Heretofore in our analysis, we have assumed rational decision-making
on the part of the sellers, but an exogenous distribution of buyer
types. Given a vector of search costs 
, such that 
denotes the cost of comparing the prices of i sellers, it is also of
interest to consider buyers as rational decision-makers, thereby
giving rise to endogenous search behavior. As mentioned previously,
rational buyers estimate the commodity's price 
that would
be obtained by searching among i sellers, and select the strategy
that minimizes 
, provided that 
; otherwise, the buyer does not search and does not
participate in the marketplace.
Before studying the decision-making processes taken by individual
buyers, it is useful to analyze the distributions of prices paid by
buyers of various types and their corresponding averages at
equilibrium. Recall that a buyer who obtains i price quotes pays
the lowest of the i prices observed. (At equilibrium, the sellers'
prices never exceed v since F(v) = 1, so a buyer is always
willing to pay the lowest price.) The cumulative distribution for the
minimal values of i independent samples taken from the distribution
f(p) is given by 
. Differentiation
with respect to p yields the probability distribution:
.
The average price for the distribution 
can be expressed as follows:
where the first equality is obtained via integration by parts, and the
second depends on the observation that 
. Combining Eqs. 3, 9, and
11 would lead to an integrand expressed purely in terms of
F. Integration over the variable F (as opposed to p) is
advantageous because F can be chosen to be equispaced, as standard
numerical integration techniques require.
Fig. 3(a) depicts sample price distributions
for buyers of various types: 
, 
, and 
, when S = 20 and 
. The
dashed lines represent the average prices 
for 
as computed by Eq. 11. The blue line
labeled Search-1, which depicts the distribution ,
is identical to the green line labeled 
in
Fig. 2(b), since
. In addition, the distributions shift toward lower
values of p for those buyers who base their buying decisions on
information pertaining to more sellers.
Fig. 3(b) depicts the average buyer prices obtained by
buyers of various types, when 
is fixed at 0.2 and 
. The various values of i (i.e., buyer types) are
listed to the right of the curves. Notice that as 
increases,
the average prices paid by those buyers who perform relatively few
searches increases rather dramatically for larger values of .
This is because is fixed, which implies that the sellers' profit
surplus is similarly fixed; thus, as more and more buyers perform
extensive searches, the average prices paid by those buyers decreases,
which causes the average prices paid by the less diligent searchers to
increase. The situation is slightly different for those buyers who
perform larger searches but do not search the entire space of sellers:
e.g., i = 10 and i = 15. These buyers initially reap the benefits
of increasing the number of buyers of type 20, but eventually their
average prices increase as well.
Given a fixed portion of the population designated as buyers of type
1, Fig. 3(b) demonstrates that searching S
sellers is a superior buyer strategy to searching 1 < i < S sellers.
Thus, there is value in performing price searches: shopbots offer
added value in markets in which there exist buyers who shop at
random.
Figure 3: (a) Buyer price distributions for 20 sellers,
with 
. (b)
Average buyer prices for various buyer types;
20 sellers, 
.
Initially, we model buyer search costs following Burdett and
Judd [2], who assume costs are linear in the number
of searches; in particular, 
, where 
are, respectively, fixed and marginal costs of obtaining
price quotes. Moreover, we assume buyers are rational decision-makers
who strive to minimize overall expenditure, and who use
(as in Eq. 11) as an estimate of . Thus, an
optimal buyer strategy satisfies:
.
At equilibrium, 
, since if 
, then all buyers
perform some degree of search, in which case all sellers charge the
competitive price r (see Eqs. 6 and 7), from
which it follows that it is in fact not rational for buyers to search
at all, leading to the contradiction that 
. Now since the
buyer cost function 
is convex, it is minimized at either a
single integer value , or two consecutive integer values
and 
. Thus, at equilibrium, either , in which case
all sellers charge the monopolistic price v, or 
and
the sellers' prices are given by the distribution
f(p). 
In the case where , by substituting Eq. 6
into Eq. 11, we obtain analytic expressions for the
average prices seen by buyers of types 1 and 2:
Fig. 4(a) plots 
(i.e., Search-1)
and 
(i.e., Search-2) as a function of 
, given
that v=1 and r=0.5. The curves decrease monotonically with ,
reflecting the fact that prices decrease on average as the degree of
search increases.
Fig. 4(b) plots the marginal benefit 
of Search-2 over Search-1, which can be computed
by subtracting Eq. 13 from 12:
Suppose that all buyers are fully informed and rational,
and therefore estimate expected marginal benefit
accurately. Then, if buyers are free to choose between the strategies
Search-1 and Search-2, the buyer population reaches
equilibrium when the marginal benefit of a second price quote exactly
balances its marginal cost. Fig. 4(b) graphically
illustrates the situation in which the marginal cost 
is finite and reasonably small; 
is valued at 0.02 and
depicted by the dotted line. There are two points of intersection on
the marginal benefit and marginal cost curves in this diagram,
representing 2 of the 3 equilibria that arise in this setting. Above
the dotted line, benefit outweighs cost; thus, it is advantageous to
search and there is momentum in the rightward direction. Below the
dotted line, cost outweighs benefit, and it is therefore not desirable
to search; hence, there is momentum in the leftward direction.
Following the direction of the arrows, we observe that the filled-in
circle that falls on the curve is a stable equilibrium, while the open
circle represents an unstable equilibrium. The unstable equilibrium
represents a boundary between two basins of attraction: a buyer
population in which the initial value of is greater than this
threshold will migrate towards the equilibrium near 
, while
one in which is initially smaller than this threshold will
migrate towards . In addition, there is a second stable
equilibrium in the lower left-hand corner of the graph (indicated by a
second filled-in circle) where , the equilibrium price is the
monopolistic price v, and 
.
Figure 4: Economy of buyers of type 1 and 2.
Buyer valuation v = 1; seller cost r = 0.5.
Before closing this section, we note that expected buyer surplus per
purchase, which we denote by 
, is defined as follows:
This quantity is of particular interest in comparing overall social welfare between markets with and without shopbots.
