Question

Externalities II [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 S=60 miles per hour when N<500, and S=30000/N mph when N≥500. Notice as N increases above 500 the speed drops. But when N is less than 500 there is no impact on speed of an extra driver.

a.) How much time does it take to travel 1 mile as a function of N? Use the speed curve. Calculate this for both N<500 and N≥500.

b.) If the opportunity cost of sitting in traffic (W) is $10 per hour, what is the average opportunity cost (AC) of driving one mile as a function of N, when N<500 and N≥500.

c.) Draw a graph with $ on the y-axis and N on the X-axis. Plot the AC of driving one mile.

The average cost is the opportunity cost the driver perceives for traveling one mile. It is effectively the marginal private cost to the driver. (That is, what we have called AC is equivalent to the MPC.) Drivers do not generally appreciate that they are part of N, and that they lower the speed/raise the costs for everyone else. While the effect on speed from one more driver may be small, it isn’t zero. To figure out the marginal social cost from driving, we first calculate the total social cost.

d.) Derive the formula for the total cost to society if there are N drivers. Determine this for both N<500 and N≥500.

e.) What is the marginal social cost (MSC) of one more driver when N<500? Remember, the marginal social cost is the change in the total social cost divided by the change in N. A little calculus (which you don’t have to do) will show that when N≥500, the MSC is 2*MPC.

f.) Plot the MSC on your graph. (Note that at N=500 the MSC curve will not be continuous.)

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 MSBP =.75-(1/6000)N. Let MSBO = .5-(1/1000)N. I’ve selected these numbers so that the results work out reasonably cleanly.

g.) 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?

h.) 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 (NM)? What is the MSB at this level of traffic during time P?

i.) What is the difference between the market level of traffic (NM) and the optimal level (N*)? What is the deadweight loss per mile from too much traffic?

j.) 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.)

k.) 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.

Answer 1

a.) When N<500, time taken to travel 1 mile is given by 0.016 hrs. When N>=500 time taken to travel 1 mile is given by N/30000 hrs. It is a function of N. The answer is arrived at using the formula for speed given by S=Distance/Time.

b.) The average opportunity cost of driving 1 mile when N<500 is given by $0.16. It comes to 10N/30000 when N exceeds 500.

d.) The formula for the total cost to the society when there are N drivers is given by 0.16N when N <500 and 10N^2/30,000 when N >=500

e.)The marginal social cost of one more driver when N<500 is given by 0.16. This is arrived at by taking the derivative of Total cost with respect to N. The same figure when N>=500 comes to 20N/30000. The working formula for this is given by dTC/dN