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Consider an implicit function, $F(e,w,a,b,i,n)=0$, whose code is:

D[w - 1/((1 - e) w), e] ==  D[1 - e, e] ((-((-a (i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) + (-1 + a^2) (-1 + e)^2 w - a (-1 + e) (i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w^2)/(a (-1 + e) i (1 - e + i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w))) - (-((-a (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1))) + (-1 + a^2) (-1 +   e) (-1 + e - i) w -  a (-1 + e) (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
    1 - n e (1/(b (a e)^(b)))^(1/(b - 1))) w^2)/( a (-1 + e) i (1 - e + i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w))))

From the above implicit function, $e$ is implicitly defined as a function of $w$, and I would like to find the values of $e$ and $w$ as the solution of $\frac{\partial e}{\partial w}=\frac{e}{w}$. And my ultimate goal is to plot each of these values of $e$ and $w$ against $a$.

First, I found the $e$ function, $e=e(w)$ numerically for various values of $a$: $a = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1$ along with the other simulation values of $n=1$, $b=0.7$, $i=0.1$. My code for this is:

Clear["Global`*"]
n = 1;
eqns[i_, b_] = {D[w - 1/((1 - e) w), e] == 
D[1 - e, e] ((-((-a (i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
         1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) + (-1 + 
         a^2) (-1 + e)^2 w - 
      a (-1 + e) (i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
         1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w^2)/(
     a (-1 + e) i (1 - e + i + (
        n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
        1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w))) - (-((-a (
       n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
       1 - n e (1/(b (a e)^(b)))^(1/(b - 1))) + (-1 + a^2) (-1 + 
         e) (-1 + e - i) w - 
      a (-1 + e) (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
       1 - n e (1/(b (a e)^(b)))^(1/(b - 1))) w^2)/(
     a (-1 + e) i (1 - e + i + (
        n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(
        1 - n e (1/(b (a e)^(b)))^(1/(b - 1)))) w)))), 0 <= i <= 1, 0 <= a <= 1, 0 <= e <= 1, w >= 0, 0 <= b <= 1};
ContourPlot[Evaluate@ Table[Simplify@    eqns[1/10, 7/10][[1]], {a, {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}}], {w, 0, 5}, {e, 0, 1}, PlotPoints -> 50, MaxRecursion -> 4, FrameLabel -> (Style[#, 14] & /@ {w, e}), RotateLabel -> False, PlotLegends -> Placed[LineLegend[{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}, 
LegendLabel -> "a"], {.7, .4}]]

And the result is: enter image description here

Now, I would like to use this $e=e(w)$ function numerically found as above to solve $\frac{\partial e}{\partial w}=\frac{e}{w}$ and use the solutions for $e$ and $w$, which would be a function of $a$, to plot each of them against $a$. The plots I'm looking for would be something like these. Please help. enter image description here enter image description here

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1 Answer 1

1
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$Version

(* "13.2.1 for Mac OS X ARM (64-bit) (January 27, 2023)" *)

Clear["Global`*"]

n = 1;
eqns[i_, b_] = {D[w - 1/((1 - e) w), e] == 
    D[1 - e, e] ((-((-a (i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - 
                   n e (1/(b (a e)^(b)))^(1/(b - 1)))) + (-1 + 
                a^2) (-1 + e)^2 w - 
             a (-1 + e) (i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - 
                   n e (1/(b (a e)^(b)))^(1/(b - 1)))) w^2)/(a (-1 + e) i (1 -
                e + i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - 
                  n e (1/(b (a e)^(b)))^(1/(b - 1)))) w))) - (-((-a (n (1 - 
                   e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - 
                 n e (1/(b (a e)^(b)))^(1/(b - 1))) + (-1 + a^2) (-1 + 
                e) (-1 + e - i) w - 
             a (-1 + e) (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - 
                 n e (1/(b (a e)^(b)))^(1/(b - 1))) w^2)/(a (-1 + e) i (1 - 
               e + i + (n (1 - e) (1/(b (a e)^(b)))^(1/(b - 1)))/(1 - 
                  n e (1/(b (a e)^(b)))^(1/(b - 1)))) w)))), 0 <= i <= 1, 
   0 <= a <= 1, 0 <= e <= 1, w >= 0, 0 <= b <= 1};

