Because people dislike commuting to work, homes closer to employment centers tend to be more expensive. The price of a home in a given employment center is $60 per day. The daily rental price for housing drops by $2.50 per mile for each mile farther from the employment center. The price of gasoline per mile of the commute is pg (which is less than $2.50). Thus, the net cost of traveling an extra mile to work is pg-2.5. Lan chooses the distance she lives from the job center, D (where D is at most 50 miles), and all other goods, A. The price of A is $1 per unit. Lan's utility function is U=(50-D)0.5A0.5, and her income is Y, which for technical reasons is between $60 and $110.
a. Is D an economic bad (the opposite of a good)? To answer this question, find ∂U/∂D.
b. Draw Lan's budget constraint.
c. Use the Lagrange method to derive Lan's demands for A and D.
d. Show that, as the price of gasoline increases, Lan chooses to live closer to the employment center. That is, show that ∂D*∂pg<0.
e. Show that, as Lan's income increases, she chooses to live closer to the employment center. Reportedly, increases in gasoline prices hit the poor especially hard because they live farther from their jobs, consume more gasoline in commuting, and spend a greater fraction of their income on gasoline. Demonstrate that as Lan's income decreases, she spends more per day on gasoline. That is, show that ∂D*∂Y<0.