Solution

Here is a brief summary of the solution.

1. Here are the steps:
   
   `g = (P_e Q_e)/M = ($850B)/($17000B) = 0.05`
   
   `U = Q_e^0.05 Q_o^0.95`
   
   `U = ((0.05*M)/P_e)^0.05 ((0.95*M)/P_o)^0.95`
   
   `U = (0.05/P_e)^0.05 (0.95/P_o)^0.95 * M`
   
   `M = U/( (0.05/P_e)^0.05 (0.95/P_o)^0.95 )`
   
   `M = U * (P_e/0.05)^0.05 (P_o/0.95)^0.95`
2. `Q_e` = 680 billion, a 20% decrease. `Q_o` = 16,150 billion, unchanged from the initial amount. Original U = `850^0.05 * 16150^0.95` = 13,939 billion; expenditure needed to get the original utility at new prices `M_3`= $17,191 billion; the CV is therefore $191 billion; revenue is 0.25*680 billion = $170 billion; the net cost of the policy (the CV less the revenue) is thus $21 billion.
3. Solving for the demand equations:
   
   `b = Q_o/Q_e = (16,150 B)/(850 B) = 19`
   
   `M = P_e*Q_e + P_o*Q_o` (budget constraint)
   
   `Q_o = 19*Q_e` (from preferences)
   
   `M = P_e*Q_e + P_o*19*Q_e`
   
   `M = (P_e+19*P_o)*Q_e`
   
   `Q_e = M/(P_e+19*P_o)` (demand for E)
   
   `Q_o = (19*M)/(P_e+19*P_o)` (demand for O)

   Check by plugging the demand equations into the budget constraint:

   `M = P_e*(M/(P_e+19*P_o)) + P_o*((19*M)/(P_e+19*P_o))`
   
   `M = (P_e*M + P_o*19*M)/(P_e+19*P_o) = (P_e + 19*P_o)/(P_e+19*P_o)*M = M` (passes check!)
4.   `Q_e`= 840 billion, a drop of 1.2%. `Q_o` = 15,951 billion, a drop of 1.2%. The percentage drops are equal as a consequence of the perfect complements preferences. Revenue is 0.25*840 billion = $210 billion. To get the original utility, consumers need to be able to purchase the original consumption bundle -- that's the point at the corner of their indifference curve. The cost of that bundle is $1.25*850 billion + $1*16,150 billion = $17,213 billion; the CV is thus $213 billion; and the net cost of the policy is $3 billion.
5. The table below summarizes the key results from the two cases. In a nutshell, if preferences are close to Cobb-Douglas, the policy would: (1) be pretty effective at reducing energy use, (2) raise considerable revenue, and (3) have a modest net cost before accounting for climate benefits. If preferences are closer to perfect complements, the policy will be: (1) largely ineffective at reducing energy consumption, but (2) will raise more revenue, and (3) will have a smaller overall cost than under Cobb-Douglas.
   
   

   Variable	CD	PC
   Impact on energy use, %	-20	-1.2
   Cost in CV, billion $	191	213
   Tax revenue raised, billion $	170	210
   Net cost before climate benefits, billion $	21	3

   Different views on preferences are thus an important part of the debate over climate policy. People in the public who instinctively believe things are closer to Cobb-Douglas (although they may not know the technical term) argue that an energy tax would work in the sense that it would reduce consumption significantly and would not be too costly in terms of CV. Put in quantitative terms, a $191 billion CV cost for a 20% reduction is a $9.55 billion per 1% reduction. People who instinctively believe things are closer to perfect complements argue that the tax would ineffective and costly in terms of CV: in this case $213 billion for a 1.2% is $178 billion per 1% reduction, or almost 20 times as costly.
   
   However, the CV alone overlooks the tax revenue, which is substantial. Taking that into account, the analysis is quite optimistic overall. Either: (1) the policy will work pretty well at lowering energy use (the CD case) or (2) its overall cost will be very low (PC). It's possible to rule out the worst possible outcome: that the policy would be both expensive (in net terms)  and ineffective.
   
   The actual situation in the US lies between the two cases but is much closer to Cobb-Douglas than perfect complements.

Addendum:

An interesting question to ask is: "How large would an energy tax have to be in order to reduce emissions by 20% when people have the second set of preferences?" That is, how much harder is it to get the same energy reduction as in part 2? It's easy to figure that out using the demand equation. Insert `Q_e=680` billion (a 20% reduction from 850 billion), `P_o=$1` and `M=$17,000` billion and then solve for `P_e`:

`E = M/(P_e + 19*P_o)`
`680 = 17000/(P_e + 19*P_o)`
`P_e + 19*P_o = 17000/680`
`P_e = 25 - 19*1`
`P_e = 6`

This says the price of energy would have to rise from $1 to $6, which implies that the tax rate would have to be 500%. The revenue and CV of such a tax would both be high:

Revenue = $5*680 million = $3,400 billion

`M_3` = $6*850 + $1*16,150 = $21,250 billion

CV = $21,250 - $17,000 = $4,250 billion

Those numbers are all in billions: the revenue and CV, in other words, are in trillions of dollars. The upshot of this is that if consumers regard energy and other goods as perfect complements, a 20% reduction in fuel use would require a very large tax and would have a much larger net cost ($850 billion) before accounting for climate benefits achieving the same reduction when consumers have Cobb-Douglas preferences.