the set [(1 — μ)θ 1 v/(μN + (1 — μ)θ 1), v] according to the
price distribution (12) and consumers randomize between searching for one price with a probabil-
ity θ1 which solves equation (14) and with the remaining probability they stay out of the market
altogether.
Overview of equilibria
Two equilibria may arise: either consumers search for one price with probability 1, or they mix
between searching and not searching. Inspection of the equations above immediately reveals that
whether consumers search more or less in equilibrium depends on three critical model parameters:
(i) the value of the purchase compared to the search cost c/v, (ii) the number of consumers with
negligible opportunity cost of time μ, and (iii) the number of firms N. The regions of parameters
for which these equilibria exist are represented in Figure 1 (in these graphs we set N = 2). The
left graph exhibits a market with many shoppers while the right one illustrates a market with just
a few of them. The decreasing curve represents Φ(θ 1; μ, N) as a function of θ 1.
5 The proof of this statement is omitted to save space.