Question

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 1500 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 CBD, as in the monocentric city model from Chapter 2. Distance is measured in blocks.

Suppose that income per household equals $25,000 per year. It’s 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 .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 consumer’s 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. We’ll see below, however, that c* must take on just the right value or else the city won’t be in equilibrium.

a)Substituting c* in place of c in the budget constraint c + 1500p = 25 - .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 15000p – 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 part (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 1500 square feet, each square block of the city has 10 households living on it. As a result, a city with a radius of x̄blocks can fit 10πx̄2 households (πx̄2 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 x̄in 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 $2000 per square block of land, so that rA = 2. Substitute c* = 15.5 into the land rent function from part (b), and compute the implied boundary of the city. Using your answer to part (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.

e)Using the equilibrium c* from (d) and the results of (a) and (b), write down the equation for the equilibrium land rent function. What is the rent per square block at the CBD (x = 0) and at the edge of the city? Graph the land rent function. How much does a household living at the edge of the city spend on commuting?

f)Suppose that the population of the city grows to 255,000. Repeat parts (c), (d), and (e) for this case (but don’t repeat the calculation involving c* = 15.5). Explain your findings. What can you say about the impact of population growth on the utility level of people in the city? The answer comes from looking at the change in c* (since housing consumption is fixed at 1500 square feet, the utility change can be inferred by simply looking at the change in bread consumption). Note that because they’re 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 x̄ the same?

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 x̄ value can’t change as rA rises (what is the reason?). Repeat parts (d), (e) for this case. Compare your answers to those in part (f).

h)Now 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), 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 don’t value the beach!) Can you give an intuitive explanation for your answer?

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 x̄ (compare the answer in (h)), and repeat parts (d) and (e). What is the impact of the height restriction on the utility of urban residents? Can you explain intuitively why this effect emerges? Does the restriction seem to be a good policy?

Answer 1

a. Denoting c* as the constant level of bread consumption, the budget constraint is c* + 1500p = 25 - .01x
     or, p = (25 - c* - 0.01x) / 1500

Taking differential of both sides, dp = -0.01dx. In other words, as x increases, p decreases at a linear rate of 0.01 (i.e. $100 per block).

b. As per the problem, profit = 15000p – 90 – r.
    In equilibrium, 15000p – 90 – r = 0,
                         or, r = 15000p – 90
                         or, r = 250 - 10c* - 0.1x – 90 (substituting the value of p from part a.)
                         or, r = 160 - 10c* - 0.1x

Taking differential of both sides, dr = -0.01dx. In other words, as x increases, r decreases at a linear rate of 0.01 (i.e. $100 per block).

c. To fit the population of 200,000 households,
      10πx̄^2 ≥ 200000
or, πx̄^2 ≥ 20000
or, x̄^2 ≥ 6366.2
or, x̄ ≥ 79.79

Therefore, to fit the entire population, the radius must be at least 80 blocks.

d. From part (b), r = 160 - 10c* - 0.1x

Therefore, 2 = 160 - 10c* - 0.1x (Since, r = 2)
               or, 2 = 160 – 155 - 0.1x
               or, 0.1x = 3
               or, x = 30

Using the answer from part c (i.e. 80), this city is not enough to fit the entire population.

To fit the entire population, a radius of at least 80 blocks is necessary. The equation is:
     2 = 160 - 10c* - 0.1x
or, 0.1x = 158 - 10c*
or, x = 1580 – 100c*

Therefore, for the city to be at least 80 blocks in radius, c* must be less than or equal to 15.

Note: This question has 9 subparts. Only the first 4 subparts have been answered.