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.


2 Answers 2


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 $t=k(n-1)$

  • $\begingroup$ Would we not take the disutility of tax into account? We're at the same solution, only difference is I assumed the dis-utility of tax to be -t*x and added that into the first equation you wrote $\endgroup$
    – Arshdeep
    Jun 15, 2017 at 7:49
  • $\begingroup$ 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 $\endgroup$
    – Henry
    Jun 15, 2017 at 8:18

I don't think you really need any of the math. The optimal Pigouvian tax on a negative externality is exactly the cost imposed on others.

Since the cost imposed on others by an individual is $k*x(i)*(n-1)$, that should be the tax. IE the tax should be $k*(n-1)$ per mile. The utility individual receive from driving miles has no bearing on the optimal tax. Doing this will put pressure on each individual to only drive the number of miles such that $f'(x_i)>k*(n-1)$

You can see that this will maximize any utility function of the form you described for any number of individuals. Its this simple because the negative utility others receive from others driving is the same for all individuals.

  • $\begingroup$ Actually, there are $(n-1)$ others negatively impacted, so the cost imposed on others would be $(n-1)k x_i$ $\endgroup$ Jul 25, 2023 at 21:48
  • $\begingroup$ Ah yes, of course, the sum of costs over all others. I added the term $\endgroup$
    – B T
    Jul 27, 2023 at 0:18

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