Suppose we have a monocentric city, but strict laws govern both living space and building heights. Every dwelling contains exactly one person, and each must have exactly 1,500 square feet of space. Every apartment building must contain exactly 10 dwellings per square block of land area. Thus, each will be the same size and height, regardless of distance to the CBD (where distance x is measured in blocks). Income per household is $25,000 per year, which we’ll express in thousands of dollars, i.e., y = 25. The transportation cost parameter is t = 0.01, so the cost of living 20 blocks from the CBD is 0.2 per year (i.e., $200). Consumers all have the same utility function u(c, q). (a) Write down the standard consumer budget constraint from lecture II, including the above parameter values for y, q, and t. (b) Since each person only consumes q and c, and q is fixed at 1,500, what must be true about c for all consumers (regardless of location) if we are to be in a spatial equilibrium? (c) Using your answer to the previous question, write down an equation for the housing price gradient in this city as a function of all other variables/parameters (i.e., it should take the form p = some stuff). What is the relationship between p and x?
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Suppose we have a monocentric city, but strict laws govern both living space and building heights. Every dwelling contains exactly one person, and each must have exactly 1,500 square feet of space. Every apartment building must contain exactly 10 dwellings per square block of land area. Thus, each will be the same size and height, regardless of distance to the CBD (where distance x is measured in blocks). Income per household is $25,000 per year, which we’ll express in thousands of dollars, i.e., y = 25. The transportation cost parameter is t = 0.01, so the cost of living 20 blocks from the CBD is 0.2 per year (i.e., $200). Consumers all have the same utility function u(c, q).
(a) Write down the standard consumer budget constraint from lecture II, including the above parameter values for y, q, and t.
(b) Since each person only consumes q and c, and q is fixed at 1,500, what must be true about c for all consumers (regardless of location) if we are to be in a spatial equilibrium?
(c) Using your answer to the previous question, write down an equation for the housing price gradient in this city as a function of all other variables/parameters (i.e., it should take the form p = some stuff). What is the relationship between p and x?
(d) Suppose that developers’ profit per square block is given by 15, 000p − 90 − r, where 15, 000p is the revenue they receive per block, 90 reflects a $90,000 fixed cost per year for building materials, and r is the land rent per block. We’ll assume that land rents adjust to ensure that this profit is zero in equilibrium, regardless of location. Use this idea to solve for land rent as a function of p. Then, plug in the equation for p that you found in part c to find the relationship between r and x. How does this behave?
(e) Since each block can house 10 people (by construction), if the city has a radius of x ̄ blocks, it can house 10πx̄2 people. If the city holds 200,000 people, how large must x ̄ be to hold them all? (You can round to the nearest integer.)
(f) Suppose now that c = 15.5 for everyone in the city, i.e., non-housing consumption is $15,500 a year. Assume also that the agricultural rent for land is rA = 2, i.e., $2,000 per block. Using these, along with the land rent function from part (d), find the implied boundary of the city x ̄. Is this large enough to house our 200,000 people? What smaller value of c would yield a sufficiently large city?
(g) Using the value of c you found above and the result from part (d), find the equilibrium land rent function. Graph this with money ($) on the y axis and distance (x) on the x axis. Label the intercept and the point where it crosses the agricultural rent function rA = 2.
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