Tax and Compensation
- 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.
- V.A.4: Calculate the revenue raised and deadweight loss by a commodity tax.
Lingli is deciding how to allocate her income between two goods: the first ($x_2$) is heating oil and the second represents her consumption on all other goods ($x_2$). Her preferences are represented by the following utility function:
$U(x_1,x_2)=(0.5x_1^{\frac{1}{4}}+0.5x_2^{\frac{1}{4}})^4$
and a standard budget constraint:
$p_1x_1 + p_2x_2 = I$.
Let $p_1 = p_2 = 2$ and $I = 200$. Note that the function $U(\cdot)$ is quasiconcave, so you don’t need to worry about second-order conditions. With these parameters, Lingli’s utility-maximizing choice is to purchase $x_1 = 50$ units of heating oil and $x_2 = 50$ units of all other goods.
Now suppose that in order to discourage the use of heating oil, the government imposes a 2 dollar excise tax on it. This increases the price of heating oil from $2$ to $4$ per unit. This tax is incredibly unpopular, so the government enacts a home heating assistance program that pays this person $100$, and hence increasing her income from $I = 200$ to $I’ = 300$.
- At first, the government only imposed the per unit tax without the subsidy. Could Lingli afford her original consumption bundle of $(50,50)$?
- Once both policies have been enacted, can Lingli afford her original consumption bundle?
- After both policies were imposed, what is Lingli’s new optimal choice and how much utility does she have? How do these compare to their counterparts before the policies were imposed? Using the graph above, answer this question as precisely as you can.
- What will be the net effect of these two policies on the government’s budget?
- Based on the utility levels of the optimal bundles before and after the change, is $100$ more or less than the compensating variation of an increase in the price of heating oil from $2$ to $4$? In a few short sentences, explain why.
- How much should the subsidy be so that Lingli ends up at the same level of utility she started at? Use the graph above to answer this question.