Consider a firm that causes pollution in a lake affecting a group of individuals. The production cost of the firm is c(q) = cq2/2, where c > 0 and q is the quantity produced by the firm, and the total cost imposed on the individuals (adding the costs suffered by each of them) is (q) = q2/2, where > 0. Suppose initially that the firm cannot bargain, trade, or otherwise compensate the individuals.

1. What is the profit maximizing choice of output by the firm?
2. What is the revenue of the firm?
3. What is the private cost of the firm?
4. What is the externality that the firm imposes over the individuals?
5. What is the total social cost?