Exercise 1: In this exercise, you will analyze building height variation in a city where buildings of different ages coexist. As in the example in Chapter 3, suppose that the city grows outward by one block each year. Block 0 is built in year 0, block 1 is built in year 1, block 2 is built in year 2, etc. Suppose also that buildings are torn down and replaced after 4 years of life. Buildings built in year 0 are replaced in year 4, buildings built in year 1 are replaced in year 5, etc.
a) Derive the city's building-age pattern in year 11, when it has a radius of 12 blocks (0-11). Once the age pattern is derived, write down the year in which buildings in each block were built.
b) Suppose that the height of buildings depends on their location and their construction date. For a given construction date, buildings farther from the CBD are shorter. At a given location, buildings built later in time are taller. Let S equal the height of a building measured in stories, and suppose that S = 5T - 2x, where T is the construction date (the number of the year in which the building was built) and x is the number of the block in which the building is located. Using this formula and the results of part (a), compute building heights in each block of the city in year 11. Graph your results.
c) Now suppose that building heights are determined by a different formula, namely, S = 5T - 5x + 10. Redo part (b) for this case.
d) Contrast the patterns shown in your graphs to the building height pattern predicted by the model from Chapter 2, where building height is continually adjusted in response to changing conditions.
144A: Urban Economics
Exercise 2: In this problem, you will solve for the equilibrium population of an idealized city experiencing rural-urban migration, following the augmented Harris-Todaro model from Chapter 3. The incomes earned in urban employment and in the rural area are y and yA, respectively, and t is commuting cost per mile. J is the number of available urban jobs.
a) Suppose the city is a rectangle 10 blocks wide with the employment center at one end (it's an island like Manhattan). The city spreads out along the length of the island to accommodate its population, with its edge located x? blocks from the employment center. Compute the city's land area in square blocks as a function of x?. Assuming that each urban resident consumes 0.001 square blocks of land, compute the amount of land needed to house the city's population L. Set the resulting expression equal to the city's land area, and solve for x? in terms of L.
b) With J jobs in the city, the chance of a resident getting one of the jobs is J/L, which makes the expected income of a city resident equal to y(J/L). The expected disposable income net of a commuting cost for a resident living at the city's boundary is then y(J/L) - tx?. The rural-urban migration equilibrium is achieved when this disposable income equals the rural income as explained in Chapter 3. Write down this equation, and substitute your solution for x? in terms of L from part (a). Then multiply through by L to get a quadratic equation that determines L.
c) Suppose that y = 10,000, yA = 2000, t = 100, and J = 30,000. Substitute these values into your equation from part (b), and use the quadratic formula to solve for L (it's the positive root). The answer gives the city's equilibrium population.
d) In equilibrium, what is the chance of getting an urban job? What is the implied unemployment rate in the city?
e) What is the distance to the edge of the city? How much does a resident living at the edge spend on commuting?
f) Suppose that y rises to 12,000. Repeat the calculations in parts (c), (d), and (e). Given an intuitive explanation of the changes in your answers to (c) and (d)