In El Carburetor, California, population 1,001, there is not much to do except to drive your car around town. Everybody in town is just like everybody else. While everybody likes to drive, everybody complains about the congestion, noise, and pollution caused by traffic. A typical resident’s utility function is U(m, d, h) = m+16d−d2−6h/1, 000, where m is the resident’s daily consumption of Big Macs, d is the number of hours per day that he, himself, drives, and h is the total amount of driving (measured in person-hours per day) done by all other residents of El Carburetor. The price of Big Macs is $1 each. Every person in El Carburetor has an income of $40 per day. To keep calculations simple, suppose it costs nothing to drive a car.
(a) If an individual believes that the amount of driving he does won’t affect the amount that others drive, how many hours per day will he choose to drive?
(b) If everybody chooses his best d, then what is the total amount h of driving by other persons?
(c) What will be the utility of each resident?
(d) If everybody drives 6 hours a day, what will be the utility level of a typical resident of El Carburetor?
(e) Suppose that the residents decided to pass a law restricting the total number of hours that anyone is allowed to drive. How much driving should everyone be allowed if the objective is to maximize the utility of the typical resident? (Rewrite the utility function, substituting 1, 000d for h, and maximize with respect to d.)
(f) The same objective could be achieved with a tax on driving. How much would the tax have to be per hour of driving?