# Solution to Exercise 7 on Multisource Pollution

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1 Peter J. Wilcoxen Economics 437 The Maxwell School Syracuse University Solution to Exercise 7 on Multisource Pollution 1 Finding the Efficient Amounts of Abatement There are two ways to find the efficient amount of abatement for each plant. Method 1 Start by constructing the overall MCA curve by combining the MCA curves for the individual plants. Here is how it goes. For efficiency, Q1, Q2 and Q3 must be set so that MCA1 = MCA2 = MCA3. Since the MCA s will be equal we can drop the subscripts and just call them MCA. Putting this into the MCA equations: (1) 1: MCA = 1,000 Q1 (2) 2: MCA = 10,000 Q2 (3) 3: MCA = (10,000/3) Q3 Solving each equation for Q in terms of MCA: (4) 1: Q1 = MCA / 1,000 (5) 2: Q2 = MCA / 10,000 (6) 3: Q3 = 3 * MCA / 10,000 Summing to get the total amount of abatement for a given MCA: (7) QT = Q1 + Q2 + Q3 (8) QT = MCA /1,000 + MCA / 10, MCA / 10,000 (9) QT = ( 1/1, /10, /10,000 ) MCA (10) QT = ( 14/10,000 ) MCA Rearranging produces the overall MCA: (11) MCA = (10,000/14) QT The efficient level of QT is the one where MBA = MCA: (12) MBA = MCA (13) 50,000 = (10,000/14) QT - 1 -

2 (14) 70 = QT Expressed in terms of the original 350 tons of emissions, the efficient reduction is thus: (15) 70/350 = 0.20 = 20% Graphically, the situation looks as follows: This approach works even when MBA is not constant, as shown in the example below. Example Problem with Sloping MBA This section is an illustration only it is not part of the assignment. Suppose that the problem had specified that the MBA was given by MBA = 71, QT. Since QT is initially unknown, it s not possible to say what the efficient MBA is right off the bat. However, the only change to the steps above is where MBA is set equal to MCA: (16) MBA = MCA (17) 71, QT = (10,000/14) QT (18) 71,000 = ( ,000/14) QT (19) 71,000 = (14,200/14) QT (20) 70 = QT Inserting this into the MBA and MCA curves to determine MBA and MCA at the efficient point: (21) MBA = 71, *70 = 50,000 (22) MCA = (10,000/14)*70 = 50,000 Once these values are known, the reduction can be allocated across plants using the steps in method A: set each plant s MCA equal to 50,000. End of Example - 2 -

3 Shifting back to the actual problem in the exercise, the remaining step is to figure out how the 70 ton reduction should be allocated across plants. That can be done by using the abatement equations derived above but inserting MCA=\$50,000 (23) 1: Q1 = 50,000 / 1,000 = 50 (24) 2: Q2 = 50,000 / 10,000 = 5 (25) 3: Q3 = 3 * 50,000 / 10,000 = 15 End of Method 1 Method 2 Because the MBA curve is completely flat in this problem, it s clear from the outset that the efficient MCA will have to be \$50,000 (no matter what the MCA curve looks like, it will hit the MBA curve where MCA=\$50,000). In that case, it s possible to skip deriving the overall MCA curve and insert the \$50,000 straight into the 3 equations immediately above. However, please keep in mind that this ONLY works when the MBA curve is perfectly flat. End of Method 2 Finally, under either approach the total cost of abatement is the sum of what each plant spends cleaning up (the area under its MCA curve): (26) 1: (1/2) * 50 * 50,000 = 1.25 million (27) 2: (1/2) * 5 * 50,000 = 125,000 (28) 3: (1/2) * 15 * 50,000 = 375,000 Total: \$1.75 million 2 Equal Percentage Reductions If each plant reduced its emissions by 20 percent, emissions would still drop by 70 tons but the overall cost would be higher. Calculating each plant s Q of abatement: (29) 1: Q1 = 0.2 * 100 = 20 (30) 2: Q2 = 0.2 * 50 = 10 (31) 3: Q3 = 0.2 * 200 = 40 The corresponding marginal costs: - 3 -

4 (32) 1: MCA1 = 1,000 * 20 = \$20,000 (33) 2: MCA2 = 10,000 * 10 = \$100,000 (34) 3: MCA3 = (10,000/3) * 40 = \$133, Areas under the MCA curves: (35) 1: (1/2) * 20 * \$20,000 = \$200,000 (36) 2: (1/2) * 10 * \$100,000 = \$500,000 (37) 3: (1/2) * 40 * \$133,333 = \$2,666,667 Total: \$3.367 million An equal reduction policy costs about twice as much as an efficient approach. 3 Emissions Tax Under an emissions tax, each plant would pay for abatement up to the point where its MCA equaled the tax. To achieve the efficient amount of abatement, the tax should be set to the value of MBA, which is \$50,000. (In general, to get the right amount of abatement, an emissions tax should be set to the dollar value of MBA at the intersection of the MCA and MBA curves.) The cost to each firm will be the sum of what it pays in abatement (from part 1) and what it pays in taxes (\$50,000 times its remaining emissions). The results are summarized in the table below: Original Amount of Remaining Abatement Emissions Abatement Emissions Cost Taxes Total Cost M 2.5 M 3.75 M K 2.25 M M K 9.25 M M Total M 14 M M 4 Comparing Policies Each of the plants would prefer the inefficient approach from part 2 because it imposes lower total costs when tax payments are taken into account. From an economic perspective, however, the part 2 approach is bad because it wastes resources: the actual cost of cleaning up pollution is twice what it should be: \$3.367 million instead of \$1.75 million. 5 Tradable Permit System A permit system should issue a total of 280 permits: enough for the emissions remaining after 70 tons have been cleaned up

5 No matter how the permits are initially distributed, the firms will trade them until their marginal abatement costs are equal: MCA1 = MCA2 = MCA3. From that we know that the permits will eventually end up distributed like the efficient emissions: 50 at plant 1, 45 at plant 2, and 185 at plant 3. As discussed in class, the equilibrium price of a permit will be equal to the efficient MCA (that is, it will equal MCA1 = MCA2 = MCA3), which is \$50,000. To equalize overall costs, we can tweak the distribution of permits so that firms that need to pay less overall will have permits to sell (plant 1), and firms that need to pay more overall will have to buy permits (plants 2 and 3). The table below summarizes the abatement costs and equalized costs across the three plants. Note: it s OK to round the equalized costs rather than carrying around the fraction. If you do that, it is a good idea to adjust the rounded numbers so that the total comes out to 1.75M; for example, it would be OK to round them to 583, 583 and 584. Amount of Abatement Equalized Abatement Cost Cost Difference M 583 1/3 K /3 K K 583 1/3 K /3 K K 583 1/3 K /3 K Total M 1.75 M 0 To lower plant 1 s costs by \$666 2/3 K, it needs to be able to sell \$666 2/3 K/\$50K = 13 1/3 permits; that is, selling 13 1/3 permits at \$50,000 each brings in \$666 2/3 K of revenue. s 2 and 3, on the other hand, need to pay more and will have to buy permits. 2 will need to be given \$458 1/3 K/\$50K, or 9 1/6 permits less than it really needs. Similarly, plant 3 should be given 4 1/6 permits less than it needs. The results are summarized below: Efficient Emissions Plus Permits to Sell Less Permits to Buy Equals Permits Allocated / / /6 35 5/ / /6 Total /3 13 1/

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