Consider a representative consumer in the dynamic model of Chapter 9. The consumer's utility function over current and future consumption is given by \"((3, c") = ln(c} + (3111(6). In the current period she has income y = $20,970, and in the future period she will have income y' = $21,900. She pays lump-sum taxes of t = $5,000 in the current period and will pay t' = $5,400 next period. Suppose j? = 0.9 and r = 0.1. Note: Not all answers will be whole numbers. Round any decimal answers to two digits. 1. {1 point) Write out the consumer's formal optimization problem. {No need to solve it yet, just write out the problem.) 2. (1 point] Given the utility function above, derive the expression for the marginal rate of substitution betWeen present and future consumption (MRSClp). J 3. {3 points) Compute the present value of lifetime income (y + g], present value of lifetime taxes {t + ], and lifetime Wealth (we}. 4. (3 points) Solve the consumer's utility n'iaxin'iization problem for c." and c". Then compute savings 3\" and determine if she is a borrower or a lender. o. (3 points) Draw and label a graph of the lifetime budget constraint, the optin'ial bundle (c',c"}, and the indifference curve. Be sure to label the axes, the intercepts, the slope of budget constraint, and the optimal decisions. 6. {3 points} Suppose current lump-sum taxes decrease to f. = $4,240 {and all else as above}. Solve for the new optimal consumption choices, (:5 and (35*, and the new savings decision, s5. 7. (2 points) Does this consumer experience an income effect when t decreases? If so, describe how it affects the optimal choices for c, e', and s. If not, explain why there is no income effect. 8. (2 points) Does this consumer experience a substitution effect when 1 decreases? If so, describe how it affects the optimal choices for c, c", and s. If not, explain why there is no substitution effect. 9. {3 points) Suppose the interest rate increases to r = 0.14 (and still assuming it = $4,240 as above). Solve for the new optimal consumption choices, cf; and eff, and the new savings decision, 3;. 10. {2 points) Does this consumer experience an income effect when 7' increases? If so, describe how it affects the optimal choices for c, e', and s. If not, explain why there is no income effect. 11. (2 points) Does this consumer experience a substitution effect when 1' increases? If so, describe how it affects the optimal choices for c, c", and s. If not, explain why there is no substitution effect