- Suppose driverless cars increase vehicle-kilometers travelled (VKT) by 30% while no new roads are built and no existing roads are widened. In the simplest model of traffic accidents that we discussed in class, what will the percentage change in accidents be if driverless cars are no skillful at avoiding accidents than today’s drivers are? (The answer is a percent.)
- Good City and Bad City are identical in all respects, except that Good City has great radio stations to listen to while driving to and from work and Bad City has horrible ones. Everybody listens to radio while driving to and from work. Each city has an identical bottleneck with no tolls. How do the cities differ (or not) on:
The number of people who drive to work?
Total schedule delay cost?
Total waiting cost?
Total time spent waiting?
- Georgiola consists of a large number of small cities, each of which has to have a TV station to broadcast patriotic songs and movies to its inhabitants (the signals are pretty weak). A TV station costs $100,000 a year to operate, including capital costs. There are no other public services.
Each city’s inhabitants live along a long narrow strip of land on one side of long road. Each inhabitant takes up 10 meters on this road, and produces everything needed for living on her small farm. She travels to the city center every day to exchange news and engage in recreation and look at the TV station. (Everything in the city center takes up essentially no space.) Over a year, an additional meter in distance to the city center adds $ .01 to her annual travel costs. Every city in Georgiola is optimally sized.
People spread out from the city center in both directions along the road.
Suppose driverless cars cut travel cost per meter from $.01 to $.005. How does the optimal city size change?
How does the aggregate land rent at the optimal city size change?
How does the size of cities in Georgiola change?
The population of Georgiola remains the same. How does the number of cities in Georgiola change?
- Explain in no more than 4 normal-sized sentences why the entertainment business is concentrated in only a few places like New York, Los Angeles, and Mumbai.
- Taking a VSL approach, what is the approximate mortality cost of COVID so far in the US, both in absolute dollar terms and percent of GDP? What about in China (dollar terms and percent of GDP)? In India (dollar terms and percent of GDP)? Show your work and identify your sources.
- How does mask-wearing affect the standard SIR model of a pandemic that we saw in class?
Start with the standard model, and assume that the fraction m of infected people wear masks during their encounters, and that the fraction n of susceptible people also wear masks.
Masks work the following way in encounters between infected and susceptible people: If the infected person is wearing a mask (properly) and the susceptible person is not, the probability that the susceptible person will be infected is 60% as great as it would be if both parties were maskless. If the infected person is not wearing mask and the susceptible person is wearing a mask, the probability that the susceptible person will be infected is 80% as great as it would be if both parties were maskless. If both parties are wearing masks, the probability that the susceptible person will be infected is 45% as great as it would be if both parties were maskless.
Everything is independent: whether the parties are maskless or not, whether they are susceptible or infected.
- Derive a new equation for the rate of change of infection as a function of m and n.
- Find the derivatives of the rate of change of infection in this equation with respect to m and n. What is the sign of these derivatives? Which is bigger in absolute value?
- What is the susceptibility fraction of the population at which “herd immunity” is reached when there are no masks? What is it with masks (as a function of m and n)?
- Draw a phase diagram and trace out the path in (S, I) space of a pandemic that begins with some share of the population infected and with the susceptible portion greater than that required for herd immunity. Do this first for no masks, and then for a pandemic with positive m and n that begins at the same point. Label the two paths.
The long answer part counts for 220 points. The short answer part counts for a total of 300 points. Each question in the short answer part counts for 60 points, and within questions, sub-questions are weighted equally. The total number of points on the exam is 520.