documentclass[11pt,a4paper]{article}
usepackage{geometry}
usepackage{tikz}
usepackage{pgfplots, relsize}
usepgfplotslibrary{polar}
usepgflibrary{shapes.geometric}
usetikzlibrary{calc}

usetikzlibrary{arrows,decorations.markings}
begin{document}

begin{center}
begin{tikzpicture}
  begin{axis}[
  unit vector ratio*=1 1 1,
   grid = none,
    axis x line = middle,
    axis y line = middle,
    xlabel={$k_t$},
    xlabel style={at=(current axis.right of origin), anchor=west},
    ylabel={$k_{t+1}$},
    ylabel style={at=(current axis.above origin), anchor=south},
    xmin = 0,
    xmax = 15,
    scale=1.5,
    enlarge y limits={rel=0.13},
    enlarge x limits={rel=0.07},
    ymin = 0,
    ymax = 10,  xtick=empty,ytick=empty,
    after end axis/.code={path (axis cs:0,0) node [anchor=north west] {0};},
  ]
    coordinate (0,0) ;
    % domain can be set individually!!
    addplot[color=red, domain=0:10,samples=100,smooth,thick] {x} node[above,pos=1] {$45^circ$};
    addplot[color=blue,domain=0:11,samples=1000,smooth, thick] {3*sqrt(x)} node[right,pos=1] {$W(s,k_{t},delta)$};
    addplot[color=Cerulean,domain=0:11,samples=1000,smooth, thick] {2*sqrt(x)} node[right,pos=1] {$W(s',k_{t},delta)$};

    addplot[mark=*,only marks, fill=white] coordinates {(9,9)} node[above, pos=1]{$A$};
    addplot[mark=*,only marks, fill=white] coordinates {(4,4)} node[above, pos=1]{$A'$};

    addplot[mark=*,only marks, fill=white] coordinates {(9,6)} node[below, pos=1]{$A_1$};
    addplot[mark=*,only marks, fill=white] coordinates {(6,6)} node[above, pos=1]{$A_2$};
    addplot[mark=*,only marks, fill=white] coordinates {(6,4.899)} node[below, pos=1]{$A_3$};
    addplot[mark=*,only marks, fill=white] coordinates {(4.899,4.899)} node[above, pos=1]{$A_4$};

    % NO output shown at all
    node (A) at (9, 9){};
    node (Ap) at (4, 4){};
    node (A1) at (9, 6){};
    node (A2) at (6, 6){};
    node (A3) at (6, 4.899){};
    node (A4) at (4.899, 4.899){};
    draw [dashed] (A) -- (A1) -- (A2) -- (A3) -- (A4);
  end{axis}

end{tikzpicture}
end{center}
end{document}