Pigouvian Taxes. (50 Points) Widgets are a necessary part of modern life, but they are also associated with pollution and pollution-related externalities. Consider the private market for widgets described by the following marginal benefit (MB) and marginal cost (PMC) curves:

=1000.1

=4+0.06.

Assume that output ( and ) varies from 0 to 1000. This problem does not explicitly require you to plot any curves - all answers (point estimates and welfare totals) can be derived using the functions provided and some basic algebra and geometry. However, I would suggest plotting them on your own to help visualize the problem and the areas you need to calculate.

1. a) Solve for the optimal private level of production, , and the optimal private market-clearing price, .
2. b) Calculate consumer and producer surplus (social welfare) for the private market outcome.
3. c) Assume that each widget that is produced generates a pollution externality that is associated with the following "Marginal Damage" function (MD): =16. Incorporate this externality into the problem and calculate the value of the externality as well as the Net Social Benefits that takes account of this externality (Net Social Benefits = CS+PS-Externality).
4. d) Assume that the government implements a tax on widgets that fully internalizes the externality. Using the SMC (PMC+MD) and a tax equal to the externality (marginal damage associated with each unit of widget production), calculate the new consumer and producer surplus, the new externality, the total tax revenue, and the net social benefits from the implementation of the tax. Is society better or worse off following the tax?
5. e) After deriving all the previous functions and making your calculation for the socially optimal outcome (part d.), you realize the widget industry has a large political lobby that has likely resulted in a tax transfer equal to a $4 per-unit tax subsidy. Using this information, adjust your results in part (d.). After building in the subsidy do you find that society still better off with your current Pigouvian tax, relative to the results in part (c.)? Are they better off relative to the "true" socially optimal outcome?