In this exercise, we will analyze the supply-demand equilibrium of a monocentric city. The standard set-up is assumed in that the city is circular, all jobs are located at a central business district (CBD) and residents commute radially to work at the CBD. Here distance is measured in blocks. The residents consume two goods: bread (c) and dwelling size (q). The price of bread is $1 and the price per square foot of housing is denoted byp.

Now for ease of calculations, we will make some special simplifying assumptions about land- use: (i) all dwellings must contain exactly 1500 square feet of floor space, so regardless of locationq= 1500, and (ii) apartment complexes must contain exactly 15,000 square feet of floor space per square block of land area, such thatQ= 15,000. These land-use restrictions, which are imposed by a zoning authority, mean that unlike the monocentric model described in the textbook, dwelling sizes and building heights do not vary with distance to the CBD.

Suppose that income per household equals $25,000 per year. It's convenient to measure money amounts in thousands of dollars, so this means thaty= 25, whereyis income. Next suppose that the commuting cost parametert= 0.01. This means that a person living 10 blocks from the CBD will spend .0110 = .1 per year (in other words, $100) getting to work.

The consumer's budget constraint isc+pq=ytx, which reduces toc+ 1500p= 25 - .01xunder the above assumptions. Since housing consumption is fixed at 1500, the only way that utilities can be equal for all urban residents is for bread consumptioncto be the same at all locations. The consumption bundle (the bread, housing combination) will then be the same at all locations, yielding equal utilities.

For c to be constant across locations, the price per square foot of housing must vary with x in a way that allows the consumer to afford a fixed amount of bread after paying his rent and his commuting cost. Letc* denote this constant level of bread consumption for each urban resident. For the moment,c* is taken as given. We'll see below, however, thatc* must take on just the right value or else the city won't be in equilibrium.

a)Substituting c* in place of c in the budget constraint c + 1500p = 25 - .01x, solve for p in terms of c* and x.The solution tells what the price per square foot must be at a given location in order for the household to be able to afford exactly c* worth of bread.How does p vary with location?

Recall that the zoning law says that each developed block must contain 15,000 square feet of floor space. Suppose that annualized cost of the building materials needed to construct this much housing is 90 (that is $90,000).

b) Profit per square block for the housing developer is equal to 15000p- 90 -r, whereris land rent per square block. In equilibrium, land rent adjusts so that this profit is identically zero. Set profit equal to zero, and solve for land rent in terms ofp. Then substitute yourp-solution from part (a) in the resulting equation. The result helps youwrite down the land rentras a function ofxandc*. How does land rent vary with location?

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 ofxblocks can fit10x2households (Note thatx2is the area of the city in square blocks; and= 3.1416).

c) Suppose the city has a population of 200,000 households.How big must its radius bexin order to fit this population?Use a calculator and round off to the nearest block.

d)In order for the city to be in equilibrium, housing developers must bid away enough land from farmers to house the population. Suppose thatc* = 15.5, which means that each household in the city consumes $15,500 worth of bread. Suppose also that farmers offer a yearly rent of $2000 per square block of land, so thatrA= 2. Substitutec* = 15.5 into the land rent function from part (b), and compute the implied boundary of the city. Using your answer to part (c),decide whether the city is big enough to house its population.If not,what is the value ofc* that leads the city to have just the right radius?

e)Using the equilibriumc* from (d) and the results of (a) and (b), write down the equation for the equilibrium land rent function. What is the rent per square block at the CBD (x= 0) and at the edge of the city?Graph the land rent functionwith rent in the vertical axis and distance x on the horizontal axis. On the same graph,plot the agricultural rentto help you clearlyindicate the city boundary.Calculate how much does a household living at the edge of the city spend on commuting.

f)Suppose that the population of the city grows to 255,000 residents. Repeat your calculation from parts (c), (d), and (e) for this case (but don't repeat the calculation involvingc* = 15.5, just use the equilibriumc*directly.). Using your calculations forx,c*,rfunction,explain your findings. What can you say about the impact of population growth on the utility level of people in the city?The answer comes from looking at the change inc* (since housing consumption is fixed at 1500 square feet, the utility change can be inferred by simply looking at the change in bread consumption). Note that because they're fixed, housing consumption doesn't fall and building heights don't rise as population increases, as happened in the model in Chapter 2.Are the effects onrandxthe same?

g)Now suppose that population is back at 200,000 (as in part (c) but thatrArises to 3 (that is, farmers now offer $3000 rent per square block). Note that, unlike in the lectures, thexvalue can't change asrArises (what is the reason?). Repeat parts (d), (e) for this case i.e.calculate c* and the r function.Compare your answers to those in part (f).

h)Still continuing with a population of 200,000 residents, suppose that instead of being located on a flat featureless plain, the CBD is located on the ocean (where the coast is perfectly straight). This means that only a half-circle of land around the CBD is available for housing. How large must the radius of this half-circle be to fit the population of 200,000 residents? Using your answer, repeat parts (d) and (e) i.e.find the newx,c*and therfunction, assuming that all parameters are back at their original values.Are people in this coastal city better or worse off than people in the inland city of parts (c) and (d)?(Assume unrealistically that people don't value the beach!) Can you give anintuitive explanationfor your answer?

i)Finally, focus again on the inland city, and suppose that the zoning authority imposes a building height restriction. This restriction limits housing square footage per block to 7500, half the previous amount. The cost of building materials per square block falls from 90 to 43 (note that the cost is less than half as much because of diminishing returns). Find thenew value ofx(compare the answer in (h)), and repeat parts (d) and (e) to find the associatedc* and r function. What is the impact of the height restriction on the utility of urban residents? Can youexplain intuitivelywhy this effect emerges?Does the restriction seem to be a good policy?