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ps2_key

# ps2_key - EEP 101/ECON 125 Prof Zilberman GSIs Alix McKim...

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EEP 101/ECON 125 Prof. Zilberman GSIs: Alix, McKim, Schoengold Suggested Solutions for Problem Set # 2 Numerical problems 1. Assume that manufacturing of PVC tubing, which produces large amounts of carcinogenic byproducts, is undertaken by two firms. The total benefits that the firms derive from releasing this cancer-causing stuff into the environment are given by: 2 1 4 1 1 1 75 X X B = and . A combination of experts from the School of Public Health and the Department of Resource Economics have assessed the marginal damage function to be: 2 2 2 2 100 X X B = X MEC 4 3 10 + = where 2 1 X X X + = . Using this information, please answer the following questions both graphically and using mathematics. a) Find the aggregate demand function of firms for carcinogens and show it on your graph. To get the aggregate demand, you first need to find what the two demand functions are. Since the equations above are given in terms of total benefit, all you need to do is differentiate them with respect to X 1 and X 2 . For firm 1 and firm 2, the respective demands are: 1 2 1 1 75 X MB = and 2 2 2 100 X MB = . The next step is to sum these things up horizontally; in order to do so, you need to take their inverses: 1 1 2 150 MB X = and 2 2 1 2 50 MB X = . The aggregate quantity demanded comes in two pieces. Up until the point where MB=75, only firm 2’s demand is taken into account, after that point, the two demand curves are summed together. You can see this by looking at the graph. Mathematically, it is defined as follows: < = 75 0 200 100 75 50 2 5 2 1 MB for MB MB for MB X You could also define these two pieces in terms of the X’s. On the graph you know that there are two parts because there is a kink at MB = 75. b) Find the amount of carcinogens released by the two firms in the absence of regulation. Label these points Q 1 and Q 2 on your graph . Without regulation, firm one will produce 150 units of pollution and firm two 50. c) What is the socially optimal level of pollution? To find the socially optimal level of pollution, we first need to resolve our aggregate demand function in terms of X. It turns out to be: X MB 5 2 80 = . We then set marginal benefit equal to the marginal externality cost, since in this problem private costs have been netted out. The optimal level is output comes from X X 4 3 5 2 10 80 + = , which is where X = 60.8 and MB =

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55.7. This point is denoted by a star on the graph. The optimal production for each firm is marked by a smiley face on each of their marginal benefit lines. d) Suppose now that the government decides to impose a system of pollution permits. They are not allowed to trade. The government is considering two different forms of permit allocation. In the first, each firm receives permission to release ½ of the socially optimal amount of pollution. The other option is to allocate the socially optimal number of permits based upon historical emissions levels. Show graphically and calculate the marginal benefit to each firm from emitting their permitted levels of pollution in each case.
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