please provide step by step solution please

Problem Set 3 1. There are two consumers, 1 and 2, and one firm. There is one private good, x, and one public good, z. The price of good x is Px = 1. Let Pz denote the price of the public good. Consumer i's preferences are given by Wi(Xi, z) = Xi + In(Z1 + z2). Consumer i's income is wi = 10. Here xi and zi denote, respectively, the quantities of the private and the public good purchased by agent i. The total amount of public good consumed by the agents is z = 21 + 22. The firm's cost of supplying z units of the public good is c(z) = 22. (a) Solve for the competitive equilibrium in the market for the public good. That is, find the equilibrium price p and the aggregate equi- librium quantity z* of the public good. What are z1 and z2 in equi- librium? (b) Represent the competitive equilibrium graphically. (c) Is z* inefficient? If so, why? What is the efficient allocation of the public good? Represent it graphically. What is the deadweight loss at the competitive equilibrium? (d) Show that there is a subsidy si to each consumer i that implements the Pareto efficient level of z. (e) How does the analysis change if the firm is a monopolist? 2. Same as 2 with the following utility functions: ul (21, z) = x1 + 2 In(21 +Z2) and u2(202, z) = 202 + In(21 + Z2)