Question

1. Pigovian Taxes

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

?B = 100 − 0.15??

PMC = 4 + 0.06??.

where ???? is the quantity demanded and ???? is the quantity supplied. Assume here that output (?? and ??) varies from 0 to 1000. After plotting these curves, use them to answer the following

questions. (This question requires graphing for several parts—you can put all of your curves on the same figure. At the very least, I would like for you to try using Excel to plot the lines.)

1. a) (5points)Generate a graph of the MB and PMC curves.

2. b) (5 points) Solve for the optimal private level of production ???? and the optimal private

market-clearing price ???? and plot these values on your graph.

c) (5 points) Calculate both the consumer and producer surplus (social welfare) for the private market outcome and indicate their location on your graph. (It may be easiest to divide the area of the graph into subsections and indicate the different values using the sums of these different areas.)

d) (10 points) Now assume that each widget that is produced generates a pollution externality that is associated with the following “marginal damage” function (MD):

???? = 0.04????.

Incorporate this externality into the problem from parts (a)-(c); plot the MD function and the new “social marginal cost curve” (SMC); and calculate the value of the externality (i.e. the total external costs associated with ??) as well as the net social benefits in the presence of this externality (net social benefits = CS + PS - External Costs).

e) (10points)Recallinclasshowwecomparedthewelfareoutcomeof????withthesocially optimal quantity ??∗. Assume that the government implements a Pigovian tax on widgets that fully internalizes this pollution externality (i.e. the tax is set in order to shift the supply curve so that ??∗ rather than ?? is the equilibrium quantity). Under this tax, calculate the new consumer and producer surplus, the new level of external costs, the total tax revenue, and the net social benefits from the implementation of the tax. Indicate the tax revenue and externality areas following the implementation of the tax using your graph. Is society better or worse off following the tax?

Answer 1

To get the same graphs as I have, I suggest you plug in values in an Excel sheet.

In the first column, enter all values from 0-1000 using Ctrl+Drag. In the first cell of the second column (suppose we want values of MB there), enter: "= 100 - (0.15*A1)". Again, press Ctrl, click and drag along the row. You'll get values of MB for the given range. In the first cell of the second column, enter "=4 + (0.06*A1)" and repeat the above process. You'll get values of PMC. Thus, you have three columns filled with necessary data. This will look like:

Once you have the data, plotting the graph is easy. Select the second column to plot MB. In the Insert tab, you'll find an option for Line Chart (under the heading Charts). The graphs are as shown:

1) For MB vs Q:
2) For PMC vs Q: