2. (20 points) The taxes paid by consumers in chapter 9 (denoted t and t') are what we call lump-sum taxes. In reality many taxes are proportional. In this question, consumers pay lump sum taxes and consumption taxes. Let's denote the consumption tax rate in the current period by r and the consumption tax rate in the future period by r. 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 + rc. The total amount of taxes paid by the consumer in the future period is t +/'&. (a) (3 pts) You can take as given that the consumer's lifetime budget constraint is - de (1+1)(1+1) + 1+ Write the expression corresponding to the slope of the consumer's lifetime budget constraint. Imagine that you want to draw the lifetime budget constraint on a graph with d on the vertical axis and c on the horizontal axis. 1 +7 C. From now on, assume that y = 200, y = 150, t = 25, r = 0,1 and r' = 4:25. (b) 4 pts) Suppose that initially, we have t; = 50 and 11 = 0.5 and that the optimal bundle selected by the consumer is = 100 and 4 = 100. I am using the subscript "1" here to refer to the initial situation that is, before the taxation policy changes). Prove that the bundle (ci = 100, 100) satisfies lifetime budget constraint (4). (c) 3 pts] Draw the consumer's budget constraint on a graph (d on the vertical axis and c on the horizontal axis) and represent the consumer's optimal consumption bundle. Note that your answer must also include an indifference curve (d) (2 pts) The total amount paid in taxes in the current period is 100 when e - 100. Now suppose that the government reduces the consumption tax rate from 173 0.5 to 12 = 0.25. Calculate the new value of (denote it that would make total tax payments in the current period equal to 100 (sull using e = 100). Show your work. (e) (4 pts On your graph from part (e), show how the change in taxation policy described in part (d) (from () to (1.2)) aflects the consumer's lifetime budget constraint. Justify why the constraint shifts the way it does on your graph. I am asking for a sketch. No need to draw a gap te scule, ( 4 pus) Given the shift in the budget constraint you illustrated in the previous part, would the consumer still select the bundle (100 100to maximize his utility? Justify your