Problem Statement: Denveria is a small island nation located in the South Pacific. The country has an unusual geographic feature: its edge is entirely ringed by a tall mountain. This means that all of the inhabitants of Denveria live inside a bowl-shaped interior of the island. The population of 10,000 is roughly split between highland inhabitants (30 percent of the total population) who live on the slopes of the mountains, and lowland inhabitants who live in the valley below, on the bottom of the fruit bowl with the pits as the highlanders say. The geographic features, combined with the settlement patterns and economic activity, have led to an unfortunate problem. Air pollution emitted from Denveria has a long residence time. In fact, the air in the lowlands can be quite stale, and there is often a brown fog lying on the ground well into the morning. This pollution problem has taken a serious toll on the health of Denveria, and particularly the health of the lowlanders, who get watery eyes, asthma attacks, and a higher incidence of emphysema. There are four sources of air pollution. The biggest single source of pollution is the timber mill in the highlands, which produces 50 units of pollution per year. The copra drying and processing activities in the highlands contribute another 20 units of pollution per year. In the lowlands the gin mill produces 20 units per year, and the hog farming produces another 10. If you do the math, you see that there are total emissions of 100 units per year. The marginal damage caused by pollution is independent of its point of origin - i.e., lowlanders suffer just as much from a unit of pollution emitted in the highlands as they do from a unit emitted in the lowlands. Similarly, it makes no difference in terms of health effects for highlanders where pollution originates.

1. The Island Council commissioned a study to examine the benefits that could be derived from reducing emissions from their current total level of 100 units. A consulting company determined that because of their different positions in the bowl, highlanders and lowlanders have very different demands for abatement of the air emissions. Using generally accepted economic methods they found that the demand functions of average individuals in the two areas are

AL = 100 - 11.6532P

AH = 100 - 58.2660P

where, AL and AH are the reductions in annual emissions, and P is the annual cost (value) of the reductions. These represent individuals willingness to pay for one year of emissions reductions in the lowlands and highlands, respectively. However, emissions reductions occur as a result of investments in long-term, or semi-permanent, abatement, with project lives of 25 years.

Your job is to:

a. Show that the social demand curve, AS, for semi-permanent reductions in annual emissions (i.e., reductions over 25 years) is AS = 100 - 0.000132PS

where PS is the NPV of the social value of making semi-permanent (25-year) reductions in annual emissions, and a seven percent (7%) discount rate is applied.

NB1: If AS = 10, then the level of emissions would be 90 units per year, for the next 25 years, the abatement is 10 units of emissions per year, accounting for the total of 100 units per year that would have occurred in the absence of abatement.

NB2: If P represents the marginal willingness to pay for reductions for one-year, then PS represents the value (or marginal willingness to pay) for 25 years of that same reduction. You may not know the numerical value of P, but you know that PS = P (P/A, 0.07, 25).

Hint: follow these steps:

(1) find the inverse demand function (willingness-to-pay) of individuals from each of the two regions,

(2) find the present value of the semipermanent stream of annual reductions,

(3) aggregate to find the inverse social demand function PS=f(QS), remembering that the first two equations above are for individual, annual demands, and

(4) solve for QS to get the demand function. Remember, you have to account for how many highlanders and lowlanders there are. Also: Recall that there is a finance tutorial posted on the Canvas site.

b. Graph the marginal social benefits, MBS, of abatement activities, using the social demand equation in part 1a.

c. What is the total social benefit of reducing emissions by 40 units, (given the social demand function in part 1a, above)?

d. What portion (as a percent or absolute number) of those benefits go to highlanders?