Compensating Variation
- II.3: Solve multivariable constrained optimization problems with Lagrangians; e.g., utility maximization, expenditure minimization, profit maximization with inputs, cost minimization.
- III.C.9: Determine the change in a consumer’s well-being from a price change. Explain why it is not given in terms of utility.
- III.C.10: Identify compensating variation, equivalent variation, and consumer surplus and when they are used.
Rohan’s preferences are represented by the following utility function: \[U(x) = x_1x_2^2\]
The price of good 1 is $p_1$, the price of good 2 is $p_2$, and his income is $I=12$. His utility maximization problem is shown in the above figure.
- Derive Rohan’s (Marshallian) demand functions and the indirect utility function. Use a Lagrangian and show your work. Suppose $p_1 = 1$ and $p_2 = 1$. Determine his optimal consumption bundle using the graph. Check this bundle algebraically.
Now suppose that the government imposes a tax of 3 dollars per unit on good 1 so Rohan now has to pay $p^\prime_1 = 4$.
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Based on the figure above, how much revenue would be raised?
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How much less utility does Rohan achieve due to the tax? Why is the decline in utility a bad measure of welfare loss?
- In the graph above, the compensation slider controls how much additional income you give Rohan in the world where they face $p^\prime_1.$ Move this slider until Rohan has the same utility as the original consumption bundle. What value did you arrive at? This is referred to as the compensating variation (CV), and it is a much better measure of the welfare cost of the tax than the reduction in utility.
- Calculate the CV algebraically, and show that it is the same value you got with the graph.
The compensating variation (CV) is the value that Rohan would have to paid to be willing to go along with the change (in this case, the excise tax). On the other hand, equivalent variation (EV) is the amount Rohan is willing to pay to undo the change.
In the graph above, you can control how much to reduce Rohan’s income in the world where they face $p_1$ using the Lump Sum Tax slider.
- Use the above graph to calculate the the EV of the 3 dollar excise tax on good 1. You can do this by considering the same implied price change (from $p_1=1$ to $p^\prime_1=4$), and seeing how large a lump sum tax is required to reduce Rohan’s utility to what they receive in the world where they face $p^\prime_1$.
- Calculate the EV algebraically, and show that it is the same value you got with the graph.