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Total abatement will be 160 this solution is optimal

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Total abatement will be 160. This solution is optimal because each firm will set the marginal benefit equal to the marginal cost as we have seen in part (a). d) Consider a cap and trade system. The problem faced by firm A is that it can either buy a permit a price p or engage in pollution abatement. If a A is the amount of permits purchased by A, it must engage in 100 - a A unit is pollution abatement. The objective is to minimize total cost of pollution abatement: min (100 - a A ) 3 + p * a A (35) FOC is given by 3(100 - a A ) 2 + p = 0 (36)
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which implies that q p/ 3 = 100 - a A (37) a A = 100 - q p/ 3 (38) Similarly, the problem of firm B is given by min (60 - a B ) 2 + p * a B (39) The FOC are - 2 (60 - a B ) + p = 0 (40) we obtain for firm B: a B = 60 - p/ 2 (41) Equilibrium requires that the demand for permits (by firm A) equals the supply of permits (by firm B): - a B = a A (42) Hence we have p/ 2 - 60 = 100 - q p/ 3 (43) which implies that the equilibrium price is given by: p = 300 (44) At the equilibrium price, we have: a A = 90 (45)
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a B = - 90 (46) Which sums up to zero as required by the market clearing condition. e) Trading of permits thus leads to an optimal provision of pollution abatement. It has the advantage that the government does not have to raise revenues to subsidize the firms. Instead there will be a transfer between firms in equilibrium which guarantees that the social optimum is implemented.
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