(calculus problem) suppose that the utility function is linear u(x)=x and that the costs of global warming are quadratic in the current stock: c(g)=g2. The transition equation is the same as in the text: gt= gt-1+c1t+c2t. The next several exercises will compute the social optimum for this case.

21.5. Write down the expression for per-period net utility.

21.6. Formulate the social optimality problem when there is just one more period of emissions left. What is the socially optimal net utility level V1(g)?

21.7. Formulate the social optimality problem when there are two periods left. What is the socially optimal consumption rule? What is the socially optimal net utility level V2(g)?

21.8. Formulate the social optimality problem when there are three periods left. What is the socially optimal consumption rule? What is the socially optimal net utility level V3(g)?

21.10. What can you infer about the general case of T remaining emission periods? What if there are an infinite number of periods? In particular, show how the global optimal consumption level depends on the inherited GHG stock g. that is, the optimal strategy is not a constant strategy.

***please answer all questions (not using AI) and clearly show what final answer is (especially for 21.6-21.10)***