maj 172006
 

Da vi i dag var i gang med specialeskrivningen, opdagede jeg desværre, at jeg havde glemt strømforsyningen til min computer ude pÃ¥ arbejde. Løsningen var at skrive videre pÃ¥ Caspers kærestes computer i notepad. Hvis I er interesseret i at se, hvordan man skriver i WinEDT/LATEX, sÃ¥ er output fra i dag tilgængeligt hvis I trykker ‘Continue reading’. Inden kritikken hagler ned over mig, sÃ¥ vil jeg gerne understrege, at det er et absolut første udkast, hvor der er lagt meget mere vægt pÃ¥ at forstÃ¥ Burdett & Judd 1983 (som er psyco svær!) frem for at have det sproglige pÃ¥ plads!

Desuden er der 2 kodefejl i det skrevne:

  • ^\tilde{p} skulle have været ^{\tilde{p}}
  • A consumer will choose to search twice if $V(q)>c skulle naturligvis have været afsluttet med et $


Der er en del af formateringen, der muligvis ikke bliver vist korrekt i html, så den virkelig nørdede kan hente tex-filen her

For de knap så nørdede kan resultatet (efter at WinEdt har bearbejdet det) ses i denne pdf-fil

%%%%%%% Skrevet i Notesblok %%%%%%%%
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

 Posted by at 23:51

  2 Responses to “The power of notepad!”

  1. Ja, ok – der var ogsÃ¥ et par andre fejl kan jeg se i pdf-filen, men ikke noget seriøst… 😉

  2. Min løsning i den slags situationer plejer at være at tage pÃ¥ café (for at læse, selvfølgelig). LATEX-Notepad-løsningen ville i hvert fald være det sidste, der ville falde mig ind. Nørder! 😉

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