Deadweight Loss and Excise Taxes

Question

I am currently reading through the 5th Edition of Economics, by Paul Krugman and Robin Wells. Right now I am learning about excise taxes and how their different aspects (government revenue, deadweight loss, reduction in consumer/producer surplus) can be represented on a supply and demand diagram.

In relation to the diagram featured in Figure 7-8 below, the book explains that “a portion of the loss to producers and consumers from the tax is not offset by a gain to the government—specifically, the two triangles B and F. The deadweight loss caused by the tax is equal to the combined area of these two triangles.”

This explanation makes a lot of sense, and is in keeping with what I have learned about deadweight loss up to this point. However, in the 'Check Your Understanding' section of this particular chapter (featured below), an answer to one of the questions seems to completely contradict this explanation. Allow me to elaborate.

Question 1f is the one that I am having trouble with.

The described supply and demand schedule and the excise tax imposed generates the supply and demand diagram featured below, where the yellow triangle represents the deadweight loss created by the imposition of the excise tax:

To clarify some of the components of this diagram, the 0.40 USD excise tax drives a wedge between the price paid by consumers and the price received by producers, equal to the value of the tax. This reduces the quantity transacted from 4 units at the equilibrium level to 2 units (represented by the vertical dashed line).

Based on the explanation of deadweight loss which I earlier cited, it seems to me that the deadweight loss here should be equal to 0.40 USD:

However, in the answers section (section S-7 of the linked PDF), it is said that the deadweight loss is actually 0.20 USD:

Though I can see the logic of subtracting the government revenue and the total surplus after the tax from the total surplus before the tax to arrive at the deadweight loss, the answer provided seems to contradict the explanation given earlier — either the deadweight loss is equal to 0.20 USD, or the deadweight loss is equal to the area of the shaded triangle (0.40 USD). From my perspective, it does not seem that these two assertions can be simultaneously true.

I hope that I have provided enough information for readers to be able to figure out whether I have made a mistake here, or whether this is actually an error in the textbook.

Answer 1

The problem is that your supply-demand graph is wrong. You assume in your graph that the demand and supply are linear functions but in this case they are step functions.

For example, in your graph there would be 2.5 demand at price $\$0.55$ but the problem clearly states that the units are sold at discrete quantities of 1. So the drawing is incorrectly modeling the situation as there should be a steps in increases in demand not a simple linear function. The same holds for supply.

If you draw your supply and demand correctly you will see that in your case deadweight loss (DWL) is not a triangle but just two squares with areas 0.10 respectively which combine into a rectangle with area 0.2 and hence total DWL will be also $\$0.2$.

This is how your problem would look visualized (pardon the bad drawing):

Where the shaded area is the DWL. The two methods will always match but you have to use the same demand and supply functions in both problems if you assume different supply and demand of course results will change (in your original drawing also consumer surplus after tax would be wrong by the way as in your original drawing it would be area of two trapezoids both having areas of 0.15, and the pre tax consumer surplus in your picture would also not be 1.2).