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Assume that the current stock of CO2 held in the atmosphere is s0 in Gt, and evolves according to: st=st1+ct where s, is the stock at the end of period t. To simplify the analysis, consider 50 -year increments, so that t=0 is today, t=1 represents the next 50 years, t=2 the following 50 years and so on. The persistence rate of CO2 in the atmosphere is (the amount remaining after 50 years). Assume that emissions are constant at et every 50 years. The total damages over a 50 year period are: Dt=0.02st2 where st is the end-of-period stock of CO2. 1. Write the stock of CO2 in the atmosphere at the end of period t=1,2,3 as a function of s0 and emissions in each period. 2. Find an expression for the marginal damage done in period t=3 from emissions released during period 1,e1. 3. Suppose that s0=4000 and emissions each period are constant at et=2000 for t=1,2,3. Also assume that =0.8. How much should society as a whole be willing to pay to reduce emissions in period 1 by a small amount, based on the damage that these emissions would do over 150 years? Assume that the social discount rate for the 50 year period is 0.6. and that marginal damages are measured at the end of each period (so that we discount t=1 once). 4. Now suppose that scientists are uncertain about the damage function. With probability 0.4, the total damages are low at: DiL=0.005st2 and with probability 0.6 the total damages are high: DtH=0.03st2 All other assumptions remain the same as above. Now how much should society as a whole be willing to pay to reduce emissions in period 1 by a small amount, based on the damage that these emissions would do over 150 years