A competitive industry produces a unit of sulphur dioxide with every unit of its output. The private marginal cost (PMC) in the industry is 2+(Q/2) when Q is the number of units of output. The social marginal benefit (SMB) of the industry's output is 38 - (7Q/2) when Q units are produced. Sulfur dioxide is a pollutant that causes marginal damage (MD) of Q/2 when Q units are produced. Show all your steps in all your calculations below. For each equation you use, explain why it is the right one to use.
a. Graph the SMB, PMC, and social marginal cost (SMC) curves for this industry.
b. Find the competitive equilibrium price Pc and quantity Qc in the industry.
c. Find the quantity of output Qe that maximizes total social surplus in the industry (taking into account all social benefits and costs of the output). Compare Qe to the equilibrium quantity Qc in part b and explain why the quantities differ.
d. Find a tax per unit of output to be paid by the producers, which would lead the competitive industry to produce in equilibrium the output level in part c.
e. What information would a government need in order to compute the tax in part d? Explain why it might be difficult for a government to obtain the necessary information.
f. Suppose that, instead of charging a tax, the gov requires all producers in the industry to buy permits, one for each unit of output they produce. The permits are perfectly divisible, so firms can buy fractional amounts of them if they wish. If the gov offers 7 perfectly divisible permits for sale, what would be the market price of output (the price that buyers in the industry would pay)?
What would be the price of a permit? Show all your work and explain your answer.
g. Suppose that in part f the gov offered the efficient number of permits for sale (the number that would maximize total social surplus in the industry). How many permits would the gov offer? What would be the resulting market price of output? What would be the price of a permit? Show all your work and explain your answer. Compare the permit price in this case to the tax in part d.
h. Now suppose that the marginal damage (MD) changes in the industry above. The SMB and PMC curves stay the same as before. MD(1) is still positive and MD is never negative, but with the new marginal damage function, the competitive equilibrium output Qc in part b is efficient (maximizes total surplus). Show how this is possible in a new graph of the SMB and PMC curves, with a new SMC curve. Explain why the competitive output is efficient with the marginal damage and SMC curve in your graph.