Tikz: wrong intersection with axis

Question

In the following code the dashed line that intersects with the y-axis is out of place and I don't know where I'm making the mistake (if there is any):

\documentclass{article}
\usepackage{pgfplots}
\usetikzlibrary{intersections,calc}

\begin{document}

\begin{tikzpicture}
    \draw [<->] (0,5) node (ejey) [left]{$\omega$} |- (5.5,0)  node (ejex)
    [right]{$l$};
    \draw [name path = ls] (0.5,0.5) coordinate (a) -- (4.5,4.5) node
    (b) [right] {$l^{s}(\omega, r)$};
    \draw [name path = ld] (0.5,4.5) coordinate (c) -- (4.5,0.5) node
    (d) [right] {$l^{d}(\omega)$};
    \path [name intersections={of=ls and ld, by=i}];
    \draw [dashed] (ejey |- i) node [left] {$\omega_{0}^{*}$} -| (i |- ejex)
    node [below] {$l_{0}^{*}$};
\end{tikzpicture}

\end{document}

Answer 1

The problem appears because your dashed line starts at the x-coordinate of the centre of the ejey node, that is at the centre of the omega. To make the line start flush with the axis, you can use

(ejey.east |- i)

instead of

(ejey |- i)

\documentclass{article}
\usepackage{pgfplots}
\usetikzlibrary{intersections,calc}

\begin{document}

\begin{tikzpicture}
    \draw [<->] (0,5) node (ejey) [left]{$\omega$} |- (5.5,0)  node (ejex) [right]{$l$};
    \draw [name path = ls] (0.5,0.5) coordinate (a) -- (4.5,4.5) node (b) [right] {$l^{s}(\omega, r)$};
    \draw [name path = ld] (0.5,4.5) coordinate (c) -- (4.5,0.5) node (d) [right] {$l^{d}(\omega)$};
    \path [name intersections={of=ls and ld, by=i}];
    \draw [dashed] (ejey.east |- i) node [left] {$\omega_{0}^{*}$} -| (i |- ejex) node [below] {$l_{0}^{*}$};
\end{tikzpicture}

\end{document}

Answer 2

\documentclass{standalone}
\usepackage{pgfplots}
\usetikzlibrary{intersections,calc}

\begin{document}

\begin{tikzpicture}
    \draw [<->] (0,5) coordinate (ejey) |- (5.5,0) coordinate (ejex);
    \node [left] at (ejey) {$\omega$};
    \node [right] at (ejex)  {$l$};
    \draw [name path = ls] (0.5,0.5) coordinate (a) -- (4.5,4.5) node    (b) [right] {$l^{s}(\omega, r)$};
    \draw [name path = ld] (0.5,4.5) coordinate (c) -- (4.5,0.5) node    (d) [right] {$l^{d}(\omega)$};
    \path [name intersections={of=ls and ld, by=i}];
    \draw [dashed] (ejey |- i) node [left] {$\omega_{0}^{*}$} -| (i |- ejex)    node [below] {$l_{0}^{*}$};
\end{tikzpicture}

\end{document}