A what is the optimal quantity to be provided of the public

Question

Alice, Beth, and Carol are the only three demanders of a pure public good. Their demand functions are as follows:

P_a = 70 - G,

P_b = 110 - G,

P_c = 130 - G.

G measures the number of units of the good and p_i is the price in dollars. The marginal cost of the public good is a constant $190.

a. What is the optimal quantity to be provided of the public good?

b. If the good is financed by a tax that collects how much each is willing to pay for the optimum quantity you calculated in (a), how much will Alice pay? Beth? Carol? Explain.

c. If instead this good is sold privately in a competitive market, how many units will be supplied if each person chooses to buy her preferred amount? Explain.

d. If the public good is not supplied at all, what is size of the deadweight loss arising from this market failure?