The assumptions are:

all dwellings are situated within apartment complexes,

all dwellings must contain exactly 1,500 square feet of floor space, re-

gardless of location, and

apartment complexes must contain exactly 15,000 square feet of floor space per square block of land area.

These land-use restrictions, which are imposed by a zoning authority, mean that dwelling sizes and building heights do not vary with distance to the central business district (CBD). Distance is measured in blocks.

Suppose that income per household equals $25, 000 per year. It is convenient to measure money amounts in thousands of dollars, so this means thaty=25, whereyis income. Next suppose that the commuting cost parametertequals 0.01. This means that a person living ten blocks from the CBD will spend 0.0110=0.1 per year (in other words, $100) getting to work.

The consumer's budget constraint isc+pq=ytx, which reduces toc+1, 500p=250.01xunder the above assumptions. Since housing con- sumption is fixed at 1,500, 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.

Forcto be constant across locations, the price per square foot of housing must vary withxin a way that allows the consumer to afford a fixed amount of bread after paying her rent and her commuting cost. Letcdenote this constant level of bread consumption for each urban resident. For the moment,cis taken as given. We'll see below, however, thatcmust take on just the right value or else the city will not be in equilibrium.

1. (a)Substitutingcin place ofcin the budget constraintc+1, 500p=250.01x, solve forpin terms ofcandx. 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 doespvary with location?
2. (b)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 90 (that is, $90, 000).
3. Profit per square block for the housing developer is equal to 15, 000p90r, 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 yourpsolution from (a) in the resulting equation. The result gives land rentras a function ofxandc. How does land rent vary with location?
4. (c)Since each square block contains 15,000 square feet of housing and each apartment has 1,500 square feet, each square block of the city has 10 households living on it. As a result, a city with a radius ofx blocks can accommodate 10x2households (x2is the area of the city in square blocks).
5. Suppose the city has a population of 200,000 households. How big must its radiusx be in order to fit this population? Use a calculator and round off to the nearest block.
6. (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 (b), and compute the implied boundary of the city. Using your answer to (c), decide whether the city is big enough to house its population. If not, adjustcuntil you find a value that leads the city to have just the right radius.
7. (e)Using the equilibriumcfrom (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?Plot the land rent function. 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. Repeat (c), (d), and (e) for this case (but don't repeat the calculation involvingc=15.5). Explain your findings. How does population growth affect the utility level of people in the city? The answer comes from looking at the change inc(since housing consumption is fixed at 1,500 square feet, the utility change can be inferred by simply looking at the change in bread consumption). Note that because they are fixed, housing consumption doesn't fall and building heights don't rise as population increases. Are the effects onrandx the same?

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

(h)Now 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 be the radius of this half-circle to fit the population of 200,000 residents? Using your answer, repeat (d) and (e), 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 (c) and (d)? (Assume unrealistically that people don't value the beach!) Can you give an intuitive explanation for 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 7,500, 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 the new value ofx (compare the answer in (h)), and repeat (d) and (e). How does the height restriction affect the utility of urban residents? Explain intuitively why this effect emerges. Does the restriction seem to be a good policy?