need step by step explanations thx

Question 2. (15 marks) In this question you will study market provision of a public good. Consider two types of household with preferences over a public good x and a composite good y, Household A: UA(x, y4) = 20x1/2 + y Household B: UB (x, y#) = 32x1/2 + yB Where households rank choices according to the total amount of the public good which is purchased privately, x = x4 + x, and their private consumption of the composite, y' for i = A, B. Households' feasible choices satisfy their budget constraints, Household A: px4 + yA = 120 Household B: px + y = 150 Finally, the industry cost function is C (x) = 4x. i) (3 marks) If each household can purchase the public good privately, find each household's inverse demand function for the public good. (This is equivalent to their marginal benefit function, how much they would be willing to pay for a marginal change in quantity of x.) ii) (6 marks) Use your answer in i) to determine which of the following allocations could be market equilibria if we use the Nash Equilibrium concept: a) {(x4, y4), (xB, y ),) = {(6.25,95), (9.75,111)} b) {(x4, y4), (xB, yB), ) = {(6.25,95), (0, 150)} ifi) (6 marks) Demonstrate that the Nash Equilibrium (market) allocation ((x4, y4), (xB, y'),) = {(0,120), (16, 86)] is inefficient by finding a planner's choice that make everyone strictly better off when the planner can allocate the sum of the households' endowments of composite (120 + 150 = 270), and pays for the conversion of composite into public good according to C(x) = 4x