Suppose you work for a robo-advisory firm and are interested in modeling your clients' consumption and saving decisions using a simple, two-period model. For all parts of this problem, assume your clients' utility functions satisfy the assumptions we discussed in class (i.e., utility is increasing and concave). Consider the case of a representative client. This client's current income is $100 and their current real wealth is zero. They have signed a contract for next year, promising them future income equal to $120. Suppose the real interest rate is 5% per year, and the client faces no borrowing constraints. a) What is the client's inter-temporal budget constraint? First, derive the budget constraint math- ematically. Then, illustrate the constraint graphically, indicating the x- and y-intercepts, the no borrowing/no saving point, and the slope of the constraint. b) Suppose in the baseline scenario described in the problem, the client's preferences are such that they are a saver. Add indifference curves to the budget constraint graph from Part (a) that are consistent with such preferences. Indicate the optimal level of consumption in each period, C and C₁, as well as the level of saving. Suppose the real interest rate falls to 2.5%. We will now consider how the robo-advisor's recommendations about saving would change. c) Compute the client's new budget constraint. Again, begin by deriving it mathematically and then illustrate it graphically, indicating the x- and y-intercepts, the no borrowing/no saving point, and the slope of the constraint. d) Will the client's optimal saving amount rise, decline, stay the same, or is the outcome uncertain? Explain your answer. e) What further information (outside of what is given in the problem) would the robo-advisory firm need to determine by how much the optimal level of saving would change? What assumption(s) would the firm need to make, and what type of data would the firm need to inform these assumptions? (This question is intentionally open-ended, so feel free to be creative in your answers.) Suppose that the real interest rate moves back to its baseline level of 5%. The firm learns that its client's borrowing is constrained, as their lender cannot verify the increase in future income. Therefore, the client can borrow only against $100 of their $120 future income. f) What is the client's new budget constraint? For this question, illustrate the constraint graphically, indicating the x- and y-intercepts, the slope of the constraint, and any other relevant points on the constraint. g) Suppose the client's preferences are the same as those in Part (b). Does the borrowing constraint make the client better off or worse off? Or are they equally well-off? Explain your answer. h) Are there any specifications of the client's preferences such that the borrowing constraint would make them better off than in the baseline (i.e., the consumption levels specified in Part (b))? Explain your answer.