Peter J Wilcoxen > PAI 723 Economics for Public Decisions

Exercise 7

Due Monday 4/3

If the United States decided to try to reduce carbon dioxide emissions in order to slow global warming, one policy it might use would be a tax on fossil fuels. A central question that has been a subject of very intense debate is how damaging a tax like that would be to the U.S. economy. Believe it or not, in part the debate really boils down to an argument about what people's preferences look like. This exercise will explore why.

Suppose people buy two goods: energy, E, and all other goods, O. In 2014, total consumer spending, M, was about $12 trillion ($12,000 billion). About $600 billion of that was on energy and $11,400 billion on everything else. To keep things manageable, we'll focus only on consumers and will ignore the fact that energy is also used by firms. In addition, suppose the price of energy, Pe, is initially $1 and that the price of everything else is also $1 (that is, we choose the units of O so that each unit is worth $1).

  1. Suppose that consumers' preferences between goods can be represented by a Cobb-Douglas utility function of the form: U = (E^0.05) * (O^0.95). (Don't worry about the fact that there are lots of individual consumers--here and in the rest of the exercise, treat the consumers as one big consumer with an expenditure of $12 trillion.) It can be shown that the demand equations for energy and other goods corresponding to these preferences are: E = 0.05*M/Pe and O = 0.95*M/Po. Use this information to show that the consumers' expenditure function is given by:

    M = U * (Pe/0.05)^0.05 * (Po/0.95) ^ 0.95

    In this context, "show" means to derive the equation via algebra.
  2. Suppose the government imposed a 25% tax on energy so that Pe rises to $1.25. How much does energy demand decrease? What happens to the demand for other goods? Calculate the compensating variation for this policy.
  3. Now suppose that the utility function in part (1) is wrong and that consumers really regard energy and other goods as perfect complements. In particular, suppose that they always choose other goods, O, to be exactly 19 times their consumption of energy, E. That is, they choose O = 19*E. Using this relationship and the budget constraint, show (via algebra) that the consumers' demand equations for E and O are:

    E = M/(Pe+19*Po)
    O = 19*M/(Pe+19*Po)

  4. Suppose the government imposes the 25% tax on energy from part (2) when consumers have the preferences from part (3). How much does energy demand decrease? What happens to the demand for other goods? What is the compensating variation of the policy? (Calculate the C.V. in this case by finding the increase in income needed to allow consumers to buy their original amount of E and O and explain why that approach is appropriate.)
  5. A lot of the debate over climate and energy policy is essentially about whether preferences look more like part (1) or part (3). Compare the results from (2) and (4) and explain why the exact form of consumer preferences is of enormous practical importance to climate change policy. Don't just write down the numbers, think about what they really mean for the policy.

Additional Information

Compensating Variation
Web note giving graphical and numerical examples of compensating variation calculations.
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Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 03/27/2017