LaTeX Forum

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\begin{document}

\begin{align*}
(2) ~ c_{i} &=(1-\tau)y_{i}+G \\
(6) ~ \hat{c_{i}}&=\hat{y_{i}}=A_{i} [\alpha k_{i}+ (1-\alpha)e_{i}]  \\
\Omega &= Var(c_{i}-\hat{c}_{i}) \\
&= Var((1-\tau)y_{i}-\hat{y}_{i}) since Var[G]=0\\
&= Var ((1-\tau)y_{i} - \hat{y}_{i} +(1-\tau)\hat{y}_{i} - (1-\tau)\hat{y}_{i}) \\
&= Var(-\tau \hat{y}_{i}) + Var [(1-\tau) (y_{i}-\hat{y}_{i})] \qquad \text{since} ~ Cov(\hat{y}_{i}, y_{i}-\hat{y}_{i}) = 0 \\
(7) ~ \Omega &= \tau^2 Var(\hat{y}_{i}) + (1-\tau)^2 Var(y_{i}-\hat{y}_{i}) \\
%\end{align*}
\intertext{Derivation of equation \ref{eq:formel11} and \ref{eq:formel12}:}
%\begin{align*}
(6) ~ \hat{c_{i}} &= \hat{y_{i}}= A_{i} [\alpha k_{i}+ (1-\alpha)e_{i}] \\
(10) ~ k_{i} &= (1-\tau_{e})\beta_{i}A_{i} \text{and} e_{i}= (1-\tau)\beta_{i}A_{i}\\
\hat{y_{i}} &= A_{i} [\alpha (1-\tau_{e})\beta_{i}A_{i} + (1-\alpha)(1-\tau)\beta_{i}A_{i}] \\
\hat{y}_{i}&=A_{i}^2\beta_{i}\alpha(1-\tau_{e})+A_{i}^2 (1-\alpha)(1-\tau) \\
(11) ~ \hat{y_{i}} &=[1-\alpha \tau_{e} -(1-\alpha)- \tau] \delta_{i} \qquad \text{since} ~ \delta \equiv A_{i}^2\beta_{i} \end{align*}
\end{document}

%
\documentclass[a4paper,12pt]{scrartcl} 
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{amsmath}

\begin{align*}
(2) ~ c_{i} &=(1-\tau)y_{i}+G \\
(6) ~ \hat{c_{i}}&=\hat{y_{i}}=A_{i} [\alpha k_{i}+ (1-\alpha)e_{i}]  \\
\Omega &= Var(c_{i}-\hat{c}_{i}) \\
&= Var((1-\tau)y_{i}-\hat{y}_{i}) since Var[G]=0\\
&= Var ((1-\tau)y_{i} - \hat{y}_{i} +(1-\tau)\hat{y}_{i} - (1-\tau)\hat{y}_{i}) \\
&= Var(-\tau \hat{y}_{i}) + Var [(1-\tau) (y_{i}-\hat{y}_{i})] \qquad \text{since} ~ Cov(\hat{y}_{i}, y_{i}-\hat{y}_{i}) = 0 \\
(7) ~ \Omega &= \tau^2 Var(\hat{y}_{i}) + (1-\tau)^2 Var(y_{i}-\hat{y}_{i}) \\
\end{align}

Derivation of equation \ref{eq:formel11} and \ref{eq:formel12}:

\begin{align*}
(6) ~ \hat{c_{i}} &= \hat{y_{i}}= A_{i} [\alpha k_{i}+ (1-\alpha)e_{i}] \\
(10) ~ k_{i} &= (1-\tau_{e})\beta_{i}A_{i} \text{and} e_{i}= (1-\tau)\beta_{i}A_{i}\\
\hat{y_{i}} &= A_{i} [\alpha (1-\tau_{e})\beta_{i}A_{i} + (1-\alpha)(1-\tau)\beta_{i}A_{i}] \\
\hat{y}_{i}&=A_{i}^2\beta_{i}\alpha(1-\tau_{e})+A_{i}^2 (1-\alpha)(1-\tau) \\
(11) ~ \hat{y_{i}} &=[1-\alpha \tau_{e} -(1-\alpha)- \tau] \delta_{i} \qquad \text{since} ~ \delta \equiv A_{i}^2\beta_{i} \end{align*}