Here is a brief summary of the solution.
- Here are the steps:
`g = (P_e Q_e)/M = ($600B)/($12000B) = 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`
- `Q_e` = 480 billion, a 20% decrease. `Q_o` = 11,400 billion, unchanged from the initial amount. Original U = `600^0.05 * 11400^0.95` = 9839 billion; expenditure needed to get the original utility at new prices `M_3`= $12,135 billion; the CV is therefore $135 billion; revenue is 0.25*480 billion = $120 billion; the net cost of the policy (the CV less the revenue) is thus $15 billion.
- Solving for the demand equations:
`b = Q_o/Q_e = (11400B)/(600B) = 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!)
- `Q_e`= 593 billion, a drop of 1.2%. `Q_o` = 11,259 billion, a drop of 1.2%. The percentage drops are equal as a consequence of the perfect complements preferences. Revenue is 0.25*593 billion = $148 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*600 billion + $1*11,400 billion = $12,150 billion; the CV is thus $150 billion; and the net cost of the policy is $2 billion.
- The table below summarizes the key results from the two cases. In a nutshell, if preferences are close to Cobb-Douglas, the policy would be effective at reducing energy use, would raise a fair amount of revenue, and would have a modest net cost before accounting for climate benefits. If preferences are closer to perfect complements, the policy will have a much smaller impact on energy consumption but it will raise more revenue and the overall cost will be smaller than under Cobb-Douglas.
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 CS. Put in quantitative terms, a $135 billion CS cost for a 20% reduction is a $6.75 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 CS: in this case $150 billion for a 1.2% is $125 billion per 1% reduction, or almost 20 times as costly).
|Impact on energy use, %
|Cost in CV, billion $
|Tax revenue raised, billion $
|Net cost before climate benefits, billion $
However, the CS calculation overlooks the tax revenue, which is substantial. Taking that into account, the analysis is surprisingly optimistic. Either the policy will work pretty well (the CD case) or 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 and ineffective.
The actual situation in the US lies between the two cases but is much closer to Cobb-Douglas than perfect complements.
An interesting extension is to ask the question: "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 480 billion (a 20% reduction from 600 billion) for `Q_e`, $1 for `P_o` and $12,000 billion for M and then solve for `P_e`:
`E = M/(P_e + 19*P_o)`
`480 = 12000/(P_e + 19*P_o)`
`P_e + 19*P_o = 12000/480`
`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*480 million = $2,400 billion
`M_3` = $6*600 + $1*11,400 = $15,000 billion
CV = $15,000 - $12,000 = $3,000 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 ($600 billion) before accounting for climate benefits achieving the same reduction when consumers have Cobb-Douglas preferences.
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Peter J Wilcoxen, The Maxwell School, Syracuse University