Question 1 (similar to Exercise 2.1 from textbook)

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 by p.

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 location q = 1500, and (ii) apartment complexes must contain exactly 15,000 square feet of floor space per square block of land area, such that Q = 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. Its convenient to measure money amounts in thousands of dollars, so this means that y = 25, where y is income. Next suppose that the commuting cost parameter t = 0.01. This means that a person living 10 blocks from the CBD will spend .01 10 = .1 per year (in other words, $100) getting to work.

The consumers budget constraint is c + pq = y tx, which reduces to c + 1500p = 25 - .01x under 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 consumption c to 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. Let c* denote this constant level of bread consumption for each urban resident. For the moment, c* is taken as given. Well see below, however, that c* must take on just the right value or else the city wont be in equilibrium.

7. g) Now suppose that population is back at 200,000 (as in part (c) but that rA rises to 3 (that is, farmers now offer $3000 rent per square block). Note that, unlike in the lectures, the x value cant change as rA rises (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).

8. 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 new x , c* and the r function, 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 dont value the beach!) Explain your economic intuition.

9. 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 the new value of x (compare the answer in (h)), and repeat parts (d) and (e) to find the associated c* and r function. What is the impact of the height restriction on the utility of urban residents? Can you explain intuitively why this effect emerges? Does it seem to be a good policy?