Solve the following question, please

3) Borrowing-savings decision with varying prices Remember Randy, wirorrr we met in lecture 15, and who was about to retire? Randy's friend Sarah will also retire next year, and she has the same income and preferences. m1 = 100,0; m3 = 20,000 U = c1422 But, as soon as she retires, Sarah is thinking of moving to Montana, where the cost of living is lower. In fact, we will assume that the cost of consumption goods in Montana is half their price in California [not an unrealistic assumption, especially if we include the cost of housing. For this problem we will let prices of both current and future consumption (pl, 332) 1wary, instead of normalizing them to 1. a] Write Sarah's budget constraint. [Hintt This will be very similar to the ori'nal budget constraint. Remember that c is a madeup good. When we normalized p1 to 1, (:1 represented both the amount you spend on good c in the present, and the number of units of it you buy. If price is not normalized, (:1 will be replaced by picl in the budget constraint, and the same for 302 .] b) Solve the utility maximizltion problem to find (:1 and c; as functions of 1711, mg, p1, 5212, and r. c] Consider interest rates 7' = 0% and r' = 40%. What will be Sarah's consumption ifshe decides to stay in California p1 = p: {but they are not set to equal 1). How much will Sarah save for next year, and how much will she spend on c1 and c; at these two interest rates? How do these amounts differ from Randyfs spending? d) Find Sarah's sayings and consumption spending at the two interest rates, this time for p1 2 2p; [i.e- if she moves to Montana} l[Zolzrrerouglas utility has the property that the price of one good does not affect consumption of the other good. That is why whether or not Sarah moves to a more affordable region next year is not relevant for her current consumption. But with many other utility functions, this is not the case. As extra practice (not to turn in), redo this problem for the following utility functions: E) U = Mil] (CDCZ) fl U=JE+~JFz