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 first step is to determine the values of `A_h` and `A_l`. That can be done by inserting the given information into each equation and solving for the parameters. For high income households:
`Q_h = A_h - B_h\cdotP^d`
`2237.5 = A_h - 150\cdot 1.75`
`2500 = A_h`
For low income households the process is similar and gives:
`1000 = A_l`
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:
`Q_m = 1M *Q_h + 1M*Q_l`
Substituting the parameters into the demand equations for `Q_h` and `Q_l` and then inserting those into the market demand and simplifying it gives:
`Q_m = 1M *(2500-150*P_d) + 1M*(1000-150*P^d)`
`Q_m = 2500M-150M*P_d + 1000M-150M*P^d`
`Q_m = 3500M-300M*P^d`
Without the tax, `Q_m` would be 2,975 million gallons. With the tax, `Q_m` drops to 2,900 million gallons. A graph of the market is shown below (see the "fine point" note at the bottom for a subtle point that doesn't affect the solution):
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 say anything directly 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 since the primary way to cut gas consumption is to drive less.
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, it's best not to make too much of that -- 0.7% and 0.9% are really very close together: they both round to 1%); (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.
Fine Point about the Market Demand Curve
Strictly speaking, the market demand curve has a kink where the price equals $6.67. That's the price at which a low income household's demand drops to zero. At higher prices, only the high income households buy gas. Working through the two cases:
Case 1: `P_d` below $6.67 (same as above)
`Q_m = 1M *Q_h + 1M*Q_l`
`Q_m = 1M *(2500-150*P_d) + 1M*(1000-150*P_d)`
`Q_m = 3500M-300M*P_d`
Case 2: `P_d` above $6.67 (no purchases by L households so `Q_l=0`)
`Q_m = 1M *Q_h + 1M*Q_l`
`Q_m = 1M *(2500-150*P_d) + 1M*(0)`
`Q_m = 2500M-150M*P_d`
The graph looks like this:
Accounting for the kink would affect the total value of CS but it doesn't affect the change in CS because the price is always below the kink. This is one reason why it's often better to calculate changes in surplus directly rather than calculating before and after values and subtracting.