short and concise answers please

just as a general help to you

"First Welfare Theorem

Suppose that there are markets and market prices for all the goods, that all the people are competitive price takers, and that each person's utility depends only on her own bundle of goods. Then any competitive equilibrium allocation is Pareto optimal. In fact, any competitive equilibrium allocation is in the core."

"Second Welfare Theorem says that any Pareto optimal point is a competitive equilibrium, given the appropriate modification of the traders' budget constraints."

In the remote planet called Jupifar, far removed from any star, it is totally dark. There are only two intelligent living beings, called Alpha and Omega. Thus, consider a pure exchange economy with these two consumers. Consumer 1 is Alpha and consumer 2 is Omega. There are two goods (matches r and lighters y). Alpha's endowment is the bundle (10, 0), while Omega's is the bundle (0, 20) -these goods are measured in hours of light for their individual ice cabins. Consumer I's utility function is u(1, y1) = min (1, 31 ), while consumer 2's is uz(12, y2) = VI2 + Vy2. (It is not clear that, due to Jupifar's atmospheric conditions, matches or lighters will work, but we will assume they do, or at least that this is what Alpha and Omega think - remember this is an economics course, not a physics or chemistry course!) (a) Represent this economy (preferences and initial endowments) in the Edgeworth box. (b) Calculate the competitive equilibrium of this pure exchange economy. You should indicate final consumption of each agent and the equilibrium prices (remember that you can normalize the price of one good to be 1, py = 1 for example). Can you check that the conclusion of the first welfare theorem holds? (c) The Interplanetary Authority does not like the allocation of goods in the existing equilibrium. In particular, it proposes that Alpha should receive the bundle (35/4,35/4) and Omega should receive the bundle (5/4, 45/4). Is the Interplanetary Authority efficiency-minded, i.e., is that proposal Paretoefficient? Could it be implemented using the competitive market system after some suitable transfer of good y? (Suppose that, for some reason, no ex-ante transfer of x is feasible for the Authority.) That is, as an illustration of the second welfare theorem, you are asked to find the exact transfer required and the new competitive equilibrium that would support the allocation being proposed