Here we consider a negative externality. Specifically, we have a market where each unit of quantity (q) produced creates h <= 1 units of harm for third parties (neither consumers nor producers). The market for q, where production occurs at constant marginal cost 1, is competitive. Consumers, each of whom has income y, have preferences represented by the utility function q(2 q/2) + x, where x is consumption of all other goods. Let p denote the price of q. Normalize the price of x to one.

(1) Assuming for the moment that the producers do not cooperate with the third parties, what is the supply curve in the q market?

(2) Assuming for the moment that the consumers do not cooperate with the third parties, what is the demand curve in the q market? The equilibrium price and quantity?

(3) What is the surplus, separately for consumers, producers, and third parties as a function of the amount q produced in the market? Surplus refers to their welfare, profits, or utility defined relative to a situation where q = 0. In your analysis, be sure to refer to the utility function.

(4) What is the quantity that maximizes aggregate surplus and how does it compare to the no-cooperation equilibrium quantity you calculated in part (2)?

(5) There is a tax t*q levied on consumers that could induce them to choose the optimal quantity you calculated in part (4). That rate t is known as the "Pigouvian tax rate." Calculate the rate and explain how it is related to the amount of third-party harm h.

(6) Suppose that the Pigouvian tax rate is in place. Are third parties still harmed by production of q? To the extent that they are harmed, does that mean that production of q is too high?

(7) How much better or worse off is the consumer with the Pigouvian tax? The third parties? What do these findings tell you about how each group would try to influence the "science" of estimating the optimal tax rate? That is, what are the various parties' costs or benefits of a government that over- or under-estimates the Pigouvian rate?

(8) Next, draw a supply-demand diagram, and ignore the demand curve for the moment. Is there a point on the supply curve that makes both consumers and third parties better off? How does this point compare to the Pigouvian-tax equilibrium?

(9) If third parties and, say, consumers wanted to coordinate to implement a point on the supply curve that is not on the demand curve, what are some of the challenges they would face? How are those challenges related to the gap (in the price dimension) between the supply curve and demand curve?

(10) Assume that the cost of overcoming the cooperation challenges is a quadratic function

of the gap between the supply and demand curves. Draw the cooperation equilibrium in the supply-demand diagram and show how it relates to the no-cooperation equilibrium.

(11) If the cooperation equilibrium cuts the quantity back halfway from the no-cooperation point to the socially optimal point, how do the deadweight costs of excess production compare to those without cooperation (and without any Pigouvian tax)?