Here are notes on the solution. Some of the algebra is omitted and the explanations are a bit terse. If you have any questions, please don't hesitate to stop by during office hours or the lab session to talk over things in detail.
The overall market demand is the sum of the demands by all 2 million households. Since there are 1 million (1M) of each type, the total quantity demanded can be written:
Qm = 1M *Qhi + 1M*Qli
Inserting the equations for Qhi and Qli and simplifying:
Qm = 1M *(2500-150*P) + 1M*(1000-150*P)
Qm = 2500M-150M*P + 1000M-150M*P
Qm = 3500M-300M*P
Expressing the coefficients 3500M and 300M in terms of billions gives:
Qm = 3.5B - 0.3B*P
Without the tax, Qm would be 2,975 million gallons. With the tax, Qm drops to 2,900 million gallons. The graph looks like this:
CS before the tax is areas A+B+C; after the tax, it is only area A; the change in CS is thus -B-C. Here are the corresponding numbers:
Lost CS = B + C
Lost CS = $0.25*2,900 million gallons + (1/2)*$0.25*(75 million gallons)
Lost CS = $725 million + $9.375 million = $734.375 million
Area B in the diagram is tax revenue. The government collects $0.25*2,900 million gallons = $725 million. Area C is lost by consumers but not gained by producers or the government. It is, therefore, deadweight loss. The dollar value of DWL is (1/2)*$0.25*75 million gallons = $9.375 million.
This step is a bit subtle. Strictly speaking, the calculations above only provide information about the amount of gasoline used -- they don't directly say anything about traffic congestion. However, in the short run it's probably reasonable to assume that a reduction in gasoline consumption will cause a proportional reduction in the amount of driving people do.
In percentage terms, overall gas consumption drops by 75/2,975 = 2.5% so it would be reasonable to assume that people will drive 2.5% less. That's a pretty small change and isn't likely to do much about congestion. To put it in perspective, that's like taking one car in every 40 off the road. (2.5 cars in 100 is the same as 1 car in 40).
The following wasn't part of the problem but you might find it interesting. Over longer periods of time, people respond more vigorously to persistently high prices. They buy smaller, more fuel-efficient vehicles, they move closer to public transportation, and they form car pools, etc. The demand elasticity, in other words, is considerably higher in the long run.
The results for each group can be computed from each demand curve. See table below:
|High Income||Low Income|
|Q before||2237.5 gal
|Q after||2200 gal
|Change in Q
||-37.5 gal||-37.5 gal|
|Change as percent||-1.7%||-5.1%|
For the high income group, the ratio is 4.69/550 = 0.0085 = 0.85%. That means that each dollar of revenue costs consumers $1.0085: the $1 of revenue plus an additional $0.0085 of deadweight loss. That's actually pretty good for a real-world tax: the loss of surplus per dollar of revenue is usually much higher: 10% or more.
For low income households, the ratio is 4.69/175 = 0.0268 = 2.68%. Raising a dollar of revenue from those folks costs them a total of $1.0268 in surplus. The DWL per dollar of revenue is thus about 3 times higher for low income households than it is for high income ones.
The ratio is more favorable (in the sense of lower DWL per dollar of revenue) for the high income group because its demand is relatively insensitive to price (the elasticity is low). There is only a small percentage change in consumption, which has the dual effects of keeping revenue high and DWL low.
Some key points that are important to mention are: (1) the tax falls more heavily on the high income group in terms of absolute revenue: they are paying about 3 times as much as the low income group ($550 vs $175); (2) the tax is slightly regressive because it falls more heavily on low income group as a percentage of income: the rich pay $550/$80,000 or 0.7 percent of their income while the poor pay $175/$20,000 or 0.9 percent (however, don't make too much of that--0.7% and 0.9% are really very close together); (3) the tax has a much larger proportional effect on consumption by the low income group: 5.1% rather than 1.7%; and (4) the DWL suffered by both groups is equal even though incomes are very different.
Supporters of the tax would be likely to argue that it raises a lot of revenue without causing very much deadweight loss and that it is not too regressive: the rich are bearing most of the burden in absolute terms. An opponent would probably play up the regressivity, arguing that the tax does disproportionate harm to low income people, both in terms of taxes paid as a share of income and in terms of the percentage change in consumption. All of those points are valid and would need to be considered by political leaders.