From Brueckner, 2.1) In this question, you will analyze the supply-demand equilibrium of a city under some special simplifying assumptions about land use. The assumptions are:

(i) all dwellings must contain exactly 1,500 square feet of floor space, regardless of location (

(ii) 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 by distance to the central business district. Additional assumptions include:

(iii) Income per household equals $25,000 per year. It is convenient to measure money amounts in thousands of dollars, so let y = 25, where y is income.

(iv) The commuting cost parameter t = 0.01. This means that a person living 10 blocks from the CBD would spend 0.01 10 = 0.1 thousand per year (or $100) getting to work.

2.1 Write down the consumer's budget constraint. (Note that from (i) housing consumption is fixed at q = 1, 500 square feet of floor space.)

2.2 If q = 1, 500, why must all consumers choose the same amount of M, the composite good? Don't solve for M; instead, just let M denote the optimal value of this composite good.

2.3 Use the budget constraint to solve for the housing price, p, in terms of M and x. The solution tells us what the price per square foot must be at a given location in order for the household to be able to afford exactly M worth of the composite good. How does p vary with location?

2.4 Suppose that the annualized cost of building materials needed to construct 15,000 square feet of floor space is 90 (i.e. $90,000). Profit per square block for the housing developer is equal to: = 15000p 90 r where r is the land rent per square block. In equilibrium, land rent adjust so that this profit is equal to zero. Using this, solve for the land rent in terms of p. Then substitute your value of p from (2.3) to solve for r as a function of x and M . How does land rent vary with location?

2.5 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 as result, a city with a radius of x blocks can accommodate 10x 2 households (where x 2 is the area of the city in square blocks). Suppose the city has a population of 200,000 households. How big must x be in order to fit this population. (Use a calculator and round off to the nearest block).

2.6 In order for the city to be in equilibrium, housing developers must bid away enough land from farmers to house the population. Suppose that M = 15.5, meaning that each household in the city consumes $15,500 worth of the composite good. Suppose also that farmers offer a yearly rent of $2,000 per block of land, so that rA = 2. Substitute M = 15.5 into the land rent function from (2.4) and compute the implied boundary of the city. Using your answer to (2.5), decide whether or not the city is big enough to house the population. If not, adjust M until you find a value that leads the city to have just the right radius.

2.7 Using the equilibrium M from (2.6) and the results of (2.3) and (2.4), write down the equation for the equilibrium land rent function. What is the land rent per square block at the CBD (x = 0)? What is the land rent per square block at x? How much does a household at the edge of the city spend on commuting?

2.8 Suppose the population of the city grows to 255,000. Repeat (2.5), (2.6), and (2.7) for this case (solving for a new value of M ). Explain your findings. How does population growth affect the utility level of people in the city? (Note that the answer comes from looking at the change in M .) Because they are fixed, housing consumption does not fall and building heights don't rise as population increases. Are the effects on r and x the same? 2.9 Now, suppose that the population is back at 200,000 (as in (2.5)) but that rA rises to 3 (that is, farmers now offer $3,000 rent per square block). Note that unlike in the lecture notes, the value of x does not change as rA rises. Why? Repeat (2.6) and (2.7) for this case, and compare your findings with those in (2.8).