Can you please explain the math in depth on how to solve part (a) of question #3 I am very confused.

(e) Is z a Pareto efficient allocation? Explain. (f) True or false: The First Fundamental Theorem of Welfare Economics indicates that a government intervention may be needed in order to ensure that Alessandra and Bruno don't end up consuming their precise initial endowments. Explain. 3. The contract curve in the Edgeworth economy (40 points total) The economy of Athe- nium is characterized by two goods, textbooks (x) and footballs (y), and two in- dividuals, the Professor and the Coach. An allocation z = (xp, yp, xc, yc) tells us how many textbooks the Professor gets, how many footballs the Professor gets, how many textbooks the Coach gets, and how many footballs the Coach gets. The Pro- fessor's utility from consuming xp textbooks and yp footballs is given by up ( Xp, yp ) = , In(xp) + - In(yp). The Coach's utility from consuming xc textbooks and yc footballs is given by uc(xc,yc) = 3 In(xc) + 5 In(yc). There are a total of 40 textbooks and 20 footballs in the economy. (a) Find an equation that describes the contract curve for this economy. The equa- tion must be written only in terms of xp and yp (no C's), and it must be written in the form "0 = ..." (b) Is z = (40, 20, 0,0) Pareto efficient? (c) Is z' = (10, 10, 30, 10) Pareto efficient? (d) Is z" = (16,5, 24, 15) Pareto efficient? (e) Is 2 = (20, 10, 20, 10) Pareto efficient? (f) Of the four allocations z through 2 that are defined above, which allocation (or allocations, in the event of a "tie") is most desirable according to the utilitar- ian social welfare function? (Note that In(0) cannot be calculated, but in this context we can assert that In(0) = -co. That is, "infinite unhappiness"!)