% \topmargin -.4in
% \oddsidemargin .25in
\documentclass[12pt]{article}
\usepackage{verbatim}
\newcommand{\comments}[1]{}
\renewcommand\floatpagefraction{.9}
\renewcommand\topfraction{.9}
\renewcommand\bottomfraction{.9}
\renewcommand\textfraction{.1}
\setcounter{totalnumber}{50}
\setcounter{topnumber}{50}
\setcounter{bottomnumber}{50}
\usepackage{hyperref} \hypersetup{breaklinks=true, pagecolor=white,
colorlinks=
true, linkcolor=black, hyperfootnotes= false, urlcolor=blue } \urlstyle{
rm}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amssymb}
\oddsidemargin -.5in
\textheight 10in
\textwidth 7in
\parskip 10pt
\reversemarginpar
\begin{document}
\parindent 24pt \parskip 10pt
\titlepage
\vspace*{-1in} \noindent \textbf{The Parking Lot Problem}
\noindent
13 November 2007. Maria Arbatskaya, Kaushik Mukhopadhaya, and Eric
Rasmusen.\\
marbats@emory.edu,kmukhop@emory.edu, Erasmuse@Indiana.edu,
http://www.rasmusen.org.
\noindent \textit{Assumptions: } $N$ drivers all have the same
preferred
arrival time, $t= T$ in a parking lot of size $K$. The value of
finding a
spot is $v$, and a player has cost $w$ per period for arriving early,
so the
loss from arriving early is $L(t) = w(T-t)$. All the players arriving
in the
period $t^{\prime}$ when the lot fills have an equal probability $p$
of
getting a spot. The cost of increasing the parking
lot
size is $c$ per spot with linear costs (or $C(K)$ more generally).
In the first best, if $N>K$ all $N$ players arrive at $T$ and $K$ of
them park in the parking lot.
Define the \textit{indifference arrival time} as $t^{*}\equiv T-v/w. $,
so a player parking then has a payoff of zero. Assume $t^{*}$ and $T$
are
even multiples of $\Delta$ so they are feasible.
\noindent
\textit{Full Observability:} A player observes all arrivals up through
period $t-\Delta$ before he makes his own decision at $t$.
\noindent \textit{Unobservability:} No player observes any other
player's
arrival.
\noindent\textbf{Proposition 1. } Under either full observability or
unobservability, as the time grid becomes infinitely fine and there
are more
drivers than parking spots ($N>K$),the players fully dissipate rents
from
the parking lot in any equilibrium.
In other words, if you build a parking lot slightly too small, you
would do
better not to build one at all.
\includegraphics[height=140mm]{parking2.pdf}
\end{document}