# Optimal taxing in case of negative externalities

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.

I would have thought the social planner might want the value of $x_i$ to be such that $f'(x_i)=k(n-1)$ at which point the marginal gain to each individual offsets the marginal total cost to the others
while individual driver $i$ might want the value of $x_i$ to be such that $f'(x_i)=t$ at which point the marginal gain offsets the marginal tax
To me that suggests the tax rate should be $t=k(n-1)$
• The social planner sees the tax collected of $tx_i$ per individual so $t \sum{x_i}$ in total. This is negative on the individuals but it can be returned as lump sums or spent on something useful, so net does not affect the social planner's view of optimal miles driven – Henry Jun 15 '17 at 8:18