Economics 1800: The Economics of Cities
Denise DiPasquale and Edward Glaeser
Problem Set 1: Understanding the Alonso-Muth-Mills Model
Due: 5PM February 15, 2019 on Canvas (NO LATE ASSIGNMENTS ACCEPTED)
General Set-Up
Assume that any fixed cost translates into an annual rental cost at a rate of 10% percent. For example, if it costs $100,000 to build a house, then it will cost $10,000 per year to service the debt and capital requirements for that building. Throughout the problem, use this rule to convert all flat numbers (like cost of building) into an annual cost.
Also assume that the annual cost of commuting “M” miles (remember there are 5280 feet per mile) to the city center is $5,000*M. Everyone works in an infinitely tall skyscraper at the city center that takes up no land. Each household has the same number of commuters. The city is a circle.
The annual value of agricultural land is $2,000 per lot. Every house costs $100,000 to build and sits on a lot of 4,000 square feet (that’s about 1/10 of an acre).
There are numerical answers to almost all of these questions, but it may be helpful to use graphs and figures to illustrate your results.
Part A (Closed City):
In this case, first assume that the population of the town is fixed at 100,000 households.
- How far will the city’s boundary extend assuming that the city is just a circle?
- Derive the annual cost of housing at the city center and at every distance M from the city. Separate that cost into land cost and structure cost.
- How will housing costs change at different point in the city if a new highway drops the price of commuting to $2,000*M?
- Who will benefit from the investment in the highway? How would you assess whether the benefits created by the highway have covered its costs? (The really ambitious could attempt to actually provide numerical answers here – but full credit will be given for reasonable verbal answers with good intuition).
- How will housing costs change if construction costs rise to $150,000 (assuming commuting costs are still $5,000*M)?
- How will those housing costs, and the edge of the city, change if land use regulations require that every house sits on 10,000 square feet (about ¼ of an acre)? Assume that construction costs are $100,000 and commuting costs are $5,000*M.
Part B (Open City):
Now, you drop the assumption that the population is fixed at 100,000. Assume that wages in the city equal $40,000 per year, and that every urban resident has the option to live in a different place, which offers zero commute times, housing costs of $10,000 annually and wages of $30,000 per year.
- What will housing costs be at the center of the city?
- What will housing costs be at a distance D from the city center?
- Using your answers to (1) and (2), calculate the size of the city and its overall population level.
- Now rederive (1)-(3) assuming that wages in the city have risen from $40,000 to $50,000 per year.
- Define “unaffordable housing” as people paying 40 percent or more of their total incomes on housing cost. What will happen to the share of people who have unaffordable housing as the income levels increase (this requires a mathematical answer)? What does the change in affordability mean for the welfare of the city (this can be answered either mathematically or verbally, and full credit is given for answers with reasonable intuition)?
- Now rederive (1)-(3) assuming that wages are still $40,000, but commuting costs have dropped to $2,000*M because of a highway.
- In this case, who benefits from the highway? How would you calculate whether the highway’s benefits exceed its costs? (Verbal answers are again sufficient for full credit, but the ambitious can do this with algebra).
Part C: Variable Density
Now assume that developers can build 1 home on 4,000 square feet for $100,000 and up to nine homes on 4,000 square feet for $2,500,000.
Keep all the basic assumptions of part B (commuting costs are $5,000*M and agricultural land is worth $2,000 annually), except please change the assumption on income, so that income in the city equals $50,000.
OPEN CITY:
- Calculate the minimum price for housing at which developers choose to build up.
- Calculate the price of housing at the city center.
- Calculate the willingness to pay for housing at every point D away from the city center (this should be just like Part B!). Use this linear function and your answer to (1) to calculate where tall buildings will be built. Recall that fixed costs are translated into rental costs at a rate of 10%.
- Assume that income in the city increases to $60,000. Recalculate (2) and (3).
- How does the ability to build up impact the city’s physical size and total population in this model? (This requires a purely verbal answer).
CLOSED CITY: Now assume that the city population is fixed at 1,000,000, and that commuting costs are 5,000*M.
- Calculate the price at the center of the city assuming that the edge is at a distance R and low-rise housing is built at the city’s edge.
- Calculate the prices closer to the city center. How far in from the city’s center should you start building skyscrapers, using the price you calculated in (6)?
- Calculate the total population as a function of the edge of the city. Start by figuring out the population implied by R if everyone lived in single family houses. Then add in the extra units that are delivered by the tall buildings in the parts of the city where you have calculated that it is optimal to build up (you are using your answer to 7 to do this). Set this function that gives population as a function of R equal to 1,000,000 and put it into the form , where A, B, and C are numbers that you have found. Then find the value of R using the quadratic formula ().
- Now that you have R, describe where skyscrapers will be built and give the equations for prices everywhere in the city.
- Discuss how the skyscraper changed the prices and population size in the closed city and compare with the open city. (This requires a purely verbal answer).