Due Wednesday 2/6
A couple of years ago the Washington metropolitan area had a large budget deficit and a severe problem with traffic congestion. One suggestion put forward was to increase the gasoline tax by 25 cents. This exercise is loosely based on that proposal.
Suppose that for the purposes of gasoline demand, Washington households can be roughly divided into two groups: those with high income (h) and those with low income (l), and that there are 1 million households of each type. The demand for gasoline by an individual household of each type is believed to be linear, as shown below, where `A` and `B` are parameters that vary between high and low income households:
`Q_h = A_h - B_h*P^d`
`Q_l = A_l - B_l*P^d`
In addition, suppose that a government agency, call it the "Bureau of Statistics," reports that the price of gasoline to buyers, `P^d`, is currently $1.75 per gallon, `Q_h` is currently 2237.5 gallons, `Q_l` is currently 737.5 gallons, and both `B_h` and `B_l` are 150 gallons per dollar.
To keep things simple, throughout the exercise you may assume that: (1) the tax increase applies to the entire metropolitan area, and (2) that people can't evade it by buying gas outside DC.
- Please use the data given above to determine the values of `A_h` and `A_l` and then show algebraically that the overall market demand for gasoline by all households in Washington (`Q_m`) is given by the following equation, where "M" indicates millions (`10^6`):
`Q_m = 3500M - 300M*P^d`
Just to confirm the units, if `P^d` were $1, `Q_m` would be 3200 million gallons.
- Now consider a $0.25 increase in the gasoline tax. You may assume that the tax causes the price of gas to buyers, `P^d`, to rise to $2.00 (in other words, the supply curve is horizontal). Draw a clearly labeled diagram of the overall market showing consumer surplus before and after the tax. Be sure to show the value of `Q_m` with and without the tax. What is the dollar value, in millions, of the change in CS?
- How much revenue will the tax raise in millions of dollars? Also, calculate the dollar value of the deadweight loss (also in millions).
- How much is the tax likely to reduce traffic congestion? In order to answer this you'll need to think about how (1) changes in gasoline consumption, which you know about from above, are likely to translate into (2) changes in the amount of driving people do, which determines congestion. You'll then need to make an assumption about that relationship in order to extend your gasoline results to a rough conclusion about congestion. Be sure to state your assumption.
- Now calculate the following items for a typical household of each type: quantity demanded before the tax increase; quantity demanded after the tax increase; additional tax revenue paid (due to the 0.25 increase); deadweight loss; and the percentage change in quantity demanded. Present the results in a table.
- The ratio of the deadweight loss from a tax to the amount of revenue it raises is often used to measure the relative efficiency of the tax: it is the deadweight loss per dollar of revenue. An ideal tax would have zero deadweight loss and thus a ratio of 0; a horribly inefficient tax might have a ratio of 0.5, which would indicate that each dollar raised in revenue caused $0.50 of deadweight loss (that is, collecting a dollar of tax revenue causes consumer surplus to drop by $1.50: $1 of taxes and $0.50 of deadweight loss. What is the ratio of deadweight loss to tax revenue for each of the two groups? Discuss why the ratio is so much more favorable for the high income group.
- Finally, suppose that a typical low income household earns $20,000 per year and a typical high income households earns $80,000. Is the tax likely to be considered fair? Please discuss. There are arguments to be made on both sides; for this exercise it's much more important to anticipate what the arguments would be than to reach a definite conclusion.
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Peter J Wilcoxen, The Maxwell School, Syracuse University