Suppose an individual $i$ has the utility function
$U= f(x(i)) - k$(sum of all $x$ with index not equal to $i$)
Where $x(i)$ denotes the miles driven by $i$, and $k$ is a positive constant.
The function $f$ is such that $f(0)=0$, and has a negative double derivative. It crosses the x axis once after increasing at $x=0$.
We're asked, that if a social planner taxes individuals on basis of $t$ dollars per mile driven, how will such an optimal tax change as we increase the number of individuals? It is given that the planner maximizes sum of all utilities.
I started out by finding the best response of individual $i$, which is $f'(x)=t$. Hence every person has the same best response, so miles driven by everyone are the same. Hence maximising utility of any person will suffice.
Max. $f(x) - f'(x)*x - k(n-1)x$
Which gives me $x=(n-1)k/f''(x)$
I don't know how to proceed further.