2. (modied from Brueckner 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 1,000 square feet of oor space, regardless of location, and (ii) apartment complexes must contain exactly 20,000 square feet of oor 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 the textbook. The city is circular, and distance (x) is measured in blocks. Suppose that income per household (y) equals $20,000 per year. It is convenient to measure money amounts in thousands of dollars, so this means that y = 20, where y is income. Next suppose that the commuting cost parameter (t) equals 0.02. This means that a person living ten blocks from the CBD will spend 0.02 x 10 = 0.2 per year (in other words, $200) getting to work. The consumers budget constraint is c + pq = y - tx, which reduces to c + 1,000p = 20 - 0.02x under the above assumptions. Since housing consumption (q) is xed at 1,000, the only way that utilities can be equal for all urban residents is for bread consumption (c) to also be the same at all locations. The consumption bundle (the bread, housing combination) will then be the same at all locations, yielding equal utilities. However, for c to be constant across locations, the price per square foot of housing (p) must vary with x in a way that allows the consumer to aord a xed amount of bread after paying his rent and his commuting cost. Let c* denote this constant level of bread consumption for each urban resident. Well see below, however, that c* must take on just the right value or else the city will not be in equilibrium. (a) If we substituting c* in place of c in the budget constraint, we get c* + 1,000p = 20 - 0.02x. First, algebraically solve for p in terms of c* and x. This will tell us what the price per square foot must be at a given location (x) in order for the household to be able to aord exactly c* worth of bread. Does p rise or fall as x increases? Recall that the zoning law says that each developed block must contain 20,000 square feet of oor space. Suppose that annualized cost of the building materials needed to construct this much housing is 100 (that is, $100,000). (b) Prot per square block for the housing developer is equal to 20,000p - 100 - r, where r is land rent per square block. Note, the rst element in this equation is the yearly rent (20,000 square feet times p price per square feet), the second element is the amortized construction costs and the third element is the rent of the land. In equilibrium, land rent adjusts so that this prot is identically zero. If not, other developers would outbid the price of land. Set prot 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 algebraically gives land rent r as a function of x and c*. Since each square block contains 20,000 square feet of housing and each apartment has 1,000 square feet, each square block of the city has 20 households living on it. As a result, a city with a radius of x blocks can accommodate 20 x2 households ( x2 is the area of the city in square blocks) (c) Suppose the city has a population of 190,000 households. How big must its radius be in order to t this population? Use a calculator and round up 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, which means that each household in the city consumes $15,000 worth of bread. Suppose also that farmers oer a yearly rent of $8,000 per square block of land, so that rA = 8. Substitute c* = 15 into the land rent function from (b), and compute the implied boundary of the city (rA is the also the urban land rent at the very edge of the city, x). Using your answer to (c), decide whether the city is big enough to house its population. If not, nd the right c* that is appropriate for the size of the city you found in (c). (e) Using the equilibrium c* from (d) and the results of (a) and (b), write down the equation for the equilibrium land rent function (r as a function of x). What is the land rent per square block at the CBD (x = 0) and at the edge of the city? (f) How much does a household at the center of the city spend on rent and how much do they spend on commuting? How much does a household at the edge of the city spend on rent and how much do they spend on commuting? Conrm that the sum of consumption, rent and commuting add up to 20 for both kinds of households. (g) Suppose that the population of the city grows to 265,000. Repeat (c), (d), and (e) for this case (but dont repeat the calculation involving c* = 15). How does population growth aect the utility level of people in the city? The answer comes from looking at the change in c* (since housing consumption is xed at 1,000 square feet, the utility change can be inferred by simply looking at the change in bread consumption). Note that because they are xed, housing consumption doesnt fall and building heights dont rise as population increases, as happened in the model in chapter 2. Are the eects on r and x the same? (h) In reality, what typically occurs in terms of the change in the makeup of the building stock when the population increases by 40%, as it is here?