\documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Monday, June 03, 2002 16:58:05} %TCIDATA{LastRevised=Tuesday, March 04, 2003 18:50:21} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=LaTeX article (bright).cst} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \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}