Externalities. 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) = pq2/2, where p > 0. (a) Suppose initially that the firm cannot bargain, trade, or otherwise compensate the individuals. Calculate the profit maximizing choice of output by the firm, the revenue of the firm, the private cost of the firm, the externality that the firm imposes over the individuals, and the total social cost. (b) Suppose now that the individuals affected by pollution in the lake own the lake, and that they organize to sell pollution permits, with a price of p, so that the firm must pay p per unit of output. Find the competitive price of pollution permits, as well as the profit max- imizing choice of output by the firm if the firm must buy pollution permits and buys them at the competitive price.