Suppose that a consumer has income y in the current period, income y' in the future period, and faces proportional taxes on consumption in the current and future periods. There are no lump-sum taxes. That is, if consumption is c in the current period and c' in the future period, the consumer pays a tax sc in the current period, and s' c' in the future period where s is the current-period tax rate on consumption, and s' is the future-period tax rate on consumption.

The government wishes to collect total tax revenue in the current and future periods, which has a present value of R. Now, suppose that the government reduces s and increases s', in such a way that it continues to collect the same present value of tax revenue R from the consumer, given the consumer's optimal choices of current-period and future-period consumptions.

(a) Write down the lifetime budget constraint of the consumer.

(b) Show that lifetime wealth is the same for the consumer, before and after the change in tax rates.

(c) What effect, if any, does the change in tax rates have on the consumer's choice of current and future consumptions, and on savings? Does Ricardian equivalence hold here? Explain why or why not.