High levels of automobile traffic in big cities are incredibly costly to society

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Externalities II (33 points) [Warning, this problem is an enhanced version of a negative
externality problem. While I break it down into a series of short steps, it still may take a
long time to figure out. Budget accordingly. Don’t skimp on your graph. Your picture will
hopefully help you understand what is going on.]
High levels of automobile traffic in big cities are incredibly costly to society. Time spent
idling in a car is time that could probably have been spent on more fruitful activities.
Multiply these costs by tens/hundreds of thousands of drivers per day and you’ve got
huge numbers. Drivers likely take into account the cost of sitting in traffic when they
decide to drive. But they do not likely take into account the fact that by driving they are
adding to the traffic congestion and slowing others down. That is, driving produces an
external cost on society, in the form of higher traffic.
The idea of “congestion pricing” is to charge drivers when they are contributing to the
traffic externalities, so as to internalize the external cost of driving. With modern
technology, having a device in your car that tracks your driving and charges you the
applicable amounts is becoming more and more feasible, and has been implemented in a
few urban areas around the world. This problem attempts to walk you through a way of
modeling congestion pricing in an effort to raise social wellbeing.
During times of low traffic, there may not be much, or any, external cost. So we should try
to incorporate that feature into our plan to charge drivers. This introduces a non-linear
feature into the model, which complicates things.

The “speed curve” gives the speed of traffic (S) as a function of the number of drivers (N)
at a time.
On certain stretch of highway, let miles per hour when N

The benefits of driving changes based on the time of day. During Peak (P) times of day
(morning or afternoon rush hour) the marginal social benefit is higher than at Off-Peak
times (O).
Let MSB​ P​ =.75-(1/6000)N. Let MSB​ O​ = .5-(1/1000)N. I’ve selected these numbers so that the
results work out reasonably cleanly.
Calculate the socially optimal level of traffic (N*) during P and O times of day. What is the
value of the MSB at the optimal level during these times?
Hopefully you recognized in the last part that during time O the socially optimal level and
the market level will be the same. During time P, what will the market level of traffic be
(N​ M​ )? What is the MSB at this level of traffic during time P?
What is the difference between the market level of traffic (N​ M​ ) and the optimal level (N*)?
What is the deadweight loss per mile from too much traffic?
What should the “congestion price” be per mile during time period P? (The congestion
price is effectively a per unit tax that brings the MPC and MSC into alignment at the
optimal level of traffic.)
Are there any unintended consequences possible from imposing a congestion price
during peak times? (Hint: remember what substitute goods are?) Write a paragraph of at
least 100 words clearly describing your thoughts.