In this problem you will analyse how optimal household's choices react in response to a change in the timing of distortionary taxes.

Consider the two-period model of the household. Suppose that the consumer has current income of y, future income of y`, and faces proportional taxes on consumption in both the current and future period. That is, if consumption is c in the current period and c` in the future period, then the consumer pays a tax tcC in the current period and t`cC in the future period. There are no lump-sum taxes. The government wishes to collect total tax revenue in the current and future periods which has a present value of G in order to nance its government purchases. Now, suppose that the government reduces tc and increases t`c in such a way that it continues to collect the same present value of tax revenue G from the consumer, given the consumer's optimal choices of current and future period consumption.

1. Write down the lifetime budget constraint of the consumer before the tax change.

Hint: As we did in class, start by writing down two budget constraints, one for the current period and one for the future period, that depend on the consumer's consumption, income, and savings.

2. Suppose that the government cuts tc and increases t`c while keeping the present value of tax collections xed. Explain whether the lifetime wealth of the consumer (i.e., we using the notation in class) is a ected by this policy change.

3. Using a graph, plot the lifetime budget constraint of the consumer before and after the change in proportional taxes. Does the slope of the lifetime budget constraint change?

4. Based on your answer to part (c), does Ricardian equivalence hold in this example? Use a diagram to illustrate your response. Explain whether the timing of proportional taxes matters in a way that the timing of lump-sum taxes does not and, if so, why.