Consider the following functions, where x = c02 emissions

Problem 1. (20 points) Consider the following functions, where x=C02 emissions (in millions of tons). i) D(x)=A-ln (x),withA>0 ii) D(x) = B - ex,with B>0 iii) D(x) = x2 iv) D(x) = ax2 + bx + c if a=-1, [7:2, 6:0 and OSXSIOO A. Which of these four functions are, and which are not, legitimate social damage functions? B. Obtain the marginal damage function for iii) and draw it in a graph where the x-axis is emissions and the y axis is marginal damages. C. Consider the marginal abatement cost function AC = g - a = g - (E x), where E = maximum or current level of emissions. What is the socially optimal level of emissions if D(x) is the one in iii)? What conditions must be imposed on the parameters of the problem for this social optimum to make sense? D. Compute the total costs to society, broken into total damages and total abatement costs, as a function of the parameters, at the socially optimal emissions in item C. above