This question has a lot of parts but will walk you through a supply-demand equilibrium of a city under some special simplifying assumptions about land use (mostly to make the math a bit easier). The assumptions are: (a) all dwellings must contain exactly 1,200 square feet of floor

space, regardless of location, and (bi) apartment complexes must contain exactly 24,000 square feet of floor space per square block of land area (20 units per block). These land-use restric- tions, 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 (so we will use the formula for a circle), and distance (x) is measured in blocks. Please don't hesitate to reach out if you need hints along the way! I'm more than happy to help and I am more interesting in you learning the model here than just putting down an answer that you don't understand. Suppose that income per household (y) equals $18,000 per year. It is convenient to measure money amounts in thousands of dollars, so this means that y = 18, where y is income. Next suppose that the commuting cost per block per year (t) equals 0.04. This means that a person living ten blocks from the CBD will spend 0.04 x 10 = 0.4 per year (in other words, $400) getting to work. The consumer's budget constraint is c + pq = y - tx, which reduces to c + 1,200p = 18 - 0.04x under the above assumptions. Since housing consumption (q) is fixed at 1,200, the only way that utilities can be equal for all urban residents (which must be true in equilibrium for prices to be stable) 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 consumption 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 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, let's keep c* as a variable. We'll see below, however, that c* must take on just the right value or else the city will not be in equilibrium. (a) Let c* be the exact amount of consumption each household has in equilibrium. If we sub- stitute c* in place of c in the budget constraint, we get c* + 1,200p = 18 - 0.04x. 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 afford 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 24,000 square feet of floor space. Suppose that annualized cost of the building materials needed to construct this much housing is 55 (that is, $55,000). (b) (Economic) Profit per square block for the housing developer is equal to 24,000p - 55 - r, where r is land rent per square block. Note, the first element in this equation is the yearly rent (25,000 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 profit is identically zero. If not, other developers would outbid the price of land. 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 algebraically gives land rent r as a function of x and c*. Since each square block contains 24,000 square feet of housing and each apartment has 1,200 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 353,430 households. How big must its radius be in order to fit this population? Use a calculator. It should be extremely close to a whole number. Use that whole number moving forward. (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* = 13, which means that each household in the city consumes $13,000 worth of bread. Suppose also that farmers offer a yearly rent of $5,000 per square block of land, so that rA = 5. Substitute c* = 13 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. (e) It shouldn't work. Solve for a value of c* that leads the city to have just the right radius found in part (c). (f) Using the equilibrium c* from (e) 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 (x = x)? (g) 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? Confirm that the sum of consumption, rent and commuting add up to 18 for both kinds of households. (h) Further, confirm that the developer is making zero economic profits at all points. At the center, how much do they collect in rent? What is their land costs and building costs? Are they the same? Do the same for the edge. (i) Suppose that the population of the city grows to 453,960. Repeat (c), (e), and (f) for this case. How does population growth affect the utility level of people in the city? The answer comes from looking at the change in c* (since housing consumption is fixed at 1,200 square feet, the utility change can be inferred by simply looking at the change in bread consump- tion). Note that because they are fixed, housing consumption doesn't fall and building heights don't rise as population increases, as happened in the model in chapter 2. Are the effects on r and x the same? (j) Here the city just gets spatially bigger, but in reality, what other changes in the makeup of the building stock might happen when the population increases by almost 30%?