Hello I need assistance with this exercise. It's about estimating the efficiency levels of pollution and taxes. Thanks

Environmental Economics 4. Since the regulator does not, a priori, know the marginal abatement cost function, the best that she can do is to use the expected value. If she were to establish a tradable permit program, how many emission Suppose that the total abatement cost function for a firm is C(e) = (3+p)q-, where q is the amount of emissions permits would she issue? (Hint: You need the emission, e, at the point where your marginal abatement controlled (that is, the q is the amount of abatement). Uncontrolled, the firm would produce 2 units of cost function and your marginal damage functions intersect in the graph above.) emissions. Thus, q-2-e, where e is emissions (in tons). The variable r is a cost component unknown to the pollution control board. All they know is that there is a 50/50 chance that it could take the value of either 1=0 or 1=4. Marginal damages from emissions is given by MD(e)=4e. 1. Write the total cost of pollution control in terms of e. This is done by substituting the equation for q into the C(e) equation. You will need to simplify this equation into a second order polynomial in e. Report 5. If, instead, she were to establish an emission tax, what tax rate would she choose? Show your work that equation here. here. (Hint: Since you know the optimal emissions from part 4, you can use one of your equations to solve for the tax.) 2. Solve for the two possible marginal abatement cost functions, -C'(e). This requires taking the derivative of the total abatement cost function from part 1 with the each value for r, and multiplying each equation 6. Suppose that after the tradable permit program or emission tax was set, it turns out that 1=4. Show the times negative one [Remember, it is -C'(e), not C'(e)]. Also calculate the expected value of the deadweight loss from both a permit program and a tax on your graph, assuming that cannot be changed. marginal abatement cost function by using the average of the two possible values for r. Report those Which instrument appears to be better? (You don't have to necessarily solve for the area of the equations here. deadweight loss triangle. If you have drawn carefully, the answer should be obvious from the graph.) 3. Graph and label the three marginal abatement cost functions you reported in 2. Graph this with 7. How could you have known which instrument was better without having to graph it or mathematically emissions, e, on the horizontal axis and dollars on the vertical axis. Graph the marginal damages from solve for the area of deadweight loss? emission on the same graph