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Consumers search to minimize the expected cost of the good, i.e. the
sum of the total search costs plus the expectde price. Given the
distribution of prices given by $F^q(p)$ the expected price paid by
a consumer who observes one price is $\int^\tilde{p}_0pdF^q(p)$
while the expected price of a consumer searching two times is
$2\int^\tilde{p}_0p(1-F^q(p))dF^q(p)$. Thus, the expected gain from
observing two prices instead of one is given by
\begin{displaymath}\begin{array}{rcl}
V&=&\int^\tilde{p}_0pdF^q(p)-2\int^\tilde{p}_0p(1-F^q(p))dF^q(p)\\
&=&-\int^\tilde{p}_0pf^q(p)dp-2\int^\tilde{p}_0pF^q(p)f^q(p)dp
\end{array}
\end{displaymath}
Integration by parts yields
\begin{displaymath}\begin{array}{rcl}
V&=&-\left[pF(p)\right]^\tilde{p}_0+\int^\tilde{p}_0F(p)dp+\left[pF(p)^2\right]^\tilde{p}_0-\int^\tilde{p}_0F(p)^2dp\\
&=&\int^\tilde{p}_0F(p)dp+-\int^\tilde{p}_0F(p)^2dp\\
\end{array}
\end{displaymath}
as $\left[pF(p)\right]^\tilde{p}_0=\left[pF(p)^2\right]^\tilde{p}_0$
since $F(0)=0$ and $F(\tilde{p})=1$. We use that
\begin{displaymath}\begin{array}{rcl}
\int^\tilde{p}_0pF^q(p)f^q(p)dp&=&\left[pF(p)^2\right]^\tilde{p}_0-\int^\tilde{p}_0(F(p)+pf(p))F(p)dp textrm{ (by integration by parts)}\\
&=&\left[pF(p)^2\right]^\tilde{p}_0-\int^\tilde{p}_0F(p)^2dp-\int^\tilde{p}_0pf(p)F(p)dp
\end{array}
\end{displaymath}
such that
\begin{displaymath}
2\int^\tilde{p}_0pF^q(p)f^q(p)dp=\left[pF(p)^2\right]^\tilde{p}_0-\int^\tilde{p}_0F(p)^2dp
\end{displaymath}
A consumer will choose to search twice if $V(q)>c and will be
indifferent between observing one or two prices only if $V(q)=c$. As
argued above no equilibrium exists where all consumers search twice,
hence, in equilibrium prices will be distributed such that $V(q)=c$.
\subsubsection{Market Equilibrium with Noisy Sequential Search}
Sequential search is when a consumer pays $c$ to observe a price
whereafter he chooses whether to shop at the lowest price observed
to date or whether to search again. When the search is noisy the
consumer observes an unknown number of prices each time he search.
Although the consumer does not know how many prices he will observe
he knows that $k$ prices are observed with probability $Q_k$, $k=1,
2, ....,$ and $\sum^\infty_{k=1}Q_k=1$.
A consumer will search as long as the cost from searching is lower
than the expected gain from searching another time. Let $z$ be the
price where the consumer is indifferent between searching again and
accepting the current lowest price. If the lowest price a consumer
has observed is higher than $z$, the consumer will search
again.\footnote{Naturally, $z\leq\tilde{p}$ before any market can
exist} That is, $z$ becomes the effective reservation price. Hence,
no firm will set prices higher than $z$ in equilibrium since there
will be no sale here. As a consequence, no consumer will search more
than once, and, hence, a share $q_1$ of the consumers will know only
one price. This is the necessary condition for the exsistence of a
dispersed price equilibrium. The $q_1$ consumers will be uninformed
about prices while the rest of the consumers will know two or more
prices, and the equilibrium will be similar to the one derived in
section \ref{BJ_nonseq}. If $q_1=1$ the monopoly price equilibrium
is the only possible equilibrium, since all consumers will be
uninformed. As all firms charge the monopoly price, no consumer is
tempted to engage in a second search. If $q_1=0$ all consumers know
at least two prices and the only possible equilibrium is the
competitive equilibrium. Note that since $q_1$ is a parameter and
not generetad endogenously when search is sequential, a firm can not
raise its price without loosing all customers.\footnote{With
nonsequential search we saw that there consumers would only search
once if all firms were charging the competitive price. This made it
possible for firms to increase prices slightly to make a positive
profit.} For $0