In problems 1-5 you will analyze the equilibrium of a city under some special simplifying assumptions about land use. The assumptions are:

1. each household has one person who commutes to the CBD,
2. all dwellings must contain exactly 1,500 square feet of floor space, regardless of location,
3. all dwellings are in apartment buildings which must contain exactly 15,000 square feet of floor space and occupy one square block of land area, i.e., there are ten dwelling units per apartment building.

Distance is measured in blocks. These land-use restrictions, which are presumably imposed by a zoning authority, mean that dwelling sizes, building heights and density do not vary with distance to the central business district. In economic terms the zoning authority doesn't allow households to substitute between housing and bread (i.e., all other goods), nor does it allow developers to substitute between land and capital. [Note: This is a very special case, but as you will see, still, quite challenging. The more general case with substitution requires considerably more mathematics.]

1. The Size of the Closed City

Since each square block contains 15,000 square feet of housing and each apartment has 1500 square feet, each square block of the city has 10 households living on it. As a result, a city with a radius of x* blocks can accommodate 10x*² households (x*² is the area of the city in square blocks). Suppose the city has a population of 250,000 households. How big must its radius be in order to fit this population?Use a calculator and round off to the nearest block.

2. Housing Prices at the Periphery of the Closed City

Recall that the zoning law says that each developed block must contain 15,000 square feet of floor space. Suppose that the annualized cost of the building materials needed to construct this much housing is $75,000. [Note: $75,000 = iK, where K is the capital needed to build 15,000 square feet of floor space. If i= 5%, then K = $75,000/.05 = $1,500,000.] The annual profit per square block for the housing developer is equal to 15000p-75000-r, where r is land rent per square block. In equilibrium profit for the developer is just equal to zero everywhere. In the absence of any other activities, land rent at x* (call it r*) is just equal to zero. What is the value of p at x*?Call this p*.

3. The Closed City's Corn Consumption Level

Using your results from above, suppose that income per household equals y=$25,000 per year. Next suppose that the commuting cost parameter t equals $10 per block.This means that a person living ten blocks from the CBD will spend 10*10=$100 per year getting to work. The consumers' utility functions are all U (c, q), where c is the quantity of bread (at $1 per loaf) consumed by the household and q is the quantity of housing per household, i.e., 1500 sq. ft. The consumers' budget constraints are all c+pq=y-tx. Under our special assumptions this reduces to c+1500p=25000-10x. Given that a household at x* faces a budget constraint, c+1500p*=25000-10x*, what is the value of c at x*? Call this c*. Explain why in equilibrium every household, regardless of location, must be consuming c*. Notice that since q is the same throughout the city, to have an equilibrium in which U is the same throughout the city, c must be the same throughout the city and everywhere equal to c*

4. The Closed City's Housing Price Function

Substituting the value of c* in place of c into the budget constraint c+1500p=25000-10x solve for p in terms of x. The solution tells what the price per square foot must be at a given location for the household to be able to afford exactly c* worth of bread. This is the city's housing price curve. How does p vary with location?

5. The Closed City's Land Rent Function

Substitute the function for p you just calculated in (4) into the zero profit constraint for developers: 0=15000p-75000-r. Now solve for r in terms of x. What is the rent per square block at the CBD (x=0)? Plot the bid rent function for land in the city.