Thanks, Gianluca! Let me clarify my goal by referring to the following graph:
Each curve is a plot of
$e$ that maximizes
$v$ as a function of
$w$ for a given
$a$, and the curve shifts rightward with an increase in
$a$. (Typo in the graph:
$s$ should be
$a$.) I would like to find an equilibrium pair of
$(e,w)$ for different level of
$a$, and that is indicated by the dot in each curve. And the equilibrium condition is
$\frac{\partial e}{\partial w} = \frac{e}{w}$, i.e. the point where the ray from the origin meets the curve tangentially.
I'm also attaching a Notebook file which contains my previous code where I used f[...] = D[v,e] instead of f[...] = ArgMax[v,e] and hence what's inside FindRoot[] was also different. As you can see in the file, this code did generate a result. But I was dubious of the shape of the plots obtained. And I was suspecting D[v,e] could be the problem since in this code, only the first-order condition, but not the second-order condition, was verified; hence we cannot guarantee whether it is maximum or minimum. That is why I have changed over to ArgMax[v,e], which however does not yield an outcome at all in the first place. I hope this clarifies what my goal is. If not, please let me know.
Thank you so much for taking time to help!
Attachments:
