Exercise 2.1 In this exercise, 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, and (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 with distance to the central business district, as in the model from chapter 2. 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 that y = 25, where y is income. Next suppose that the commuting cost parameter t equals 0.01. This means that a person living ten blocks from the CBD will spend 0.01 × 10 = 0.1 per year (in other words, $100) getting to work. The consumer’s budget constraint is c + pq = y – tx, which reduces to c + 1,500p = 25–0.01x under the above assumptions. Since housing consumption is fixed at 1,500, 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. We’ll see below, however, that c* must take on just the right value or else the city will not be in equilibrium.
(a) Substituting c* in place of c in the budget constraint c + 1,500p = 25–0.01x, 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 15,000p – 90 – r, where r is 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 of p. Then substitute your p solution from (a) in the resulting equation. The result gives land rent r as a function of x and c*. How does land rent vary with location? 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 of x blocks can accommodate 10π2 households (π2 is the area of the city in square blocks).
(c) Suppose the city has a population of 200,000 households. How big must its radius be in 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 that c* = 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 $2,000 per square block of land, so that rA = 2. Substitute c* = 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, adjust c* until you find a value that leads the city to have just the right radius.
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