Exercise 3

Due Monday 2/13

Used automobile batteries can be a serious environmental problem. If they are dumped into landfills, the lead in the batteries can leach out and contaminate nearby aquifers. To explore this issue, consider the following scenario. A particular town has 100,000 people, each of whose demand for car batteries per year is given by P= 200–300Qi. In addition, suppose that batteries are produced at a constant marginal cost of $50, and the interest rate is 5%. Finally, please note that because the demand is batteries per year, Qi does not have to be an integer – if it were 1.5, for example, that would be 3 batteries every two years.

1. Please derive the market demand curve for batteries in a given year. Using it, determine the market equilibrium price and quantity of batteries.
2. Now let’s consider the disposal problem. Suppose that each battery is used for 6 years (years 0 through 5) and then thrown in a landfill. After it has been in the landfill for 15 more years (i.e., in year 21), there is a 25% chance that the lead in it will begin to leach out. Once it begins leaching, the lead causes damage every year forever. The magnitude of the damage is not certain, however. With 90% probability, it is only $0.10 per year (ten cents) but there is a 10% chance it will be $10.00 per year. Calculate the expected present value of the damages associated with the purchase of a battery. Be sure to show all your work!
3. Using your answer to part (2), find the efficient price and quantity of new batteries. If we were to achieve this by taxing batteries, what should the tax rate be? What will be the total dollar effect of this on consumer surplus? On government revenue? On the externality problem? What is the overall welfare gain?