ImplicitD was introduced in version 13.1

eqn[a_, e_, w_] = FullSimplify[ImplicitD[eqns[1/10, 7/10][[1]], e, w] == e/w];

ContourPlot[
 Evaluate@Simplify[
   Table[eqn[a, e, w], {w, 1, 5}]],
 {a, 0, 1}, {e, 0, 1},
 FrameLabel -> (Style[#, 14] & /@
    {HoldForm[a], HoldForm[e]}),
 RotateLabel -> False,
 PlotPoints -> 35,
 MaxRecursion -> 4,
 PlotLegends -> Placed[
   LineLegend[Range[1, 5],
    LegendLabel -> Style[HoldForm[w], 14]],
   {.2, .25}],
 WorkingPrecision -> 15]

enter image description here

ContourPlot[
 Evaluate@Simplify[
   Table[eqn[a, e, w], {e, 2/5, 4/5, 1/10}]],
 {a, 0, 1}, {w, 0, 5},
 FrameLabel -> (Style[#, 14] & /@
    {HoldForm[a], HoldForm[w]}),
 RotateLabel -> False,
 PlotPoints -> 35,
 MaxRecursion -> 4,
 PlotLegends ->
  LineLegend[Range[0.4, 0.8, 0.1],
   LegendLabel -> Style[HoldForm[e], 14]],
 WorkingPrecision -> 15]

enter image description here

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7
  • $\begingroup$ Thanks, Bob, for your help. But it seems my question was misunderstood. Here is a short version of what I'm looking for: Given $e=e(w)$ numerically derived as in the first figure, I would like to find $e$ and $w$ that solve $e_w = e/w$ for varying values of $a \in [0,1]$ while the rest of the parameter values are fixed. Finally, I would like to plot each of those $e$ and $w$ against $a \in [0,1]$. The plots will look something like my second and third figures. $\endgroup$
    – ppp
    Jun 28, 2023 at 2:23
  • $\begingroup$ Note: Each of the solution pairs of $e$ and $w$ corresponding to $s \in [0,1]$ would be found graphically in my first figure at a point where the ray from the origin meets at a tangent point on each of the curve therein. $\endgroup$
    – ppp
    Jun 28, 2023 at 2:27
  • $\begingroup$ Hello Bob, Thanks for your answer post. May I ask for your additional help on my follow-up comments I made above? Many thanks in advance! $\endgroup$
    – ppp
    Jun 29, 2023 at 5:46
  • $\begingroup$ Your equation and the subsequent D[e, w] == e/w have three variables, i.e., {a, e, w}. In any 2D plot, the third variable will be represented by a family of curves, not a single curve as you suggest in your plots of e vs. a and w vs. a. The ContourPlots that I show are for the equation D[e, w] == e/w. What I have shown is how I would approach the problem. $\endgroup$
    – Bob Hanlon
    Jun 29, 2023 at 19:32
  • $\begingroup$ Bob, thanks for your explanation. Let me clarify: $a$ is an exogenous parameter, while $e$ and $w$ are variables endogenously determined within the model. I'm interested in how the determination of $e$ and $w$ would depend on $a$. That is, for each level of $a \in [0,1]$, there is a unique pair of (e,w). With this relation, I would like to plot $e$ and $w$ against $a$ but separately, i.e. a plot of $e$ against $a$ and another plot of $w$ against $a$. I hope this clarifies. I would really appreciate your help once more! $\endgroup$
    – ppp
    Jul 5, 2023 at 3:42

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