After you graduate, you get a job in a small city where you have taken your sister’s offer of living in her apartment. Your job pays you $20 per hour and you have up to 60 hours per week available. The problem is you also have to get to work.
A: Your sister’s place is actually pretty close to work — so you could lease a car and pay a total (including insurance and gas) of $100 per week to get to work, spending essentially no time commuting. Alternatively, you could use the city’s sparse bus system—but unfortunately there is no direct bus line to your place of work and you would have to change buses a few times to get there. This would take approximately 5 hours per week.
(a) Now suppose that you do not consider time spent commuting as “leisure” — and you don’t consider money spent on transportation as “consumption”. On a graph with “leisure net of commuting time” on the horizontal axis and “consumption dollars net of commuting costs” on the vertical, illustrate your budget constraint if you choose the bus and a separate budget constraint if you choose to lease the car.
(b) Do you prefer the bus to the car?
(c) Suppose that before you get to town you find out that a typo had been made in your offer letter and your actual wage is $10 per hour instead of $20 per hour. How does your answer change?
(d) After a few weeks, your employer discovers just how good you are and gives you a raise to $25 per hour. What mode of transportation do you take now?
(e) Illustrate in a graph (not directly derived from what you have done so far) the relationship between wage on the horizontal axis and the most you’d be willing to pay for the leased car.
(f) If the government taxes gasoline and thus increases the cost of driving a leased cars (while keeping buses running for free), predict what will happen to the demand for bus service and indicate what types of workers will be the source of the change in demand.
(g) What happens if the government improves bus service by reducing the time one needs to spend to get from one place to the other?
B: Now suppose your tastes were given by u(c, ℓ) = cαℓ(1−α) , where c is consumption dollars net of commuting expenses and ℓ is leisure consumption net of time spent commuting. Suppose your leisure endowment is L and your wage is w.
(a) Derive consumption and leisure demand assuming you lease a car that costs you $Y per week which therefore implies no commuting time.
(b) Next, derive your demand for consumption and leisure assuming you take the bus instead, with the bus costing no money but taking T hours per week from your leisure.
(c) Express the indirect utility of leasing the car as a function of Y.
(d) Express your indirect utility of taking the bus as a function of T.
(e) Using the indirect utility functions, determine the relationship between Y and T that would keep you indifferent between taking the bus and leasing the car. Is your answer consistent with the relationship you illustrated in A (e) and your conclusions in A (f ) and A (g)?
(f) Could you have skipped all these steps and derived this relationship directly from the budget constraints? Why or why not?