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. 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.
Is D an economic bad (the opposite of a good)?
b. Draw Lan's budget constraint.
c. Derive Lan's demand functions for A and D: The graph.
c. Derive Lan's demand functions for A and D: The Lagrange method.
d. Show that as the price of gasoline increases, Lan chooses to live closer to the employment center.
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 ("For Many Low-Income Workers, High Gasoline Prices Take a Toll," Wall Street Journal, July 12, 2004, A1). Demonstrate that as Lan's income decreases, she spends more per day on gasoline.That is show that ?  D*/Y < 0.