2. (20 points) The taxes paid by consumers in chapter 9 (denoted t and ') 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 2 and the consumption tax rate in the future period by z'. 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' + a'd'. (a) (3 pts! You can take as given that the consumer's lifetime budget constraint is 1+1 (1+r)(1+x) 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 + girly t) -C. From now on, assume that y=200, v' = 150, t' = 25, r = 0.1 and x = 0.25. (b) [4 pts) Suppose that initially, we have t1 = 50 and 21 = 0.5 and that the optimal bundle selected by the consumer is 6 = 100 and 41 = 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 ( 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