B Question [25 points] Mease write all your derivations and answers on the box after each question. (Rounded off to two decimal places where necessary) Consider the basic urban model. Assume that (i) all dwellings must contain exactly q. square feet of building heights not varying with distance to the central busines s district (CBD). Distance is meaSince housing consumption is fixed at q, the only way that utiliti es can be equa dents is for bread consumption c to be the same at all locations. Suppose that the bread consur c=15, the income per household is y=25 per year, and the commuting cost parameter is t=0.01. (a) Let denote p as the rental price per square foot of floor space. Write down the budget constraint (a) Let denote p as the rental price per square foot of floor space. Write down the budget constraint of a consumer who lives x away from the CBD. Then, solve for p in terms of q and x. How does vary with location? Recall that each developed block must contain 15,000 square feet of floor space. Suppose that ar alized cost of the building materials needed to construct this much housing is 3 . (b) Write down the profit per square block for the housing developer in terms of p and r, where r is developers' land rent per square block. (c) In equilibrium, land rent adjusts so that the profit in (b) 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 q. How does land rent vary with location? Since each square block contains 15,000 square feet of housing and each apartment has q square feet, each square block of the city has 15,000/q households living on it. Suppose the city is on an island and is a rectangle 400 blocks wide with the CBD at one end. The city spreads out along the length of the island to accommodate its population, with its edge located x blocks from the CBD. As a result, the city can accommodate (15,000/q)400x households ( 400x is the area of the city in square blocks). (d) Suppose the city has a population of 200,000 households. Solve for its length x in terms of q such that the city fits this population. (e) Substitute the x from (d) in the land rent function from (c) to compute the land rent at the boundary in terms of q. (f) Suppose that farmers offer a yearly rent of rA=2 per square block of land. Compute the equilibrium q by equating the developers' land rent to the farmers' land rent. Then, compute the implied boundary of the city x. B Question [25 points] Mease write all your derivations and answers on the box after each question. (Rounded off to two decimal places where necessary) Consider the basic urban model. Assume that (i) all dwellings must contain exactly q. square feet of building heights not varying with distance to the central busines s district (CBD). Distance is meaSince housing consumption is fixed at q, the only way that utiliti es can be equa dents is for bread consumption c to be the same at all locations. Suppose that the bread consur c=15, the income per household is y=25 per year, and the commuting cost parameter is t=0.01. (a) Let denote p as the rental price per square foot of floor space. Write down the budget constraint (a) Let denote p as the rental price per square foot of floor space. Write down the budget constraint of a consumer who lives x away from the CBD. Then, solve for p in terms of q and x. How does vary with location? Recall that each developed block must contain 15,000 square feet of floor space. Suppose that ar alized cost of the building materials needed to construct this much housing is 3 . (b) Write down the profit per square block for the housing developer in terms of p and r, where r is developers' land rent per square block. (c) In equilibrium, land rent adjusts so that the profit in (b) 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 q. How does land rent vary with location? Since each square block contains 15,000 square feet of housing and each apartment has q square feet, each square block of the city has 15,000/q households living on it. Suppose the city is on an island and is a rectangle 400 blocks wide with the CBD at one end. The city spreads out along the length of the island to accommodate its population, with its edge located x blocks from the CBD. As a result, the city can accommodate (15,000/q)400x households ( 400x is the area of the city in square blocks). (d) Suppose the city has a population of 200,000 households. Solve for its length x in terms of q such that the city fits this population. (e) Substitute the x from (d) in the land rent function from (c) to compute the land rent at the boundary in terms of q. (f) Suppose that farmers offer a yearly rent of rA=2 per square block of land. Compute the equilibrium q by equating the developers' land rent to the farmers' land rent. Then, compute the implied boundary of the city x