In this question, consumers pay lump-sum taxes and consumption taxes. Let's denote the consumption tax rate in the current period by x and the consumption tax rate in the future period by x'. As usual, y and y' denote the consumer's income in the current and future periods, respectively.

The total amount of taxes paid by the consumer in the current period is t + xc.

The total amount of taxes paid by the consumer in the future period is t' +x'c'.

a: You can take as given that the consumer's lifetime budget constraint is