Here is a brief summary of the solution.
- Here are the steps:
U = E^0.05 * O^0.95
U = (0.05*M/Pe)^0.05 * (0.95*M/Po)^0.95
U = (0.05/Pe)^0.05 * (0.95/Po)^0.95 * M
U/( (0.05/Pe)^0.05 * (0.95/Po)^0.95 ) = M
M = U * (Pe/0.05)^0.05 * (Po/0.95)^0.95
- E = 480 billion, a 20% decrease. O = 11,400 billion, unchanged from the initial amount. Original U = 600^0.05 * 11,400^0.95 = 9839 billion; expenditure needed to get the original utility at new prices (using the expenditure function above) = $12,135 billion; CV is therefore $135 billion.
- Solving for the demand equations:
M = Pe*E + Po*O (budget constraint)
O = 19*E (from preferences)
M = Pe*E + Po*19*E
M = (Pe+19*Po)*E
E = M/(Pe+19*Po) (demand for E)
O = 19*M/(Pe+19*Po) (demand for O)
Check by plugging the demand equations into the budget constraint:
M = Pe*(M/(Pe+19*Po)) + Po*(19*M/(Pe+19*Po))
M = (Pe*M + Po*19*M)/(Pe+19*Po)
M = M*(Pe+Po*19)/(Pe+19*Po) = M (passes check!)
- E = 593 billion, a drop of 1.2%. O = 11,259 billion, a drop of 1.2%. The percentage drops are equal as a consequence of the perfect complements preferences. 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.
- If preferences are more like the first case, where consumers are willing to trade off energy and other goods, the energy tax would be effective at reducing fuel use (E drops by 20%) and would reduce welfare by $135 billion. However, if consumers preferences are closer to the perfect complements case, the tax would be more costly (a CV of $150 billion) and much less effective at reducing fuel use (E only drops by 1.2%).
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 (here $135 billion 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 this case $150/1.2% is $125 billion per 1% reduction, or almost 20 times as costly).
The actual situation in the US lies between the two cases but is much closer to Cobb-Douglas than perfect complements. Also, real energy tax policies usually involve reductions in other tax rates, which lessens the CV of the policy.
An interesting extension to this exercise 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 E, $1 for Po and $12,000 billion for M and then solve for Pe:
E = M/(Pe + 19*Po)
480 = 12,000/(Pe + 19*Po)
Pe + 19*Po = 12,000/480
Pe = 25 - 19*1
Pe = 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 CV of such a tax would be high:
M3 = $6*600 + $1*11,400 = $15,000 billion
CV = $15,000 - $12,000 = $3,000 billion
Those numbers are all in billions: the CV, in other words, is $3 trillion. 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 gigantic tax and would have a very adverse affect on welfare (the large CV).
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Peter J Wilcoxen, The Maxwell School, Syracuse University