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[...]
  (upper left.west|-upper right.north) -- 
    node [font=\sffamily\large, align=center, left=5pt] {\Centerstack{\emph{Cond.}\\ \emph{GE}}:}
  (upper left.west|-lower.south)
[...]

\documentclass{report}
\usepackage{mathtools}% lädt amsmath
\usepackage{amssymb}% lädt amsfonts
\usepackage{relsize}
\usepackage[usestackEOL]{stackengine} 
\usepackage{empheq}
\usepackage{tikz}
\usepackage{showframe}

\usetikzlibrary{decorations.pathreplacing,tikzmark}

\begin{document}
\begin{empheq}[left=\hspace{-1cm}\Centerstack{\emph{Full}\\\emph{Endowment}\\\emph{GE}}: \empheqlbrace]{align}
  \hspace{1.6cm}\tikzmarknode{upper left}{\textit{Direct}}\,\,\mathit{PE}: 
  \Biggl\{ \mathlarger{X{ij}N} &= \mathlarger{\tikzmarknode{upper right}{\frac{Y_iE_j}{Y}} \left(\frac{t{ij}}{\Pi_iP_j}\right)^{\mkern-5mu 1-\sigma}} \label{eq:AvW2GE}\\
    \nonumber \\
    \mathlarger{\Pi_i^{1-\sigma}} &= \mathlarger{\sum_j \left(\frac{t{ij}}{P_j}\right)^{\mkern-5mu 1-\sigma} \frac{E_j}{Y}} \label{eq:AvW2GE}\\
    \nonumber\\
    \mathlarger{P_j^{1-\sigma}} &= \mathlarger{\tikzmarknode{lower}{\sum_i} \left(\frac{t{ij}}{\Pi_i}\right)^{\mkern-5mu 1-\sigma} \frac{Y_i}{Y}} \label{eq:AvW3GE} \\
    \nonumber\\
    \mathlarger{p_i} &= \mathlarger{\left(\frac{Yi}{Y}\right)^{1/1-\sigma} \frac{1}{\alpha_i \Pi_i}} \label{eq:AvW4GE} \\
    \nonumber\\
    \mathlarger{E_i} &= \mathlarger{\varphi_i Y_i} = \mathlarger{\varphi_i p_i Q_i} \label{eq:AvW5GE}
\end{empheq}
\begin{tikzpicture}[overlay, remember picture]
\draw [
  decorate, 
  decoration={brace, mirror, amplitude=5pt}, 
  very thick,
  transform canvas={xshift=-2pt}
]
  (upper left.west|-upper right.north) -- 
    node [font=\sffamily\itshape\large, align=center, left=8pt] {Cond.\\GE} 
  (upper left.west|-lower.south)
;
\end{tikzpicture}
\end{document}

\documentclass{report}
\usepackage{amsmath, amsfonts, amssymb}
\usepackage{relsize}
\usepackage[usestackEOL]{stackengine} 
\usepackage{empheq}

\begin{document}

\begin{empheq}[left=\hskip -3.5cm\Centerstack{\em Full \\ \em Endowment \\ \em GE}: \empheqlbrace]{align}
  \textit{Direct}\,\, \mathit{PE}: \Biggl\{ \mathlarger{X_{ij}^N} &= \mathlarger{\frac{Y_iE_j}{Y}\left(\frac{t_{ij}}{\Pi_iP_j}\right)^{\mkern-5mu 1-\sigma}} \label{eq:AvW2GE}\\
    \nonumber \\
    \mathlarger{\Pi_i^{1-\sigma}} &= \mathlarger{\sum_j \left(\frac{t_{ij}}{P_j}\right)^{\mkern-5mu 1-\sigma} \frac{E_j}{Y}} \label{eq:AvW2GE}\\
    \nonumber\\
    \mathlarger{P_j^{1-\sigma}} &= \mathlarger{\sum_i \left(\frac{t_{ij}}{\Pi_i}\right)^{\mkern-5mu 1-\sigma} \frac{Y_i}{Y}} \label{eq:AvW3GE} \\
    \nonumber\\
    \mathlarger{p_i} &= \mathlarger{\left(\frac{Yi}{Y}\right)^{1/1-\sigma} \frac{1}{\alpha_i \Pi_i}} \label{eq:AvW4GE} \\
    \nonumber\\
    \mathlarger{E_i} &= \mathlarger{\varphi_i Y_i} = \mathlarger{\varphi_i p_i Q_i} \label{eq:AvW5GE}
\end{empheq}

\end{document}