Multiple graphs in TikZ with intersections and naming

You already loaded the intersections library, but then just didn't make use of it. Here one possible way to achieve the desired result. For details please have a look at the comments in the code.

% used PGFPlots v1.14
\documentclass[border=5pt]{standalone}
\usepackage{pgfplots}
    \usetikzlibrary{intersections}
    % use this `compat' level or higher so that TikZ coordinates don't have to be prefixed
    % with `axis cs:'
    \pgfplotsset{compat=1.11}
\begin{document}
\begin{tikzpicture}[
    % define a style for the dots
    dot/.style={
        draw=black,
        fill=blue!90,
        circle,
        minimum size=3pt,
        inner sep=0pt,
        solid,
    },
]
    \begin{axis}[
        xmin=-1,
        xmax=12,
        ymin=-0.5,
        ymax=5,
        axis lines=center,
        ticks=none,
        xlabel={$k_t$},
        ylabel={$f(k_t)$},
        xlabel style={below right},
        ylabel style={above left},
        % (moved common `addplot' options here)
        smooth,
        domain=0:10,
        samples=101,
        no markers,
    ]
        % to use the `intersections' library you have to give a `name path' to path that
        % should be used for intersection calculations
        \addplot+ [green,thick,name path=sqrt] {sqrt(x)}
            node [below,pos=1,color=black] {$f(k_t)$}
        ;
        \addplot+ [black,thick,name path=mod sqrt] {2/3*sqrt(x)}
            node [below,pos=1,color=black] {$s f(k_t)$}
        ;
        \addplot+ [red,thick,name path=line,samples=2]   {2/5*x}
            node [above,pos=1,black] {$\delta k_t$}
        ;

        % find the intersection of the modified sqrt function and the straight line,
        % draw a circle on it and store the coordinate
        % (here we have to use the second found intersection point, the first one is at (0,0))
        \fill [name intersections={of=mod sqrt and line}] (intersection-2) circle (1.5pt)
            coordinate (a)
        ;
        % draw an invisible vertical path at the x coordinate of the found intersection point
        \path [name path=vertical line]
            (a |- 0,\pgfkeysvalueof{/pgfplots/ymin}) -- (a |- 0,\pgfkeysvalueof{/pgfplots/ymax});
        % now find the other intersection point with the sqrt function and again store the
        % the found coordinate
        \path [name intersections={of=sqrt and vertical line}] (intersection-1) circle
            coordinate (b)
        ;
        % draw the dots (using the above defined style) and labels
        \draw [dashed]
            (b -| 0,0)      node [dot,label=left:$y^*$]  {}
            -- (b)          node [dot]                   {}
            -- (b |- 0,0)   node [dot,label=below:$k^*$] {}
        ; 
    \end{axis}
\end{tikzpicture}
\end{document}

Pure TikZ solution:

\documentclass[tikz, border=5pt]{standalone}
\usetikzlibrary{arrows.meta, intersections}

\begin{document}
    \begin{tikzpicture}[yscale=2,
         > = Stealth,
dot/.style = {circle, draw=black, solid, 
              fill=#1, % default blue!50
              minimum size=3pt, inner sep=0pt,
              node contents={}},
dot/.default = blue!50,
domain=0:5, samples=100,
                        ]
% axis
\draw[->] (-0.6,0) -- (6,0.0) node [below left] {$k_t$};
\draw[->] (0,-0.2) -- (0,2.5) node [below left] {$f(k_t)$};
\coordinate (O) at (0,0);
% curves
\draw[green,thick,name path=sqrt green] plot (\x, {sqrt(\x)})
            node [right,text=black] {$f(k_t)$};
\draw[black,thick,name path=sqrt black] plot (\x, {2*sqrt(\x)/3})
            node [right,text=black] {$s f(k_t)$};
\draw[red,thick,name path=line red] plot (\x, 2*\x/5)
            node [right,text=black] {$\delta k_t$};
% intersection sqrt black and line red
\path[name intersections={of=line red and sqrt black, by={not used,a}}]
    (a) node[dot=black];
% draw an invisible vertical path through coordinate a
\path[name path=vertical line]
            (a |- O) -- + (0,2);
% intersection sqrt green and vertical line
\path [name intersections={of=sqrt green and vertical line, by={bb}}]
    (bb) node (b) [dot=blue!50];
% draw the dashed lines snd dots on axis
\draw[dashed] (b) -- (b -| O) node [dot,label=left:$y^*$]
              (b) -- (b |- O) node [dot,label=below:$k^*$];
    \end{tikzpicture}
\end{document}

A PSTricks solution:

\documentclass{article}

\usepackage{pst-plot}
\usepackage{xfp}

% constants
\def\xA{2/3}
\def\xB{1}
\def\xC{2/5}
\def\xEnd{9.8}

\def\point{(\xA)^2/(\xC)^2}

\def\fA(#1){(\xA)*sqrt(#1)}
\def\fB(#1){(\xB)*sqrt(#1)}
\def\fC(#1){(\xC)*(#1)}

\psset{
  algebraic,
  plotstyle = curve,
  plotpoints = 500
}

\begin{document}

\begin{pspicture}%
  (-0.5,-0.5)%
  (\fpeval{\xEnd+1.15},\fpeval{max(\fA(\xEnd),\fB(\xEnd),\fC(\xEnd))+0.55})
  \pnodes(\fpeval{\point},0){A}%
         (\fpeval{\point},\fpeval{\fB(\point)}){B}%
         (0,\fpeval{\fB(\point)}){C}
  \psaxes[labels = none]{->}%
    (0,0)(-0.5,-0.5)(\fpeval{\xEnd+0.7},\fpeval{\fC(\xEnd)+0.55})
    [$k_{t}$,135][$f(k_{t})$,315]
  \psplot[linecolor = black]{0}{\xEnd}{\fA(x)}
  \uput[0](\xEnd,\fpeval{\fA(\xEnd)}){$sf(k_{t})$}
  \psplot[linecolor = green!70!black]{0}{\xEnd}{\fB(x)}
  \uput[0](\xEnd,\fpeval{\fB(\xEnd)}){$f(k_{t})$}
  \psplot[linecolor = red!70!black]{0}{\xEnd}{\fC(x)}
  \uput[0](\xEnd,\fpeval{\fC(\xEnd)}){$\delta k_{t}$}
  \psline[linestyle = dashed](A)(B)(C)
  \psdots[dotsize = 3pt 2, dotstyle = o, fillcolor = blue!60](A)(B)(C)
  \psdot(\fpeval{\point},\fpeval{\fC(\point)})
  \uput[270](A){$k^{\ast}$}
  \uput[180](C){$y^{\ast}$}
\end{pspicture}

\end{document}

All you have to do is change the values of the constants and the drawing will be adjusted automatically.