1. the article. "In Hong Kong. A shift from Central to Kowloon." In no more than 2-3 sentences. give a brief explanation of how this article illustrates ONE aspect of either the basic urban spatial model or the modifications to the urban spatial model.
2. Suppose that there are two income groups in a city: high-income households with income = yH and low-income households with income = yL. Assume that like the basic urban spatial modeL consumers maximize utility, which depends on the quantity of housing (q) and all other goods (c). subject to their budget constraint. Also, assume all jobs are located at the CBD.
In this case, we will assume that the cost of commuting consists of a flat amount (t) per commuting distance plus an amount that reflects the cost in lost wages from commuting - in other words. that commuting reduces the amount of time an individual has to work by some fraction of δ hours. So. total commuting costs are then δ + t times the commute distance (x).
The consumer budget constraint is an equation where income minus the cost of commuting is equal to the cost of housing plus the costs of all other goods: where the price of housing per unit is p and the price of all other goods is normalized to 1.
At some distance (R) from the CBD. the price that high-income households will pay per square fret will equal the amount that low-income households will pay.
a) How do housing prices vary as we move away from the CBD?
b) Use the budget constraints for a low-income household and a high-income household. to solve for equations for pHandpt Now. use calculus to derive the slope of each price equation in terms of x.
c) What do we know about qH and qL at x^? What do we also know about cH and cL at x^?
d) Compare the cost of commuting for high-income and low-income households at x^.
e) What assumption would be needed for high-income households to locate closer to the CBD and low-income households to locate farther away? Briefly explain.
3. This problem will look at rural-urban migration. The urban income is y and the rural income is yA. The city edge is x- blocks from the CBD and is a function of the city's population L. the cost of commuting per distance (x from the CDB) is a fixed amount as. and J is the number of available urban jobs. Just a reminder. the consumer budget constraint in the city is c + pq = y - tx and utility of all households (people in the city and outside the city) depends on housing (g) and all other goods (c).
a) Suppose that population density is constant at 10 housing units (or 10 people - since each household has one person) per square block and that x is distance from the CBD in blocks and that. as in the basic urban spatial model, the city is a circle. If the population exactly fits the city. solve for the equation where/ is a function of L.
b) What happens to the size of the city (x-) if population (L) increases? Use the equation from a): also explain using concepts we learned in class.
c) Ignoring the possibility of unemployment. write out an equation that expresses the relationship between urban net income at S and rural income if the region is in locational equilibrium and no migration will take place from rural to urban areas. Make sure you use substitute x with the equation for .1 from part a).
d) In part c). why don't we have to worry about utility functions?
e) Suppose that y=10,000, yA=2,000, and t=100. calculate the population of the city (L) when the city is in locational equilibrium. (Round to the nearest whole number).
f) With J jobs. the chance of a resident getting one of the jobs is J/L. Now, write out an equation when the region is in locational equilibrium but where rural residents consider expected net income at the city edge when deciding whether or not to migrate. Assume that an urban resident pays the commuting cost (perhaps to look for a job if he/she doesn't have one) regardless of whether he she has a job.
g) Assume we are in the world depicted in part f). If the number of jobs increases. what will happen to L and the size of the city? Explain what is happening using the concepts
4. Provide a brief (no more than 2 to 3 sentences) explanation of why buildings further away from the CBD might be taller than those near the CBD. Use the concepts from class.