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%TCIDATA{Created=Monday, June 03, 2002 16:58:05}
%TCIDATA{LastRevised=Tuesday, March 04, 2003 18:50:21}
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\begin{document}
\section{A Simple Improvement}
\bigskip This section was deleted from the primer paper.
\bigskip
Briefly consider a price decrease in alternative $1$, $p_{1}^{1}0$ and equal to $p_{1}^{o}-p_{1}^{1}$. If in group $B$,
the expenditure level required to keep the individual at his original
utility level, $u^{o}$, is $y^{o}+(p_{1}^{1}-p_{1}^{o})=\mu 0$. In terms of the required levels of expenditures,
an individual in group $C$ or $D$ will require expenditures less than $y^{o}$
and greater than $\mu $.
As with the deterioration, our expectation of the level of expenditures
required to make an individual whole can be decomposed into a number of terms%
\[
E[m]=c_{A}+c_{B}+c_{C}+c_{D}
\]%
All individuals in group $A$ require the same expenditure level, $y^{0}$, to
make them whole in the new state, so,%
\[
c_{A}=\Pr (\text{in }A:y^{o})y^{o}
\]%
where\footnote{%
Note that this probability is calculated at the new, lower price, for $p_{1}$%
. That is, the probability that an individual will not choose alternative $1$
at either price is the probability that they will not choose it at the lower
price.Contrast this with Equation \ref{2-11}, which was for a price increase.%
}
\begin{equation}
\Pr (\text{in }%
A:y^{o})=1-P(1:y^{o},y^{o},y^{o},p_{1}^{1},p_{2}^{o},p_{3}^{o})
\end{equation}%
Likewise, all individuals in group $B$ require the same expenditure level, $%
\mu $, to make them whole in the new state, so%
\begin{eqnarray*}
c_{B} &=&\Pr (\text{in }B\text{: }\mu )\mu \\
&=&P(1:\mu ,y^{o},y^{o},p_{1}^{1},p_{2}^{o},p_{3}^{o})
\end{eqnarray*}%
\bigskip If the individual chooses an alternative at the old price, he will
continue to choose the alternative after the price has decreased.\footnote{%
Note that $P(1:\mu
,y^{o},y^{o},p_{1}^{1},p_{2}^{o},p_{3}^{o})=P(1:y^{o},y^{o},y^{o},p_{1}^{o},p_{2}^{o},p_{3}^{o})
$ and is the probability that the individual chooses alternative $1$ at both
its old and new lower price.}
For group $C$,
\begin{equation}
c_{C}=-\int_{>\mu }^{\mu }^{0
\end{equation}%
which is positive, as required, but closer to zero than to $\$2$ because $%
58\%$ have a $cv$ of zero and only $12.5\%$ have a $cv$ of $\$2$.
\end{document}